Introduction to Formal Logic (Learn how to argue)

1. Logical arguments, premises and conclusions: although there everybody and welcome to my first ever a crash course and formal logic In this first part of the first series, we're gonna cover logic, arguments and premises and conclusions all explained simply by do you have to warn you? One thing that won't be simple about this course is it is a crash course. We're going to covering about two lectures worth material in each one of these video presentations. So hold on tight for that now. The first course, I'm gonna cover logic basics. That's just basic argument analysis, fallacies, category logic and proposition all logic, including truth table and natural deduction methods. And that should be enough for any college level introduction to logic. But later on, I'll offer an advanced course and logic covering probabilities, quantified motile and first order quantified modal logic. Aristotle's the person we have to credit with first formalizing logic as a discipline, and he also created nearly every other discipline that we study in. The university is almost from scratch. It's been said about Aristotle. He may have been the last person on earth who knew everything there was to know in his lifetime arguments as we're going to study them are not heated exchanges or personal assaults, and by the same token, they're not merely disagreements and opinions or the automatic contradicting of an opponent or one's opponent's position, even though the term argument is sometimes used that way in the vernacular. And one person who exploits this use or understanding of the term argument is Charles Schulz and Peanuts cartoon Siri's Lucy Astro. Dwight likes Beethoven better than her, and, well, shorter comes a little bit argumentative. And that sort of argumentation does not leave room for discussion. Lucy complaints. Linus is another person you always itching for some verbal combat. When asked if it's a beautiful day, Lieutenant Linus replies with a lot of combative questions. And what he says, last of all is pretty interesting. A good fanatic is always ready for an argument. Now that may be true, but a good fanatic is not ready for an argument in the sense that we're gonna define the term. This is a logic course, and logic is the science of argument evaluation. We put arguments under the microscope of logic and see if those arguments stand up or whether they work in order to do that The first thing we're gonna have to do before study logic is to get an understanding of what argument is an argument is a set of statements. In other words, one statement by itself never constitutes an argument. Then unique thing about these sets of statements is this. Some of those statements called the premises, claim to be support or reasons for another in the batch. So you get these sorts of relationships of evidential support holding between statements. The statements that give evidence are called the premises and statements that receive the support from the premises. On the opposite end of those arrows, those statements are called the conclusion. Well, now that we've defined the term arguments, it's time to move on and define statements and more detail, statements or sentence is capable of being true or false. For example, if all cats are said to be vicious, that statement would be false. But notice we have a sentence here capable of taking a truth value, and similarly, somebody might say that some old men are grumpy and that sentence is more than likely true , since we just limited it to some old men. But again, statements are a unique type of sentence. Not every sentence or utterance qualifies a statement. There are certain very meaningful sentences or utterances such as Where's my milk? Yeah, we or get me a sandwich. Perfectly meaningful but not statements. Why? Well, as a general rule, questions, exclamations, imperatives and commands cannot take truth values because they don't assert anything about the way the world is. Therefore, they cannot be true or false, and they cannot be statements, and they cannot service premises or conclusions and arguments. As we have defined the term argument well, so far, I've given you a lot of important terms, The most important of which, on this little cheat sheet that you can look back on is the term inference. That's the reasoning process of an argument. It's whatever type of reasoning gets you from the premises or to the conclusion, and we'll study that more detail later on. For example, consider the following batch of statements all film stars or celebrities. Halle Berry is a film star, and Halle Berry is a celebrity. You can divide these statements up into premises and a conclusion that could be reached upon them, and as it turns out in this case, all of our statements turnout is true, but that is not always the case. Consider the following batch of statements. How about some film stars? Airmen and Cameron Diaz is a film star. Therefore, Cameron Diaz is a man. Well, if you these air premises and conclusions. What we find out is that the premises statements turnout true, and the conclusion statement turns out to be false. The point here is this. In this sort of case, something seems to have gone wrong with the inference, the leap from the premise, statements to the conclusion statements and what went right or wrong in these inferences is the subject matter for logic in this course. But before we study logic in any detail, we gotta get more clear on how to distinguish conclusions and premises. One helpful hint is indicator words, and just about any good logic textbook is going to give you a list. Something like this. Therefore, accordingly entails that wherefore don't memorize the list. Just get the basic principle that these are conclusion indicators. They tell you that somebody's about to state the conclusion of their argument. For example, somebody might say tortured prisoners will say anything toe relieve their pain. And consequently, torture is not a reliable method for obtaining information from prisoners. Now. That term consequently tells you that the persons about to state the conclusion that they've reached upon their reasons or premises probably given earlier and conclusion indicators aren't your only helpful tool. There are also premise indicators, reason indicator terms such as since in that seeing that, as indicated by all of those sorts of terms, tell you that the person is about to present reasons for a particular point of view that they hold. So, for example, parents should never shake a crying baby. Why reached that conclusion? Since the baby's delicate body and brain might be easily traumatised since indicates a reason or premise. Statement is about to be given now regarding reasons. Watch out for this term McDonald test fatty foods. For this reason, I should go to subway in this sort of paragraph. Reason indicates that a conclusion I should go to subway is about to be reached. But by way of contrast, I should go to subway for the reason that McDonald's has fatty foods well here. Reason indicates that a premise is about to be offered So for those of you might be a little confused. Let me clear this little matter up what somebody says. For this reason, this looks back to a reason already given and says that a conclusion is about to be offered on the basis of reasons already given previously, however, for the reason that looks forward to a reason that is about to be given. So that is a premise indicator. There's some other helpful tips when you're dealing with paragraphs and conversations in everyday life. One indicator word may signal more than one premise. For example, since my companies in the red and I'm not seeing any hope of recovery, I should file for bankruptcy. Technically speaking, you could divide this up into three statements. My companies in the red. I'm not seeing any hope of recovery. The conclusion statement is I should file for bankruptcy. Two premises or two reasons were given for that conclusion, and also, sometimes you'll find that there are no indicators in a passage. And if that happens, suspect the conclusion was offered up front. Maybe I've just been playing Frogger too much, but I came up with this example. I shouldn't cross the road. It's rush hour in. The last two frogs were flattened. Notice, no conclusion and no premise indicators here. But the conclusion I shouldn't cross the road receives support from two other statements. Well, as I promised from the outset of this video, in each video, you're gonna receive about two college level lectures in logic. And this is where I start Lecture number two. Basically, we're gonna cover the topic of Is it an argument? Because I find that logic students, once they get started distinguishing conclusions and premises and making out inferences. They tend to find arguments in every conversation or paragraph that they read, and that's not the case. We need to distinguish between the paragraphs and conversations that contain arguments and those that don't. There are two conditions. For an argument. You have to have a set of statements, the premises claiming to present reasons and a claim that there's a conclusion that has been supported. Two claims. Now the first claim. The factual claim is not something that logic evaluates the truth. Value the premises. That's just something you have to figure out on your own. Logic evaluates the support or the inference claim, but The point is, you have to have two claims on the table toe. Have an argument. A factual claim or set thereof without the inference claim amounts to a non argument. So to spot arguments and non arguments, you have to learn to spot inferences and spot non inferences. So we have two lessons to cover here real quickly. First, spotting inferences. Inferences can be explicit when they use premise or conclusion indicators that makes things easy. Since my companies in the red and I'm not seeing any hope of recovery, I should file for bankruptcy. The premise indicator word there gives away the inference, so use those indicator words if you find them in a passage. But watch out for them. I have a few caveats that need to add later on. Otherwise, the inference can be implicit or example. In the case of the frog crossing the road, the reader had to catch the inference because no indicator words were offered. So arguments may lack indicator words, and in a second we're going to see a few non arguments that might even have the indicator words. The point is, this don't use indicator words as a crutch for spotting inferences and such. If you think that you have an implicit inference on the table, insert therefore in front of whatever you think. The conclusion is, if the passage makes sense, chances are you are dealing with an argument in the case of the Frog, it's rush hour in the last two frogs were flattened. Therefore, I shouldn't cross the road. Makes perfect sense. Therefore, we can tell the inference was there all along. It was just implicit. Now consider this. As Einstein developed his relativity theory, he was unknowingly laying the groundwork for quantum theory. And since Einstein published his theories, quantum theory has enjoyed many successes. Looks like we have premise indicators indicated in the red right? Well, that would be wrong as an sense are not being used here to indicate premises or reasons. They are time indicators as means at the same time and since means subsequent to the time. Consequently, there is no inference really being made here and no argument now. Contrast that with the passage like this, since quantum theory was not likely to develop without Einstein's work. He deserves some credit for the theory, despite the fact that he actually disliked it interesting bit of trivia, right? Well, we're being asked to infer here since is being used as a premise indicator, and the conclusion here is that Einstein deserves credit for a theory that he really didn't like. Well, now let's talk about non inferences, specifically four types of non inferences that are commonly mistaken for arguments. If you want a more detailed exposition of this, I recommend Patrick Hurley's concise introduction to logic. But in the meantime, I'm gonna abbreviate that sort of conversation drastically and just cover the four main areas where students are tempted to make mistakes. Instruction does not amount to offering of inference. Whether the instruction is negative, like a warning, the US cannot keep running up its deficit without winding up in the same financial catastrophes Greece or a positive bit of the device. If you want to keep the US economically secure, we ought to first cut reliance on foreign fuels. The temptation on the part of students is to twist these bits of advice and warning into premise by premise. Arguments like premise. The U. S. Should not allow itself to wind up like Greece and premise if it runs up its deficit, it will wind up so and there's a conclusion that follows and the same sort of thing happens . In the case of the positive bit of advice, the U. S should not allow itself to lose economic security. And economic security is best preserved, plausibly through less reliance on foreign fuels. And a conclusion follows that the U. S. Should decrease its reliance now. For a lot of students, this might be tricky because they're thinking that the Warner the adviser would certainly like the arguments that we spelled out premise my premise and that surely they were assuming something of this form of reasoning or logic when they offered their warnings and bits of advice. But how do you know, for example, that the advisor thinks that less foreign fuel dependence is the best solution to our economic security problems? You're attributing a premise to the arguer that they do really didn't argue, and this is a tricky point. Warnings and advice usually are based on a pre supposed some reasons, but that does not mean that the adviser offered or stated them. So when somebody offers you warning or bits of advice, that doesn't mean that they're offering you an argument per se and a similar point holds with respect to expressing opinions. Take a look at the lover's quarrel below your always nagging me for spending time with my friends and your just jealous his interlocutor response. No, you just don't want spend time working on the marriage. Now the temptation is to attribute an argument premise by premise to the argue hours. In this case, your accusations are merely based on jealousy of my friends and premise. Such accusations, so based, are unfair, and the conclusion is your accusations are unfair. But wait a minute. The interlocutor in these cases I shouldn't call them an Arguer really didn't offer a premise by premise argument like this. Did they again? Just because you think the speaker does should or must hold a set of reasons for what they say or express does not mean that they have those reasons. You're just guessing at that, and it certainly doesn't mean that they've offered them and that they've offered some sort of argument. Well, case in point is nearly all political talk, whether you get it from the right wing or the left wing. What you see in the so called political debates is generally not really anything in the way of argumentation, but rather an elaborate spelling out of one's views on a political topic. And if you'd like to get some exercise on making the distinction, just go to the opinion section of your local newspaper. Often times you find people doing argumentation, and sometimes they're just expressing in an elaborate way their point of view on the perfecter particular topic. So to be able to make the distinction within that sort of section, the paper is a very important skill to develop. Now let's talk about information giving that does not amount to inference offering. Consider reports, reports Just give you information. The temptation on the part of students is to see a slant or a gist, or to sense that the reporters driving at a conclusion and then to assume that the reporter has offered an argument. For example, if somebody says sales of assault guns have increased drastically, critics of assault rifles say the following and recent school shootings have resulted in well, obviously this person seems to have an axe to grind their presenting a lot of negative information, but notice they didn't state a conclusion off the base of this information, and hence they did not offer an argument. And our sense of the term expositions are another form of information giving their just lengthy talks around a topic. And if you confuse these for arguments, you probably gave into the temptation to confuse the topic sentence of the talk with a conclusion. But that's not the same thing. If somebody says Eggs Benedict is a delightful, an impressive dish, you could make easily. And this is how you poached the eggs and prepare the muffins. And here's how to make the sauce. They're not arguing you into the position that Eggs Benedict is a delightful, impressive dish. They're just giving you a lot of instructions on exposition on exactly how Eggs Benedict is made and illustrations are expositions that are littered with examples. These could be very confusing because the temptation is to confuse an assumption that the speaker makes, followed by some examples with a controversial conclusion that has argued for from various instances, these air very different cases compared the following Suppose somebody says there's many, many types of screwdriver, including the Phillips head, the Flathead Hex, even the early Robertson's head. Now this person is not trying to argue you into the position that there are many, many types of screwdrivers. Unless, of course, you've started the conversation just debating that sort of issue, which I doubt that anybody would do. It just doesn't seem like a controversial conclusion. They're just giving you an at exposition with examples. Now, by contrast, in a war in peace rally if they said war by itself can never resolve conflicts. Now that sounds controversial. The person may choose, then toe back up their claim with instances like the Civil War bringing decades of division World War ending with the treaty that brought World War Two. Consequently, the Cold War turmoil and Korea and Vietnam. Clearly, this person is arguing from instances, and that's a different thing altogether from illustrating a topic sentence. So bear in mind topic sentences are not conclusions, and there's a temptation to confuse topic sentences with conclusions that needs to be resisted. There are various connections between ideas that do not amount to offering an inference, and this could be confusing because inferences involves some relationship between ideas contained in the premises and ideas contained in the conclusion. But there's other ways of connecting ideas that do not involve inference. Explanations are a very popular one. That's where you try to shed light on some sort of event or phenomena, and usually you haven't explain. And, um, here's a hint. That's the thing you're dumb about. And then you have the explanations, which answers, Why is that thing happening? But that doesn't mean that the explain and, um and explain ans constitute conclusions or premises. Consider a great clap or a classic case. The case of Sir Isaac Newton, who had a certain explain and, um, why do ocean tides roll in at night? He gives the explanations. There's a gravitational force that holds between the moon and the waters. Now the temptation on the part of students is sometimes to take the explain and, um, and treat it as a conclusion, because if there is a gravitational force holding between the moon and the waters, it would follow that ocean tides would roll in at night. But notice nobody was really confused as to whether ocean tides rolled in at night. We didn't know why it was happening, but nobody had to be argued into that as a conclusion this was not an argument. Now explanations are like arguments, reason giving activities. The explanations gives you a reason why the explain and amiss happening even consider this example. Lucy asked. Why don't you ever call me Que t? Now? There's an interesting explain. And, um, and Schroeder inherently gives her enhancer or explanations that she doesn't like, because he doesn't think she's very cute. And Lucy complains that she hates reasons. Now. Explanations do give a reason for the explain. And, UM, that's true, but it's not. The same relationship is a premise and conclusion relationship. Now here's a difficulty. Sometimes, if an explanation is good and it covers a lot of ground, doesn't that argue for its truth? And the answer is yes, sometimes. But notice something here. First of all, when that sort of thing happens, Usually what you have is not an argument for the explain. And, um, you have an argument for the explanation ends. The answer. Why? Because the answer why is so good now when that happens, that sometimes called an inference to the best explanation. We'll cover that in the next lesson, but the point for now is explanations by themselves do not amount to inferences or arguments. And another connection between ideas is just the straight board. If this then that the conditional statement you say if a then b A is said to be sufficient for B and B is necessarily to follow upon a Now, the temptation on the part of students is to take these conditional statements and treat a as apprentice and be as a conclusion. Is that right? Well, consider this example. If I'm taller than shack, then I'm taller than you. Isn't that right? But what are you gonna conclude on the basis of that? Well, I'm not asking you to reach a conclusion. It's probably the case that if I'm taller than shack, I am taller than you. But I'm not saying that I'm taller than shack, and I certainly am not trying to infer that I'm taller than anybody listening to this lecture. It's just a connection between two ideas that probably does hold. Now, here we encounter another difficulty that you're gonna have to sort through in later lessons. Conditional sentences are often important. Parts of argument. You're gonna find them his premises very often, and we're gonna go in detail on that in future lessons, however, the point for now is that by themselves, conditional sentences do not constitute arguments. And that's kind of the lesson with respect to explanations and conditional is that could be important parts of arguments. But that does not make them arguments in themselves. And if you can remember that, you won't get tripped up in spot inferences when there really are none. Well, I've given you a whole lot of material for now. Wait for my exercises on this lesson and has promised you've done in less than 20 minutes to college level classes in logic. So congratulations feel free to review and wait on my next logic lesson. Thanks again, everybody. 2. Deductive and Inductive Inferences: Well, everybody And welcome back to my crash course and formal logic. In the second part, we're gonna be studying inferences both deductive and inductive. And for those of you who muscled through the first part of this course, you know that this is genuinely a crash course. We're gonna cover about two lectures worth material in one single web address to give your self time to rewind and review. Now unless number one we learned that logic is the science of argument evaluation. And we defined arguments and its constituents, such as premises and conclusions and inferences. Very precisely. And just as importantly, we looked at the topic as to whether or not a passage even contained an argument or not, which was pretty much equivalent whether or not the passage or conversation involved in inference or an inference claim or not. But now we're ready to go into more detail. The two key elements of an argument are always are the premises, true are and to what degree do they support the conclusion and logic analyzes the second of these two key issues. Now, that means that logic is actually the science rather of support than truth. What we want to know is whether the premises support a conclusion, whether not those premises. Air True is a different issue. Alternatively, you may say that logic is the science that evaluates inferences instead of truth. Logic cannot tell you whether the premises or conclusion of an argument is true. It can evaluate the link up between the premises and the conclusions. Now this departs from a usage of the term logic that you may find in everyday talk. Maybe one that was picked up from Mr Spock's considered the argument all Vulcans or cannibals. Captain Kirk is a Vulcan. Therefore, Captain Kirk is a cannibal. If Spot thinks that this argument is illogical, it's because he's using this term logical as a synonym for rational. But we're gonna re define that term. What I want to consider is whether or not the conclusion would follow. If Captain Kirk was actually, as the premises assert him to be, let's take a look. The first premise said that all Vulcans were cannibals, right? So I'll just scoot the bulk in class inside the cannibals class. And as the second premise says, Captain Kirk is a Vulcan. Does it follow the captain Kirk is a cannibal Noticed by illustrating the premises, I was forced to illustrate the conclusion. So in a sense, this argument is logical against the sense in which it's logical is one that even spotted appreciate. He'd at least have to admit that the premises already contained the conclusion to illustrate it a different way. If somebody says there's a squirrel in my shoebox and furthermore, they add that the shoeboxes in their closet do you know whether or not there's a squirrel in their closet? Well, the answer is already given way right. What we've encountered here are two examples of a type of logical support that's referred to as deductive because there's no possibility of the conclusion being false in the premises air. True, that's because the conclusion is somehow contained in the premises implicitly already. But there's another type of logical support that we need to investigate. Inductive levels of support involved premises that merely raised the odds of a conclusion, although the conclusion is allowed to go beyond with the premises explicitly warrant. Now I'm gonna just these definitions in a second. But let's just examine types right now, since philosophers like to use illustrations involving Ravens. I'll not depart from that venerable tradition. Let's take a look at some deductive support for a conclusion. How about Ravens? One through? Four are all black. And let's add that Ravens one through four were each distinct. And we cataloged four distinct animals. Doesn't it follow that there are at least four black Ravens? In fact, that conclusion follows deducted Lee. It can't be false if the premises air true, but you can't count every animal under sun. So typically all you get is inductive support. Used to may say that Ravens want to for all black. And if you're a really diligent scientists, maybe Ravens 5 to 10,000 were black and all those being distinct animals. You can reach the conclusion that all Ravens are black Onley through a leap in logic. You get some evidence for this conclusion 10,000 bits of evidence, in fact. But the conclusion does not strictly follow from the premises to go back to a deductive type of support. What if he said Ravens one through four were black, and we added that 5 to 10,000 were black and assuming that each raving counted was distinct and new premise Onley 10,000 Ravens exist. Alright, now, clearly it follows that all Ravens are black after all, we counted every single one of them and catalogued. Um But by way of contrast, if somebody said merely that Ravens one through four were all black Ravens 5 to 10,000 were black, assuming each was a distinct animal. And new premise, less than 11,000 Ravens exist. Where does that prove? Well, it lends a lot of support, though not proof to the conclusion that all Ravens are black. After all, there's just 1000 left that we didn't catalogue. You get the idea, right? I'm gonna put a supporter meter to the right. In the first example, we achieved the maximum degree of support for our conclusion that we could get the premises back. The conclusion in such a way that the conclusion cannot possibly be false, that the premises air true. But in the second example in which there were 10,000 Ravens counted, well, we got some evidence or support for the claim that all Ravens are black. Although our support a meter has to drop quite a bit. And in the case of ah, Ravens being counted, and we added a premise that Onley 10,000 Ravens exist, assuming that we counted them all. The conclusion then is supported to the maximum degree once more. But if there were 11,000 Ravens and we only counted 10,000 the notice on my supported meter , I dropped the degree of support down just a little bit to leave a little bit of room for error, even on the assumption that our premises were true. That's how support works. You get the picture right. There's two types of logical support, deductive and inductive and quite a variety of degrees of strength thereof. I do want you to bear in mind and please pardon my pun about bears and minds. This distinction does not amount to the distinction between a good versus bad argument or a strong versus weak argument. All I've pointed out so far is that support for a conclusion comes in two forms. And if you ever want to attack somebody's argument, be sure to understand and make your attack appropriate to the degree of support. The argument intended to offer its conclusion. Having said that after we know how to recognize arguments like them, the last lesson. We have to figure out which type of argument we're dealing with in any situation. One little bit of help is the use of indicator words when people say necessarily certainly are absolutely before their conclusion. Clearly, they're trying to do a deduction. In other cases, they may used the term probable, improbable, plausible, implausible in those cases, what they're trying to say is that the premises made the conclusion more likely. That's an attempt at induction. But once again we have a problem. In the last lesson, we learned that indicator words were not reliable in distinguishing arguments and non argument's premises versus conclusions. So can you rely on them to distinguish deduction and inductive support? And for my two cents, the answer is this. No way you can. For one thing, missing indicator words, as in the last lesson, are the norm. And that's true when you're dealing with an argument of Paragraph one, a premise by premise form. For example, we used on example involving a frog early on who reason that he shouldn't cross a road now . That conclusion was reached inductive Lee on the basis of two premises, but notice are hypothetical frog never said it's merely highly probable. I shouldn't cross the road. That little phrase was left out and generally, But there's a more important reason we have to take a look at. There are such things as bad deductions. A person may say a conclusion necessarily follows when in fact it does not. And Aristotle, who first developed deductive logic by looking at how categories of things related to each other catalogue good versus bad deductions on the basis of this. So the idea here is that is, that instead of defining arguments in terms of two types of logical support, we should define them instead in terms of types of claims. In the case of a deductive argument, there's a claim by the arguer argument that there's no possibility of the conclusion being falls from the premises air. True now, whether or not that claim holds up under scrutiny, that's the job of logic to tell us. And similarly, for inductive arguments, there's a claim that premises raised the odds of the conclusion. For example, Aristotle said, all men are mortal. Socrates is a man. It follows deductive lee that Socrates is mortal notice. We're dealing with three different categories. Men, Mortals and Socrates and category logic is Aristotle's Baby. But the point here is that this deduction is successful. But now consider this spoof By Woody Allen. All men are mortal. Socrates is a man. Therefore, all men are Socrates. The question is, Is Woody Allen doing a deduction? Well, the problem is, if this is a deduction, it's a bad one. We call those fallacious deductions. The conclusion did not follow. Not even. Probably, however, we night need to redefine deduction so that it does not refer to cases in which the conclusion follows. Necessarily. Rather will say the conclusion was supposed to necessarily follow. Clearly, Woody Allen intended to be doing a deduction. The deduction is just a failed one. So again, we're gonna define arguments, an argument types in terms of inference, claims now returning to our original topic, how do we distinguish whether we're dealing with an inductive or deductive argument? A second ways to look at the general former style of argumentation. Now, when people are relating categories to one another, like Socrates and Woody Allen a second ago, or if they're just appealing to math or word meanings or semantics and other forms will discover later. They're always doing deductions. By way of contrast, there are certain forms of argument that will cover right now that are always going to be inductive attempts at inference. So let's cover several inductive forms. And by the way, this list is by no means exhaustive. That's covered generalization, analogies, arguments from signs, causal inferences, appeals to authority and predictions. And if you need a little acronym to remember my list of inductive argument forms, I suggest the acronym Gas Cap G A S C ap First, Let's cover generalizations. Since we already talked about Ravens, generalizations involved arguments that move from a knowledge of a sample to knowledge of a whole group. In the case of Snoopy, he tends to be very bad at these sorts of arguments. He says. Dogs been howling at the moon for over 5000 years, and the moon still hasn't moved and dogs are still dogs. That proves something. I don't know what actually what it does is instead of proving something it inductive. Lee supports the generalized claim that every evening that dogs halat the moon are gonna be evenings where nothing is accomplished. Another popular type of inductive argument form is the analogy. That's when you in for a further similarity between two things based on knowledge or premises, about known similarities. It's actually very important certain fields of science. So, for example, somebody would say all and only the rats that ate Agent X got sick and recognized by analogy that humans and rats have very many relevant similarities and draw the conclusion from these two premises that if I a human eat Agent X, I'll get sick. However, this is an inductive argument. How do you know that humans don't have special immunities that rats lack? Or perhaps the rats in the study were already sick to begin with? That would certainly weaken our inference to the conclusion. However, in this case, I think we have enough information in our premises to draw the conclusion that we shouldn't be eating that Agent X right. Inductive arguments can be very persuasive. I should also hasten to mention that arguments from analogy are very important for moral thinking. Here's a little cartoon illustrating or perhaps suggesting an argument that the mascot, the Cleveland Indians, is probably not ethically appropriate by analogy. Would you use a different ethnic group to be the mascot of the Cleveland team, and I use this argument. This is actually a store located in Columbia, Missouri, not far from where I did my doctor research, liquor, guns and ammo. No joke. Well, by analogy, if, ah, liquor, guns and ammo, it were to be perfectly ethically responsible. Wouldn't monkeys matches in dynamite? Be an equally legal and ethical name for a store? I think the two standard fall together on the basis of similarities, and you may even find an illogical thinking in the Bible. Some of you may recall story where Jesus heals the withered hand of a man and does so on the Sabbath. When he's challenged on it, he claims people would have done the exact same thing if it had been an animal stuck in a ditch they'd have helped out on the Sabbath. So there's some Anna logical thinking going on. Whether or not Jesus was arguing from analogy or not in that early chapter of Mark's gospel , arguments based on Sign are another very popular form of inductive inference. That argument proceeds from knowledge of a sign to knowledge of the thing that is symbolized by the sign. For example, you see a wavy road sign and you infer that there's gonna be curves in the road ahead. The jump from the sign to the things signified involves an inductive or merely probabilistic inference. The same thing goes when you look at your speedometer and see a certain reading and infer that you are actually going that speed or even at your timepiece. You see what it signifies, and you infer that it is, in fact, that time of day. Would you see that these involved some sort of inductive risk? All you have to ask yourself is, How do you know, for example, that a jokester didn't put the road sign there to fool passersby? And how do you know that? Your speedometer and timepiece air working correctly? Typically, we don't double check that constantly, and that's the nature of induction. There's always a risk of moving from truth falsehood when you move from the premise to the conclusion. Now, causal arguments are very popular form of inductive argument and a very important one. They occur when you move from knowledge of a cause to knowledge of in effect or conversely , from premises involving knowledge of effects to knowledge is of a cause. This latter case I promised to come back to in an earlier lecture. Knowledge of an effect and taking that is leap to knowledge of a cause is sometimes called inference to the best explanation. It happens an awful lot, especially in sciences. For example, Sir Isaac Newton had a certain effect, like the ocean tides rolling in at night. That was an explain and, um, something we were actually quite dumb about until he offered the answer or explanations that it was due to a gravitational force. That's the cause of the ocean tides rolling in the gravitational force holding between the moon and the waters. So the move from a effect to end cause is very important in science. And actually, it's kind of important in comedy to Gary Larson likes to give you a single panel cartoon strips and see if you can look at the effect here and guess the cause. If you can't, then you don't get the joke. Here's another example effect right in front of you. Can you guess the cause of this and others? Comics like to make fun of people who are really bad at making this sort of inference. Lorna suggests here that doesn't it seem strange that the cleaners would shrink all of your pants? Her husband apparently had the wrong explain ans for his explain. And, um, and I guess that's just due to a self serving bias. And, ah, on occasion, we're all guilty of it. Now. Arguments from authority involved cases in which you make an inference based on those statements or authority of a authority in the field, say science or a witness to an event that you were not present to see. Now, in those cases, you are making an inductive leap. It's amazing how often we do this How, for example, do you know that water is made of H 20 or that the ocean tides role in due to gravity? How do you know when the pyramids were built, or how do you know when the Constitution was written? Even your local news reports and the inference that you make towards situations where they occurring in the world are cases in which you rest. Your inference on the presumption that the authority can be trusted Predictions are another case in which people take inductive leaps in logic premises deal with some present or past knowledge and move beyond that to knowledge of the future. Even something as simple as the claim I will die someday is based on knowledge that is fallible. I just know that everybody up to now has died. And perhaps we could be saved by science or by some religious intervention. And we won't die someday. We're making an inductive leap. We moved from knowledge of the past to knowledge of the future. So why would you want to stick your neck out like that? I guess George Santayana said it best. Those who do not learn from the past are condemned to repeat it. So I guess in general, Peanuts characters just are not very good at induction there. Inductive skeptics. I guess they don't trust generalizations and don't trust predictions. But for people who do such, I guess there's a risk to be taken Well, congratulations. You've learned a whole lot about deduction and induction and different types of inference and the claims they make. But I want to go into more detail here. Justus. We close inference claims of the most important element of logic. They are decided by logic. And if the inference claim is wrong, the reasoning from premises to conclusions is a bad form of reasoning. The argument is junk, and we actually use the strength of the claim of support to distinguish deduction and induction argument types. I've come up with an acronym to help students wrap their minds about what's involved in various inference claims. A. In terms a NTS ants refers to additional premises and whether or not they can add support in a deductive inference claim that should not be permitted. Button and inductive inference claim it's always permitted. So in our example of deductive support for the conclusion that there were at least four black Ravens are premises that we already catalogued for black ones raised our support to the maximum possible degree. You just can't get better than 100%. But we also had an example involving 10,000 catalogue Ravens. Now you can add more support your conclusion that all Ravens are black. It's just a simple is going out and counting a few more Ravens and similarly, we had a study in which rats were examined and on the basis of their similarities, we found out that we shouldn't be eating Agent X if they got sick, Perhaps we will. We had a lot of support. Broad conclusion, but it could get better. We could answer the following questions. How do we know that humans don't have special immunities that the rats lacked? Or what if the rats in the study were already sick? Answer these questions and include those answers as premises, for example, that Agent X effects organs, humans and rats both have, and that the rats in the study were in perfectly good health. Add those his premises more support for your conclusion that you will get sick if you eat Agent X Now the end in Aunt Stands for novel information In the conclusion in a deductive inference claim, there's a claim that there's no new information in the conclusion that wasn't present in the premises. Inductive inference is, or sometimes called amply a tive because they allow and actually require there to be novel information in the conclusion. So when the premises in the conclusion form a seamless bond, you have deduction inductive arguments with ones in which the premises and conclusion have a gap between them. That informational gap is one of the hallmarks of an inductive inference claim. Let's look at our deductive examples. Captain Kirk, we said, was a firm to be a cannibal. Why, on the basis of two premises, we said that Vulcans were all cannibals is a first premise. And then he said that Captain Crook was a Vulcan. Now there's no new information offered. If I say to you that Captain Kirk is a cannibal, that information was already implicit in the premises. Are conclusion. Just teased it out. Made it explicit. Similarly, if all men are mortal in Socrates is a man. The information that Socrates is immortal is already contained in those premises. And similarly, when there was a firm to be a squirrel in a shoebox and we asked where the shoebox was. Well, in that case, when we found out that was our closet, you really didn't need me to tell you whether or not there was a squirrel in our closet to answer that question. And the information in that answer is already contained in the two premises I gave you in black in an argument from signs. By way of contrast, you actually have to go beyond the information that the sign says thus. And so, if you're gonna reach the conclusion that reality is thus and so and for that reason, the degree of support that you can achieve in an inductive cases market, they're going to be lower than the degree of support you can get from deductions. Now the tea and ants is very important. We're gonna spend a great deal the course discussing this, whether or not there's a truth preserving structure or form. Deductive arguments are supposed to be truth preserving inductive arguments or not what I mean. But truth preserving is that the argument is set up or structured so that truth in the premises will be funneled if there is any truth in the premises straight down in the conclusion, and you can take that to the bank. So take a look at the example from Aristotle. It doesn't matter that we're talking about men being mortal. We could just as well have talked about men being Greeks. If all men are Greeks and Socrates is a man, Socrates would have to be Greek again. It's not the semantics of mortality or Greek hood. It's the shape of the argument that's doing the work. Same thing holds for Socrates. It's not a matter of Socrates. We could have talked about Aristotle if all men are Greeks and Aristotle's a man, Aristotle's a Greek. Similarly, our argument involving Ravens doesn't have anything to do with them. In particular, we could just as easily have talked about bears. It bears one through four, all black. There's at least four black bears, and speaking of which, is there anything special about blackness? How about brown This? It bears one through four all brown and their distinct bears that follows their at least four brown bears. Again, deductions have to do with types of inference based on form. And, lastly, does support admitted degrees. What's involved in the inference claim in deduction and induction? Deductive inference claims do not allow support to admit of degree inductive arguments. Claims always do so. The way to think about it is that when you're dealing with deductive arguments, the support is there or not. It's like a light switch on or off all or nothing on the support of meter. So, for example, the claims that Norman or mortal and Socrates is a man 100% that support the conclusion that Socrates is immortal and you can't get better than 100% degree of support. But what about the spoof by Woody Allen? All men are mortal on Socrates is a man. To what extent or degree did he support the claim that all men are Socrates? And the issue here is that since he was attempting a deduction, his degree of support failed completely. If it failed it all. Well, thank you guys once again for muscling through another tough lesson this time on inference claims. If you need a review, just remember your aunts. And here's a little cheat sheet, just in case you need to go back and do some re memorizing. In the meantime, that's all. For now. Wait for my exercises in my next logic lesson. In the meantime, master the ants and you'll know the difference between inductive and deductive. Inference. Claims will see you next time. Thanks for mastering 20 minutes of logic material. Bye bye 3. Validity, Soundness and Cogency: Well, hello there. Once again, everybody, and welcome back to my crash course and formal logic in this third partner, Siri's. We're gonna be studying the crucial concepts of validity, strength, soundness and coach Insee. So you won't watch this video several times. I think, however, this is a crash course. Hopefully, you'll be reviewing the material presented in each of these lectures more than once Now. Speaking of review, now would be a very nice time for review because you've been covering a whole bunch of lessons in the past two sections of this course, which will be relevant to this third section. Of course, it's quite a mouthful to go back over. But we said early on that there were two conditions of an argument. You have to have a set of premises or reasons, and those factual claims are not determined by logic. Rather, there's a second claim. The inference claim the claim that support has been offered, and whether or not support really is there is decided by logic. Now it immediately follows that a set of factual claims without an inference claim amounts to a non argument. We said that in less than one, whereas a set of factual claims with the claim of support or an inference that's when you get argumentation and our sense of the term argument. So logic is actually the science of support, whether or not the premises back the conclusion, not the science of truth, Alternatively said, Logic is the science that tells us whether or not premises and conclusions enjoy the right inference or the right hook up between one another. Whether or not the premises or conclusions are actually true is a different matter altogether. So we said what that respected this appropriateness of, ah, hookup between premises and conclusions. There's more than one way that could be linked that could be linked, deducted lee or inductive. And we also give you an acronym so that you could memorize the difference between the inference claim that argue hours making is the arguer allowing that additional premises can add more support to their conclusion? Or are they allowing that there's novel information? The conclusion that wasn't there on the premises is the arguer saying that there's a structure to the argument that preserves truth on the way from the premises of the conclusion, or are they allowing that support admits of degree, as opposed to, say, being an all or nothing matter. All of these issues go into inference claims now inference. Claims are very important to logically said and argumentation that the most important element of an argument decided by logic and if they're wrong, junk. The argument and the strength of such a claim for an argument divided our arguments into inductive and deductive styles or types of arguments. So bear in mind, we said early on, this distinction between inductive and deductive is not the same as a distinction between good and bad or strong and weak arguments always said so far on this course, is that support can come in two forms, and attacks on arguments must be appropriate that the degree of support that they purported to give to their conclusions well, so by now you're probably wondering them. What is the relevance of all this? What does this inductive deductive talk have to do with evaluating arguments? Well, that's really the subject for this third part in our logic. Siri, there are two key elements to any argument Are the premise is true? Yes, but the crucial point is to what degree do those premises support the conclusion? That's what logics there to help you with. So initially, degree of support is the key to evaluating arguments. We're gonna use this little flow chart. Once you know that an argument is set forward, you want to know whether it's inductive or deductive. That is to say, You want to know the type of argument that you're dealing with and then move on to the key of degree, whether or not a person's claiming an inductive or deductive degree of strength for their conclusion. That is the issue of the inferences strength was the degree of support claimed actually achieved. Now, once you evaluate in inferences strength on Lee after that, can you go on to talk about whether or not it's factual claims were true? So once you go to this flow chart, you're gonna have three simple steps. What is the type of the argument, the inferences, strength and Onley? Lastly, whether or not it's factual claims held up now, logic, we said, can help you evaluate argument types and inferences strengths. But whether or not the premises air, true or false, that's up to you to decide the key point for now is this. The first and most important issue with respect to argument. Evaluation is evaluating its reasoning and inference, and whether or not it's factual claims were correct. Is the secondary concern the lowest rung on our little flow chart? So if you want to know what the relevance of all this is, and what does the inductive deductive distinction have to do with evaluating arguments? That's what gets us running on our flow chart. Basically, once you know that an argument has been presented, it's all a game uplink. Oh, really? Figure out which direction you wind up on this chart. And obviously, as you look at the bottom, cogent and sound arguments are the ones that we want to put some weight into him by into. So once you know you've got an argument on hand, check to see if it's deductive and whether it's degree of support was the strongest claimed . Lastly, check its premises to see if it was sound, and the same thing goes when you're dealing with left hand form of this particular flow chart. What you want to do, make sure you're dealing with an argument, and if it's of the inductive variety make sure strong inductive inference was present. Then finally, check the arguments factual claims to see if it was, in fact, cogent. Now I want to talk in more detail about what happens on the right hand side of our little flow chart, as we just saw. I want to talk about deductive inferences before I just talked about inferences of that sort. Being successful or not, or being fallacious or not here, I want precise my language. I want to talk about validity versus in validity. A valid deductive argument is the successful deductive inference. That's when the deductive inference claim holds up. It is impossible for the conclusion Be false, given that the premises air true. An invalid deductive argument is one in which the deductive inference claim did not hold up . There remains a possibility of a conclusion being false, even assuming the truth of the premises. The thing is, if you have a valid argument, it has a valid form or shape. It's set up so that the truth of the premises will be funneled into the conclusion, so to speak. That's what we call such arguments genuinely truth preserving They do not lose truth on the travel, from premises to conclusions. In the case of an invalid argument or one that has an incorrect or fallacious form, the truth can be lost on the way from premises to a conclusion. Now, sometimes you might get lucky. Maybe you have true premises, and it just happens to be the case that you wind up at a true conclusion. But that's not in virtue of the arguments shape. That's just a matter of luck, because invalid argument forms are such that if you'll pardon the pun, they can't handle the truth. So there are several principles governing validity and by the same token, governing in validity. First of all, no middle ground. As we said earlier, validity is all or nothing, and validity is not a matter of the truth values the premises of the conclusions. It's rather just a matter of the relationship between the premises and the conclusion. So here's a valid argument that has true premises and true conclusion. That's a nice case if all men are mortal, as the first premise claims, and Socrates is a man, as the second premise claims, Socrates is indeed mortal. But validity does not require true premises. In a true conclusion, even though this structure is a very good one, you could have a valid argument, which you had false premises and a false conclusion like this one. All television networks are terrorist organizations. NBC is a television network. Does it follow that NBC is a terrorist organization, actually, yet would basically that would happen for all the same reasons as in the previous argument . If all television networks fall within the category of terrorist organizations and NBC, at least this premise is true is a television network that puts NBC squarely in the blue container of terrorist organizations again. Valid argument. And it doesn't matter whether the premises air true or false. It's a matter of the structure of this argument. Now here's an argument that's invalid, but it has true premises. In a true conclusion, if all banks or financial institutions will put our red circle here right within the category, The Blue Circle and premise to says Wells Fargo, is a financial institution. But what does that tell us? It tell us what tells us that Wells Fargo goes somewhere in the blue? Does the conclusion follow that Wells Fargo is in the Red Bank category that was not secured by these premises. Now, this may be a little confusing. We have a true conclusion here. Is the argument still junk? Just because the reasoning or the inference was a bad or invalid one? And the answer here is yes. This junk argument could just as easily have led us to a falsehood in virtue of its bad form or bad shape. So if it does lead us to the truth, pretty much were in the position of the proverbial blind squirrel getting lucky and finding a nut look at that argument once more. It's invalid, and it has true premises in a true conclusion. Now notice what happens if I swap out the term banks, financial institutions and Wells Fargo. How about instead squirrels, animals and Socrates? Same type of argument. I just swapped out the terms in the red, blue and black, respectively. If all squirrels are animals and Socrates is an animal, does it follow that Socrates is a squirrel? Well, once again, this illustrates the point that these sorts of arguments are invalid precisely because they're unreliable. So deductive arguments remember, are valid or invalid degree of support is 100% or zero. Trust him or don't there's no middle ground now. Soundness applies to a deductive argument when two conditions are met. Basically didn't have a sound argument. You have to have a valid argument with all true premises Now. In those cases, you have a great argument. An unsound argument is either invalid. It has a false premise where may fail on both counts. But soundness requires that both of the key elements of a deductive argument be satisfactorily met. Strong inference were appropriate inference and true premise. Now, in order to see how are new and precise language applies the arguments, let's go back to an illustration that I've been using throughout this lecture series all Vulcans or cannibals. And Captain Kirk is a Balkan. Is Captain Kirk a cannibal? Is there something illogical about this argument? Well, let's use more precise language now that we have it in hand. What we asked ourselves is whether Captain Kirk, if he was as described as the premises describe him, would be, as the conclusion describes him as well. And the answer that we came up with there was, yes, if all Vulcans, air cannibals and Captain Kirk is a Vulcan that would automatically make him a cannibal. So there is a sense in which the logic of this argument does hold up. It is a valid argument, we would say. But Spot can still take on bridge and say, This argument is unsound. And the reason is the argument contains false premises. Actually, both premises in that argument were false. Now let's take a look at the different example. What if we say all Vulcans air rational and Captain Kirk is rational now that for those you buy into Star Trek lore are true premises, right? But what about the conclusion that Captain Kirk is a Vulcan? Is there something illogical about this argument? Let's take a better look. Suppose we say that all Vulcans air rational, and then we say, Captain Kirk is a rational being. Well, what happens then? If Captain Kirk is a rational being, that doesn't tell me whether or not he's a Balkan. He could be anywhere in that red canister, and actually he's in the part that does not include Vulcan's. So, in this case, again, the argument is unsound, but for a different reason. The premises air true But the logic does not lead to the conclusion the logic is deducted. Lee invalid. See how that works? Now here's a question for you. Can a ballot argument have a false conclusion? I want you to think about that. A second. Ah, valid argument. Yes, can have a false conclusion, just in case it has false premises that you started with. But can a sound argument have a false conclusion? No, it cannot. Sound arguments always have true conclusions, so be careful not to call an argument sound unless you've looked at that conclusion really carefully. Sound arguments always have true conclusions because of two factors. The premises are true, it's admitted, and the structure of the argument is such that truth of the premises will be funneled down into the conclusion. You put those two points together, and that gets you to the result that sound arguments in virtue of the true premises and the true or truth preserving structure, the argument will always have true conclusions. Now let's work down the left hand side of the flow chart. We're investigating earlier inductive inference is, once you know you're dealing with an argument and you know it's inductive What is the inference claim? The inference claim would be that the conclusion follows probably upon the premises. If that claim holds up, you call the inference a strong one. That means it's improbable for the conclusion to be false, given truth. The premises. On the other hand, if the inference claim does not hold up than it's a week inductive argument, that means the conclusion does not follow from the premises, even though it's claimed to. So there's different principles that govern inductive strength. First of all, there are degrees of strength and weakness. It's not an all or nothing issue. But as with validity, the strength of the argument depends solely on a relationship between the premises and the conclusions. But lastly and most importantly for us right now, additional premises will tend to strengthen or weaken the inference. So the principle of total evidence that I'll talk about later applies uniquely. In the case of inductive arguments, consider this. This barrel contains 100 apples. Three apples selected at random were found to be ripe. Can we reach the conclusion that probably all 100 apples are ripe? This is a case of a week inductive inference now, by way of contrast, if this barrel contains 100 apples and 80 apples were selected at random and found the right, then we have a pretty strong inference to the claim that all 100 apples were ripe. So in the first case, what we have to say is that we have a low degree of inductive strength, and in the second case, we'd say the degree of inductive strength was rather high. Now, strength of an argument does not depend upon truth of the premises or truth of the conclusion. Again, it's strictly a matter of the relationship of the information in the premises and conclusions. Consider this. All dinosaur bones found to this day have been at least 50 million years old. Therefore, probably the next dinosaur bone to be found will be at least 50 million years old. That's a strong argument, and the conclusions and premises are all true now. Here's another strong argument, but with false premises in a false conclusion, all comets found to this day have contained intelligent life forms. Therefore, probably the next comet to be found will contain intelligent life. Now you may be looking at that and wondering why does that count as a strong inference? Well, for exactly the same reason is the previous example there the same style of argument all found this day have been a certain way. Therefore, probably the next will. So if the inference in the case of the dinosaurs was a strong inference, then you should see that the case of the Comets involves a strong inference as well. So strength of inference has nothing to do with truth or falsity the premises. It has to do with the relationship of the information found in the to. Now here's a weak argument that has all true premises and a true conclusion. Over the past five decades, the national debt has increased dramatically and the US will probably remain a leader in world trade for the next decade. Now the premise is true, and the conclusion is true. But what's the relevance of one to the other? Here we have a week link between the two and consequently not a very good argument. So Coach Insee is a term kind of analogous to soundness that applies to an argument of the inductive sort when it is strong and when it has all true premises. If an argument of the inductive sort is strong and has all true premises, you call it cogent and it becomes a NCO geant if it fails on one or both of these points, either by being weak or by having false premises, maybe both. Now one last thing, because inductive inference is, have a different principle, namely, the principle that additional premises can add or weaken support. We have a new principle that only applies in the inductive cases. That's called the principle of total evidence. Consider this. Swimming in the Caribbean is usually lots of fun premise. Today the water is warm, the surf gentle and maybe as a further premise on this beach, there's no dangerous currents. Seems like we have a lot of evidence for the claim that it be fun to go swimming today, but you might find out that there's extra information regarding what's going on in these waters that would undermine that inference. So here's my two cents on what goes on in these sorts of cases, and it is the standard view on these scenario. Inductive arguments that violate the principle of total evidence automatically count as weak arguments now That means that if you overlook evidence when you're filling out your premises, that would have weakened the leap from premises to conclusions. If you overlook that evidence, your argument argument is automatically counted as a bad one in virtue of the weakness, the lack of strength in the inference now that should make sense to you if you think about it. Was the Caribbean swim argument a cogent argument? Well, that should be straight forward to answer. Absolutely not. It couldn't be cogent, because Coach Insee requires both a strong inference and it requires true premises. But did that argument have all true premises? Yes, it did, and that's a good reason for not starting out with the truth. Values the premises as your standard for evaluating arguments. Look at the inference. First, the overall similarity between soundness and coach Insee is this. For an argument account of those sorts, you want a correct inference. Basically want the inference claim that was made to hold up and you want all true premises and unsound or a NCO gin Arguments are the ones in which there's a bad inference. Basically, the inference claim being made did not hold up or you have false premises. Maybe both. So again, when just to review when you're evaluating arguments, check out whether or not they're inductive or deductive. Now, once you figure out the arguments type, then you can move on to decide whether or not the degree of strength claimed for the inference actually held up or failed to hold up. And if it did hold up now you're in a position last of all to check out the truth, value the premises. And if you do that, hopefully you'll only end up accepting the good arguments, which are, respectively, the cogent arguments in the inductive case and the sound ones in the deductive case. So there are three issues that you have to work through, and logic is there to help you with the 1st 2 evaluating argument types and whether or not the inference claims hold up now. Once you do that, as promised, it's all the little plane. Co game is an argument. If so, maybe deductive Good inference. If yes, then check the premises. And if they hold up, you've got a sound argument, and a similar form of reasoning happens in the inductive case. If you follow these steps. Hopefully, you'll never be taken in by bad arguments. So that's all for now. It's quite a mouthful. So wait for my logic exercises on this video. And more importantly, my next lesson. You've got quite a lot to digest here. And if you ever watch one of my videos several times in a row, this is the one that I recommend for you. We'll see you guys next time and we'll study fallacies. See you then. Bye bye. 4. Fallacies of Relevance: Hello, everybody. And welcome back to my crash course and formal logic in this section, I'm going to study formal and informal fallacies because by now you've learned an awful lot about arguments, deductive and inductive, and we're gonna study the techniques of analyzing those argument types in just a few lessons. But first, a little detour. Now fallacy is a type of defect in an argument other than the falsity of its premises. And that defect always generates bad inferences. The defects were gonna study can be of the form or informal variety. Now, when I talk about a formal fallacy, the term formal has nothing to do with formal wear. Formal occasions. It has to do with argument forms. We said earlier arguments sometimes have shapes, so a formal defect is a defect in structure, and those types of fallacies are only found in deductive arguments. And in order to find them, all you have to do is look at the form or shape of the argument and see if it's a correct form. By way of contrast, and informal fallacy is not a defect pertaining to form. Rather, there's some type of diversion or vagueness, some kind of illicit assumption that's throwing off your inference. In order to detect those, you've got to actually examine the content, not just the shape of the argument. Consider this all bullfights or violent events and all A Q Shins are executions are violent events to true premises. Does it follow that all bullfights or executions? Well, maybe opponents of bullfights would like to think so. But consider this. We could swap out the term execution, swap it out for the term boxing matches all bullfights or violent events. All boxing matches are violent events. Therefore, does it follow that all bullfights or boxing matches? Now that seems ridiculous. Bullfights are violent events and executions or violent events, but it does not follow in either these cases, actually that all bullfights or executions or that all bullfights or boxing matches. What we have here is an argument shape all they are be in all CRB, and we tried to reach the conclusion that all a R C that means we've got a bad shape toe. Our arguments and that type of defect will always give you a bad deductive inference. That's a formal fallacy. Now consider this. If apes are intelligent, than apes can solve puzzles and apes can solve puzzles. Does it follow that apes are intelligent? Well, this argument has the structure. If a then B and B is satisfied, should be moved to the conclusion that a now consider this example. If Tom Cruise is a physicist, then he can solve puzzles and he can solve puzzles should be moved to the conclusion that he's a physicist noticed here we have the exact same structure of argument, but in this case we reached a ludicrous conclusion on the basis of two premises. What went wrong? Well, what went wrong was a problem in each case involving bad shape or forms of arguments. In either case, if a then b b therefore a or all Air B all CRB therefore all a r c. These are bad argument shapes or forms, and they're to be rejected any time you find them. Now informal fallacies are different. Consider this. If the Brooklyn Bridge is made of atoms and secondly, Adams are invisible. Does it follow that the Brooklyn Bridge is invisible? No, it doesn't. But what went wrong with our inference? What happened here is that we moved from a property of the parts. The Adams and we moved to a property of the whole the bridge, and sometimes that sort of moved from parts. The whole is justified, and sometimes it's not. In this case, though, we've committed a fallacy. Now here's another one. A chess players, a person. Does that mean that a bad chess player is a bad person? Well, now what do we do with this sort of inference? The problem here is we're using two different definitions of the word bad. Bad could mean without moral character, like a bad person and bad comedian lacking competence, which means a bad chess players, not a competent chess player. We've equivocated upon our term. Now you notice that in the last two examples, there were no defects in the shape or form of an argument. But rather we had some illicit assumptions going on now going to study those sorts of things in detail. So let's talk about informal fallacies, starting with fallacies of relevance. Now these sorts of fallacies work by appealing to your emotions and or by negatively characterizing somebody you might dislike or appealing delays in his pride, superstition, anything like that. So that you will accept the conclusion. The way I like to characterize it is that you have two sides, your brain, one side that deals with reason, logic, control, science and those sorts of things, the analytic matters and the other side that deals with intuition, creativity and passion. What I think is happening here is that the left hand side of the brain and the right hand side the brain aren't doing their respective jobs. So what happens is that you get premises that are logically irrelevant to a conclusion. But they start to appear psychologically relevant because part of your brain that deals with things like intuition and emotion and things like that starts to get involved, making your inference for you first fallacy up in our list of studied the appeal to force argument. Um, add back alum. Actually, it means the appeal to the stick that occurs when an arguer it motivates you to make an inference by some sort of psychological or physical threat of harm, rather than by logical connections between premises and conclusions themselves. For example, somebody may say spoiling smoking spoils her looks or for every bush but vote God kills a kitten Now there's a very clear and explicit appeal to not logic, but actually a sense of irrational fear to motivate a conclusion. The conclusion being you shouldn't vote for Bush. Well, what about this? Intelligent design debates and evolution debates are getting off the ground really heavily still. But what about people who say that if we allow intelligent design in the classroom, will have to allow flat Earth, geography or different theories of sex ed and burn witches and use leeches in medicine? Now that seems like an argument. Mad Back Alum. They're using a psychological threat of harm to motivate the conclusion that we shouldn't teach people alternative theories of human origins. And that's not fair. Now in this cartoon, Charlie Brown is running away from one of his friends who says, I get you Charlie Brown, I'll get you and I'll knock your block off and Charlie Brown target around and stops Or wait a minute. We can't carry on like this. We have no right to act this way. Actually, the world's filled with problems. And if we Children can't solve are relatively minor problems, how can we expect the adults to, and wacky takes it right on the jaw, she explains. I had to hit him. Quick is beginning to make sense now, Thankfully, for us. Most of the time, when you find an argument a Mad Back alum, it appeals to an imaginary threat of harm or an imaginary threat of force. In this case, however, Charlie Brown got the Rio variety. Now you're probably feeling like you've gotten the idea of the idea of an appeal to force and why it's a bad inference generator. But I want to note here that some arguments involving worry and fear are not fallacies. There are such things as arguments from or four reasonable concern if somebody says to your friends or natural born jokers and tomorrow ST Patrick's Day. Therefore, you shouldn't wear green because you'll get pinched a lot. Well, that seems like a good argument for the claim that if you don't wear green, you are going to get pinched a lot. Even though there's worry involved in this argument, that's not what backs the inference or justifies it. If somebody tells you you've been late to work three times this week in your boss fired the last person who's late four times Therefore, you'll probably be fired unless your attendance improves. This looks like a good inductive argument for the conclusion in question, and whether or not worry is involved is not a concern here. What happens is that with the two sides of your brain or doing their work respectively, one side is saying on the emotional end, I'm scared and the other side governing reason and logic says Yeah, and based on the evidence you probably should be no fallacy is committed in those cases. Closely related the appeal to pity argument. Um, ad misericorde um, occurs when an arguer attempts to motivate the inference simply by a pity, evoking now that is a fallacy and bad inference making. So, for example, if somebody tries to argue that they shouldn't be found guilty of driving through a school zone at a reckless speed on the following grounds? Well, after all, it's not fair. I was late to work. I lose my job and I have four kids now. This guy is sympathy mongering. He's guilty as sin, probably, and once again, you got to be careful. Not every argument that evokes pity is a fallacy. Sometimes very good and sound arguments invoke pity, but the inference was not generated simply on the basis of pity. So if you see an argument that you ought to be contributing to world aid or something like that, this is an argument from compassion that may evoke pity on your part. But the inferences were justified by logic alone, and the same thing holds in court cases in which you find mitigating circumstances like the age of the defendant and knowledge of the relevant laws. Inappropriateness of sentences. If you find arguments in this that evoke sympathy on your part, double check to see if the logic backs your inferences, which you may find is that in your brain you're getting some sniffles out of a case that ought to write Leah according to the reasonable side of your brain. Be evoking some sympathy on independent and logical grounds. Appeals to the people a very popular argument, Um, at popular. We always have a need socially to be like to be loved, to be admired, to belong. But if we make inferences based on those needs alone were committing fallacies. Schroeder is a good example of this actually is a good example of how to avoid it now for the number one hit song across the nation. And when Schroeder listens, he says, this nation's in sad shape. That's probably the right inference. Whether or not a songs popular does not determine whether it's not the song was actually worthwhile or worth listening to at Popular's could be of the direct or the indirect variety. Now, when it happens directly, an arguer is in the context of all his friends or a group of people. And the group of people gets excited through emotions, and a person tries to just go with that crowd that they're part of. This serves in cases of mob mentality when people make speeches at conventions, things of that nature. When this happened, you often time times find emotional accusations. Communist, you right wing fashion hater, something that gets people's emotions in the mob charged up so that individuals in the mob are tempted just to go along with the flow. The indirect variety is much more common now. Look at Calvin here, turning on the television and just asking it to pander to him. He's being addressed separately by the television, but typically the television will remind him that he has a relationship to the crowd and that he wants to win their exception. That will get Calvin here to make bad inferences. So when people were reminded that there are part of a crowd, even if that crowd is absent, well, then you've got the indirect approach. Ah, person could be reminded about this through advertisements thinking that if they do what the advertisement says, like buy a product, then they're going to be loved and accepted, just like the people in the advertisements are. That's a clear case of an indirect add popular argument. You often find this indirect, add popular mused in advertisements, because advertisers have to address individuals in their homes or individually away from crowds. Thes air advertisements are often bad inferences to the claim that you ought to be buying our product. Now. Bandwagon ing is another and related form of indirect at popular. Basically, the idea is, if you don't get on with everybody else or get on the bandwagon, then you're going to be left behind and nobody wants to be left out. Were just social animals that way. So if you get that sort of inference or motivation for an inference you're making a bad one . An informal fallacy, we said, and appeals to vanity or snobbery occur especially in advertisements. When someone associates the product with someone admired or pursuit or imitated so that they'll make you think that if you buy their product, you too, will be admired and pursued. Check out this example. If a little girl wants to be like Britney Spears, go out and buy some milk and drink it. Actually, I like the conclusion of that argument. But the motivation for the conclusion is fallacious. So remember Calvin, when the television address is, um, panders to you, it's typically reminding you that you are a member of a larger crowd and trying to make you make inferences on the basis of that sub rational and sub logical motivation. Now let's talk about arguments against the person, the ad hominem argument there, three sub varieties, the abusive circumstantial and, to quote, okay, argument. What they all have in common is that they involve to argue hours. Somebody makes an argument, and the second person tries to get around that argument. So it's a diversion strategy. Get around the argument and attack that first argue hours. Person directly. Ad hominem is not a legitimate debate tactic. It is not a legitimate way of responding to an argument. If you want to respond to an argument attack, it's factual claims or the strength of its inference, not the person who made the argument. The ad hominem abusive is pretty common. That happens when, instead of responding to the person's argument, you just pour out verbal abuse on the person and their character, say bad things about them. So if you respond to a person's argument by name calling, calling them a trader coward, Communist fascist will. Then if they've made an argument and you try to divert this way, you're committing the ad hominum fallacy. In the abusive variety, the ad hominem circumstantial is much more common dough. You don't pour out bore verbal abuse, but you respond to an opponent trying to discredit them by associating with the them with certain circumstances or showing that they have certain predispositions to argue the way they did. Basically, you place Sigmund Freud and try to psychologically analyze your opponent. This is a very common diversionary tactic, so if you're in the context of a debate and your opponent has presented an argument, and then you turn around and try to analyze what's going on inside their head. Chances are you are committing the ad hominem circumstantial fallacy respond toe arguments when their president don't respond to the arguer or make unsubstantiated psychological claims about their person. The tu quo que fallacy or the YouTube fallacy occurs when an arguer tries to defend themselves by saying, Yeah, but my opponent is Justus guilty now Notice. This is sometimes called the two wrongs. Make a right fallacy. If your opponent is wrong, that doesn't automatically make you right. If you want to risk the fallacy. Here is when you assume that by attacking them, you've defended yourself. So in this case, you have a political cartoon with the GOP attacks the Democrats by a U two fallacy. In this Garfield cartoon, for example, Garfield is busy attacking normal. Now let's assume that Garfield has made an argument that you're too cute and cute, is tasteless and cute, rots the intellect. But Normal commits the fallacy by saying, What's so hot about ugly? You got a problem with your personal appearance to Garfield, and whether or not that's the case, it doesn't really defend Nirmal against the original argument. Normal has to do better. I don't have any easy pneumonic device for remembering the Tu quo que fallacy or the u two fallacy. But maybe this little slide will help you out. If so, all the better. Their songs are really just three chords long all the darn time. I'm not trying to say here that you can never attack a person or make an ad hominem attack like somebody's in Agra or somebody is not doing what they need to do for their kids. What I'm saying is an ad hominum fallacy is committed as a Dodge to get around somebody's arguments. Now I'm not ruling out simply presenting negative reports on somebody. Sometimes that's perfectly rational thing to do. For example, John Wayne gacey, I think, is a perfectly evil person on the basis of the premises that he killed a lot of young, innocent men and hid their bodies in the crawl space of his house. Now those observations are relevant to the conclusion of what kind of person John Wayne Gacey is. Furthermore, observations about a person may be relevant to whether or not a person actually did something or committed a crime. It may be relevant to promises and testimony as to whether or not you should put any weight in them. When you attack promises and testimony, or you attack a person's guilt or innocence on the basis of these observations, no fallacy is bidding being committed. Remember, the fallacy only occurs when you try to dodge people's arguments by going after the person directly. Now, straw manning or the strongman fallacy occurs when you distort or make a straw man out of your opponent and there views by attacking the view that you've distorted. You try to reach the conclusion that the opponents original position has been destroyed. You set up a straw man and bang gave it a thrashing. Before you go attacking anybody's position, it really is your obligation to try and understand their position as best you possibly can to avoid misrepresentation and simplification. If you go ahead and misrepresented, simplify their view like well, if you try to simplify evolutionary theory to make evolution a nice target for your attacks , you knock down a straw man and claimed victory. Those three steps always commit the straw man fallacy, but the creationists are not the only ones committing this fallacy. Try this one. Evolutionists also often accused creationists of just saying hypothesis. Darwin was wrong. Experiment. Read your Holy Bible and here's the results. Now I know that the creationists are wrong, but this is not the way that they arguing. It is not their view. It's very unfair to them to misrepresent them this way. Now Straw Manning galore happens in these sorts of debates. Consider this cartoon, somebody says. Teach an alternative theory to evolution and somebody distorts the position. To say what you mean is teach the alternative toe. Everything teach alchemy as opposed to chemistry. Teach Magic is opposed to science. Teach Astrology is opposed to astronomy. Clearly, the original position that alternative theories of origins ought to be explored is being distorted maliciously to make it an easy thing to attack. The same thing happens in this vaguely disguised attack on the A, C. L. U. And one panel, the A C O. U is depicted as defending people's expression of religious opinion in the right hand side . Hypocritically, they're depicted as being against displays of religious expression. Now wait a minute is the A c o use position. Really that all and no positions on religion ought to be publicly expressed. No, there's a difference between personal and public displays. And so the a c o used a position is being distorted to make them a straw man. And in this case, you have evolution. Being attacked by somebody who depicts evolutionary theory ist is basically saying we don't know any of the details yet, but we really know we evolved now. That's not the evolutionist position. They think we know some of the details in fact, a great many of them. But their position is being depicted as a position arguing from ignorance, and that's not fair. If you want to study Straw manning more, just go to your favorite political talk show program and you'll find tons of straw manning happening in the political arena. But let's move on missing the point of the ignore a show. And she is a term devised by Aristotle, meaning ignorance of the proof. And the reason is the premises of the arguer argument does support a conclusion. So what goes wrong in these arguments? What happens is that the arguer draws a different or related conclusion that was not supported by the premises. If you're gonna criticize people who make this mistake, be able to identify the correct conclusion, the correct conclusion is the one that genuinely did follow from the premises. Consider this political cartoon a sociopath, a gun control advocate in a law abiding citizen or sitting in a bar and sociopath pulls a gun on the two. What is the gun advocate do? He advocates more restrictive gun control laws? Looks like we have premises that would back that sort of conclusion, but he advocates them on the law abiding citizen. Now the law abiding citizen, according to this cartoonist, ought to say, Hey, you drew the wrong conclusion from the fact that we just discovered red herrings are another type of diversion. Eri fallacy. The arguer tries to divert the subject to a different and sometimes suddenly related subject, and it assumes here's the fallacy that by settling the second issue that one diverted to their settled the original issue that was brought up. So did you order the plastic bags ass Dilbert? Well, they take two weeks for delivery notice. There was a shift in the conversation, Dilbert says. I see, you've avoided my actual question in favor of an imaginary one, he says. Now I'm fantasizing about ripping your mustache off, and the guy says, Well, I hear that a lot again diversionary or red herring tactics to avoid arguments or issues and since I do advocate some form of evolutionary theory at kind of hesitate to bring up this example. But in the first panel, somebody brings up a scientific challenge to evolution. What is the red herring do diverse the topic to whether or not we should have separation of church and state. The third panel illustrates the right response. Reaffirm the original challenge, and this illustrates that the red herring could be committed in two different ways. There is the subtle technique. In the last cartoon we looked at, there was a subtle change in subject. We stayed on the topic of evolution and the relationship of religion and science. But the change in topic was a subtle one so that the reader or listener wouldn't catch it. The alternative way to commit the red herring is just use some glitzy, eye catching topic guaranteed to distract because people would rather talk about things like sex crime and disaster than a genuine intellectual topic. Just about any day, just a couple last notes on the red herring. The term red herring actually is, comes from Ah, hunting dog history. If you had a good hunting dog and you trained them, they should not be distracted by powerful smells like red herrings. If, for example, they were supposed to be hunting a rabbit, and the same thing goes for you as a logician, if you're supposed be hunting down good arguments in good positions, don't get distracted by either subtle changes in topic or distracted by a glitzy and eye catching ones. Don't let the red herring take you off the original trail. Red herrings are a very common hot seat maneuver. They occur precisely when a person can't get around the original argument or topic and would like to move on to a different one. Hold people to the fire. But those of you who need help reviewing here's a little short list that you can refer to to memorize the seven fallacies and their subtypes that we have discussed today. Well, that's all for now. Thank you for listening to about 20 minutes of ah logic material. So far, so good weight from exercises and my next logic lesson on other types of informal fallacies will see you then. 5. Fallacies of Weak Induction: Well, everybody and welcome back my crash course. In formal logic, as promised, we're going to study some more fallacies before we start moving onto techniques and ah, technical issues in deductive logic. Now here, we're gonna study fallacies of week induction, and this is gonna be a kind of shorter lesson than normal because I think we conducted fallacies are quite a mouthful of material on themselves. So let's get underway now. We all know there's a danger to inductive arguments. The danger is you're gonna leap off from truth into a falsehood. There's always that danger even when your premises air. True, if you do an inductive argument, the question is, when does that become an irrational leap? So what you want to do is avoid falsehoods and stay safely within the secure. Ah, shelter of your true premise is the only way you're absolutely safe when you do. That is if you stick to deductive valid arguments. However, if you stick your neck out beyond your true premises and take a risk, you're doing induction and sometimes that's reasonable. The question is, how far out should you stick your neck, stick it out too far and you're committing a fallacy, a fallacy of week induction. Now we studied a variety of inductive argument forms, including generalizations and allergies, causal inferences, and I gave you an acronym gas cap to remember them by when those become relevant here, I'll bring the topic back up. But for now, let's take a note, too, that fallacies of week induction are unique and that the premises genuinely do landis. Some support the conclusion, however. The fallacy lies because there's some type of weak inference. The link between the premises and the conclusion is not strong enough. And sometimes when that happens, the inference is supplemented with some emotional motivation when the logical support is actually very low. For the conclusion, let's talk about arguments from authority. That's when we base our conclusion on premises regarding some of presumed authority or witness. And we added early on that it's amazing how much we trust these sorts of arguments on matters of science, chemistry, even on matters of history and our local news reports. But there is such a thing as appeal and appeal to an unqualified authority. The argument TEM ad very Kundi um, occurs when the cited authority or witness lacks credibility. They may lack expertise. Consider how many people get their political opinions from rock stars and celebrities. For example, those air unqualified authorities. If there ever world some and sometimes persons may have a bias to mitt misinform or a motive to distort the fax, Lucy says. Destroyer, You don't believe me, do you? It's a scientific fact that girls are smarter than boys and guests. Who discovered it? You guessed it. The women's scientists. I think my reaction here would be the same as Schroeder's. And sometimes a witness say in a courtroom may have lacked the ability to perceive a recall . Well, somebody who lacks the requisite mental or perceptual faculties. In those cases, your witness constitutes an unqualified authority. Now consider how often this sort of appeal is made in advertisements. Can Gwyneth Paltrow, Cameron Diaz really speak on behalf of the Union of concerned scientists and how much you're gonna take on the authority of Whoopi Goldberg regarding what hotel changed? Attend. Now here's some important considerations. How much authority do you really need to give somebody a good amount of inductive support for conclusion on simple matters, Justin Eyewitness will suffice or a common person's opinion, say, and that commonly holds in courtroom situations. But on matters of academic import, you may need a high degree of education, maybe a PhD in a field to speak with any authority. And secondly, even then, make sure that the witness or authority is speaking from the consensus of scholars in the field that they're not some sort of maverick in the field who has a unique opinion disagreeing with everybody else's. The appeal to ignorance is another type of weak inductive fallacy. The argument, um, ad ignore ation occurs when somebody says there's nothing that's been proven one way or another about something, So let's draw a definite conclusion one way or the other. For example, no one's disproved it. Therefore, it must be true, or somebody says, I have no proof for it, so it must be false. Very often, these sorts of appeals constitute fallacies. So, for example, if two people are having a theological dispute, saying you can't prove there's a God implicitly saying, therefore there isn't one, or if somebody says you just try to prove there isn't one trying to argue to the conclusion that therefore there must be nothing follows from either person's argument, they're both committing appeals to ignorance. I'm reminded of Ah Ben Benjamin Franklin's statement and Poor Richard's Almanac many a long dispute among the divines that theologians maybe thus abridged it is. So it is not so. It is so it is not so. A classic example of this sort of fallacy came in the 19th century, when people doubted that the Old Testament references to the hit tight empire genuinely referred to anything historical. Well, what happened was later on, in the last century, the 20th century, people actually found the hit tights. And now I guess you can do your PhD and hit cytology if you want to. So the absence of evidence for the hit tight empire did not genuinely constitute much evidence of absence. Now, by way of contrast, we could say no one's proven. There's life on Mars and there's no one's proven There's life on the Moon. Can we draw a definite conclusion from that? Well, I'd like to think so, but what makes the difference now consider this. If qualified, researchers in their area cannot find any evidence for a phenomena, say life on the Moon or life on Mars after sufficient investigation, then you have good inductive support for the claim that that phenomena does not exist. So in the case of the hit tights, apparently there was not enough investigation done. And I want to note that it's not always necessarily necessary to be an expert if I want to give you an argument that there's no squirrel in my closet. Well, I'm no expert on squirrels, but all I have to do is look around my closet a little bit. Now in a courtroom procedure, there's something called a presumption of innocence until proven guilty. Don't confuse this with argument, um, at ignore ation. What's going on here is a presumption of innocence, not an argument for innocence until proven. So let's talk a bit about generalisations. That's when you move from knowledge of a sample to knowledge about the whole group. Early on, we said that Snoopy had trouble with this because even though dogs been howling at the moon for over thousands of years in the moon hasn't moved in, dogs are still dogs. He thinks it proves something, but he can't seem to draw the proper generalization. Well, sometimes people have too much of a talent for generalisations. They make generalizations even when the sample size taken is too small or alternatively, when it's unrepresentative notice. When people do this thanks and can sometimes end up with racial prejudices and bias because they haven't experienced another culture or race enough. And when that happens, the degree of support for the conclusions they draw goes down drastically. So here's the question for you. What is too small? Quote unquote of a sample. Sometimes I'd say you could get yourself into needle and haystack scenarios. What constitutes two small often depends on how tough it is deploying the thing you're looking for. Now consider this. How big the survey. The need to discover what percentage of people in your college or mail will mail nous and gender characteristics. Air Not exactly Needles in a haystack. They're not hard to find. A small random survey should suffice. But for a more rare thing, like how many are libertarians? You're gonna need a larger survey or your estimate will end up too low and too skewed. And that's especially true if you try to figure out how many of them have relatives with pancreatic cancer, because that's really rare. Unless you do a large enough and random enough survey, you're gonna end up with a zero estimate, and that's probably wrong. So whenever you don't find what you're looking forward to quickly, don't be surprised if what you're looking for is rare. Now, the lesson here is that when you're looking for a common feature, a small random survey will suffice to give you a good inductive inference. But at the more rare your feature, the larger a sample you have to take. And there's a second issue here. What if your survey ends up non random? The classic cases? When the Chicago daily Tribune said that Dewey defeated Truman in an election when Truman had actually won, what happened was that Dewey, uh, was heavily favored in a poll that was skewed towards the Republican Party. So the lesson here is large samples to commit the fallacies, even when they taking a large amount of numbers. That happens due to something called biased statistics or biased polling, and it's related to the fallacy of hasty generalization. Now let's talk about causal arguments that happens when you move from knowledge of a cause to knowledge of effect or conversely, and more commonly, vice versa. As we said in an earlier lecture. Now there are problems detecting true causes that most people don't take into account. Oftentimes, it's hard to tell when to phenomenon related, because the cause and effect could have a great deal of time than between them. And relatedness comes in degrees. There could be a lot of causes for an effect or a lot of effects of a cause, and humans have motivations to exaggerate or misinform on causal claims. We all have them, and statistical correlations between two things may reveal little. Maybe the two things being correlated are causally related to one another. Or maybe there's some third factor involved that we haven't considered yet. Consider this false calls. Fallacies can come in a variety of forms in which the imagine causal connection doesn't exist, and the conclusion that one thing caused another is reached by week induction. So therefore, sub varieties the post hoc ergo Procter hoc after this, Therefore, because of this is probably the most famous. That's ah happens when a person takes the temporal order of events as the only relevant consideration. After all, causes proceed effects in time. Does that mean that you can detect an effect any time that you see a cause happening? Previously in time? Well, that usually attends superstitions. For example, people would say, Hey, every time that I went on to the baseball field, my lucky rabbit's foot, I had a good game. Well, yes, one thing proceeded the other the bringing of the rabbit's foot to the having a good game that doesn't make the to causally related. Ever noticed that every time you put garlic beside your window sill, you have a night that doesn't include vampire attacks? Wonder if those two are causally related? Well, what about this? Ah, what a sight Does this thing ever erupt? Note. We keep it pacified with virgins. Human sacrifices? Yep, well, that's barbaric. Well, it's been 5000 years, and we've never had an eruption. Lady responds, while you got a point, actually, post hoc. Ergo, Procter Hoc post putting the virgins in. We had non eruptions. Therefore, because of putting the virgins in, we had not eruptions. Well, consider this classic Peanuts cartoon Siri's. When it starts to rain, Charlie Brown runs away, but Linus stands out in the rain and says rain, rain go away and come again some other day the rain stops and he quickly runs home, obviously under the impression that somehow he has rain powers that were not previously discovered post hoc. Therefore, Procter Hoc, often confused with this is the non cause of pro cause of fallacy that sometimes called the correlation equals causation, Fallacy. Now that what happens when you take something to be the cause when it really is not. But the mistake is based on something other than time succession, some sort of correlation that doesn't necessarily involve time that this policy often occurs when people try to make inferences to the best explanation. In much the same way that Sir Isaac Newton made an explanation of the fact that ocean tides roll in at night due to a gravitational force, some people make inferences to the not best explanation. We said, like when your pants don't fit and you suddenly come to the conclusion that the dry cleaners shrink all of them. Although this example could actually be called a post hoc ergo Procter hoc fallacy. Maybe a better example is this Dad, why do you store your socks? Ah, flatter rolled up and he says flat, she says. Well, this study says that as people grow older, they tend to store their socks that way. And the wife responds. Maybe you feel younger if you rolled your socks up. And he says, In here I've wasted my time exercising and eating Oh, brand. There is no causal relationship between how you roll your socks and how you grow older, loose, Snoopy says. Sometimes if you stare at the back door long enough, supper comes out early. Sometimes it works, and sometimes it doesn't, especially if you do it every night. Another example of a fallacy that's maybe non causa, pro causa or post hoc ergo, Procter Hawk. But the example you get to your right ever noticed that the number of people watching is directly proportional to the stupidity of your actions, a non cause of pro cause of fallacy here, whether or not people are watching, doesn't cause you to end up looking stupid. Now, the fallacy of oversimplified cause occurs when somebody takes out a multitude of causes and selects just one, as if it were the sole cause. I wish I could discover the reason for sumo's weight problem. Yeah, gravity is one thing, but that usually reflects an agenda on the or an axe to grind on the part of the arguer. Consider this. Some people like to explain. The rise in teen violence is due to a rise in violence and video games. Is there Ah, coincidence there? I would say this may be the rise in violence in video games is one small cause within an ocean of causes that explain the rise in teen violence. But if somebody picks it out as if it were the sole cause, they obviously have an axe to grind against video games. Similarly, the gambler's fallacy commits a causal fallacy. It rests on the supposition that independent results in a game of chance, or somehow causally related, like somebody saying, I haven't rolled seven and several turns, so the next role is looking really good. Actually, whether or not you've had luck in the previous turns doesn't affect your odds on the next one, so you can commit this fallacy in sports. If you think you have a one in three batting average and you've missed two pitches, does that mean that the next pitch is certain to go out of the park. No chances are just means you've got a one in three chance still of hitting the next pitch . The pitches and your luck there on are not causally related to one another. Well, consider a person who works in sales and said, I average one and four doors a sale. How come I haven't gotten a sale in the last eight? That's because the results here are random. You're gonna get a string of bad luck from time to time, and whether or not you haven't sale on the next door is not affected by your luck on the previous doors. Typically now, the slippery slope fallacy occurs when somebody alleges that a chain reaction of events will take place when there's not a sufficient reason to think that a chain reaction will occur. The idea is basically that there's an emotional conviction, the arguer, that you should never take that first step or knock that first domino because you'll set off a chain reaction that started with an innocent step and finally ending in disaster. The argument typically runs to a conclusion that you ought not take the first step. So, for example, people think sometimes that this got us into the Vietnam War. People on the basis of the supposed domino theory said that if you allow one nation in Southeast Asia to fall to communism, then pretty soon all of Southeast Asia will be lost to communism. Or some people argue that you ought not drink that first drink or smoke that first joint because, after all, it's just one step on the chip away downhill. I'm not really sure if the author of this cartoon is four against gay marriage, but whatever position they take, apparently they see a ah slope of some sort of causal connection between the legalization of homosexual activity on one end all the way. Toothy, open acceptance of homosexual marriage is on the other. I suggest there's a mistake here. Whatever led to the leaders that legalization of homosexual marriages, there's got to be a lot of cultural influences. Playing into that, I'd suggest there's a fallacy of oversimplified cause here and also perhaps a slippery slope. Fallacy now analogies, we said earlier. Our occur when you argue for further similarity on the basis of known similarities. I give you an example earlier involving a rat lab study in which all the agents rats that ate Agent X got sick and on the basis of relevant similarities between humans and rats is humbling is that may be to think about. We argued that if we humans eat Agent X, we'll get sick. But there is a fallacious form of analogy. It's called the week and sometimes called the false analogy that occurs when there are too few similarities between the two things being considered. For example, when Charlie Brown suggests the Linus, why don't you let me find a substitute for your blanket like a dish towel or something? And Linus argues that that's a bad analogy. Would you give a starving dog a rubber, Boni said, or when relevant dissimilarities have been overlooked between the two things being compared ? Now, as an illustration of this latter case relevant dissimilarities? I did read a Garfield cartoon in which Garfield posited, Ah, that we should answer him this. Why is it the case that when they say a man has the mind of a child, they lock him up? While it's also the case that Children are allowed to run free in the streets, well, what do you guys think? Um, I hope you can find some relevant dissimilarities between the two. Obviously, there are quite a large number of similarities between Children and adults with diminished mental capacities. But what about the fact that grown adults with those diminished capacities are physically stronger than Children? Or what about the fact they can pass for mentally adept persons or persons that hold normal rights of grown adults more easily than Children? I think there are some dissimilarities that justify us and making different judgments with respect to those two cases. So this issue regarding number of similarities and making sure you don't have any relevant dis similarities suggests a two step procedure with handling analogies. In general, when you have two things being compared, find the attributes that the two things being compared have in common. And if the first thing is known to have a further attribute, let's call that Z, and you know that the similarities between the two objects account for Z in that first object. Then, if it's a complete account, go ahead and assume that attributes Zia's. I just called it was had in the second, so again, the basic idea if you have two phenomena or objects, say the red one in the blue one here and they have similar properties ABC through whatever h and you know that in the object to the left, illustrated in red those properties a through H completely account for properties e in the red object or phenomena that go ahead moved to the conclusion that the object of phenomena in the blue here also has properties. E. Make sure that a through H, though, is a complete explanation of Z and that you did not overlook relevant dissimilarities. But really, it just comes down to this numbers of similarities and making sure that you didn't overlook relevant distinction. Ah, when you do those two things, chances are you're only gonna wind up with the good analogies and not the bad or flawed ones. Well, I think that's enough material for now. In the next section, we're gonna cover fallacies of ah, variety of sorts. But fallacies of week induction, I think, covers enough material for one whole lesson even though this is shorter than my normal lessons. So hang on for now for my ah exercises regarding fallacies of week induction and better still hold on for my next logic lesson, which will cover the remaining fallacies that we have yet to study. Thanks. Everybody will see you next time. 6. Fallacies of Presumption, Ambiguity and Grammatical Analogy: Well, everybody and welcome back once again, as promised. This is our third and final lesson in informal fallacies in this crash course. In formal logic, in this lesson, we're going to study fallacies, galore, fallacies of presumption, ambiguity and grammatical analogy. So hold onto your hats and after this will get on the technical matters in deductive logic . Now we're to study first fallacies of presumption that happens when you presume something that you were supposed to be proving in your conclusion they include such things is begging the question complex question, false dichotomy and suppressed evidence. Now, begging the question is a very popular phrase. It's actually in the Latin petition Principe, or sometimes called circular reasoning. You talk yourself in a circle, and this way you create an illusion that you provide support for a conclusion when really your premises include that conclusion pretty overtly already. So this can happen in one of several ways. Now the conclusion could be just the same thing as a premises, maybe just restated in different words. When that happens, sometimes people call this strongly begging the question. You said the exact same thing in your conclusion as you did in your premises. So, for example, if we say we know that mass creates gravity because dense planets have more gravity, well, how do you know which planets are more dense? October says They have more gravity. Well, Dogbert says, That's circular reasoning. And Gilbert says, I prefer to think of it as having no loose ends. This is a case of strongly begging the question now. Sometimes you have a conclusion that's not the same thing as a premise, but your premises require a perhaps unstated premise e premise that presumes what you're trying to prove. This is sometimes called weekly, begging the question because nobody who was willing to buy into your conclusion would have likely given into your premise whether it was stated or not. So, for example, if somebody says well, second Timothy, 3 16 says the Bible is inspired, and consequently we can say that we know that the Bible is inspired. Well noticed. The premise is not quite the same thing as the conclusion, but there's an unstated point here. The unstated premises that all claims in the Bible, including the one mentioned in the premise, are true claims now. Who would believe that unless they already believed the conclusion at the bottom that's weekly. Begging the question. Now, another way. The baking of the question can occurs if a person talks themselves into a circle, but a rather large circle. This is harder to spot, like if an employer says nice resume. But I need another reference in the applicant says Joe can do that for me. But how do I know Jill is trustworthy in the applicant says, Well, I can vouch for her notice. This person's really just used his own trustworthiness to prove his own trustworthiness, just in a not so explicit or straightforward way. So if you talk yourself into a circle, even if that circle is rather large and your premises turn themselves back upon the conclusion and vice versa, you're still begging the question. Petesch, Eo Principe against a person. Now these cases of begging the question need to be distinguished from cases we're gonna examine later. If you're not giving this illusion or fakery of support for your conclusion, there's really no fallacy. Forgive my misspelling here of determined inference, but it's sometimes called an immediate inference, and we'll study this in deductive logic. For example, if somebody says some birds or things that live in Antarctica. Well, that just means there's something in this overlapping category between birds and things that live in Antarctica. Something like, Well, for visualization sake, this. So on the green circle, you have things that live in Antarctica on the red circle, you have birds. And consequently, you can say that some things that live in an article are birds. Now that move from one claim to the other is pretty much a begging of the question and sense. But really, what we're doing here is just making an immediate inference from one claim to another. And similarly, if somebody says the prizes behind door number one or Door Number two and you say, Well, that means it's find to switch the terms Door number two or door number one. You're not trying to prove anything here. It's just a immediate following or an immediate inference from the claim that you started with so bear in mind the differences. In some cases, people approved doing some bakery, giving the illusion of support. If there is no such illusion, and you're just pointing out a logical fact that one claim is equivalent to another your not really committing a fallacy. Now let's talk about complex questions sometimes called by Aristotle of many questions. Fallacy. That's what a person asked two or more questions as a single one. And the first question presumes the existence of a background condition by tricking somebody into answering the question. They're trying to trap you into admitting the existence of the background condition that that complex question assumes the classic case of complex question is a question. Have you stopped beating your wife yet? Notice whether or not the guy answers. Yes, sir. Answers. No, he'd be admitting to something like I previously beat my wife. Or How about this question? What if a cop's chasing the guy says, Where did you hide dope? How do you answer that anyway, that you answer the question, you're admitting that you got doping, you hit it and here's one of my favorites. How would you like me to kick yours? Ever find a good way to answer that question? It kind of presumes that you would like for them to kick yours right? There are many questions involved in each of these scenarios, and they're asked under the guise of one question. Now, complex questions need to be distinguished from leading questions that sometimes happens in a courtroom when and attorneys say, is questioning somebody on the stand and tries to give them information or prod them towards a certain answer. Now that happens. When that happens, there are no logical fallacies. But you can say that the person in question cheating by giving the answerer some information anyways. So leading questions are a distinct topic, from complex questions moving on false dichotomy, sometimes called the either or fallacy or the fallacy of false dilemma. Very popular name for this fallacy occurs when you give to unlikely alternatives and present them as though they're the only alternatives available. Now. The arguer, in that case, just eliminates the undesirable alternative, which actually could be either one in this case and leaves the desirable one as the conclusion. Now the illusion here is that the alternatives that are under question exhaust all your possibilities so that evidence against one counts as evidence for the remaining one. Now, arguing from dilemmas is a pretty common form of reasoning and arguing. If somebody says the price is behind door number one door number two, and you know it's not behind door number two. Common sense tells you that you should say the prize is probably behind door number one. However, there are cases in which the alternatives given or completely unlikely. Sometimes in political debates, people say you're either for us or against us. Well, wait a minute. What about the alternative of being agnostic or undecided? Or what if somebody says, you either believe a certain doctrine or religious point of view and somebody says You have to believe it or disbelieve it again. This form of dilemma eliminates the possibility that there could be neutrality or skepticism. There's 1/3 option here that's being overlooked now, those you who are in romantic relationships like this example. Suppose a guy says to his Ah, lady friend, if we really loved each other, we'd be sleeping together by now. Well, she might disagree with that notice. In this case, if then, sentences show that they are common forms of presenting dilemmas. If this, then that means if not, then no. In other words, if we're not sleeping together, then we don't really love each other. Well, are those the only two options? Maybe she loves him and just wants to wait a while now. Notice the following are bad dilemmas for us or against us, for example, or believer or disbeliever. But there are such things as fair dilemmas. What about believer or not? Well, that not a believer. Canvases both skepticism and disbelief. Belief in the opposite of the claim of the believer, or you're either for us or against us. That looks like an unfair dilemma. But you're either for us or you're not for us. Well, yeah, those were the only two options available, so watch out for dilemmas the either or premises, and see if they really do exhaust all your possibilities. In these cases, we got some pretty fair choices. Now. The fallacy of suppressed evidence, sometimes called the fallacy of special pleading occurs and inductive arguments when an argument or a arguer ignores evidence that would either lead to a different conclusion or drastically undermined the original inference. And these are special cases that apply to inductive arguments, and I'll explain why you remember that inductive arguments are cogent Onley if they're strong and have all true premises. We covered that in an earlier lesson, and we used an example in which a person was deciding whether or not they should go swimming today because it would be fun. And they have lots of evidence that swimming today would be fun, but they overlooked some shark dangers in the water. Now, how to handle this? My two cents and the common view is this inductive arguments that violate the principle of total evidence, taking into account all relevant data before drawing the inductive conclusion count as weak evidence so we could have just as well covered, suppressed evidence under the fallacies of week induction. But we'll cover them here under fallacies of present presumption. Now, one common way of committing this fallacy is to ignore events or things that happened over time and just site events from the distant past that supports your view. If somebody says no war, however large, has ever destroyed life on Earth, and therefore the next war will not. Well, that overlooks the fact that we have developed some incredible weapons of mass destruction since the last major world war, or about quoting out of context, taking passages out of context from the Bible of the Constitution, or common way of suppressing the evidence. The evidence that's being suppressed is the context of the quote. Try this. For example. You shouldn't have long hair guys, right? Doesn't the Bible say so? Doesn't the very nature of things teach you that if a man has long hair, it is a disgrace to him? Well, look at the context of that passage. The context of passage was whether or not women should pray with their heads unveiled. It was a cultural custom back in the day for women to do that such thing. And Paul said we had no such custom, and neither do the churches of God Now. Matthew Henry, an older commentary on the Bible, said, Christian religion sanctioned national customs like women praying a certain way and then having their hair a certain way when they're not against the great principles of truth and holiness. Affected singularities or applications received no countenance from the Bible on I think Matthew Henry's right here. Ask yourselves people. If people are in certain conservative churches, air so uptight about men having long hair, why aren't they equally uptight about women praying with their heads uncovered? If you want to see people quoting out of context. Go to movie reviews, Boy, that's an incredible place to find this sort of fallacy. There's a blurb that said Hysterical and Entertaining about Bruce Willis is latest die hard movie. Actually, in the context, it says, the action is fast paced, hysterically over produced and surprisingly entertaining, a pout as realistic as a Road runner cartoon. Notice how you take a little bit out of context suppressed the evidence of context in this case, and you get an entirely different blurb. In this case conclusion. The fallacies of ambiguity occur when there's some sort of ambiguity, and that means multiple legitimate meanings in the premises or conclusions. These cover fallacies of equivocation. Inevitably, for example, boy Louis, everything that's bad for you is good, and everything that's good for you is bad. Looks like good and bad are being swapped out for their meanings. Diets are hard, he goes on to say, and Louis, I think, is justifiably confused when he says, Wait, I'm trying to follow your logic. Does that mean that easy and hard are equivalent as well again? Ah, an ambiguity was introduced into this conversation. Now there's lots of terms that could take on multiple meanings such as bats or batter. Or how about if you find out that your grandparent's are rockin, wouldn't that give you a little bit of a surprise? By now, you're probably figuring out there are a lot of jokes that turn upon ambiguity, specifically ambiguities in a single word or phrase those air called equivocations, like the old joke about the Buddhist who said The hot dog vendor make me one with everything. Do you mean one, as in numerically one and the same as what you mean one of them hot dogs? Well, try this corny joke. What do you get when you cross a river with a canoe and you get to the other side? Notice the word being equivocated on hears the word cross. It could mean either mixed together or it could mean go from one point to another. So obviously there's gonna be a lot of jokes that you're just not going to get like the peasants are revolting. Do you mean rebelling or you mean they're disgusting? You're not gonna get the joke if you don't catch the equivocation. I'd like nothing better than your pecan pie, Loretta, so I'll have nothing or try this joke on for size. I love this movie has a twist at the end. Twist is in an alternative ending or unexpected ending, or Grimi. Every chubby checker movie has a twist at the end twist, as in dance. But there are such things as fallacies of equivocation, and that happens when the equivocation occurs within the context of an argument. Like we said previously, a chess player as a person and a bad chess player is a bad person. What we noted in a previous lesson that bad could be mean without moral character, or it can mean lacking competence in a certain field or sport like chess. So in this case we have an argument that turns on an equivocation, and in this case it is a fallacy of equivocation. The Christian philosopher Norman Geisler in his book Come Let us Reason gives a good example of this. I he's not very happy with bad arguments, even when they're offered on behalf of his own religious views. So, for example, if somebody were to say, people believe in the miracles of science, so why on earth don't they go on to believe the miracles of the Bible. Wait a minute. There's an argument here, but it turns on an equivocation miracle. Kamina Supernatural event Orca means something that's just astounding. So the Bible talks about things that seem to be in some sense, supernatural science talks about miracles loosely, that it performs just meanings. It's amazing what we can pull off with science now. Whenever you think in equivocations President, you should be able to spot the term that's being equivocated upon and tell the arguer the two distinct senses in which they are using the term in equivocating. But if you think there's an ambiguity in an argument you can't located in a single word, maybe the arguer misinterpreted and ambiguity due to sentence structure as a whole. Now that often happens due to grammar or poor punctuation. And if you have a contract to be sure, read every sentence carefully because some sentences in the contract may take on multiple meanings if it isn't worded very precisely and again, and families can be a good source of humor. Try this example. While writing to Gettysburg on a scrap of brown paper, Lincoln wrote his most famous speech. I think you know what that means, I'll guarantee you probably don't think that Lincoln was riding on a scrap of brown paper, but the sentence structures ambiguous enough to give it a second meaning. Another example that I got from merely Sammon's book introduction. Logic in critical thinking involves the 18 hundreds or rather 18th century evangelist John Wesley, who wrote in his journal that he note, whilst meditating upon the 23rd Psalm, did he kneel on the 23rd Psalm? Seems little in pious, doesn't it noticed the sentence structure is ambiguous. Now this one's you're gonna have to think about a little bit. The guards and prisoners who refuse to join in the prison break were tied up and left behind. Now look for an infidelity here. What is this sentence saying were the guards who refused to join in the prison break as opposed to guards who did join in the prison break? While obviously, the guards are one group of people and prisoners who refuse to join in the prison break our second. But those who refuse to join the prison break that phrase could modify or describe guards or prisoners for a while that the sentence structure tells you not consider this example from Maryland Boss Savant, widely considered the smartest woman around due to her high I Q score. The anthropologist went to a road area and took photographs of native women, but they weren't developed. Well, wait a minute. What wasn't developed the photographs or the native women? Are we just talking about a primitive tribe here again, this sentence structure is pretty ambiguous now. Ambiguity is not the same thing is vagueness. I'll just put that out immediately. Ambiguities occur when you have to crystal clear, distinct meanings of terms or sentences that are being used in arguments or in conversations. Vagueness occurs when you have just one imprecise or unclear meaning. Now, when terms are imprecise, unclear. A fallacy is not being committed, although there are sometimes problems of vagueness that are very similar to problems of ambiguity. Now vagueness can create problems such as cliche mongering and weasel wording. That happens when a person starts talking and they have the impression that they're giving you information. But really, they're not saying anything specific at all. Calvin gets a little mileage out of this. I used to hate writing assignments, but now I enjoy them. I realised that the purpose of writing is to inflate weak ideas, obscure poor reasoning and inhibit Claire A. T. With a little practice. Your writing can be intimidating and an Impenetrable fog. His report is the dynamics of Inter being and Mon a logical imperatives, and Dick and Jane. A study and psychic trans relational gender modes. Academia is not fit for Calvin. We want to avoid such imprecision in our talk. Generally or else wise, we're gonna get just a bunch of talk. That's not worth having. Now let's move on to our last batch of fallacies, fallacies of grammatical analogy called fallacies of grammatical analogy because the arguments bear a grammatical or structural similarity and are sometimes confused. Bus four Good arguments, basically the fallacies of grammatical analogy. Just like the last category. Fallacies of ambiguity really just have to major subdivisions in them. In this case, fallacies of grammatical analogy consist of mainly fallacies of composition and division, which are very nearly the same fallacy. Now, the previous lesson. We used the example of the Brooklyn Bridge being made of Adams and Adams being invisible. Does that follow that the Brooklyn Bridge is invisible? Where did we illicitly moved from a property of the parts Adams to a property the whole. Now you can see why this is called a fallacy of grammatical analogy. Sometimes you are legitimately permitted to move from properties or features of the parts to properties of the whole. Just when you're able to do that is ah well, it's not a easy matter to nail down some fallacies division. You divide property of the whole and divide it down to the parts. In a fallacy of composition. You illegitimately go from features the parts to feature the whole. For example, the Brooklyn Bridge example was one the fallacy of composition or for person said. Each element in the recipe on making is delicious. Therefore, the dishes, the whole will be delicious. They're running a risk of the fallacy of composition. But did they really commit that fallacy in general, how can you tell? An erroneous from and legitimate transference is of properties from parts to hold. A vice versa. I'm afraid there are no easy answers here. If you're looking at an argument and you see a division or a combination of properties from parts to hold, a vice versa, you're gonna have to use your background knowledge to figure out whether it was a legitimate tactic. And generally you have the background knowledge that you need to do that. For example, if somebody says each brick in the wall is over 12 ounces, can't you just move to the conclusion that the wall is gonna be over 12 ounces? Yes, over 12 ounces is going to distribute really easily whether and if each brick in the walls under £12 under £12 is a property that is not going to distribute in that case. Or how about this? If the wall is over one foot tall, well, does it follow that each brick in the wall is over one foot tall? That property does not distribute. But if the walls physical, the property of physicality probably does distribute. What if you find out that the campus population is 50% female? Does 50% female distribute down to each member of the campus population? I'm joking, Of course. This ah, 50% femaleness is a property that does not thus distribute. Now here's a joke for you. Does reasonableness distribute are well, Suppose there's Ah, church that makes the following plan. We're gonna build a new church that seems reasonable, will build it on the side of the old church. Reasonable materials from the old church will be used to build the new one Very reasonable . And we will continue to use the old church until the new one is built independently. Considered number four is reasonable, all combined. Reasonable. The reasonableness did not combine very well. Basically, if you put all these together and think that the whole plan is reasonable, you're committing a composition fallacy and consider an all star team. If each member of the team is a great player, will the team be a great team? Well, not necessarily. You run the risk of committing a fallacy. Ofcom composition. Here. The same thing goes, if you have a great team and you automatically assume that each member is a great player, that runs the risk of a fallacy of division. Now a little bit of help in discerning thes sorts of fallacies. You need to know the difference between talking about a class as a whole like when somebody says please air numerous and talking about them distributive Lee Uh, fleas are small. Well, that's talking about fleas, each member of the class or in the case of Wales, you could say, for example, that whales are endangered doesn't mean that each and every member of that class is endangered. It's just a statement about the classes. A whole whales are mammals, however well, the class of whales is probably not a mammal, but each individual whale is definitely a mammal. So that's a case of distributive reference. Well, I don't mean to burden you with that last example. Just memorize your composition and division fallacies and other fallacies in this lesson, and we'll see you next time for our lessons. In deductive logic. Thanks for hanging around, but by 7. Categorical Statements: Well, everybody and welcome back my crash course and formal logic in this section, we're gonna begin our study of category logic. This would be a briefer lesson than most that I've offer online. We're just gonna cover categorical proposition. Now. Aristotle is the person credited with inventing the discipline of logic. He did that way back in the fourth century BC pretty impressive. But he thought the proper subject for logic to examine was categories and their relations to one another. That's not uncontradicted claim, but it motivates ah lot of his other philosophy that I'll have to study present on in other video lectures. So a categorical proposition is a proposition that relates to categories or classes indicating how they do or don't share members. For example, if somebody says professional athletes eat healthy diets, there's two classes involved here, professional athletes and those who eat healthy diets. I'll return to this example later. Or, for example, skateboarders are not allowed on public sidewalks. Well, you got skateboarders on the one hand, and you have persons not allowed on public sidewalks and the other again there to classes or categories being mentioned, or to give another example. One of my computers is mount functioning well, you got a kind of small category your computers and malfunctioning things again to classes or categories. In each case or not, every rainbow has a pot of gold. Sad but true. Well, that that's relating is the class of rain bows to the class of things that possess pots of gold or Steven Spielberg shoots blockbusters that be a category Steven Spielberg that would just have one member in it and person to shoot blockbusters is, of course, a slightly larger but still rather small category. What these all have in common is that each of them has a subject term that represents the subject class. For example, in our skateboarders illustration, it was the class of skateboarders. And then you have some predicated term a term that represents the predicate class, something that you predicated of your subject persons not allowed on public sidewalks in this case. Typically, you the, uh, a characteristic or property P is being predicated of some subject in each case. So when we talk, it looks as though we're always saying something about something, and that's pretty much to say. We're always predicated about some subject In fact, even in this sentence here, you could translate this sentence as all times that we speak are times when we predicated about some subject. So the president example is no exception to the rule. In fact, Aristotle thought that every proposition could be reformulated as some type of s is p categorical statement. He may not be right about that. There are some exceptions to the rule, but it's amazing how many of our propositions that we speak of an everyday talk to be reformulated into this subject predicate form. Now, doing that type of translating into a subject predicate form is actually a skill that needs to be developed. There's exactly four types of categorical prop proposition or class relationships, and Aristotle laid them out. There's some that say the whole class s is included in the class P. Some say the whole of the subject class is excluded from the predicate class. And then you have situations where a part of the S class is included in the P class and again times when there's a part of the S class that's excluded from the P Class. So when we looked at our example involving athletes we saw a case in which the whole s classes being included in the P Class. The person clearly wanted to say that when it comes to athletes and healthy eaters, all the athletes are in the category of healthy eaters would just block out this Ah little section in the red to illustrate that fact. And when we talk about skateboarders, what we found was the person was probably trying to assert that no skateboarders are things or persons allowed on public sidewalks. So when we want to illustrate that, we just make sure that we block out the section in the middle to illustrate that there is none in that overlap, know skateboarders and that overlapping area. Or when it was asserted that one of my computers is malfunctioning, it just means that one of my computers is in the malfunctioning things category. We usually to note that by pushed just putting an X in the middle to mark the spot without one thing is. And when it was asserted that not all rainbows air things with pots of gold, while we could have made a stronger claim, we all know that there's none of them with a pot of gold. But that proposition asserts that there's at least one rainbow. That's a thing without a pot of gold. And since I want to illustrate this properly, I better get that about a lot of gold, right? We usually illustrate that fact or that proposition with an X in the rainbow circles. Well, so we need to say a little bit about standard form, standard forms of proposition that expresses one of these four relationships while with complete clarity. For example, it's gonna be a substance, uh, substitution instance of either all srp no srp, some SRP and some s or not p. So, by way of contrast, all S are not P is not a statement in standard form because it lacks the clarity that these four crystal clear propositions have. So it could be translated as no SRP and alternatively, is some s are not P. For example. All prisoners are not sad and all prisoners are not free. No notice that has the form. All are not in both cases, but in the first example all prisoners air Not sad. That just means that some prisoners air Not sad. If somebody says all prisoners are not free. Well, that means all prisoners lack the property of freedom. None of them are free. So notice again. This form does not have the crystal clarity that the four examples we saw previously enjoys . So we need to talk a little bit more about thes standard forms. Now, these four examples are the only ones that we're gonna do. You use in this section on categorical logic in this class. But let's take the components here first you have qualifiers. That's probably most noticeable thing all know or some denotes how much of the subject class that you're gonna be talking about. And it tells you how much of the subject classes included in the predicate class and the Coppola. Well, that's really just a functional, uh, term. It's just linking up the subject and predicate class and remember to use on Lee are or are not other things, like is or has those will not be used in this course were strictly going to stick with are and are not a SAARC opulence. So let's try an example. Suppose somebody says all persons in the rock band U two are persons capable of playing bass bass guitar you can divide this up into our crystal clear standard. Formerly easily Qualifier is gonna be all. And the subject term is gonna be members of you to copulate are and the predicate term persons capable of playing guitar or rather playing the bass guitar. I need to be more clear. And when I point out these terms and highlight them in different colors, you can easily see how this sentence is divided up into all the elements that we just discussed. So going back to the examples that we used earlier, somebody says professional athletes eat healthy diets. What they probably intend to say is that all professional athletes are persons who eat healthy diets. Quantify was gonna be All this subject term will be professional athletes Coppola are and predicate term persons who eat healthy diets. You have to do a little translating to get to this point, but it's, ah, the talent that develops with time or if somebody says no skateboarders or persons allowed in public parks, well, that's the proper trance. Standard form translation of skateboarders are not allowed in public part on public sidewalks. Rather example I'll just highlight those terms where they show up in the sentence. You can see that has all four. The elements that we just discussed are when we talked about standard form quantify air a subject class copy. 11. A predicate class, that's all there is to it. Or if somebody says one of my computers is malfunctioning. Well, that would mean some computers of mine are malfunctioning things. You notice we had to finagle the terms there to make it work out dramatically. But the quantum fire is gonna be some. The subject term will be computers of mine are will be the Coppola and the Predator term is gonna be malfunctioning things. And again, if I highlight those in that sentence, you can see all for the elements. Show up in our standard forms the translation of our original sentence. And previously we said not every rainbow has a pot of gold. Well, you have to do somethin. A going to get the standard form translation of that. But it clearly means to exclude some rain bows from the class of things that have pots of gold. So the proper translation is some rain. Bows are not things with pots of gold, and the quantify will be some the subject term rain bows the Coppola are not and the predicate term things with pots of gold. And if I highlight them here in the in the original sentence, you can see each of those four terms. And this time is it. It's a unique situation. The Coppola is different from all the other examples we looked at. This is the one time where you're allowed to use. The Coppola are not well. I'll tell you what, You guys have covered an awful lot of logic in my previous lessons. About 20 minutes and every lesson. I promised you a short one this time. So that's all for now. Wait on my for my exercises and my next logic lesson which will be coming up soon. Take care. 8. Squares of Opposition and their features: Hello, everybody. And welcome back to my crash course and formal logic. For those of you who have made it this far, I gotta admire your diligence in this section. We're going to start to study squares of opposition in their features. And we're gonna move on to start doing some genuine category logic and just a few more lessons. So now remember, quality and quantity are the Onley attributes of categorical clerical positions, categorical propositions. There is no such thing as a qualifier. I find students introducing that term and often so bear it in mind. Let's talk about quality and quantity a little bit. Now the quality of a sentence tells you whether it's affirmative, which is to assert class membership were negative, which denies class membership. So all SRP and some SRP are your affirmative sentences because they are basically telling you that some s are included in the P category now no SRP and some s or not P. Obviously those air negative. You can tell by the words No and not in the Coppola on the ladder. Example. Quantity. On the other hand, either asserts universal regarding every member of a class or its particular regards at least one member of a class. So all that, sir P and no S R p r your universals because they either university, assert or universally deny membership in a class. By way of contrast, some SRP in some s are not p those air particular. Obviously that words some gives it away, right? So when I lay out our propositions with the universal's at the top and the particulars at the bottom notice, I put the negatives over to the right and the positives over to the viewers left. You can tell that they kind of form a square, and people have recognized that for quite a number of years they've actually given names to these sentence types. These categorical sentence types, the universal affirmative is, well, that's designated with the letter a the universal negative with the letter e eyes or particular affirmative. And those are your particular negatives. Now you're probably looking at AEI and oh, wondering where's why? Well, I'm gonna tell you why. Well, as you know, Aristotle was the one who developed logic. But most of Aristotle's work was lost to the Western world until it was recovered in the high middle ages. In the meantime, all they had to work with was his logic. And since the people the lingua franca of the day was Latin, here's what you get. Ah, the universals and the particulars in their designated spots. But going up to down the affirmative sentences Latin term a firmo has the 1st 2000 It's that well designated those sentence types and you can see the Latin term, for I deny the verb. Nay, go on the right. And so basically Ah, this is, ah, relic of the well Latin language that was used in the academies of the Middle Ages. So let's return to our top. When people started to examine this table, they found certain relationships held between certain sentences for Aristotle and for the scholars of the Middle Ages. If you knew that a was false, that directly implied that the universal negative was true. Indicated here in the red and green for you. And if the universal negative is true, well, if no SRP that directly denies that some SRP and actually they do another relationship if they if they saw that some S r P is false, they drew the conclusion that some s or not p for reasons that will go into in the next lecture. But for now, just memorizing the order of the table is important. So you got a couple of ways here you can remember your vows conveniently space spelled out A E I and oh, I find that most students find that helpful. Or you can remember your Latin firmo and neg O. So this table pretty well summarises everything they covered so far. You have the propositions, the four categorical sentence types it will be using in this course, each designated with a letter name and each distinguished in terms of their quantity and quality. No sentence in our list has the exact same quantity, and quality is another, and it's a good way to keep everything nice and well. Now I need to talk a little bit about distribution. That's a technical term pertaining to subjects and predicates and propositions. A term is said to be distributed just in case it makes an assertion about every member of that class. And remember, categorical sentences always involved two classes. 1 may be distributed and the other not so. If it doesn't tell you something about every member of a class, then it's going to be called undistributed. So this table basically summarizes everything that we've studied so far and what lies ahead . You noticed. I just added one new column to the table, one for distribution. Turns out that the A sentence is all that, Sir P distributes the subject. And that's pretty obvious why that would be the case. Now that's e sentences. No. SRP distributes both the subject and predicate. And obviously, if you know that some S R p the I sentence that destroyed distributes neither the subject or predicates, it doesn't really tell us. It only tells us about what's the case for one subject. And you think the same thing would happen with the O sentences? Some s are not P. As it turns out, they distribute the predicated on. I'm gonna explain why that's the case here. In just a minute, let's go over an explanation for each Suppose you know that all s r p everything in the subject classes corralled, so to speak by this blacking out into the predicate class. Well, if all srp we certainly do know something about every s there, they're all p. However, we can't tell whether or not all the peas are or are not s is I'll illustrate that Remember , in our example of all athletes are healthy eaters. That tells us about all the athletes out there. What it doesn't tell us about is all the healthy eaters. Surely there have to be. Even if that statements true, there have to be a lot of healthy eaters out there besides the athletes. That would be the remainder of us who take care of ourselves. So let's let's talk a little bit about e sentences. The ones that say that no SRP, if no srp You know something about every member of the s class and about every member of the P class Every s fails to be a P, and every P fails to be in s. So that's what happens when you block out this section here in the middle. You can see now that nothing's allowed toe have mutual membership in both classes at the same time. So in our example involving skateboarders, we said that Ah, no skateboarders or persons allowed in city on public sidewalks. That tells you a lot about the skateboarders. But it also tells you about people who are allowed Teoh be on public sidewalks. But what you know about them is they all failed to be skateboarding. Now the ice sentences give us the least amount of information With respect to S and P. We we really know nothing about every s and every P. All we know is that there's one member marked with the X here in the center, which is a member of both. But there may or may not be s is that are not peas or vice versa. You may recall the example that I gave earlier involving a malfunctioning computer. If you know that one of your computers or something, that is my computer or a computer of mine is a malfunctioning thing, that certainly doesn't tell you about whether or not you got other computers and whether or not they're malfunctioning in it, it certainly doesn't tell us about every malfunctioning thing on the planet. It's just one bit of information that says there's a member that occupies the position in both categories. Now let's talk about the most difficult case the O sentences. Some s are not P. We do know something about every piece in that sentence, every P fails to be identical to at least one s. Now, that may sound a little bit pedantic, but it's actually an important point. And I'll explain that alternatively, you could think about it as Class P fails to completely encompass s. So no matter what, P is not gonna be sufficient to completely envelope or cover all the members of s and another way to think about it. If you recall our example involving rainbows and pots of gold, not every rainbow has a pot of gold, which means some rain bows are not things that have pots of gold. Now, for all that tells us, we can't tell whether it s completely encompasses p or not. Maybe rainbows are the only thing on the planet that has product pots of gold in it. I haven't seen a pot of gold my lifetime, but for all that sentence tells me, rainbows might be the only place to find him. Okay, by now I'm starting to even board myself. So, uh, you might be wondering why care about distribution wise is an important concept to wrap our minds around. Well, distribution tells you how much information about a class, whether the subject or predicate that the categorical proposition contains. And the reason why that's important is that it measures how much you can deduce from those conclusions. In other words, we start putting together a logical arguments. You need to have as much information on the class and the premises as you have in the conclusion, because one principle of deduction that we studied earlier is that the conclusion is never allowed to go beyond the premises. It just teases out information that was already there. So, for example, if you have a term that's distributed in your conclusion, it had better have been distributed at least once in your premises. Well, Patrick Curly's logic book gives us some pneumonic devices that might be helpful for you and nominal lists. And right here, uh, unprepared students never passed. That basically stands for universals, distribute subjects and net on negatives, distribute predicates, but the one I like better is any student. Learning bees is not on probation. That's an easy way to remember that AIDS district a sentences distribute the subject. East sentences distribute both the subject and predicate I sentences distribute neither, and the O sentences while they distribute the predicated we just explained. So I hope that was helpful for you. Um, memorized some principles of distribution. They'll help you with your logic later on on. Otherwise, I hope this video is giving you a more firm grasp of the principles of ah, logic. Next up, we're going to start studying immediate inferences. 9. Squares of Opposition Continued: although there again, everybody. And welcome back to this crash course and formal logic in this section, we're going to study squares of opposition in some detail, and it will be helpful for us to make our inferences are first inferences and deductive logic. Now, first up, we're going to study the modern, sometimes called the Boolean square of opposition. Even though air estoppel had developed the earliest forms of categorical logic and squares of opposition formulated in the Middle Ages were based on his work. I'm going to cover Bull first because he revised his area stop views quite a bit, and, uh, makes this square a lot simpler. So what squares of opposition do is they illustrate relations between categorical propositions. That's our A Z is eyes and nose, and and sometimes the first square that we're going to study is called the Boolean Square of Opposition or, alternatively, the modern square of opposition. So for the time being, let's bring in our A zis i's and O's and take a closer look at what George Boole contributed to logic. Now the only relationship on this table that George Boole recognized was the relationship of contradictory is the A's contradicted Theo's and the ease contradicted the I, and it's actually not too hard to see why that's the case. If you put the sentences right over top of one another, you have one sentence putting an X precisely where the sentence above has a space blocked out, indicating that nothing should be there. The same thing goes for the A sentences and the O sentences. Now two statements are contradictory. If and only F one and only one congee true. And this is important for making inference. Inference is, if one is true, then necessarily the other one has to be false and vice versa. So that's to say they necessarily have opposite truth values. So the bully and square only recognizes contradictory Zall. The other relations along the square are what we call logically undetermined. That means it's not gonna be able to make any inference claims across the top row or the bottom row, and similarly from the top to the bottom. Now, that may all seem simple enough, but actually it hides a very important logical problem that I'm going to discuss in some detail here. What Aristotle, um Booth disagreed about was whether or not universal propositions. That's the A and e sentences whether they made existence claims for Aristotle to definite Yes, for him if you say all lesser P, that means there is something in the category of s and all of those air p. And similarly, if you say no SRP, that means there are things in the category of s and none of these are p A. George Boole, by way of contrast, much later said no. Those sentences lack what would later be called existential import. All that, sir, P just means hypothetically. If there was an S than it would be a p and no, SRP just means hypothetically. If there was an s, it would not be a peak. So you can see it better if we look Take a look at some specific claims. All wear wolves or monsters or no unicorns have two horns and Gandalf is powerful. All these things failed to exist. But for bull, all you're saying when you say all wear wolves are monsters is hypothetically. If there were aware wolf he would be in the monster class. And when you say that no unicorns have two horns what you're saying is no unicorns would be in the two horned animal class. And when you say Gandalf is powerful hypothetically, if anything was identical to Gandalf, it would be a powerful thing. So all these come count is true for Bull because they don't make any existentially claims. Aristotle by way of contrast. Well, he's not gonna allow you have an empty subject class. So if you say all wear wolves are monsters, then well, where do you have to put him? You're gonna have to put him right in that middle area. And if you say no unicorns have two horns, Aristotle's gonna assume there's something in your unicorn class. I'll make a comment on that in just a moment. But it would have to be over here to the left and similarly with the Gandalf claim things work out pretty well the same way So Aristotle was not gonna let you have an empty subject class to talk about. Now, on the subject of whether or not particular sentences, the I's and O's make existence claims both Aristotle on Buhler and perfect agreement with one another, both assert that the IRS knows make existence claim. If you say it state, And I proposition that some SRP you mean there's at least one s that exists. And it is also a P. So Aristotle and you'll have a common understanding of what those sentences were going to be like. And if you meant ah state. Oh, proposition that means at least one s exists, and that s is not a P, so the way that they write out their circles if Aristotle ever engaged in that sort of thing, he seems to not use that technique, but their circles would look the same way. So it might be wondering out what's all the fuss. Aristotle. Well, some of you have seen this painting before, depicting philosophers back in Plato's Academy. Actually, ah, Socrates was the teacher of Plato, and Plato is the teacher of Aristotle, Plato and Aristotle or depicted here players. Plato was significantly older than Aristotle, And so he's the older gentleman to the viewers left. You may notice that our painter here has perhaps worked himself into the painting as well as many other notable philosophers from history. Well, Plato had a unique cosmology. It's ah, interesting that he's holding the two mayo, which lays out his Ah, cosmology and theory of creation. For Plato, the ideal was the most riel. So if you want to talk about things like circularity, justice and other things, it's the ideal. Or some people say, Uh, ideal forms are located in Plato's heaven. That gives existence to the various circles that we see all around us. And the same thing holds for things like humanity for justice, etcetera. So in his view, all we ever see are dim reflections of of an ideal, and that holds for pretty much anything that you look at. Uh, there's no perfect circle in existence. There's no perfect justice. There's no of ideal form of humanity. But we all have these sorts of properties, and we see them in the world around us because in some sense the things in the world around us participate in that ideal form that Plato was talking about. So part of Aristotle's project was to radically depart from his teacher. Plato's views on this, instead of having the concrete circles in our universe exist in virtue of some ideal or heavenly form. He wants to reverse that direction and make sure that any sorts of abstract objects are categories that we talk about and notice. I use the word category. Any categories that you talk about owe their existence to concrete objects in the world around us. And if you think about it like that, you can see why he wouldn't want there to be any truth in a proposition of a universal sort . If unless there was something that made it true, some concrete object in the world around us. So Plato's project is to emphasize ideals and, uh, and their role in creating the world that we see around us. Aristotle keeps things down to earth with particulars, so you can see why Aristotle is not gonna be able to accept boules. Table of opposition. Um, booze table leaves the relationships between the A and the e sentence at the top of the table logically undetermined, which means they could, for all logic, toes us. They could both be true at the same time. But watch what happens if that occurs automatically. That gave you an empty subject class. And similarly, if the I in the O sentences were both allowed to be false, well, because the I in the o. R. Contradictory to a and e then That means if they're both false, the A and the E would have to be true once more. And that gets us back in the same problem. So a and E cannot both be true. And thus I n o cannot both be false. Aristotle once again wants to keep us from talking about empty subject classes. A Ziff. Those subject classes just hung in the air like Plato's forms or something that effect well . So you can tell from where I put X is in the i e and the A sentences on this chart that no longer can we say that that relationship between the two is logically undetermined. And the reason for that is each sentence puts an ex right in the spot that the other sentence has blacked out. So this logically under determined relationship has to give way to the claim that both both cannot be true at the same time. And we can't say that the sentence that the bottom is logically determined thigh and the O sentences. The reason for that is if each of them are false, well, then, in virtue of the fact that they're contradictory to the sentences at the top. That would make both of the senses of the top true once more and we'd be back into the original problem. So we'll have to put a new relationship with the bottom. Let's just say both cannot be false and the same thing goes for those ah sentences at the sides. You can tell if the sentence at the top, like in a sentence, is true. The ice sentence has to follow from it. And if the east sentence is true, then the over sentence pretty much follows because they put X is right in the spot where well, where the other sentence put it. And consequently, if the sentence is at the bottom are false, then they're gonna prove the falsity of sentence at the top because the sentence of the top involves the same existentially claim. Just so. So let's do a little bit more investigating. All these relationships have ah names just like we put the term contradictory through the center, the sentences that the top are called contrary. If somebody is contrary to you, you both can't be right at the same time. And at the bottom we have something called a sub contrary, there's no good reason for naming it. That, except that you know it is sub like a submarine is underwater. This is underneath the contrary, relations and the relationships down the side. Whether you move up up the chart or down the chart or refer to a sub alternation So we'll talk about those in a bit more detail. So essentially you can prove more with Aristotle Square. Since he has the no empty subject class assumption, a lot follows from that Take a look at the example of the bottom If a is true that on Aristotle Square it immediately follows that he is false because they're contrary to one another. It also follows that eyes True because a involves an existential claim that I was making. And it also follows that Theo sentence is false in virtue of being contradictory to a the only relation that bull would have justified from the truth of a would have been the falsity of Oh, so we get a lot more inference is validated on Aristotle Square and again try this example If the I sentence is false well, then a involves the same existentially assumption that I make so a would have to be false as well. E would be true in virtue of being contradictory. Toe I. It's going to take the opposite truth value and look at Theo sentence because you can't have falsity in both of the sentences on bottom than if one is false. The and the other one has to be true again. The only inference that bull would have justified from all this is the move from the falsity of I to the truth of E. Now not every inference is gonna be validated on air establish table. There are fallacies in Aristotelian immediate inferences that you can make on this table, for example, their names for bad moves along the Aristotelian Square. They all tend to have the word illicit. When somebody tries to dio to use a contrary relationship between the A and the E and they do so badly, that's called illicit Contrary. That's when you move from the truth and falsity of Universal sentenced to universal sentence and that occurs. Sometimes people assume that just because both can't be true that they can't both be false , and that's not the case and ah, illicit sub contrary. That's just when you do this sub, you try to use the sub contrary relationship inappropriately. When you move from the truth and falsity of particulars. Particulars remember they can't. They can't both be false at the bottom of the table. But that doesn't rule out than both being true. So watch out that you'll make a mistake in that assumption. And then, of course, for the last relationship we just studied a list that sub alternation is when you move from the truth or falsity of Universal's to the truth or falsity of particulars or vice versa. Now, not every inference on the Air City Lian table is going to be justified. Do some practice with these if it's false, that all comic books or work of art can. We claim then that no comic book is a work of art who noticed, uh, both sent This moves from the falsehood oven a sentence to the truth of an e sentence. There's nothing in Aristotle's logic that rules out both the A and the E being false. And similarly, if some wars are morally justifiable events does it follow from that claim alone that some wars are not morally justifiable? Events notice the person moving from an I sentenced to an O sentence. Now we know that the sentences at the bottom of Aristotle's table they can't both be false . But there's nothing that rules out, both of them being true at the same time. So the move from the truth of one to the truth, the other is an illicit move. Illicit sub contrary. And there's one last important thing that we need to cover. Um, the absence of a formal fallacy is not sufficient for validity and Aristotle system, and maybe this is ah detriment to his system. But there are existentialists use that can invalidate an inference, and they have to do with something other than mere forms of arguments. Take a look at this. All unicorns are animals, some unicorns or animals. Now that looks like a move from an A to a nice sentence. But we want to lose arguments like that and say that the conclusion did not follow. After all, unicorns don't exist, but all cats are animals. Therefore, some cats or animals notice the's. Two arguments have the exact same form. Aristotle wants to keep the arguments at the bottom and lose the ones like the Unicorns example. That means he has to introduce something into a system other than a formal elements, because these arguments have the same form existentialists use. The fact that unicorns don't exist can invalidate an inference for air established system. And it's ah, it's an extra. It's an issue other than form that probably is a detriment to his logic. So in summary, the's squares could be used to determine whether, in immediate inference, and I've used that term before. Basically, it's an argument from a single premise to a conclusion, and you can use these squares to tell if those arguments are valid. Now. Arguments that are valid from the Boolean standpoint are called unconditionally valid because they don't have to depend upon finding out whether the items talked about in your subject classes in your universal statements actually do exist. And Aristotle, by way of contrast, along the outer making inferences around the outer edge of the squares may be validated. But you have to make sure that they don't commit any existential problems making incorrect existence claims because a lot of those inference is located in the brown along. My diagram at the top depends vitally on the legitimacy of those existentially assumptions . So if you, uh, you can use it Aristotle's table. But those arguments are called conditionally valid. They're only going to be valid on the condition that the things talked about in the subject classes of your premises when your premises include Universal's that those things really do exist. So consider these arguments and try to figure out if they're valid and on which square it's false that all private eyes are masters of disguise. Therefore, some private eyes are not masters of disguise. Notice I moved from the falsity of a sentence to the truth oven. Oh, sentence. Now, if you stop to think about it, those two sentences have to take opposite truth values. That's one of the lessons we learned from Bull. We don't need Aristotle's arguments to prove that to us. Now consider the argument on the bottom. No meteor showers are daytime events that asserts an e sentence. Therefore, it's false that all meteor showers are daytime events. Well, since ah, those two claims air contrary, but only on Air Stoffel Square. Then we have to say yes, but we're gonna have to say that it's conditionally valid because it's using an inference that's only validated on Aristotle Square of opposition. So when you, ah, move from one of those claims to the other, make sure that the objects talked about in your subject class exists. Consider this argument all cell phones or wireless devices. Therefore, some cell phones or wireless devices. Well, that seems like a good influence, but only based on the claim that we know cell phones actually do exist. So from the bullion standpoint, this argument would commit the existential fallacy. This is a form of argument, the move from aged eyes that he does not accept because he's willing to permit empty sub classes. So for rule, the premise lacks existentially import. But the conclusion snuck it in. The premise did not include the claim that cell phones actually existed, and we need that assumption to get the inference to go through. So here's my little cheat sheet for you, the Boolean Square of opposition at the top and Aristotle's Square at the bottom. Feel free to look back and refer to this at any time, and, uh, let's move on to the last issues we're gonna cover. - Let's try a few applications. Try this one. All cats or animals. Does it follow that some cats or animals? Step one. Is this valid on Bull Square? No, Doesn't we said because his premise lacks existentially import. But Aristotle recognizes existence claims involved in the universal sentences. So last step. You have to check. Do the cats really exist? Yes, they do. You can stop. This argument was valid after all. - Well , try this. No witches who fly on broomsticks or cowardly women does it follow that Some witches who fly on broomsticks are not cowardly women. Well, here we have a move from an e sentence to an O sentence. And of course, step number one Bulls Square will not justify that. By way of contrast, Aristotle Square Step two will justify it. Are there any witches out there? Let me add the caveat ones who actually fly on broomsticks. The answer to that question is absolutely no. So this argument is still in Balad and same thing goes for no wizards with magical powers are malevolent beings. So it is false that all wizards with magical powers are malevolent beings. So here we have a move from an e sentence to the falsehood oven a sentence. Now, that's an interesting move, Wolf Step number one. We know that bull is not gonna justify it. The ease and the A's are logically unconnected. In his view, will Aristotle justify it? Well, he won't allow both sentences to be true at the top of the table to be true. So you can move from the truth of one to the falsehood of the other. So Aristotle Square will justify this inference. However, by way of contrast Ah, we need to go to step three and check and see if there are any existential issues involved . There are no such things as wizards. Well, at least not ones with magical powers. Well, I think you've mastered everything that you need to know for now, about a categorical logic. Congratulations. You're gonna be doing immediate inferences. Uh, pardon the pun almost immediately. So stick around for more logic lessons and I'll have another video for you here soon. Take care 10. Conversion, Obversion and Contraposition: Well, welcome. Once again, everybody, it's time for our next lesson. The logic in this part we're gonna study Ah, another form of immediate inference. We covered some immediate inferences in the last lesson in this lesson, I want to cover a conversion aversion and contra position. Categorical sentence equivalents occur when two statements have necessarily the same truth value. In a sense, it's the opposite of how we define the term. Contradictory, contradictory sentences necessarily have the opposite truth value one from the other. Now there are three methods in, ah category logic for attaining these equivalents. There's conversion and you can remember that by remembering that the after the prefix con you get an e and an I immediately after those work on E and I sentences contra position. You can remember because after the con that comes up front, the A and the oh are the next vows. Contra position is a technique that's gonna work for A and O sentences, and lastly will study ob version, which works for all the sentence types. So that's a pretty helpful one to have in your pocket. So, essentially, by the time we've done, we're done studying these three methods. You'll have two methods for any sentence type to obtain, Ah logically equivalent versions of that particular sentence. That is to say, you have two ways of transforming that sentence into something that says the exact same thing. That's pretty much what we're gonna do. We're gonna learn, Ah, how to take any categorical sentence and do at least two operations up on it so that you come up with a total of three sentences that pretty much say the exact same thing, but in different ways. That's a nice skill to have in your pocket Now when we do these operations, one thing I need to point out, we've already learned how to make inferences immediate inferences from a single premise to a conclusion, using the bully in and the Air City Lian squares of opposition. What you need to know is that these techniques were going to study now, work on Bulls Square of opposition. And since Aristotle's Square just expands upon Bulls Square, well, that means that all these operations will be valid in his system of logic as well. So that's useful thing to know. So once again, we'll bring in our familiar a zis I's and O's more importantly, will bring in the ah, green and red circles that I've used to illustrate the information contained in those categorical sentences. And let's get this going, Let's start first with can conversion because that's the simplest and that works on the ease in the eyes. Now, conversion is the simplest cause. All you do is you swap out p for s and vice versa. So if no srp, no PRS and if some srp some prs And if you notice the e and I sentences, say the same exact thing about S and P. So I guess it really shouldn't be surprising that you could swap one for the other. Well, how about an illustration? You get us started. Let's start by doing conversion on an eye. If some birds or things native to Antarctica, Well, you notice that X in this circle is ah, right there in the middle of the two circles and you could swap one back and forth. It doesn't really make a difference just as long as you keep that X in the middle. So let's put a bird there, and I will let the red circle represent birds and We'll let the green circle represent things native to Antarctica, really kind of worried that I put an animal in there that I probably shouldn't have. But, hey, at least they're cold climate animals, right? So if some birds or things native to Antarctica, pretty clearly something's native Antarctica are birds. We just said the same thing about the green circle that we said about the red circle and vice versa. Now let's try conversion on an e sentence. No, dogs are cats. That's a good one to try. So what that tells us is that nothing is in the overlapping category between the two. And if that's the case, no cats or dogs just that simple. What it tells us really is that the category of dogs and cats are completely distinct and with no common ground in between them. And how about another one? Ah, involving an e sentence? Ah, the great poet John Dunne once said that no man is an island well, doesn't pretty well follow from that that no island is a man. Yeah, what that tells us is that there's just no common ground between the two islands, and men are distinct categories with no overlapping members. That's important to notice that conversion does not work with the A sentences or the O sentences. So, for example, if all ladies air humans and I'm pretty sure they are woman does it follow that all humans are ladies? No, not at all. Uh, that's an illicit conversion. And if some humans are not ladies, there's an o sentence for you. Does it follow that some ladies air, not humans. I'm sure some feminists out there would take exception to that. What this illustrates is that you can't do conversion on A and O sentences. In fact, the fallacy of elicit conversion occurs. It's actually a formal fallacy in logic, as opposed to the informal fallacies we studied and less than four. When you make these sorts of inference is that we just looked at So by way of review, conversion works on me and I sentences and it occurs when you just flip flop the places with a subject in the predicate term occur and just to give you a couple more examples Ah, If some firemen are brave persons, can you figure out this one? Yeah, some brave persons air fireman. And if no used car salesmen are reliable persons. Well, no reliable persons are used car salesman. So let's take a look at the other sentence types that we have to deal with and more importantly, the information contained in those sentences. And once again remember, uh, it's kind of strange that the sentences that bear the contradictory relationship to one another should have the same operation here that would render logical equivalents. But that is the case. And having said that, this move I want to talk about contra position that works with the A sentences in the O sentences. What that happens is you switch the subject and predicate term, and in that sense, you could say you start out by doing conversion, But then after you do that, you change each term to its complement. And a complement of a term is the set of all things to which the term does not apply. And typically you get a compliment just by putting the expression non right in front of your class term. So, for dogs, the complement of dogs would be everything that isn't a dog or, more simply, just non dogs. So you got two steps here, do Ah, conversion. And you're already familiar that and then change each of our terms to its complement. So So you have all that Sarpy Step number one all PRS, then all non p r non s. It's really that simple. Or in the case of the, uh oh sentences, it gets a little bit more complicated because you're gonna have double negatives popping up all the time. But some s are not P. You Then you just say some PR not s, and then you change each term to its complement. Some non PR, not non s. Those oh sentences get really tricky. Usually you use ah, you find contra positives used more when you're doing dealing with a sentences. Well, how about a few examples to help all the apples or fruit you noticed? I just moved all the apples into the fruit category. And what that means is, if it ain't a fruit, it ain't an apple. All non fruit are non apples. So anything outside of that green circle, so to speak, can't be an apple. That's our fruit circle. How about another Contra position on? Ah, in a sense, all bulls are cows, right? I just got one more bull into the cal category here. And what that means is today Nakao attainable. We'll all non cows or non bulls. Perhaps I should have used the term cattle instead of cows, but you get the point. So Contra position, we said, works on the A sentences in the O sentences when we switched the subject and predicate term simply doing conversion. And we make sure to change the each term to its compliment. Because if you don't do that, we're gonna commit the fallacy of illicit conversion that we talked about earlier. So, for example, if all humans are mortals, what does that imply to you? Probably means that if it ain't mortal, then it's not human. All non mortals are non humans. Well, how about some runners are not vegetarians? This one gets a little involved. How about some non vegetarians are not non runners? No. All those negatives were really trip you up. So one other formal fallacy that we want to avoid committing is the fallacy of illicit contra position. And I use the ladies in humans example again just to make the point. So take us a premise. No ladies are non humans on. That sounds true. Therefore, can I draw the conclusion that no humans are non ladies? Well, no. Um, there's plenty of humans that are non ladies. All the guys, by the way, notice in this example. When I switched the term non humans to its complement rather than double up on the non, I just dropped non that was already present. You don't want let those ah non sneak up stack up on you, but looking at the example involving an I sentence some humans or non ladies Quite true. All the guys does it follow that? Some ladies are non humans. Ah, once again as a false false conclusion. And it also illustrates once again that ah, when you switch the term non ladies to its complement, you want to just switch the to the term ladies. That's the A set of all things to which the term non ladies does not apply. Now it's time to work on ob version, which is interesting because that's the one method of getting logical sentence equivalents out of all the sentence types that we've got. But it also has two steps, and these ones are a little bit more complicated. First you have to change the quality of the sentence. So you change in a sentence to an e sentence or vice versa, and you change and I sentence to an O sentence or vice versa. Basically, you move left to right along this Ah, this table, and then you change the predicated to its complement. Um, so this is the one technique where you don't swap the position of the S and P, and that's important to remember. So let's take a look at a few examples. So let's give each of these a try. Let's start up in the upper left hand corner first. All S R P. What wouldn't change that to a negative? So no S r p. And then we're gonna change p to its complement. So no s are non p. And that makes sense, right? If all the srp, none of them fail to BP, let's go over. Take a look at the East sentence. No, s r. P going to change that to in a style sentence that all s r p but then changed the predicate to its complement. So all the s are non p and again when you look at that graph. It makes perfect sense. All the yes, in this particular case are kept away from the green being in the green circle. Now, to the bottom left some s R P. Actually, this one's kind of easy. We're gonna have to change it into an O sentence. And then that means we're gonna have to change the copulate, and then we're gonna add a non to the predicate. So some s are not non P. It just looks like a double negation right there, doesn't it? And lastly, that's do the O sentence. Some s are not P. Well, we're gonna change that to a sum S r p. And then we'll change p to its complement again. That's pretty simple. Some s are non P. How about a few more exercises? Uh, how about some of these sentences? Some of my birthday gifts here in Tennessee. All cows eat grass. How about all my farts? Air smelly? That's not a confession on my part. It's just a example. How about some drunk stumble? Some bears are not tasty. Now, First we have to change these into their categorical sentence forms and then we conform The operations that we want to perform on them. So first example up, some of my birthday gifts are in Tennessee now. First thing we want to do, turn that into our ah, very clear categorical sentence form. So some birthday gifts, um, mine are things located in Tennessee. Now, once we have our crystal clear, categorical sentence form in hand, you noticed that that's an I sentence. Let's always do ob version first and get that out of the way. So the obverse of that is some birthday gifts of mine are not non things located in Tennessee. A little awkward, isn't it? But it actually does work out. And ah, more importantly, we know that we can do conversion on this sentence. So if some birthday gifts a mine are things located in Tennessee well, some of the things located in Tennessee are birthday gifts of mine. Get the exercise. Let's do a few more of them. How about this one? All cows eat grass, Mom, that's We're gonna convert that into an a sentence, of course. So all cows are will say grass eaters. Now, what can we do with this? Well, obviously ob version works with everything So we're gonna change the all to a No. And they were gonna change Grass eaters to its complement. Will be wind up with No cows are non grass eaters. And that's true. And it's logically equivalent to our first statement. The other thing we can do, we can do contra position on this so we'll swap the up subject and predicate term and change boat to their compliment. So all non grass eaters or non cows if it ain't eating, eating grass yet and a cow, and that's logically equivalent to our first statement. Okay, this will be a fun one. All my farts or smelly don't laugh. All years or two. I have to invent a special category here. How about farts? Oh, mine are smelly things. Okay, now we have our categorical sentence form in hand. What can we do within a sentence? Well, obviously OB version can work, so let's do the obverse first will change the all to a no and retained smelly things to its complement. What do you wind up with? Yep, That was easy. Enough OB versions. Not too hard. Once you get the hang of it now, Contra position also works on a style sentences so we can swap the subject and predicate term and change each to their compliment. So all non smelly things are non fort, so mine if it don't smell, it ain't my fart. How about this one? Some drunk stumble? Well, we like a copier there, and stumble has to be changed into a category. So some drunks are stumble. Er's well, what we do with this it's an I sentence. Obv er zhan is an easy one. We know that we can do that. We're gonna have to change it into an O sentence and then change tumblers to its complement . So we're gonna wind up with a lot of negatives, but it makes sense that's logically equivalent to our original sentence. Now, one other thing we can do simple conversion. So if some drunks or stumble er's some stumbles air drunk, get the exercise. Well, this will be the toughest one. Uh, well, during these exercises on the O sentences is oftentimes awkward, but just follow the steps. Some bears are not tasty. Well need to turn tasty into a class of things. Some bears are not tasty things. All right, First up is obv, Ergin. Ah, we're gonna change the copulate here. And then we're gonna put a non on through the category of tasty things so that we designate the complement of the predicate class. And when you do that, that was pretty straightforward. Some bears are non tasty things. This one's gonna be a bit more tricky. Contra position works on oh, sentences. So we're gonna have to take bears and tasty things and swap their positions. And then we're gonna have to put nonce on each one of them. Can you see how many negatives were gonna wind up with that? But just matter following the rules, some non tasty things or not non bears. If you think about that, that is just a really complicated Where it way of saying some beers, air, not tasty things. All right, you've done all of these exercises. Congratulations. Here's another set for you have to do on your own. How about this group? Not all wines are sweet. Noticed? That's the denial oven A sentence. So you want to go to the contradictory of Ah, the A sentence. How about no friend of Oprah's is a friend of mine. No pickles or sweet. Some talk show hosts are not intelligent. True enough. And no stop sign is green work these the same way that you did the last set of exercises and you get plenty of experience with OB version, Controversial contra position and conversion. All right, that's all for now. Wait from or exercises in my next logic lesson we're about to move into some Ah categorical syllogism is that some hefty logical material and not long after that will be handling even mawr upper division stuff, so stay tuned. 11. Categorical Syllogisms, Terms, Mood and Figure: Well, everybody and welcome back to my crash course and formal logic in this part. We're going to study the crucial topic of categorical Cilla Chism's. We're finally going to take categorical sentences in batches and try to draw conclusions from them now. Last lesson was kind of short, but this lesson I'm definitely going to keep my promise to give you to college level lectures in one video. So hold onto your seats. Now. Syllogism is a very broad term. It refers to any argument containing two premises and a conclusion. Categorical syllogism, by contrast, is a very specific term. It's a syllogism, but each of the three sentences, premises and conclusion have to be categorical suit sentences. Telestrate premise, premise, conclusion syllogism, by contrast, categorical sentence categorical sentence to a conclusion of a categorical sentence. That's your categorical syllogism. So just to give you a concrete example, all men are mortal. Socrates is a man. Socrates is mortal. That is a categorical syllogism. Now it may be a little difficult to see on the surface of it, but these are all categorical sentences of the either the A, e i o or O type. We just have to take all men are mortal and swap it out for all men are mortal beings because mortal beings with designated category And as for Socrates, is a man have out instead all persons identical to Socrates. That way we have an appropriate subject class in that second premise and just take these two switches and put them into your conclusion instead of Socrates is immortal. All persons identical to Socrates are mortal beings. You can see that we're moving from a sentence in a sentence to in a sentence that makes this a categorical syllogism. Now let's take a look. These terms that occur in the categorical syllogism men, mortals and Socrates there kind of links in a chain. And the way those terms hook up makes all the difference in the world to whether an argument works or not. There's a term that occurs twice in the premises, a crucial link that's called the middle term again. Thinking of these terms as links in a chain, there's something special about the one that occurs twice among the premises that is sort of a middle, a link up that hooks up the other two terms in the syllogism. Those other two terms are hooked up in the conclusion. In an interesting way, those terms are called the major, major term and minor term, and they occur in the conclusion. So look again. The term men occurs twice in the premises, and the term mortal and Socrates, which also occur on the premises, are linked up in the conclusion. It's the middle term that's doing that hook up. Oh, let's ah, get a closer look here by deleting some stuff. Take a look at those other terms. The other term mortal beings is called the major term. It has occurs one time the premises and once in the conclusion as the predicate Remember, the major term is the predicate of the conclusion. Now there's also another term that occurs within the premises. One time it's the term persons identical to Socrates here. That term occurs one time in the conclusion as the subject. So the minor term of the argument which gets linked up to the major term through the middle term is the term that occurs one time. The conclusion as a subject. So you have it again, three terms each occurring two times in the argument. But where those terms occur makes all the difference is to whether they count is major minor and, of course, the crucial middle. Now we also have something addition to the major and minor terms. In a categorical syllogism. We have the major and minor premises because the major term occurs only once in the premises, you can spot it as the major premise. And because the minor term occurs only one time the premises wherever that term occurs, it's gonna be called the minor premise again. That's just because, in contrast to the middle term, thes terms occur only once in the premises. Hence they condense IG Nate, the premises is major a minor in virtue of that fact. Now, now, when all the previous conditions for a categorical syllogism are met and the major premise is listed first as, of course, major things probably should be. Then the categorical syllogism is said to be in standard form, and I'm gonna talk a little bit here about standard form. So take some notes now, standard form, categorical syllogism, czar, all that we're going to need to analyze from here on. It's all we're going to study, and it may not be obvious to you as to why that's the case. Look at the argument to the left that we've been studying for quite a while. It's pretty much the same as the argument to the right, isn't it? I just swapped out to the premises, one for the other, but they're the same premises. So if one argument is valid, so is the other. And yet one of these is approved according to the standards of standard form and the others corrupt. Why in the world think that? And on top of the fact that you just wind up with the same argument when you transform a nonstandard into a standard form, it's kind of tedious. Consider this example. All watercolors or paintings. Some watercolors are masterpieces. It does follow from this that some paintings are masterpieces. But masterpieces, as you see from the conclusion, is the major term. So we got to swap out and put that major premise as the 1st 1 Well, that seems tedious, in addition to the fact that it just winds us up with the exact same argument. So why even do that? While sticking to this non standard form reduces the number of moods of arguments that we're gonna have to analyze. And when I refer to moods, I'm talking about the order of sentence types. When a syllogism is in standard form now, you could define mood a little differently just to be any string of sentence types with our Socrates Men Immortal argument. We end up with a premise, followed by an a premise, followed by an ache for a conclusion. Star Sentence types for this argument, and the mood thereof would be a or consider the watercolors example. We had a particular premise first and ah, Universal premise. Second, followed by a particular conclusion and I ai. But this argument, in contrast, of the last one, we give us different moods. Teoh Analyze to see if it was valid if we allowed different premise orders. Notice that if we swapped out the green box in the red box there, we'd end up with an A I. I moved for the argument. Why not just stick with the major forms, or rather, standard forms? And that way we'll have fewer moods to analyze, and we try to figure out which arguments are valid and which ones are not. It's a good argument for sticking with standard form. Let's review thus far. If you have two premises and a conclusion that's a syllogism by definition, and if they're all categorical sentences involved, then you have a categorical syllogism. And if the major premises listed first, followed by the minor than the conclusion, you have one that isn't standard form, and then you can analyze it according to its mood. That is the order of sentence types in argument. But in order to tell whether or not of categorical syllogism is valid, we're going to need to know more specifically whether or not that mood of a standard form categorical syllogism is valid depends upon figure. If you know the arguments in standard form, you know, roughly where the major minor middle terms are gonna show up. You know the major terms gonna be in premise one in the conclusion, for example, but figure nails down where these terms show up, and that's when we know exactly what type of argument we're dealing with him, whether it's valid or not. To illustrate the need, we use an example before moving watercolors being masterpieces and paintings. Now that this is an I A I form as we said before now, does the validity of this argument show us that I A My mood of a standard form categorical syllogism will always be valid. Try this. Ah, Emphasize, I'm gonna swap of the positions of the middle and minor term Middle of major term in the first and second premise, respectively. Some flying animals air animal mammals. I'm sorry. All elephants are mammals. So does it prove that some elephants are flying animals? I don't think that conclusion follows, but again, we're dealing with an eye ai mood, obviously, to find out which I A eyes are valid or not, we need to nail down with a middle term occurs and whether it occurs in the proper spot in each of the premises. How are we going to do that? Well, here, I'm gonna call in a little help from a big friend. Not a personal friend at all. In fact, not even a logician. Multiple time, Mr Olympia Jay Cutler may and look at the size of this guy's back. It's a guy who ah, maybe likes toe work on his figure quite a bit. So for those are in the mood to show off their figures. Uh, let's use this guy as, ah illustration. Rather a pneumonic device. Remember what figures there are for us to memorize. Now there are exactly four figures places that the middle term could show up. Assuming that the predicate of the conclusion will be in the first premise of the argument , the middle term is in figure. One could be listed first and then listed second in premise number two and so wanted so forth. These pretty well exhaust all the ways that the middle term could show up in the premises of an argument. How in the world is Jay Cutler gonna help us remember? Figure 123 and four. Well, maybe this isn't helpful for you, but it sure helps me out an awful lot. So thank you, Jay Cutler. So now what we've done is we've pinpointed where that middle term is going to show up in any given premise, and that's what we needed to do to get around that problem with the Flying Elephants. Example earlier, Right? So now we can tell which arguments categorical syllogism is that is, are going to be valid or we're not. That's some good news, but there's also some bad news. The good news is now that we know can pinpoint where the middle term is. Thanks to figures, we can take an inventory of all the valid categorical Cilla Chism's. What use is that, though you're gonna memorize them all wealth. Here comes some more good and bad news in the Middle Ages, they did come up with an method to memorize all the valid moods given a figure. Here you go, people. Somebody made a list of names in the Latin so that under each figure just listed, you could tell which mood is going to be a valid one under figure one. Barbara is a famous example. A is valid on one. The mood e a E is also valid and so on and so forth. So all you have to do guys to memorize which mood is valid on which figure is to memorized Barbara Celery and Dari far eo Barbary. So on and so forth all the way through. Isn't that easy? Heck, no, it's not easy. To make matters worse, this covers all the arguments or standard form syllogism that are valid notice. There's a distinction here. The ones at the top are very different from the ones at the bottom. The ones at the top are unconditionally valid. Those are the ones that are valid, given Boolean assumptions, namely that Universal's premises have no existentially import. But there's the air statist illion. Categorical syllogism is the one that are valid on the assumption that the Universal's carry existentially import. So in addition to memorizing this list of names, you're gonna have to memorize the distinctions between what's valid on which assumption. Oh, boy, how can we make this easy? Um, I guess if you want to make use of mood for figure and standard form to tell which arguments are valid or not, you've got a lot of memorizing ahead of you, don't you? Well, let me help you out here. There's a problem and a benefit to this little example listed here in the gray. There's a certain form E i O. That is valid on every single figure, and that's what I'm going to make use of to revise this very old and time honored pneumonic device. If it deserves to be called a pneumonic device at all, who wants to memorize this list of Latin names for my two cents. You shouldn't do it. It's obsolete, and I'm gonna come up with something better. The thing that I think is better I call the neighbors pneumonic those ah, for, uh, figures and moods that I listed in the gray. They all take the pattern or mood e i o e i o is valid on every single one of the figures that we studied. I'm gonna make use of that to come up with a superior pneumonic device that which been has been used throughout the ages. So to get our new project going, I'm gonna need a little help. My friends Barbra Streisand, known for her political singing and movie career and also Alfonzo Roberto, who's been known for his roles in classic sitcoms And many of you watched and last. The Orlando Bloom, who is probably best known for his role, was like a loss in The Lord of the Rings. Siri's notice. The neat thing is, each of these persons has a name that corresponds. Ah, modern and not a Latin name that corresponds to some of the valid moods on each of the figures. The A A a 00 and the O ao What if we can use this in order to memorize our valid moods? Given a figure, think of this first. Barbas Neighbors, Maiming feared. I'm not asking me why Barbara has neighbors who fear maiming. Just remember that sprays Barbas neighbors naming fear to notice. I'm using the word neighbors here. Look at figure number one and the moods that are valid on it. Under Boolean assumptions, barbers, neighbors, maiming, feared covers each and every one of these in the English. Not with those ridiculous lists of Latin names. You're one step towards getting all the valid moods given. Figures. Ah, one through four. Let's see if we can make the rest of those leaps will make that next little jump. With Alfonzo's help, Alfonzo's neighbors apparently don't like toe have jobs. Don't ask me why, but Alfonzo's neighbors feared careers just as much as barbers. Neighbors naming, feared. So if you can remember that Alfonzo's neighbors feared careers, you're just one step closer to memorizing all of the valid moods. Given a figure, only this time it's gonna be figure number two on figure number two. Alfonzo's neighbors may feared careers corresponds in terms of the vowels in it to the vowels listed in each one. The Latin names and figure. Two. Congratulations. You're one step closer. Can we make the last two leaps to figure three and figure four? In fact, we're gonna do both of them at the same time. With a little help from Orlando Bloom, Orlando's neighbors disdain, maiming and disdain careers for neighbors. I got a whole bunch of people showing disdain towards careers up in my little ah picture here. I hope you can remember it. It's goofy and just goofy enough. Be remembered. But if you can remember that Orlando's neighbors disdain, maiming and just staying careers for neighbors, then you've got all the way towards memorizing the valid moods. Given figures which figures that handle Figure three and four together noticed that Figure three has four valid moods on bullion assumption, and Figure four has three valid moods. So while this isn't a perfect example or illustration are pneumonic device, Orlando's neighbors disdain naming corresponds to all the vowels and listed under Figure three and the names, thereof and disdain careers. Four neighbors as long as you can list four as not a extra word. After all, what sort of valid mood would have just 10 f o r. Give it to that and you have staying careers, neighbors. And that corresponds in terms of its vows. Toe all of the valid moods under figure for congratulations. You're done. So remember, instead of Barbara Accelerant, Darif Aereo and all this other stuff that comes under the green portion that is the portion of this diagram that lists valid Boolean moods. It's just taking a different route. How about instead, barbers Neighbors, maiming, feared Alfonzo's neighbors for your careers. Orlando's neighbors distain naming into staying careers for neighbors. There you go. You're done now. Isn't that pneumonic? A little silly? Ah, I guess we can answer that question forthrightly. It's about a silly as it comes. Barbers. Neighbors naming, feared Alfonzo's neighbors fear careers. Orlando's neighbors disdain, naming and disdain careers for neighbors. Uh, I just memorized all the valid moods, and it's not a silly is trying remember all those Latin lady names? Plus, you have just a good memory in Latin and one last big note. There's another advantage to my ah pneumonic device. Apart from that pneumonic, that's better than the pneumonic device where you have memorised all those latte for any word that I use in that pneumonic device, for example. Ah, word feared. It's used to denote that E a is valid on a particular, uh, figure. And I never repeat any other word where e is gonna be valid. Any place you find feared is where e a is gonna be valid. Any place you don't find the word feared e a is invalid. And that goes for every word and every possible combination of vowels in the pneumonic device. Pretty helpful, huh? Well, we still have one last issue here. My little pneumonic device. Just talk to you how to memorize all the moods, given a figure that are valid under Boolean assumptions. But what about all those that are valid under Air City Illion assumptions noticed that in these Latin names, when you look at the bowels and read, you start off with A and E or E and a or a and a universal's, and then you move to I's and O's for conclusions. In other words, the universal's must have existential import because you're reaching a conclusion. That's an iron oh sentence. And those are known to have existentially import Is there any way for memorizing these? I want to point out one thing that may be helpful in memory, Captain or Fernando in his plane. And he's being chased by some other pilots. Captain Fernando, huh? Likes to go upwards with this plane. All these other pilots air trying to attain his trajectory. If they're trying to attain Hernandez trajectory, then remember this. There are only three valid Aristotelian moods for any figure, if you go back to all those names that were listed under the Aristotelian valid moods in the yellow in the previous slide, you only found three combinations of vowels that worked a a i e A O and e o are the only combinations that are ever going to turn out valid regardless of figure. Hope this pneumonic device turns out helpful for you. Well, I hope this has been a helpful lesson for you. It certainly helps you memorize the concepts of categorical syllogism, mood figure and all the other related concepts. And it gives you a helpful pneumonic device to take all that knowledge and put it to some practical use. So just stick to it and you're gonna memorize a whole lot and get good days in your college courses, and you're gonna do so really quickly. In the meantime, as usual, hang around for my exercises on this lesson and my next logic lesson, which will be coming up soon. Thanks, guys. Bye bye. 12. Venn Diagrams: Hello, everybody. And welcome back my crash course In formal logic, this time we're gonna be studying the very popular topic event diagrams, and it's gonna be explained very simply so. The basic idea is to prove the validity of categorical syllogism and the neat thing about Venn diagrams, as you don't have to worry about whether or not the categorical syllogism is in standard form or not. Then diagrams will prove validity or in validity, either which way and the other need thing as that Venn diagrams demonstrate validity and in validity in a visual way with a picture. Basically, what you want to show is that if you're forced to depict the premises that is to visually show the premises is being true, Then you are forced to depict the conclusion or not forced in the case of in validity to depict the conclusion is true as well. Now, John Archibald, then, is the person who is responsible for developing these sorts of techniques. You might want to look at the dates on this guy 18 34 to 1923. Basically, he's a long way away. From the time of Aristotle, you might wanna ask yourself if these techniques were not by used by Aristotle to prove validity versus in validity. Exactly how did Aristotle do it? And that's a topic for another time. Now remember what we said about the terms and a categorical syllogism they feature or warm like links in a chain. And there's a term in the chain. The middle that links up their remaining to the middle term links up the major and minor terms, the major minor terms occurring in the conclusion. So again, this idea the middle term is a linking term really comes into view pictorially in a Venn diagram. Here you have the basic outline the minor term over to your left, the major term to you, right in the middle term at the bottom. And sometimes it's Ah, this sort of model is flipped over on its head so that the middle terms on top, But you always get this sort of three ring circus going on. I guess now the thing is, you could express a whole lot with this particular diagram. For example, I could say that everything in the middle minor term classes in the middle two term class like that or I could say that everything in the major term classes in the middle term class like so maybe nothing in the minor term class is in the middle term class or nothing in the major term class in the middle term class. So since the middle term class always shows up in our premise sees, these sorts of ability to express these sorts of relationships is really important. Now I'm gonna use these numbers and my Venn diagrams in orderto talk about what we're doing with the Venn diagram. Strictly speaking, you don't need to put these in your homework assignment or your exercises, though for now, we're gonna keep it simple. We're going to stick to the Boolean standpoint and that is that A and E sentences. The Universal's have no existentially import. And then we'll go ahead and move into Air City illion versions of this Venn diagram scenario later on in this lesson. But for now, keep stick with bull. He's the most popular in the received view with with respect to how universal sentences ought to be interpreted anyways, so this is gonna be our very first Venn diagram. For the very first argument, we're gonna diagram with it All firemen are brave things. And no brave thing is made of stone. Does it follow that nothing made of stone is a fireman? Let's find out. So here's our argument and I put a Venn diagram down to your left notice. I also put a circle over to your right. I'm gonna make use of that here in a second. Although, strictly speaking, you don't need it. I'm just using it for educational purposes. Now, first of all, I'm gonna take a red, blue and green here. Teoh indicate which terms or which Obviously I want to use brave things is my middle term the blue circle. The bottom fireman is gonna be our major term. And the minor term is in the green things made of stone. Now, what is the the premises? I'm gonna diagram over in the Venn diagram, but I just want to take a minute over here to your right to diagram the conclusion. No thing made of Stone is a fireman. Here you go. Just means we have to block out what's in the middle. Now look back at your Venn diagram. What that means is we want to see if the premises force us to block out sections two and five. See that we want to make sure that the premises, by diagramming them, forced us to diagram the conclusion, which I've already diagrammed in a separate diagram over to your right. So it's start with premise. One. All fireman are brave things or persons. There you go. That means we have to block out two and three so that everything in the red circles in the blue And if no brave thing is made of stone, then I have to block out foreign five because nothing can be in the overlap between the green and blue circle. Now, is this argument valid? Look at the diagram. Were we forced to block out two and five, and that's exactly it. You noticed that by diagramming the premises, we were forced to diagram the information in the conclusion. That's what proves the validity of this argument. Let's take another example. No penguins are Olympic athletes. No Olympic athletes are cold blooded. So does it follow that no penguins are cold blooded. We got two negatives and we're trying to draw negative. It's an interesting case. Let's give it a shot. Well, once again just for education sake, I went ahead and ah, highlighted in the blue, or rather, change the thought to a blue for the middle term, meaning Olympic athletes. Penguin is our minor term, and the cold blooded things is our major term. Now again, over to the right, I just diagrammed the conclusion. No penguins are cold blooded things. Once again, we want to see if the premises force us to block out the middle section between the green and red circles at Sections two and five. Let's start with premise one. No penguins are Olympic athletes, so nothing can be in the overlap between the blue and the green. There we go. Form five blocked out. Now let's go with premise to know Olympic athletes are cold blooded, so that means we have to block out the section between the blue and red circle, part of its blocked out. Already, Section five is blocked out. Let's go ahead and block out six as well Now, were we forced to diagram the conclusion? No, we were not. There's information in the conclusion that was not contained in the premises. Section two should have been blocked out if we were had genuinely been forced to diagram the conclusion That means this argument is not valid. Let's try our old standby. All men are mortal and ah, Socrates is a man or all things. I didn't call the Socrates or men does it follow that all things identical to Socrates or mortal Well, by now you already know. Definitely this is a valid argument we want. What we want to see is whether or not then diagrams can help demonstrate that So here's our argument. All men are mortal things and all things identical to Socrates are men. Men is the middle term and mortal things. Is the major term Socrates being the minor term? Now? Once again, you don't have to do this but over to the right and made a small diagram where I'm gonna diagram the conclusion. All things identical to Socrates are mortal. That's the conclusion. So go ahead and I block out all the section of the green that's not in the Red Sea. That so what we want to see on our Venn diagram when we diagram our premises is whether or not we're forced to block out sections one and four. So let's start with premise one. All men are mortal things. Well, that means that four and seven are gonna have to be blocked out because everything the blue has to be in the red circle. That's try Premise to now all things identical to Socrates are men, so everything in the green has to be in the blue. There you go. Now, where we forced to diagram the conclusion? The conclusion just said that we had toe fill in places before. And that means the argument is valid by diagramming the premises over on the left hand side , we were forced to diagram the information in the conclusion which I conveniently put over to your rights. You keep it in your eyes view. Well, do you feel like you're getting the hang of this? Well, we've already worked with some arguments, but we that involved the universal claims, the A sentences and the e sentences. But now we have to worry about the other senses, the eyes in the ohs. How do we work with those sorts of claims when it comes to these diagrams? Well, one of the most convenient ways is going to be with a floating X. If you say, for example, something in the minor term class is in the middle term class. Well, you don't really know exactly where you just know that there's something in that overlap something in sections, either four or five. So we use a floating X to say there's at least one thing in the minor term class. It's in the middle term class, although we may not know where or taken oh sense. For example, some major term is not in the middle term something in the major term class that's not in the middle term class. How do you diagram that? Well, that must mean that there's something up in the red circle that's not in the blue, but you've got to places for that Sections two and three, respectively. And there has to be just one thing that we diagram because that's all the, uh, that's all. The premise here asserts something. I'll just use a floating X. That little floater means that X maybe in two or three. We know he's in there somewhere, so floating exes give you a way to diagram these sorts of term claims. Now that could be a little bit confusing and uh, I want to simplify this because in, well, most arguments you're not going to need a floating X. You're gonna be able to get away with one X. Typically, here's a hint. If you're dealing with an argument rather than going with floating exes to diagram your your claims about I and no sentences start by diagramming the universal premise first and one of those little exes and the floater is not gonna be necessary. I'll give you an example. So nothing in the middle classes in the major class and something in the minor class is in the middle class. Does it follow that? Something in the minor class is not in the major class. Sorry to be a so abstract here. I know probably many of you want to go back to Fireman, Penguins and Socrates, but I'm just gonna try and illustrate some points about exes. The main thing here is diagram. The universal premise. First, nothing is in the overlap between the blue and the red. It says that premise. Now the second premise says something is in the overlap between the minor and middle terms . So here's my exes. There's plenty of overlap there right now. Notice, though one of these exes is strictly unnecessary. It can't be because we already had a premise that says nothing exists in that area. So consequently, we're left with just one X in this scenario. Now look at the conclusion Something in the minor class is not in the major. Do we have an X in the green circle that's not in the red circle? Yes, we do. This argument is valid. Let's try another example to show how diagramming Universal's first gets. Rid typically of the need for floating X is now something in the major term classes in the middle term classes. Premise one. The second premise says everything in the middle term classes in the minor term class. We want to see if it follows that some things in the minor term class and in the major term class at the same time. So let's start with the second premise this time everything in the major middle term classes in the minor term class I just shaded in their appropriate area. Now we're gonna move to the first premise here something in the major term class. The red circle is in the blue. Do I need a floating X here. Well, obviously not one of these excesses strictly irrelevant. The one on top does not at all all illustrate the truth of a premise. We need to get rid of that thing. So the only relevant X that we could put in this diagram to illustrate the first premise that some things in the red and the blue circle the same time it's gonna be this one. So, consequently, would we get something in the minor term classes in the major term class? Indeed. So what does that prove? It proves that this argument is valid. Diagramming the premises forced us to diagram the conclusion. And that's how Then diagrams go so clearly when you have I and, oh, sentences to deal with. You really want to have a universal premise to diagram first, cause it helps you eliminate some of those floating excess typically. But you may be asking yourself, what if I don't have a universal? Well, in that case, the argument is invalid. Aristotle was the first to notice. In the absence of a universal premise, no conclusion can be drawn. Need to remember that now, if you still need to put in a floating X. I need to remind you of something every argument that you can ever diagram that requires a floating X. If you need to do make a floating X because you're not sure where the X has to go. According to a premise, your argument is invalid. Those floaters will invalidate arguments constantly. That's a second thing to remember. I'll illustrate that point with this argument. No Olympic athletes are cold blooded. Some penguins are not Olympic athletes. Does it follow that some penguins are not cold blooded? No, it's not exactly obvious, says it. So let's give a dia a diagram to show whether or not this argument is valid. Now what the conclusion says over to the right, some penguins are not cold blooded things. There's an X in the green circle that's not in the red circle in the minor term class. That is not in the major term class. So let's start with premise one. No Olympic athletes are cold blooded, so I'll block out five and six the overlap between the blue and red. And if some penguins are not Olympic athletes, there's got to be something in that green circle that's not in the blue. But where do you want to put it? You want to put it in section one? Or Section Two could be there or that something could be here either. Which way? We have something green penguin, so to speak. That's not in the Olympic athlete category. So what we got here? Did we demonstrate the truth of our conclusion? The conclusion said that we had to put something in the green circle that was not in the red but for all the premises. Told us that, uh, that something mentioned in Premise two could have been in section two of our Venn diagram . So once again, we got a little problem here. What's that prove? It proves the argument is invalid. The thing is, this little X that I have jumping about in section two of the Venn diagram if we had just diagrammed him all by his lonesome, then we would have had a perfect picture of the truth of the premises. And yet we would not have the truth of the conclusion. There would not be an X in the green circle that was not in the red. So that means there is a way for the premises to be true. And the conclusion balls. And as long as there is at least one way, the argument is invalid. What you need to remember at all times is that validity does not simply amount to ah possibility for true premises and a true conclusion. That's not what we mean. Obviously, in the last example we could have if we wanted diagram the truth of the premises in the conclusion at once, all we would have had to do is put the X and Section one, and that would have shown true premises in a true conclusion. But what validity means, really, is that it should not be possible at all for the premises to be true in the conclusion. False. And what we found was there was one possibility. It was when we put the X in Section two. So always look for the invalidating X. That's how you deal with I's and O's when you're doing them and then diagrams again. In our last example, we had a floating X, and one of the exes would have made the premises and conclusions about true it once and one of the exes would have made the premise is true, but the conclusion. False? That's the X were looking for. Try to see if there's a way to paint a picture of in validity of the argument. So if you can remember that, always try to invalidate if you can and see if you're forced to validate the argument anyways, then you've got the basic idea of a Venn diagram. So we got a couple hints here for dealing with the I and O senses the particulars. First of all, there's got to be a universal premise somewhere. So always diagram that first, and the second thing to remember is, whenever you have a floating X, you're gonna have an invalid argument because there's going to be at least one of those exes that's gonna invalidate the argument. Now we have one last thing I've already covered. Boolean approaches, defend diagrams and remember most logic. Textbooks will emphasize those because Bull is received interpretation of universal sentences or he has the received interpretation of them. But Aristotelian assumptions proved useful in certain cases. When we know that there's something that exists in our subject class, let's see if we can ah use Venn diagrams to illustrate Aristotelian validity. Remember our last lesson? We looked at different figures in standard form. For all the arguments that could possibly be made out of categorical sentences or at least all the syllogism that could be made out of him, it was a pretty complicated list. The top portion represents the arguments that are valid on a Boolean assumptions or Boolean interpretations of universals and the bottom on air their city lian assumptions. Now all the arguments that are valid for bull or valid unconditionally. Bairstow Teeley In our arguments valid on Air City Lian assumptions are called valid Conditionally, The condition is there has to be a non empty subject class and they basically all we need to know is whether or not the subject class mentioned in the conclusion is empty or not. So you might want to ask yourself, Is this going to get tough? Switching from Buddha, Aristotle and I've got good news for you. It's actually remarkably easy. Teoh, move from a Boolean um demonstration of validity with a Venn diagrams to an heiress to Teeley in a demonstration. Basically, all you need to do is follow the same procedure you'd until now But once you're done with everything, once you're done trying to prove the argument is or is not valid on Boolean interpretations of Universal's Go Ahead and Putting X in the conclusion of the subject, Class of one's not there already now. So here's an example of three moods. AI are valid on figures 13 and four. Let's try out this one and I'm gonna use Figure one is an illustration. All man mammals are animals and all tigers are mammals. Does it follow that some tigers are animals? Let's use the Ben diagram to prove it. So the conclusion says there's an X in the overlap between the green and red circles that see if that happens. Premise number one. All mammals mammals are animals. Ah, block in foreign. Seven. To illustrate that premise, Number two all tigers are mammals Saw block in premise one and two. Now notice for bull. That's the end of the game. And the conclusion does not follow for Bull because no, uh, existential conclusion can follow from universal premises. But the last step, we said, is we need to put something in the green circle to satisfy Arrow Aristotle's requirement that we not talk about empty subject classes, and there he is now, once we satisfy the Aristotelian Assumption or the Air City Lian demand, Weren't we forced to draw the conclusion that what that proves is that this argument is valid for Aristotle? Even if it wasn't for Bull? Let's try another one. Let's take a I but notice A. I is not valid on figure to even for Aristotle. Let's show that that's the case. I'm gonna have to just this example a little bit that we use before. Okay, So bear with me. I'm gonna give an example in which all mammals are animals and all tigers are animals. And see if it follows that all some tigers are mammals, by the way, for convenience. Since I had to switch up this argument and now the middle term is in red. Let me go ahead and change our color scheme here a little bit. There we go. Same argument, different colors. Just want to make sure that we keep our middle term in the blue. So the conclusion is that some tigers are mammals. Illustrated the right. Let's start with Premise one on our Venn diagram. All mammals are animals will block out sections two and three. Now all tigers are animals that forces us to block out sections one and two, although two has already blocked out. Now, obviously, the argument is invalid for Bull. But if Aristotle insists that we talked about subjects that exist, we have to put an X in the green circle. But where does it go? Were we forced to put an X in the overlap between green and red? It doesn't look like it. Consequently, this argument is invalid for Aristotle. Now some of you may be confused as to why this is such a bad argument, because types, some tigers really are mammals. The reason is this form of argument could just as easily have given us the following argument. All circus clowns are animals. All politicians are animals. Does it follow that some politicians air circus clowns? No, although they sometimes actually act clownish Lee. But that's a separate issue Now. Remember the three step short list to be used in a previous lecture? First step is always to check to see if an argument is valid on Boolean interpretations of universals, if so valid unconditionally. Otherwise you want to ask, Is it valid? Assuming that we have of subject class member of for the conclusion in this case, if no invalid, unconditionally, just like the last example we looked at. But we did find an argument just before the last example two examples back in which we did have a valid argument assuming that there were subject class members and finally then just asked where we, uh correct. And assuming that there were subjects in that class, does the Aristotelian assumption hold up? Well, in the case of the tigers and mammals and things of that nature, it did hold up. Have these two examples back. And if not, then the argument is invalid anyways. Well, that's all on Venn diagrams and everything you need to know. So awake from exercises on my next logic lessen the next logic lesson is going to the last time that recover Categorical syllogism. So then, after that, we're gonna move on to proposition a logic and that'll be the end of your first course and logic. Congratulations for making it this far. We'll see you next time 13. Rules for Categorical Syllogisms: well, though, that everybody and welcome back to my crash course in formal logic in this part, we're going to cover rules for categorical Cilla Chism's their technical rules. But if you memorize them to get him under your belt, you'll be able to spot immediately by looking what is and is not a valid categorical syllogism. This is gonna be a pretty short lesson, but it'll be pretty technical, too. So hold onto your hats. And, ah, if you have trouble with this lesson, don't worry. You've learned a lot of skills so far, more than enough for an introductory level college class. Now, the four rules that we're gonna learn in this apply to, uh so each of these rules Presupposes already that you're not gonna be committing the existential fallacy if you want to remember another rule called The Rule of Existentially fallacy. All that means is don't draw particular premise conclusions from universal premises, at least if you're doing Boolean logic. But you could treat that as 1/5 rule if you want. I'm just going to treat this lesson is covering four rules Now. As we said before, when you're doing a deduction, the conclusion is already to be contained somehow within your premises. So that tells you already that if the conclusion is thus contained than if your conclusion says a lot about some term your premises or better to say a whole lot about that term as well. A second thing you need to remember is that you better have a strong middle term, a strong link up term, so I'll return to be in just a second. Let's focus on a If your conclusion says a lot about something, your premises of better do so, too. You guys remember this, uh, issue distribution? Ah, which proposition distributes which term a propositions distribute the subject? E distributes both I distributes none, and oh, distributes the predicate. That last one was the hardest one to remember. And I would also confessed in the lesson in which be covered this topic, that it was a pretty boring one. Why should we care about it? Well, the problem. The issue is really that distribution measures how much information about a class or a term , the proposition using that term or class contains. So that's a measure of how much you can deduce from that. So let's go into more detail on this topic. So as I said before, if you say a lot about a term in the conclusion that is you distribute it in the conclusion , then you'd have better to say a lot about it in the premises. That is. You two better distributed it in the premises at least once. So look, this argument all eighties air human beings and some gentlemen are not ladies, does it follow that? Some gentlemen are not human beings? Boy, I hope not. Because those premises air true, but the conclusion looks ridiculously false. So what went wrong here? The problem is the term human beings in the conclusion it is distributed here, remember? Oh, sentences distribute their predicates Now. Was human beings distributed anywhere in the premises? It could only only have been distributed in premise one, but it's not a sentences. Do not distribute their predicates. Or consider this argument. No, gentlemen or ladies and all ladies air human beings up to true premises. Once again, Conclusion no human beings or gentlemen. Now that sounds completely false. What happened here? Well, it looks as though we have a term again human beings that is distributed in the conclusion was that term distributed in premise number two? No, it wasn't. Now there are names for these sorts of fallacies, and well, whenever you distributing the conclusion. But you didn't distribute determine the premises you're doing something illicit, So that seems like an appropriate way to describe the fallacy. But whether or not it's an illicit major or minor depends upon what's going on. The conclusion. In the conclusion of the first argument, the major term failed to be distributed among the premises, and in the second argument, it was the minor term human beings that failed to be distributed among the premises. So whether it's illicit major a minor. Well, look at the conclusion in orderto find those terms off to find the appropriate term to use . And if you're ever in doubt, you can always go to the old standby the Venn diagrams. So I'm gonna go ahead and diagram the first argument. All ladies air human beings, the limb foreign seven. Some gentlemen are not Ladies. Oops, floating X. That's a dead giveaway that we've got an invalid argument to recall that let's go ahead and move down. Argument number two no gentlemen or ladies. So block in five and six and all ladies or human beings. So block in seven and six. Does it follow that? No. Human beings are gentlemen. Is there any room left for there to be something in the green circle in the red circle the same time? Yes, there is. I just highlighted it in section two. The argument is invalid. Fallacy of illicit major fallacy of illicit minor. Here's an example so that you can get some practice. How about this? All the NBA players or athletes and all athletes are healthy people. Does it follow that all healthy people or N B A. Players? Well, the premises looked through in the conclusion fall. So you should suspect in validity which of the two fallacies we studied so far. Do you think this commits? I'll give you a second. All right, Time's up. The only thing that should come to mind is there's only one thing distributed in that conclusion. Write a sentence, is distribute their subjects and was healthy people distributed in premise number two. It absolutely wasn't. That's the fallacy of illicit minor. Here's another one to give you a little bit of practice. All mountain goats are mammals. True enough. And some mountain climbers are not mountain goats. Well, I've given you a picture of one over to the right. Does it follow that some mountain climbers air, not mammals, That conclusions Very likely false. So go ahead and try to figure out what went wrong with this argument. And your time is up. Look at the conclusion. What got distributed there, if you remember, Oh, sentences distribute their predicates. And was that distributed among the premises? It was not distributed in premise number one. This is the fallacy of illicit major. And if you're ever in doubt, remember, you can always jump back to your old standby. The Venn diagram. It'll do the job. Justus. Well, telling village validity from in validity when it comes to categorical Cilla Chism's Well, I promised you a two of rules regarding distribution. We know that if the conclusion is contained in the premises, a Your if your conclusion says a lot about some term, your premises had better do so too. Otherwise, you'll commit the fallacy of illicit major or illicit minor. Secondly, though your your premises had better have a strong link up or middle term. In other words, just to put it bluntly, you had to distribute your middle at least once in the premises. Recall again the links in Ah, the terms and a categorical syllogism function. Kind of like links. And that being the case, the middle term is a crucial toe linking up. The other two might even think of it. Something like this. So try this argument on for size and remember the rule. The premises must distribute the middle term at least once. That makes the middle term at least strong enough in those premises to make a good tight bond. So to speak, between the major and minor term in the conclusion. All ladies air, human beings and all gentlemen are human beings. Does it follow that all ladies or gentlemen? Well, you know better than that. What went wrong here? Notice that in each case we have a sentences for premises and a sentences distribute their subjects, not they're predicated. Consequently, we have a bad argument. Due to the fallacy of undistributed middle, the middle was never distributed, not even once among the premises. And if you want, you can always use your old standby. The Venn diagram. So if we say all ladies air, human beings, I'll go ahead and block in sections one and two. And all gentlemen are human beings I block in sections two and three. Does it follow that all, ladies or gentlemen? No, it did not. We still have a section four right here that I just highlighted. That shows there's place for, ah, ladies who are not gentlemen, in fact. But that's a very populated section for well, that concludes all of our lessons regarding distribution. Remember the two rules. If you're going to say a lot about a term distributed term and the conclusion, make sure you've done it at least once in the premises and make sure the middle is distributed at least once in the premises. Now moving on to talk about positive and negatives, the quality of sentences Negatives are hard to draw. Any conclusions from from two negatives First rule. Nothing ever follows. You can't get a conclusion from totally negative premises. And if you have one negative in your ah premises, you can only conclude a negative. Now, that's an interesting rule from a combination of positive negative premise. You get a negative conclusion every single time. So we can abbreviate the rules or rather illustrate them. Kind of like this from two negatives. You get nothing. You get nothing from nothing but for any combination. Negatives follow. So when negatives do get involved in the argument, you're either gonna end up with nothing or negatives. And that's just the way negatives work when their premises try this one. No, ladies or gentlemen, some humans are not Ladies and e sentence for a premise and an O sentence. Does it follow that some gentlemen's are not humans? It may seem plausible to think this follows because you're drawing and negative conclusion , but from two negative premises, you can't even get a negative conclusion. You get nothing from nothing there. And if you want me to prove it once again, we can go to our old standby. The Venn diagram. No, ladies or gentlemen will say so. Block out four and five, and some humans are not. Ladies. Looks like we got a floating exits going to show up there. And floating exes are always characteristic of invalid arguments. However, what happens if you get a negative premise and you mix it with a positive premise. Well, if you have one negative premise, you get a negative conclusion, and vice versa. So basically one negative premise and a negative conclusion are synonymous with one another . You can either have a scenario like this or scenario like this, but that's the Those are the only two ways that you're gonna wind up with negative conclusion so we can summarize the four rules. If I just add one more little brief. Here, you do get, ah, positive conclusion, but only when you get to positive premises. And if you could memorize this chart, you've memorized pretty much rules three and four. So let's try this. The upshot, that of ah, number Rule number four is that you don't get negatives from positives. How about this? Instead of saying all n b A players or athletes and all athletes are healthy, therefore, Oh, uh, healthy people R N b A. Players which we saw before committed the fallacy of illicit minor. Let's give another shot. Let's wipe out this conclusion and try for another. Some healthy people are not n b A players. Does that follow from our premises, not one bit again that commits the fallacy of trying to get a negative from two positives from positives. You only get a positive. So everything that I had to say in this lecture can be summarized here. In a nutshell. Assuming that there's no existential fallacy to worry about, you just got four issues to worry about, which can really be summarized in to the first rule is to make sure the major minor middle terms were distributed appropriately. When it comes to the major miners. Make sure that if they're distributed in the conclusion they were distributed in the premises and make sure the middle term got distributed in the premises at least once, no matter what and the second rule regarding positive and negative premises, make sure this is important. Make sure a negative conclusion is drawn. If and only if exactly. One negative premise was in the argument that pretty much summarises the entire chart that I gave you before about positive negative premises and how they work, see if a negative conclusion was drawn if and only if there was exactly one negative premise in the argument. Now you've got a lot of ways now underneath your belt to test for validity, assuming no existential fallacy is committed in these four rules are obeyed. The argument is valid, so check to see these rules were obeyed. And then you can basically assume the argument valid, regardless of whether you've done about Ah Venn diagram or not, isn't that convenient? So if you get really handy of these four rules, you can eyeball an argument and know whether it's valid or not. But if you get tired of these technical methods, you can always the lean back on our standbys. I gave you the neighbors pneumonic previously, and then diagrams were the subject matter of our last lesson, and now you have formal rules to study. So you have three techniques for telling whether or not arguments valid, the neighbors pneumonic. You could do a Venn diagram where you can just check to make sure the four formal rules that we studied the second ago were all followed. If so, the argument was valid. Well, I told you this would be a short lesson, but boy, there's a lot of technical material bill material today. Digest is, they're not, but it's getting too late at night, I'm afraid. Well, that's all for now. Wait for my exercises in my next logic lesson. In the end, keep logic chopping. Take care 14. Propositional Logic, Symbols and Functions: Well, everybody. And welcome back to my crash course. In formal logic, this is an exciting lesson. We're gonna start proposition a logic, including symbols and functions. So this is gonna include a lot of material. Hold onto your hats. Okay. Now, Aristotle thought that logic should be about categories and their relationships to one another. There's an alternative to this view, Chris. A piss of solely was a Greek stoic philosopher who thought that instead of being about categories, logic should be about propositions and their relationships to one another. What's the difference? Consider this. Categories do not take truth values. Only propositions do. If I would say to you true or false elephants, well, that can't be true or false. Obviously, neither can the term mammals be true or false. If I say to you elephants are mammals, then you have a proposition that can take a truth value. And elephants are reptiles, by way of contrast, may take the opposite truth value, But again, we're dealing with propositions at that point. So when air Stuttle, Ian, logician XYZ and people doing category logic used letters, they tend to designate categories all air, be some P or Q those peas, cues A's and B's stand for categories. When you do proposition a logic, you designate entire propositions with these capital letters. Now, after Christmas and Aristotle, what happened? Well, we're gonna have to do a little review of history. Well, the truth is, apart from the work of, Ah, Live minutes until the time of Frega, nothing really seemed a happen and logic. Aristotle dominated logic until the time of Frega and Russell and later Vic in Stein. Let's talk about these individuals for just a moment now. Prega was a German mathematician and philosopher. He was a mathematician who thought that he could reduce mathematics to deductive logic, which would allow him to show in logic, which mathematical proofs worked and which mathematical proofs did not. That's called low GIs is, um, by the way, he needed more than Aristotle because Aristotle Systems, just not powerful enough to prove all the proofs of mathematics, is also responsible for meeting a young victim, Stein and sending victims Stein to study with Bertrand Russell for our purposes. Remember, he helped develop the concept of a truth function. Now Russell, by way of contrast, was an English philosopher. He could appreciate Frega because his nanny actually was a German speaking lady. German was his first language, and he was also an expert mathematician. So being an expert math and speaking German, he could understand and appreciate Frega. Also, he was a logical Adam ist who believe that truth of a system of thought depends on the truth of its components. And when we start to build up the truth of compound sentences from the truth of the parts, you'll understand the importance of that for logic. Also, he's important for developing quantify IRS ritual study in another class. And then we have victims time. One of the most important philosophers of the 20th century. He was sent by Frega to study under Russell, and he developed two systems of philosophy in his life. Hence, the early in later victims, Stein and he developed the concept of truth tables. Early on, he even referred to ah, logic as the scaffolding of the universe. So you might want to look at all the truth tables that we study and noticed that they actually do look like scaffolds into which little truths fit. But that's a subject for later. And let's talk about validity in both systems. Aristotle's system and proposition a logic validity is a matter of form, however, in the context of Proposition a logic form. Recognition is facilitated by operators or connective. We're gonna learn five of them. The first is the tilde. It's a negating symbol. It means it is not the case that or it is false, that you have a dot, which means and a wedge, which means or or if you prefer either or you have a horseshoe, which means this implies that we just say if then and we have the triple bar, which is the equivalent symbol, we're going to study each of these in detail now. Simple statements are statements that don't contain any other as a component. For example, famous people tend to be rich, Mark Twain wrote. Huckleberry Thin paratroopers must overcome fear of heights, and Socrates was the philosopher. You just represent those with those capital letters we mentioned just a second ago F, m, p or S. Why did we choose those? Well, because they make the proposition easy to remember. F could be famous People tend to be rich. M convene, Mark Twain wrote Huckleberry Finn, P Paratroopers must overcome fear of heights. S Socrates is a philosopher. It's really kind of subjective. What letter do you choose to represent the entire proposition Now? Compound statements contained at least one other. Simple as a component. It's not the case that money grows on trees. New Yorkers on the East Coast L. A. On the West. Either the Democrats will win or the Republicans will win. If every person contributes just a little weaken end world poverty will resort to war if and only if all peaceful options have been used. Now notice I'm marked in the black. Some connective is here were connecting propositions. There's several propositions here. You can take a truth value. We can say with the first case. It's not the case that M money grows on trees. New York on that east in L. A. On the West Could be n and l. Democrats will win or Republicans de or are and so on and so forth. Now, when we abbreviate thes further, we can use our connective and make form recognition even simpler. Tilda M or n dot l and so on. And these sorts of things will help us look Locate the general form of an argument that we're examining. Now let's take some notes on the negation symbol, that tilde symbol. There's ways you can and cannot use this particular symbol. For one thing, you can't put it behind what it negates. You have to put it in front, and you can't use it to connect two propositions. It is our only one place connective that we have, and you can use it immediately after an operator. For example, G dot tota h means G is true in H is false, and you can use it to negate a compound. The last example says it's not the case that both G and H R Truth. This brings up an important concept. The concept of a main operator. The main operator is the operator that governs the most components in the sentence. In the first example here, right, it's the DOT. Basically, you have a conjunction of the claim that G is true and H is false. In the second example, it's the negation. We have a negated a compound sentence of g dot h. My students don't have trouble spotting the main operator, but if you ever do have trouble spotting it. Remember, it never has this set of parentheses around it. In the first example, DOT does not have a set of parentheses around it. However, neither does the tilde in the first example. So point number two, if two operators have no sets of parentheses and closing them, the negation is not the main operator. The only way the negation could be the main operators, like an example to when it stands outside of a set of parentheses. Now these statements are all negations, and I invite you to use the rules we just studied to try and figure out why the negation is theme main operator. Typically, it's because it has no set of parentheses and closing it now. Parentheses are important because they distinguish between different interpretations of a sentence. A dot be wedge C is not clear. It's ambiguous between a and B or true or else see is or else you could mean a is true. And also you get either beer. See, these are not the same things. Consider the following example. A Lois Lane can fly be Batman. Comply. See Superman can fly. The first interpretation with the prince sees around a M B is true in virtue of the fact that C is the case with Second, where a is asserted to be true. And also beer See is false because Lois Lane can't fly now when you translate sentences from English into symbolism, watch out for things that will help you out. Watch out, for example, for commas. Lois Lane can fly and Batman can fly. Comma probably means the 1st 2 propositions go to get there and watch out for words like either or both. Either Lois Lane can fly and Batman can fly Comma horse Superman comply. We could at least have used to comma there. It brackets off those 1st 2 propositions as things that need to go in parentheses together and watch out for a single predicated in conjunction with one or more subjects. For example, Lois Lane and Batman can fly it. We have two subjects. Lois Lane and Batman combined with the predicate can fly. That's another dead giveaway. They belong in parentheses together. Well, now that we talked about parentheses, let's talk about Tilda is in this expression the total effects only the A is false or else B is true in this expression since it comes right in front of a parentheses, the tilde affects the entire expression on the inside. He needed distinguish between those two. Case. So it's not the case that Air B is a little bit ambiguous. Let it be the case that you treat it as governing the tota as governing the A. The rule is negations Governor Onley What immediately follows unless you have reason to interpret it. Otherwise, that's a good translation tip. Now let's talk about the DOT When you have the dot, it translates. And also But however, yet emeralds are green, for example, and Rubies air red e dot are emeralds are green, but Ruby's air read. It still asserts two propositions e dot R and emeralds, agreeing, however, Rubies, air red, same difference. So whenever you have these sorts of translations, the English textbook translations always both and both emeralds or green and Rubies Air red . Little dramatically awkward, Yes, but that's the standard translation. Now be careful. Superman and Batman are heroes is a compound. It means Superman is a hero and Batman is a superhero. Now, Batman and Superman are allies, has the same grammatical structure ah, predicated assigned to both Superman and Batman. But don't translate that as two sentences. Superman is an ally and Batman as an ally because you lose the implication that they are allies of one another. Now all of these statements are conjunctions. I invite you to look through each one and figure out why the DOT is the main connective, but notice it is enclosed by the least amount of parentheses in all cases, except for the first and in the first case, it wins by default. Now the disjunction of the wedge translates either or unless, for example, the Democrats will win or the Republicans will. Either. The Democrats will win of the Republicans. Will Democrats will win unless the Republicans win? Unless the Democrats, when the Republicans will it's all d wedge, are each case now the textbook translations always either or sometimes that can be grammatically awkward. But not in this case. I either the Democrats will win or the Republicans will win the wedge are or in this example, it's snowing and aspen or in Golden. Either it's snowing in Aspen or in Golden. It's snowing and aspen unless it's snowing in golden, and unless it's snowing in Aspen, it's snowing and golden. The textbook translation in each case is gonna be either it's snowing in Aspen or it is snowing in golden and either or connecting two distinct propositions. You could abbreviate it a wedge G very easily. Now all these statements are dis junctions, and I invite you to go through and look and figure out why the wedges opposed to other connective is like the tilde would be the main connective in each one of these cases. Here's a good point of which to mention Day Morgan's rule name for the 19th century logician Augustus de Morgan. He discovered that in negated conjunction, Tilda s dot T and parentheses is equivalent to a disjunction consisting of two negations. And in a similar way, if you have a negated disjunction that amounts to a negation of our conjunction. Rather of two negated sentences were gonna go into more detail on this when we discover more about natural deduction. But it's worth noting here for translation sake. Now we come to one of most difficult connective is the material implication or horseshoes symbol, translating if then Onley. If in case that given that and many, many others, the textbook translations Always gonna be If this then that if antecedent then consequent there. Two parts. This is sufficient for that. If sufficient conditions are met, other conditions necessarily follow. If you have trouble remembering this, here's a little demonic device for you. Some people like to use the acronym son s s implies in sufficient conditions imply necessary conditions and decedent implies consequent. But this this helps you remember sufficient implies necessary notice. This is not an equivalence claim P implies. Q is not the equivalent of Q implies P. It's the only connective that we've gotten so far, where you can't swap out one for the other letter and getting equivalent claim position is important with this connective. So all these statements air conditional is in the main operator is the horseshoe. Go ahead and take a little moment to familiarize yourself with the reason why the horseshoe is enclosed by the least number of set of parentheses. No sets of prints sees essentially. And in the first example, it's the only other option besides the tilt of to be a vein operator. Now the material by conditional the triple bar abbreviate sis. If and only if or is unnecessary and condition for textbook translations. Always if, and only if, basically, it's to condition ALS and one symbol. If somebody says a if B, that means that a will follow a Palm B or B implies a A only if B means that a implies B A . If, and only if B abbreviate is both of the preceding claims, it means a goes to B and B goes to a it's ah, it's a lot of information really quickly. One way of thinking about it is that condition ALS or material implication is a one way street. Thean click ation goes one way by conditions, by way of contrast, are a two way street. The implication goes in both directions. Now I know each of these statements. The triple bar is the main connective again. Take a look and try to figure out why that's the case. And before we go any further, we do need to talk about the concept of a well formed formula. The wolf, their syntactic correct arrangements of symbols. For example, a horseshoes toe wedge B is not well formed. You can't really read that, cause you have to connect is right next to each other to to place connective. That is, however, the example in the green reads perfectly easily. And in the next example, a B and then print sees Wedge See is wrong on ah, lot of counts. You need some connective between a B and the parentheses. However, the longer example underneath it reads perfectly well, and that's despite the length of the sentence in question. And in this example, we have a tilde in front of a wedge and the red. You just can't read these sorts of things unless they are well formed. Well, now you know a little bit about proposition logic and are five symbols that we're gonna use to facilitate argument recognition. We need to go in a little bit more into detail on the functions of these symbols. So the idea of functions may be a little exciting to some, and maybe not as much to others, but nonetheless, we're gonna have a little bit of fun with them. The idea of a function is sometimes characterized the idea of a black box. It's just a predictable output for any given input. For example, here I have the doubling function, whatever number I put in my little black box or my function just doubles it. Some people think about him as sets of sentences like the ordered pair 2 to 4, 5 to 10 etcetera. This set of sets and so on could be the doubling function. So Frago was one of the people who helped develop this and apply it to the idea of truth and victim Stein, help us compute thes with tables. We're gonna use both concepts here toe get our minds more around our connective. So when you have the negating function, if you put in a nephew, get out a T. It's basically the flip flop function. Basically, whatever goes in comes out the opposite truth value in with one out with another. Now proposition logic is sometimes called truth functional because the truth of a compound expression is always an output of the truth. Value of the simple sentences you put in and the mapping job done by our definitions are functional definitions of the logical operators that we just gave the five logical operators. It didn't have to be that way. We could have had a truth be a function of the meaning of sentences or the length or complexity of sentences or the context in which they were spoken. Nope. It's just gonna come down in this case to the truth values of the propositions involved. That's the only thing are Black Box is gonna operate on. It's Our negating function is pretty simple. He just put in one value and you get out the opposite. The negating function has a map to it, so to speak. You could say that whatever goes in, you get the opposite truth value for what comes out again. This is a truth function. What if somebody said p dot Q or P and Q. With both claims are false and that your input well, then you expressed a falsehood. And in fact, if you first claim is false and your second one's true, you still expressed a falsehood because ah p dot que says that both propositions are true. The only way you're going to get a truth out of this little box is if you put in to truce like this last example. So the dot function or the conjunction function. If you prefer a permit, me gives you a true just. In the first case on this map where P is true and Q is true. What about the disjunction? What about the wedge function? The either or well, if both of the things you said either this or that or false to falsehoods, well, that should give you a falsehood for an output, right? And if only one of the two things you said was the case, Well, that's pretty much all you claimed, right? So a truth and falsehood combined should be able to give you an output of a true what if you have both elements of your disjunction being true? In that case, we're gonna go ahead and say, If you said either P or Q and both P and Q turned out to be true, you still stated the truth. Now that may seem a little bit weird, especially at top example. Can that possibly be right? If you say P Wedge Q. And both are true. Um, can that be the case? I guess it depends on context, but we're going to say yes in this case has to be admitted that there are some uses of the either or in the vernacular, that defy this sort of usage. If somebody says Either the train will arrive on track a or track be. They probably mean to exclude the idea that it's both the case, that it will arrive on Track A. Or it will arrive on Track B, where somebody says either the Democrats will win where the Republicans will win the House . Obviously, they're probably trying to rule out the idea of both it being the case that the Republicans will win and the Democrats will win. On the other hand, there are some cases in which theme or inclusive sense that we've used will be the case. If Wiley Coyote says, either I move faster, he gets away. You might say to yourself, That's true Bailly, But maybe you'll move faster and he'll get away. Nonetheless, both could be the case. So it has to be admitted that sometimes we do use the sense of either or that excludes this first possibility and says that only one could be true, the that sometimes called the exclusive disjunction. But in our case, we're going to stick with this table and if we want to symbolize an exclusive disjunction, would just say a wedge B and also the case that Well, not a and B at the same time. Now we have the material implication where if you put into falsehood, you get out the truth, falsehood and the truth will give you a truth as well. The only time with the material implication of the horseshoe that you get a fall. So it is a few. The antecedent is true and the consequences false. Now that's a little tricky, but here's the table for material implication and you're just gonna have to memorize the fact that Pius, when P is true and Q is false, you get a falsehood true in all other cases. And remember the material indication symbol is the only one where the position of the two propositions matters, whether or not there in front or behind the horseshoe. When you get falsehood for the consequent and falsehood for the antecedent, you get a truth nonetheless or if you get falsehood for the unseeded and truth for the consequent and still the truth The only time you're gonna get a false for an output is when truth goes in for the antecedent and falsehood for the consequent Any other time you end up with wind up with the truth. So member memorize this table. The output is always going to be true, unless the ants seed it was true and the consequent was false. It's very difficult to make this material implication or horseshoes symbol. Truth functional. The problem is, if we stick with the rule that I've just given you in the colored letters above with this divergence from ordinary uses because it neglects relevance of the ant seeding and consequence, relevance is not a truth functional concept. All we have to do to figure out or compute P horseshoe Q. Is figure out the truth of P and the Truth of Q. Whether or not the content of P and Q relate to one another in any normal way is a different issue altogether. Now, finally, I want to talk a little bit about by condition ALS again. This is one of those cases in which the order doesn't really matter if you get foot to falsehoods. Well, then you were true to say that the two propositions had the same truth value. By way of contrast, if the two truth values diverge it all, you end up with a falsehood, and that goes for each of our ordered pairs here, so we can give you a little map of the by conditional function pretty simply as follows. The by conditional is true when the propositions have the same truth value. False. Otherwise, using the rules that we just discovered, you can easily compute even the most complex sentences. Just follow these simple steps, assigned truth or value to the individual sentence letters and then put truth or falsity underneath. The tilde is immediately next to them. Compute for the operators joining those letters and then work from the tilde is on are rather from the parentheses on outwards. Let's go ahead and try a few examples. Let's try to compute this one. First of all, let's start by putting our truth and falsity underneath each sentence letter. Assuming A, B and C are true and D e and F for false, that looks like this. Now we can easily compute the wedge claim next because that has to be true, given our definition of the wedge. And finally, we're gonna put in F under the main connective. Why? Because the wedge claim proceeding it was true, and what came afterwards was false. the rule for the horseshoe, says That's the only time the horseshoe gets a falsehood under it. Let's try a more complex example. Assuming a B and C your true. This is what we wind up with and D E and F for false. Now we can easily compute the connective immediately connecting any two sentence letters, for example, that will be dot c gives us that we're gonna get a truth there. And true for e horseshoes to a cause e is false and a is true. Now, what does that wind us up with At the end of the day? The roof of the horseshoe says this is gonna count is true because everything to the left of the horse she was true and everything to the right of the horseshoe was true as well. Now, this may seem a little intimidating, but don't go ahead. Take it by the steps. But your truth values under each proposition, a letter and then put your truth values flip flopping the truth values of each negated proposition. After that, you can start to work on your to place. Connective de wedge f is going to be false because D and f for false. And since we have a tilde out in front of the parentheses there, that's gonna flip flop the value of our wedge claim. Now be horse wedges. To tilt A is gonna be true because B is true until they is false. That's gonna give us a true true for the dot claim for everything inside the brackets. Now, over on the other end, the horseshoe claim is gonna end up true because F was false and so was till to see. But then the tilde on the outside of those prince he's gonna flip flop that value the main connective, the horseshoe thus winds up false. We had a true conjunction on the left hand of the horseshoe that implied a false negation on the right hand of the horseshoe. Try. This one looks intimidating, but let's just take it by the numbers. Put our truth values underneath our individual proposition all variables and then underneath the till does will go ahead and put the opposite truth value each and every time . All done with that. Let's work on our to place connective. The triple board gets a teak is each slide on each proposition. Eat side is true. We could put a falsehood for for the dot But then we'll go ahead and put a true for the horseshoe over here cause a was true and so is Tilda d. And over here we're gonna put a falsehood because, see, and he had a different truth values that's going to give us a true for the wedge. And we're gonna flip flop that for the tilde outside the brackets. Tilda, outside of here, it's gonna get a true and consequently a true underneath the dot and that's gonna wind us up with a falsehood at the end of the day because we got different values on each side of the triple bar. Makes sense. They don't get any more complex than this. But go ahead and just put your sentence letters. Ah, it's assigned them truth and falsity and go ahead and put truth and falsity underneath. The tilde is immediately preceding your sentence Letters. After that, you could just work on things like the to place connective like this and dot c dot till the e is gonna be true, whereas a dot until the B will be false, because A was true and till to be was false after that Because you have, ah, negation outside the parentheses. You're gonna flip flop the value of that negation. Now you have a horse you claim here that you can say is true because the true conjunction led to a true negation. On the other end of the spectrum, you have a B wedge d Ah, that's a true claim. But you have a tooled outside the parentheses, which is gonna flip flop that value. And over here on the opposite end, ah, false claim till the sea wedges to a false E two falsehoods gives you a falsehood. But that means that our triple bars true, because each side of the Triple bar had the same truth value. What does that give us for our horseshoe in the middle? That gives us a truth. But since everything inside here's in the brackets governed by a tilde, well, that means we're gonna flip flop the entire proposition to a falsehood. See how that works. Well, you guys have done a whole lot of work here, and congratulations on mastering your operators and how to use them to compute the truth of compound sentences are operators are truth functional. In that sense, we'll see you guys. Next time we'll put these to work on truth tables. Take care. 15. Truth Tables for Propositions: Well, hello there, everybody. And welcome back to your crash course in formal logic in this section were actually going to do some truth tables. Now I hope you've gotten the last lesson of your belt and you know, your logical operators and how they work all their truth. Functional definitions. Why do we need a truth table? Well, when I gave you the truth values of simple components, you were able to compute the truth value. The compound was actually very easy. If I tell you that a is true D is false and e is false. Once you know the definitions of the wedge and horseshoe here, you can easily compute for the whole. And that's do the 11 mapping function of our logical operators and how we define them. Truth, functionally. However, when you're not given each and every one of these truth values, you need a systematic way to examine. And I have to emphasize this every possible combination of truth and falsity with simple components, sentences, all the truth assignments that could be given to those individual letters. So here's how you're going to do a truth table Number one. Figure out how many rows from top to bottom you're gonna need in the table. Those rows represent possibilities of combinations of true and false values for any given sentence letter, and then find all the possible combinations of those truth values T and F for the simple propositions and under each one of those possible combinations, compute for the whole compound. That's essentially what a truth table is. Let's start with the first issue. How many rows do you need from top to bottom in your table? That is gonna be how many possible combinations of truth value assignments are there? Given the number of simple sentences you're dealing with, the number of rows you're gonna need from top to bottom is two to the end power where N is the number of simple propositions you're dealing with Now you may be wondering why to, specifically to the end. And that's because you have two truth values, uh, t and F No in betweens. Consequently, two to the end power is going to give you all the possible combinations of teas and f's for your compound sentence. Just taken example. How about a the number of rows for two simple propositions in a compound like if we use this one, for example, a wedge to tell Toby horseshoes to be Well, the answer is gonna be. There's four possible combinations cause two to the second power is four. So there's four possible combinations of truth values. Possibility one is that a is true and B is true. Second possibility is that a is true and B is false. The third possibility A is false and B is the true one or possibility for that. They may both be false, but that's pretty, well, all your possibilities. Let's try another one. How about a compound with three simple sentence components? Well, to to the third Power is equal to two times two times two, which is equal to eight. So you can see how this sort of thing adds up. Or how about one with four simple sentence components? As you can guess, that's gonna be 16. Now let's move on to the second step, exhausting all the possible truth value combinations underneath. Well, given all your simple propositions, take the total number of rows you came up with from top to bottom on your table. Cut it in half, make the first half of the rose for the simple proposition. True. The second half false. Let me give you an example. Poser computing for the formula of up in the upper right hand corner notice I took the number of rows and our truth tables underneath the simple proposition letters to your left . I made half the rose true and half the rose. False. This time I went to on to off. After you've done that for the next column, you go by half of whatever you did before since we went by twos. I'm gonna go by ones or by instead of by halves by quarters. If there were another column for simple propositions like if we had three simple sentence variables, then I'd go by eighths, etcetera. So instead going two on two off I go one on one off, all the way down. And then I'm pretty much done in this table because I only have two simple proposition all variables. Now we're ready for the tough stuff. Computing for the compound that I hadn't over in the upper right hand corner of the last slide used the rules from the previous section and remember, compute for only one connective at a time. One connective per column. Work your way on up to the whole proposition. Remember, if you take it things in baby steps, then you won't make mistakes. So to calculate for the truth of ah, Third column in the example used till Toby and use the rules for the tota that you know from the last lesson. So, for example, to work my way up to the whole compound, I need to first handle that tilde. Then I can handle the wedge, and then finally, I can handle that horseshoe. Go in natural order like you solved for other compounds in the last section exercises and go one connective at a time per column up and down and you won't make mistakes. For example, I can just use column B to calculate the next column. The total flip flops, the truth values involved. So obviously I'm gonna go put a true ever. There's a false and vice versa. Now I'm in a position to handle that wedge by using the two columns that have outlined here in the purple. The only time the wedge is going to get a false is when you have two falsehoods assigned to it. And the only time that happens is in row number three. Now, finally, we get to the last column. I need to know where is there. Ah, row from left to right where the a wedge told a B is true, but the B is false so far is like until that happens in two places. Notice. I do have some things reversed here, but these two columns number two and number four. Rather, those two rows are the only times where you get a true in the A Wedge told a B column and a falsehood in the B column to Aunt Seeding. False consequent. That's how it works and there's are finished truth table. All right, so what exactly have we proven with R Truth table? Well, all statements fall in exactly one of the following groups. They're either gonna be a tautology, and tautology means necessarily true. It's gonna have straight True's down its column or the compound statement may be self contradictory. That is necessarily false. All efs. That means all F's under every possibility. Every row underneath it's column. Now, the contingent sentences are the ones where it could possibly be one way or possibly be the other. It all depends on what the value of the simple ah compound are. Simple expressions are now that that case you have teas and efs underneath the column, so every sentence will fall into exactly one of these categories. Into which category did the last sentence reanalyze fall? Well, look over to the far right and look up and down the column and you'll find that the A wedge till to be horseshoeing to be is a contingent sentence. Whether or not it's true or false depends upon what values are contingent Lee assigned to the A's and B's involved. When you see that mixture of truce and falsities in a column, you see it know that a proposition ah, compound proposition is a contingent proposition. Let's take a look at the different one. I set up a truth table here again with just two simple sentence variables on I've set us upto compute step by step the formula all the way in the upper right, just handling one variable to time. So let's go to head and handle the G horseshoeing toe. H column. Then we'll go ahead and handle the DOT claim and then we'll handle another horseshoe and build our way on up to the truth of the compound. At the very end. Let's begin by using columns one and two to calculate column three on the plate is that I can see where you get a true aunt seeded and a false consequent is in row number two. Now we're gonna use to these two rows in the purple to calculate the dot claim what we only put true's in the yellow box when we find truth assigned to the G horseshoeing, the H and truth assigned G. And again, that happens twice. And now, finally, we're gonna move on to calculate the horseshoe. In which places do you see that long com pound expression G horseshoe h dot g. True where H turns out to be false. Do you see any? Not me. I haven't found one place where that turns out to be false. Consequently, we're gonna have to say that that sentence that we just examined is a tautology. It is necessarily true virtue. True, in virtue of its form and logical structure alone. Not true, in virtue of assignments of truth values to the simple parts and I have set up a truth table once again with just two simple sentence letters. That means four rows from top to bottom, uh, set you up here to calculate for the expression in the upper right hand box. That's a big one. Just take it step by step one operator at a time that we start off with the tilde G column . And that's just a Z Z is flip flopping the values given to G in each case instead of two on two off to off to want fine. We come to the Total H column instead a t f t f all the way down f t f t calculate for the wedge. What we have to do is find the rows in which both dis chunks are false and mark a false there. That only happens in the last row as far as I can tell. And then we're gonna handle the conjunction. The dot the dot only gets a true when both con junks air True. Looks to me looking up those purple columns. It only happens one place row. For now, we're in a position to calculate for the triple Bar. We just have to use these two columns that we calculated and try to find a place where both uh, sides of the triple bar turn out to have the same value, each side having either true's or each side having falls. But but look is, though we never get that they necessarily have opposite truth values. Consequently, we get EFS straight up and down the column. So what does that prove? That proves that we're dealing with a self contradictory sentence. That sentence compound sentence E is necessarily false. It has efs under every possibility that is EFS and every row in its column. Now. That's how you use truth tables to analyze statements. But you can also compare two different statements to each other. The way to do that is to check them to see if they're logically equivalent. They may always have the same value. Maybe the two sentences or contradictory, always having the opposite value. Now here's another way you can go about it. You can see if sentences are consistent versus inconsistent. That means is there or is there not a possibility of both? The sentence is turning out to be true. Thes air two different approaches to examining sentences and seeing how they relate to one another. But this time you're looking at to propositions at a time. Now, two sentences are logically equivalent. They're gonna have the same values in each row. So, for example, K implying l and tilted l implying till decay. Actually, this is an interesting form of contra position, and it does yield logical equivalence. Let's prove it with the truth table. I notice I have set up your truth table here. It's very simple. There's only two simple sentence letters to deal with, So we're gonna have four rows from top to bottom. And this time we're gonna world build our way up to the two columns to the viewers. Right? And once we build up to those, we'll take a look at how those columns compared to one another first, to find out step by step what each connective amounts to. Let's start with the tilde as instead of true false, true false forget false truthful is true. And for the K well, we just flipped. I'll use given decay earlier instead of two. Wanted to off to off to one. Now where do we find K implies l. It's only false in row two, where Kay was true and l was false. And what values to be get here. Well, can you find Rome? Which total Ellis True. But Tilda K. Is false, can find one, and it happens to be the exact same row is, uh, in the previous, uh, column what we find here. We found the exact same truth values in the blue on every single row. That means the two statements being compared are logically equivalent. They get the same results under every single possibility. Every single possible truth assignment to the parts. Simple parts of the sentences. So what have we proven about the last two sentences? We've proven they are logically equivalent, always having the same truth value on our truth. Table proved it. We want to find an example of contradictory statements. Those that have the opposite truth value necessarily. For example, how about K implies l. But how about when somebody says K is true and Ellis false? And that seems like a straight up contradiction. Right? So once again, conveniently enough, I've set up your table here for you. Step by step Column by column. So we deal with one connective at a time and build up to the last two columns where the two statements being compared, uh, are so keeping things Temple. Let's go ahead and compute for Tota l said of T F T f all the way down f t f t. Then we need to find the value of K implies l. It only turns out false and row two because that's the only role you get a truth assigned decay and a falsehood assigned to L. And finally, we can use thes two columns in the purple to compute the conjunction of k dot tilda L. Onley put a true there when you find truth assigned to both columns, and I only found that in road number two. Now take a look at these two columns anywhere. One has a t, the other has an F and where the other hasn f, the other has a T. They always have opposite truth values, and that proves that they are contradictory. They will always have the opposite values. They necessarily take opposite truth values. Now let's take a look at the second distinction with respect to comparing two sentences of the time, the consistent versus inconsistent distinction. It just boils down to one question. Is there or is there not at least one row where both statements turned out? True, If the answer is yes, then the sentences air consistent. If there isn't at least one row, then they cannot be true. At the same time, they're inconsistent. So let's take a look at this example. K wedge l and k dot L, uh, set. Makes for a pretty easy truth table. If he asked me, I'll compute very quickly for K wedge. L only gonna put through there when at least one of the dis chunks is true. They both turned out false and Row four. So I put in f there. And how about for k dot L running my finger up and down the purple columns? I find that there's only gonna be a true in a row number one because that's only time that K is true. And Ellis true. So what does this prove? Well, as you can tell and row number one, we have a possibility of both sentences turning out truth same time they both turn out true precisely when K is true and l is true. What's that enough to prove, even though the other three rows underneath, where one is true and the other false or both are false. Don't really prove what we want to this one row with the top proves logical consistency. So that's an example in which two sentences are consistent. It is possible that is, there is a row in which both are true at the same time. So let's try an example involving inconsistency. How about K triple bars to L and K is true, but Ellis false? Now that seems like, given our definition of the triple bar, we ought to have a contradiction between these two sentences. Well, here's the table, taking it step by step until we can compare the two columns at the very end. First thing you want to do is handle that tilde so we don't skip steps. So we're just gonna flip flop the values under the El College. I'm instead of t f T f. We get f T f T. Next, we're going to get the Triple Bar column handled by taking a look at the column for K and column for El. We only put truth use where they have the same values like both true and row one and both true and or both false and row for now. Finally, we can handle the column for the conjunction, and we just need to run our fingers up and down to purple columns here, and we find out the conjunction turns out to be true. Just under one case. That's an road number two. So what does this demonstrate? Well, can you find a row here in which they both sentences turn out to be true? You can find one where they both turn out to be false. That's row number three, but that's not what we're looking for. Can they both be true at the same time? So the answer to whether they're consistent or inconsistent is no. There's no possibility that is no row into which both statements turn out to be true. So two statements are inconsistent if, and only if you cannot find that roe or possibility. If you stop to think about it, there's a variety of relationships between the inconsistent, consistent relation and the equivalent contradictory relations. Every pair of statements is gonna be consistent or not. That seems pretty straightforward, but some consistent statements end up being logically equivalent to one another. Some are not some inconsistent statements, maybe contradictory to one another. Or they may be logically equivalent. For example, inconsistent statements could have efs straight up and down each one of the columns that be a case in which they were logically equivalent. And yet nonetheless, they were not contradictory to one another because they didn't take opposite truth values. They had f straight all over the place. So oftentimes pairs of statements are classified as equivalent of contradictory before they're being labeled as consistent and inconsistent. But it really is up to you. Is Thea logician here? Well, that's about all For now. Thanks for watching this and wait for my logic exercises. Now you know how to do basic truth tables and how to use them to analyze sentences and what types of sentences you're dealing with and how to compare two sentences to one another. In the next lesson, we're gonna get down to testing arguments with truth tables. And that's where the fund released gets rolling. So we'll see you next time. Take care 16. Truth Tables for Arguments: well, other everybody Welcome back my crash course. In formal logic, this is a very important lesson. Finally, we're gonna test arguments with true tables, tell if they're valid or invalid, and using do that, using all the skills we developed in the previous lessons. So, essentially the moment we've all been waiting for. Finally, we get to do something like we did with then diagrams for categorical logic. Now we're going to do truth tables for Proposition A logic Now, validity means there's no possibility of all true premises in a false conclusion in validity is the opposite. There's at least one such possibility now, since the rose on Our Truth table running left to right represent possibilities. We can show validity with two tables. It's basically a three step process. Number one symbolize your arguments consistently, using simple, uh, capital letters to represent simple propositions than draw a single truth table containing a column for each the premises and a conclusion on when you're done doing that, you just look left to right along the roads to see if you can find a row where all the premises air. True in the conclusion. False. If there is such a row, the arguments invalid. If not, it's a valid argument. Of course, we're gonna need to do some examples in order to get our minds wrapped around this. How about this? It will invest. If we invest more money and police forces, crime will decrease. But we won't invest such money. Therefore, crime will not decrease. Let's be consistent with what we let each letter represents. I'm gonna let em represent investing more money and CB crime decreasing. And since we won't invest more money, does that mean that crime will not decrease again? Let's do the truth table. Now, remember, when we let set up the rose for our truth tables, you want to take the number two to the power of however many sentence variables were working with. In this case, we're dealing with two sentence variables and we'll have a four road truth table. Then I'm gonna sign truth values to in false and columns for the simple sentence letters by using the half on half off method kind of like so for the first sentence letter half on to tease half off to false is. And in the next column I cut that two on two off down Do one on one off T ftf straight down . Now, we need a column to handle that horseshoe, and I'm only gonna put in F And we need a column, of course, to handle the tilde. And lastly will need a column to handle the till to see. For the conclusion. Notice a column for each of the premises in a column for the conclusion. Now, using the 1st 2 columns I can calculate for the M horseshoeing c E. It's just gonna get in f where M is true and F c is false now for till two AM I just use the column and blew over to the far left, and I switched their values to straight to the opposite. No problem there. Same thing happens with C now. Once I've done this, I'm pretty well done with my truth table. Real question is, what am I going to do with it? What I want to do is I want to look and see if there's a row left to right where all the premises turned out. True in the conclusion. False. Let's reexamine that table. Now Here's our truth table. I'm gonna go ahead and highlight the premise sees columns here in the green. Do you see any rose from which they're both? All the premises turned out true. Well, I see three and four are the world's. They turn out true. I'm gonna go ahead and highlight here The conclusion column. Before I go any further, I'm gonna put stars. That's my typical technique beside the Rose, where all the premises air True. Then I take a look at the conclusion column notice there is one row where the premises turned out True in the conclusion. False. And when I see that sort of thing what I usually do is I circle a star Stars mean here's the grows where all the premises air True. Then I put a circle around one. If I find a row in which the conclusion is false and that invalidates the argument, there is a way. Call it a row. Call it a possibility off all true premises in a false conclusion that is, by definition, an in valid argument. Let's do another example. Just wrap your mind around the concept of voter election fraud occurs. American people will not respect their leaders. And if they don't respect their leaders. Then the national security will be weakened. Does it follow that if voter election fraud occurs, the national security will be weakened. Well, I'm gonna use f for fraud. Occurring and tilt are for people not respecting until they are implies, W according to the second premise. Consequently, if fraud occurs, will national security be weakened? This is the argument we're gonna test now. I'm gonna need a larger truth table here. Since I have three Proposition all variables, I'm gonna need to have an eight row truth table. And that means I'm gonna go half on half off for the first few rows and then I'm gonna go. Ah, half of what I did in the previous column. Switch to 1/2 of that next. Eventually, I wanna have to go down to eighths. So here's what we're going to first for the F column. I'm gonna behalf on That's four on four off for that column. And then after I'm done doing that, I'm going to switch to two on two off, see how that works. And then since I cut four down to two two is gonna go down to one there have set up my table with all my propositions assigned. Every possible combination of truce and false is now I need to fill out the rest of the table. Fertility are I'm gonna have to just flip flop the values in the column. So I'm gonna get F to efs and two teas all the way down for this column. I need to find where f is true. And our is Tilda are rather is false. And I see him rose one and two. That happens. There's the only ones I think I need to really pay attention to since I get mostly efs in the column for F anyways, So moving on, can you find a row where total are is true and w is false? Well, I can see that four and I could see it in eight and all the rest of the time it turns out true. And now, finally, for our conclusion column can you find Rose in which f is true and w is false? I see that happening in two and four. So I put efs in those columns all the me like they turn out fine and here we go again. We've done a column for every premise and the conclusions. So now we need to look and see. Is there is there a row with all true premises and a false conclusion? Well, once again, here's our truth table, and here are two premises highlighted and green at the top. Now down here, I see in rose 56 and seven, the premises turned out true, both of them. So let's take a look at the conclusion column. After, of course, we start with Rose, where all our premises turned out. True. Now, in that conclusion column highlighted here in yellow, it looks to me like we always got true's every time the premises were true. So what does that mean? That means the argument is valid. There was no way for the premises to be true and the conclusion false. There's no row or possibility of true premises in a false conclusion that is, by definition, a valid argument. Now let's take on a real monster just for practice's sake. How about this argument, Lotus four premises and a conclusion. Now this is going to be very difficult. We're gonna have to take it by the numbers and work our way through it carefully. But remember, truth tables, air easy. As long as you follow the steps, you're not gonna get tripped up with this. So first thing to notice is we got four Proposition all variables in that argument. And that means we're gonna have to have a truth table that is 16 rows long. That's the longest truth table we tried in these lessons and we set up all the possible values of combinations Rather of true and false is we're gonna have to cut in tow halves. Quite an awful lot. In fact, to get down to 16th I'd have to say a very awful lot, but it's nothing we can't handle. We'll just set up a column for each one of our proposition all variables, and will go half on half off. After that, we'll go by. Quarters after that, will divide down 2/8 and after that will divide on down to 16th. It's not really going to be hard as long as we just take it by the numbers. So first things first half on half off means a true eight falls for a now. Since we went by eight, we have to go by fours. and column are. So let's go ahead and put four on four off, four on four off. Then after that, we're gonna divide down into half of that instead of four on four off. We're gonna go with two on two off, and then finally, you always end on your last proposition. Variable with one on one off, one on, one off, all the way down. So here's the argument. We have to start making columns for our premises and conclusions, so let's start building up towards it. First off, I want to get those. Tilda is have a way. They're the easiest to handle. So any value I had for say, I could just flip flop it and get the values for tilt A right. So eight off eight on and for C same thing instead of tea ftf we're gonna have Efty efty for Till the end flip flop to off to on all the way down. That was pretty easy. Now I need to start building up towards those two place connective XYZ and I'll just lay out in advance exactly what my columns are gonna look like. I know when you need a column for tilt a wedge are and I'm pretty well set up toe fill out this column. Now look at this second premise in this argument. It's a conjunction, but it's negated. I need to handle the conjunction first, and then I need a column for the negation thereof. So I need to columns that will look like this and dot till to see and then a column over to the right of that that negates that whole conjunction. And then I can goto our horseshoeing to see that's our third premise. Finally see horseshoes to till the end and our conclusion at the very end. Now it's just a matter of playing it out by the numbers. I can get this column a until two a wedge are by just using the orange tilt A columns only time where I get falsehoods for both A and all till today and our eyes in these four rows all the rest turned out true. And as for this, I'm only gonna put truce in the rose in which n antill to see turnout True, that's one time in every four and all the rest are gonna get efs now. The next column is the negation of this one. So that's gonna be just Azizi is flip flopping the values that we just gave to end out till to see. So far, so good right now as far as our horseshoeing to see Take a look at the columns and blue I only see true for our and false for C in rows two and four and also in 10 and in 12. So consequently, all the other rose turnout true by definition of the material conditional. Same thing goes for this column. I want to see where c is true until the end is false. And I get that in 159 and 13 other. Otherwise, the conditional sentence turns out to be okay. True. And for the conclusion a wedge. See? Well, we only gonna put falsehoods where both of these turn out false. And that's pretty rare by comparison. All the rest of time they turned out true. Now we've got a column for each of our premises and a column for the conclusion. So what are you going to do now? Same thing is always look for the row where all the premises air true and the conclusion is false. look for the invalidating road that is now. Here's our truth table once again and thes. Don't let your eyes fool you. These are your premises, wrote columns. Don't be fooled into thinking that the column that's not highlighted there in the middle of the n dot tilda see is a premise column. It is not. These are the columns highlighted in yellow that we want to pay attention to. Now I put stars look at the numbers over to your left. I put stars wherever we had all true premises starring those roses Usually how I get started working on my arguments Finally over to the far right I've highlighted in the conclusion column The what conclusions turned out to be on each of those rose number three , number 11 number 15 and number 16. Do you see a row in which all the premises turned out true and the conclusion false. There you got it in the Red Star. Very bottom right hand corner. There is a way, namely when a are in and see are all false for the premises to be true in the conclusion False. That means this argument is not valid. So what you're always doing is you're looking for the invalidating row on an argument. You want to see if there's a way to call it a row. Call it a possibility off all two premises and a false conclusion. If you find such away or a row or possibility, you have invalidated your argument. Otherwise thin. The argument account is valid now remember, just take these truth tables in baby steps. They could be long and tedious, but that's about all there is to a truth table. Just remembering that truth tables are easy and they're only gonna get hard for you if you start to skip steps or tried to tackle columns to swiftly and maybe by handling, to connect IBS at a time. That's the only time that they're going to get hard, that you're not gonna be intimidated by any argument that you have to do a truth table for their all about pretty much the same. So be patient, be methodical, and the rest is going to be just child's play for you. Well, this has been a relatively short lesson by my standards. Truth tables are an important part of proposition logic, especially truth tables for arguments I'll give you more exercises later on. But basically this this lesson covers everything you need to know. Basically, all you need to remember create a column for all the premises and the conclusion. And once you've created such columns, you just need to know how to interpret your truth table. Look for a row left to right with all two premises in a false conclusion. If you find it, the argument is invalid. If it's if you can't find any such thing, the arguments valid. So that's all for now. Wait for my logic lessons coming up later, and I'll give you some exercises on this particular lesson in future appendices to this lesson. Thank you for your patience and time. Have fun. 17. Indirect Truth Tables: Well, hello there. Once again. And welcome back to my crash course In formal logic, this time we're gonna test arguments with indirect truth tables. Very valuable time saving technique. Now, the basic idea is that some arguments contains so many proposition all variables that it's just not convenient to do a truth table would be too long. So take a look at the argument that we tried in the last lesson. Now that argument had four Proposition all variables, and he said, That's gonna come to a grand total of 16 rows in your truth table. Then we use the half on half off method and basically was set up all our teas and F's in the appropriate column. It was a very time consuming task, but if we took it by the numbers, it all worked out on that table. Ended up looking like this. I've highlighted the ah columns with the premise sees in the yellow and over in the orange . I pointed out each of the rose that had, through the conclusions of, well, the rose that had all true premises and notice. We found right at the bottom an invalid dating row one that I've just marked with star here . Road number 16 is the one that proves the argument is not valid. So what that proves is that there is a way or a row or possibility of all true premises in a false conclusion. But it took us a long time to get there basic idea of a truth table. We want to get to the point. We should have just been looking for that row 16 and seeing if it showed up in the truth table or if it would have shown up if we had constructed the entire truth table. So would there have been a line on that table? If we had done the entire thing, that would have had all true premises in a false conclusion, the strategy is gonna be for this hunt for the invalidating row. I want you to assume the argument is invalid and if you are led to a contradiction than the argument was obviously not invalid, it was a valid argument. But if you get no contradiction than there is going to be a consistent row in a truth table that will have an invalidating set of truth values assigned to the premises and conclusion . So look for that contradiction. Try to prove the argument is invalid, and if you can't, then you obviously have a valid argument. So let's take a look at this example to premises and a conclusion. Step number one assumed the conclusions false, and the premise is true. That is putting f under the main connective or, if there are no connective, is put it with the F underneath the sentence sentence variable or a t in the case of the premises. So I went ahead and did that in each main connective of each premise. In conclusion, the second step. We want to see what values the variables and operators must have when you fill them in, see what you're forced to do. Well, be consistent. Remember, an inconsistency means you've either assigned a sentence that are both true and false on the same row and assigning it out or assigning an operator value that violates the operators truth table definition. So I'm gonna go ahead and put a false underneath. Be as you can see right here and look at that conclusion. I can put a T underneath the sea and an f underneath the eggs That's the only time the horseshoe turns out false. Now I have to be consistent. So I go ahead and put the same letters that I put for B and A underneath the sentence letters in the remaining premise. Now I can also fill in the value for Tilda A. If a was assigned to false. And I can also put a T underneath the wedge in that first premise because the sea was all it takes to make that true. Now, after you filled out in his many teas and f's possible interpret the results. How do you do that? Well, this truth table goes that the argument is invalid. There would be a way row possibility of the premises being true in the conclusion. False. I have constructed that. Oh, very consistently. So we know that if we did do a truth table for this argument in the road that a is assigned false B is a sign falls and see is assigned a true That's the row in which we're gonna get all true premises and a false conclusion. So we would have found the invalidating growth. We have done an ordinary truth table. Let's take a look at a second example here. I'm gonna put true's underneath eat the main beneath the main connective Zoran The third premise of your argument, which is just a a I'll just put a true underneath that now, since I put a true in a and one spot Look at the circle. I have to put a true in that spot as well, don't I? So going down to the this premise step to fill in all that you can for this, uh, conclusion. And there's only one way for that to be true. It's gonna be if till to see is true and D is false. Now that automatically tells me what to put under C and D right here they put what we put false is underneath them and finally, what we're gonna put under be what we're trying to make. The second premise Truby implies D so that tells you that we need to make be false. If we made it true, that would make that premise false. Having done it there, I have to do it here. And that gives me something for the wedge. It's actually pretty interesting. We've got a problem here. Do you guys see it. Is there a contradiction? Look at premise number one. If that horseshoe claim is gonna be true, if a is true, then the disjunction would have to be true as well. That violates our definition for the horseshoe. So we have a contradiction in this argument. The argument is valid so we could not would not be there would not be away roar possibility of the premises being true and a false conclusion when we tried to assume that we were lead into contradictions. And that's proof of validity when our attempt to make the argument look invalid winds us up , contradicting ourselves. So affirming the premises in denying the conclusion would violate our definition of the horseshoe. So there would be no invalidating row on our truth table if we had written the entire thing out longhand. Now there is a caveat here. A difficulty with indirect truth tables. One row of truce and false is won't always suffice. Look at this conclusion a dot till to be in this argument. Now I want to make that conclusion false. Let's see if I can do so consistently. But the problem is I'm not forced to sign any particular valuables to the variables. So when that happens, I can look at all the ways the premises could be true. It once where I can go the opposite route and look at all the ways that conclusion could be false. And the 2nd 1 is the technique I'm gonna show you because it's usually the easier one. So we know that there are three ways a conjunction could be false. And I assign values to the variables accordingly notice that allows me to handle the be in this argument as well. Now, if I handled that, I can certainly go and look at all my premises under each possibility of each of these three. And find out what values to assign days and bees here. I'll just assign be the same values that I did way down at the end. Same thing goes for the A's, and I can handle that Tota on the A in the very first premise. So what have you found here? Let's take a look at each one of these. So in this revision of step for they gave you earlier. If any one of the three rows is consistence than we would have found that row on an ordinary truth value table. One inconsistent line invalidates the argument. So the next question is gonna be can we find one consistent row, one consistent way of affirming the premises in denying the conclusion? Well, we got a problem right here. Uh, we assigned a truth to this premise and that violates our roof of the horseshoe when the antecedent is true and the consequence false. It should received a value of false. So that one's out of the question. How about this row? Well, it looks as though we got a true and decedent in a false conclusion. Once again, that's going to take that one out of the running. And here, once again, we violated the rule for the horseshoe. It doesn't look like we could find any consistent way to deny the, uh to affirm the premises and deny the conclusion. So that means the argument was ballot After all, we got ourselves into contradictions when we from the premises and denied the conclusion. Now we need to make an important note here. Noticed in the last example, we were forced along each step of our ah assignments of truth values to the operators and the variables. This indirect truth table that you're looking at now fails to prove that the arguments valid. Why is that? Let's start from the beginning. We put in our truce for the premises and falsehood for the conclusion. Step number one. We have to take what we put under A and C and put it underneath all the other variables. Now what do we do now? Here's the big problem. We could put a T underneath the B and then we get all truth for, uh, you get it perfectly a perfectly consistent row. Here's where to truth tables begin to deviate. In this example, there's a difference. The problem was that somewhere in this truth table that you're looking at right now, ah person arbitrarily assigned falsity gov to get a contradiction and dust approved validity. That's cheating. Validity only is proven by an indirect truth tables when Proposition all rules force us to a contradiction, not when our choices forced the contradiction. So never make an arbitrary truth or false Steve value assignment. When consistency is still at ST, go ahead and make sure you were forced along every step of your doing your indirect truth table. So you see the basic idea that truth, indirect truth tables can help us handle argument That would have too many proposition all variables to do a standard truth table for. But let's go ahead and take one last example. Let's go ahead and handle this argument that we had in the previous lesson. Rather than do that monstrous 16 road truth table notice. I just put truth and falsity values that were that you started in director of table with underneath each of the main connective for each of the premises and conclusions. Now, obviously, at the end there in order for a wedge, see to turn out false, there's only one way that that can happen. Having done that, I can put false is underneath the A's and C's elsewhere. Now the important thing here is to look for what you are forced to do. Here's a few things that I can point out. First of all, we're gonna need a false underneath that dot In other words, we need to make sure that ah well, if till to see is true. And the only way for that conjunction to turn out false is we haven't f underneath the end , and And that's gonna tell you immediately what to put under that tota And that, uh, the fourth premise. Now our is gonna have to be false that we're gonna get our horseshoe to see to turn out to be true. And that means I have to assigned the same value down here. So this argument is invalid. Uh, we got a perfectly consistent assignment of truth values in this argument. We have not violated any rules for the connective. So we know that when a r n and see or false that the arguments gonna be invalid. And that's exactly what we found on Rose 16 of our truth table. So basically found our invalidating row in a much shorter, much more convenient fashion. Now, are there other uses for direct truth tables? Well, how about we use them to show a single complex sentences Tata, Legace or self contradictory? We use truth tables to do that before we can shorten our techniques here, or making sure that a pair of sentences is consistent or inconsistent with one another. Let's take a look. Now the strategy for disproving tautology zor proving them is pretty much to assume that the sentence could possibly be false if you lead to a contradiction than the sentence could not possibly be false. The sentences, a tautology. But if you're not led to any contradiction than the sentences not, let's take a look at a few of strategies. Here's one that we tried earlier. We found out that this complex sentence way towards your viewers, right was a tautology. So one thing that we could do is just try to assume that this sentence is false and see for lead to contradictions. Well, what are we forced to do here? That's the key question. Now I'm gonna have to put a true under the conjunction and a false under the consequence of this horseshoe. And if that's gonna be the case, both of the con junks of that true conjunction have to be true. Including G now preassigned true to G and false toe h. Obviously I have to give them the same values over here. But notice what that does here that forces me to put a false underneath the horseshoe. Where is that proof? Well, it proves we don't have a true conjunction in the antecedent at all. That means we've got a contradiction of our rules for the conjunction the dot. So you may be thinking there is no way the sentence could be false. We've got lead to contradictions. So no role on a truth table could possibly give it an F. And if that's the case, ah, full truth table would have proven it to be a tautology if we had Britain out the entire table, which we saved ourselves. The inconvenience of doing a similar method holds for proving or disproving that a sentence is self contradictory. Just assume the sentence can possibly be true. That shouldn't be the case for a self contradiction. So if you are led to a contradiction in your view that the sentence can possibly be true, the sentence was genuinely self contradictory. If you're not lead to such a contradiction, the sentence is not. Let's take a look. Now here's a sentence way over to your viewers, right that we found to be a contradiction that a wedge, a G wedge H was equivalent to not G and not h. That doesn't sound like it should be true. It all we gave it all F's in its column. Alternatively, we could look for what we're forced to do. Now. There are two ways for this. Ah, by conditional to turn out to be true. One way is if both sides were, uh both the elements of the by conditional or true. Another way is if they're both falls, let's take a look at each one starting with row one. I know that if this conjunctivitis lame is true, I got to give a tilted G and a tilted H. And that means that ah, underneath the G and the agent put efs. But notice what that does on this side of the argument. The by conditional, I just violated the rule for the wedge by making both districts false. So that one, uh, leads to a contradiction. And alternatively, if, ah, I go to the second row, I'm forced to put efs under the G N H. That helps me know where to put him over here. And then I'm gonna be led to True's for the remaining elements notice on this side. I just I just violated the rule for the conjunction. I put a false where both of the elements of the conjunction were true. So I got lead to contradictions importantly in each case. So that means that anyway, if you try to make this by conditional true, you get lead to contradictions. That means it is self contradictory. So once again we have a much more convenient way of handling these sorts of sentence. You could say there's just no way that sent the sentence could be true, so there'd be no role on the truth table that would ever give it a T. So a full truth table would have proven a self contradiction. Again, we just saved ourselves the inconvenience of doing the whole table well, so much for showing whether or not sentences or to autologous or self contradictory. Let's ask yourself second question. Can we show that a pair or a set of sentences are consistent or inconsistent? Let's give the shot. Well, what you have to do is you have to assume all the sentences can possibly be true at once, and if you're led to a contradiction, the sentences air inconsistent if you're led to no contradiction. The sentences were consistent after all, and you just showed how they could be all true at the same time. So let's give it a shot. Notice we did the truth table for ah, the last two columns over to your viewers, right, in which we proved that two sentences were actually inconsistent with one another. There is no row with a boat. Not be true at once. Well, let's take it by the numbers. What are we forced to do? I'm gonna put teas underneath each sentence and treat them all like their own one row. Now, underneath the conjunction, I'm forced to put a T under the K in a T under the tota. That's obviously gonna make the l false. So if I put true underneath K and L underneath falsehood over on your viewers left just to be consistent. What am I forced to? I'm forced to a contradiction here. The roof of the horseshoe states that when a k is true and Ellis false, this horseshoe claim should have turned out false. So I violated one of the rules for our logical operators. And hence this shows that the two sentences are genuinely inconsistent. So thinking about it once again there's no way both the sentences could be true. So no role on the truth stable would have given both a t at the same time. So full truth table would have proven the sentences inconsistent if we had taken the time to do the entire table. Now there's one other advantage to indirect truth tables. Sometimes you want to test the consistency of several propositions at once and to see if they're all consistent with one another. The truth table for that sort of thing could get absolutely huge. I'm gonna show you guys had to do it and then gonna give you some exercises to prompt you along. So here's two sets of sentences. I'm gonna go ahead and put my truce underneath each and every one of the main operators. We want to see if we can make them all true it once. Now the first example I have to tell you in advance it will turn out and consistent. You will be forced to a contradiction with second set has Well, you're gonna need three rose on that indirect truth table. But there will be at least one line in which there will be no contradiction that you're forced to. So I'm gonna give you guys some prompts and see if you can guess the values that I'm gonna put in, um, you should be forced each in every step of the way. I'm just gonna show you which steps to take. But one thing I do have to do before we get started because the second batch of sentences contains only horseshoes and wedges. We're gonna need more rose for that particular table. In fact, we're gonna need three rows total. And I'm just gonna choose arbitrarily one sentence to work with. I could have started with another one. You guys ready to get started? I'm gonna give you I'm gonna point out where I'd like you to put a truth value, and you have to figure out exactly which value goes not assigned area. You should be forced to that conclusion every single time. - What do you see Any inconsistencies in any of the rose on this truth table? Well, I can point out one pretty quickly. We assigned a true when both dis junks were false. That's the only time that a disjunction supposed to count is false. And here we violated the rule for the horseshoe it down to the very last row we assigned a true when we had a true and seed int and a false consequent But they can all be true it once there is no inconsistencies to be found in row number one. So on our truth table, we would have gotten a true for every single one of these sentences when a was false, be was true and see was true. Well, it looks like you guys have memorized quite a bit of material today. How to do indirect truth tables for arguments to to prove tautology is and self contradictions how to show sentences consistent or inconsistent with one another. So I think we'll wait until next time for our next logic lesson and I'll see you then take care. 18. Natural Deduction in Propositional Logic: hello once again and welcome back to a crash course in formal logic, this time we're gonna study natural deduction systems in Proposition a logic, a very effective way of proving validity of arguments without doing truth. Tables of the director indirect sort. Now an inconvenient truth is that truth tables can get very long that there's too many simple sentences involved. We saw an example earlier that involved four Proposition all variables. We wound up with a truth table that looked like this noticed once again. I've highlighted all the premise columns and I've also highlighted in the orange where the conclusion turned out true or false, we had to get down the line 16 before we could invalidate the argument. Now you may think that we can solve this problem by going with indirect truth tables, and we found that that was an effective way for handling the argument that we just looked. What we did was we provide to see how we could be forced to try and figure out a way in which the premises could all be true and the conclusion still turned out false. And if we could find that, then we invalidated the argument, and we found precisely the row on the truth table. The row where a are in and see are all assigned false, in which we would have invalidated your argument it on a direct truth table. But there's another important backed indirect. True tables could get very long. If the conclusion is very complex notice, I'll just make this conclusion a lot more complex and try to imagine how you would even set up or begin your indirect truth table. You'd have to find all the ways in which this conclusion could be false. It's just not an effective system. So there has to be a better way. And the better way is a natural deduction system in Proposition A logic. Now, what is natural deduction? Natural deduction is a method for establishing validity of an argument by moving from premises to conclusions through steps and this steps in the proof, we're gonna be justified by rules of inference. We're going to study two broad categories of rules of inference. Notice we're still going to need truth tables to prove in validity of arguments and sometimes the indirect truth tables or the best way to go for that. But to proven argument. Ballot. Natural deduction is the way to go. So basically, you're going to start with premises. And instead of moving towards the final conclusion directly, you deduce an intermediate conclusion and maybe another intermediate conclusion and use the both of them to derive the final conclusion through a rule of inference. For example, if I take A or B Tilda A than I could deduce an intermediate conclusion, namely, I could deduce that be must be the case. And if I took the premise till to see dot de well, clearly, if that's the case, then Till does he has to be true. And I deduced from those two intermediate conclusions that till to see dot b is the case. That's the general idea. But we need rules to justify those blue arrows and which moves are appropriate and which are inappropriate were invalid. So in order to get our natural deduction system rolling, we need to prove ah, well, we need to show which steps are gonna be justified and by which rules of inference they will be justified by. There are two broad categories of rules of inference. The first of the rules of implication And that's where you drive a new line or step in your proof from one or two of the previous lines that imply it and the rules of replacement. They permit deprivation of lines or steps in a proof because basically the new liner step says the exact same thing. It's logically equivalent to the information in one of the previous lines. One way to think of it is that implication is a one way street. When you move from one or two lines and approved to something that's implied by them, that implication may go one way, but not in the reverse. When you have equivalent claims, well, you can go back and forward either which direction, For example, if I have a dot p b well, that logically implies a, but you can't move from the plane the A and just arbitrarily moved to the claim that a dot B is true. However, if you have the claim a dot B and you move the claim that b dot a. Obviously those say the exact same things. You can move back and forth between them. That's an equivalence. Or if I take looking back up that are gray signs at the top A is equivalent to be. And I have another premise that says Till to a Well, obviously, since A and B take the same truth values, I should move down the road to the claim that B is false. But from the claim that B is false by that claim alone, I can't go the reverse direction and moved to any claims about A without the help of other premises. So we're gonna study these two rules of inference in detail. We're going to start in this lesson with the rules of implication and will move to the rules of replacement in another lesson. Now, the fact of the matter is, there is no substitute for just simply memorizing the rules. So get out your memory cap and get to work. Initially, the memorizing is tedious work, but the good news is the rules are very intuitive. The hardest part is sometimes just remembering the names for the rules. Once you get the memorizing done, it's gonna pay off for you in the long run. And the other good news is that most textbooks used the exact same rules and typically the same names for those rules. I've used Patrick Curly's textbook a lot and merrily Sammons textbook standard introduction to logic and critical thinking. Another one that I've used is Ah, Ko Peas, Introduction to Logic and ah, also kind of like a Goto Carter cuddle, Carter's book and an introduction to logic and for more advanced courses. I've taught motile logic out of Kenneth Kenneth Conan Dykes book. I can tell you this much. All these books use nearly the exact same rules and nearly the exact same names for all of them. You're not gonna find much deviation and logic textbooks, so let's study the rules of implication. First, we're gonna study four of the harder rules, and then I'll get three of the simpler ones now. Cicero was a famous Roman order who said he wasn't very impressed with logic. He said that he thought his dog could do it. He's thought his dog had mastered the disjunctive syllogism. If a dog is chasing a rabbit down trails and he goes, he loses sight of the rabbit and he sees a fork in the road. The dog can sniff down one trail, and if he doesn't catch the scent immediately, charge off without sniffing down the other. Cicero thought that that meant that he was mastering the concept that if p or Q is the case , then if you knock out P, that leaves only one option left. Cue the dog master disjunctive syllogism. I'm pretty sure we can all master it then, but it has very many forms. And ah, you have to be a little bit, ah, proficient and logic to see that the following example is another example. Disjunctive syllogism. The second premise basically put a stopper on the very first dis junked of the first premise. That just leaves one option remaining. This is also a version of disjunctive syllogism at work. We'll take this example. Doesn't matter which order the premises. Aaron. If the disjunctive premise comes second, then as long as one premise puts the splotch on and one of the dis junks while there can be only one disjunctive remaining and this get kind of complex, sometimes notice. I've got an example here in which we have till to be a dis jumped and then a long, complex proposition after it as another diss junked. And in the second premise, I have double negation at work well. I'm still denying the first disjunctive the first premise, and that still leaves only one option remaining again. That's another version of disjunctive syllogism. It works in all these cases. Well, time to move on its study. Another rule of implication. The hypothetical syllogism we've studied syllogism is before categorical syllogism, hypothetical syllogism czar a bit different. The hypothetical syllogism says that if Proposition P takes you to queue and Proposition Que takes you to R. P is sufficient to get you to our after all, that may remind you of, ah, how he studied category logic. Basically, what we have instead is a situation where P is connected to Q and Q is connected to our so in a sense, P is already connected to our only. Now we're dealing with propositions instead of dealing with categories as we did before. Now take a look at this example. A implies a conjunction, and that conjunction implies total h. Does it follow that a is sufficient to take you to a church? Yes, it does. You still have that middle proposition that would hook up term between A and H in these examples, and that's basically what's doing your work getting you the proper link up between a until the H when we can try this example on for sidedness notice that till 2 a.m. is implied by one proposition until 2 a.m. implies another proposition. Consequently, that makes Tilda M ah, good link up term Between the propositions and question, we can tell from these two ah premises that the disjunction in of C and K leads to the disjunction of R N s or taken even longer. Example. Can you find the link up term here? Well, I'll help you out a little bit. It's right here. L implies in There's no rule that says that Ah, horseshoe claim cannot be the sort of middle proposition that links up to other propositions through hypothetical syllogism. And the two propositions dust linked up are like so all examples of hypothetical syllogism that work. Now be careful not to confuse hypothetical syllogism with this invalid version thereof. If p implies Q and are also implies Q. That tells you nothing as to whether or not peeing our imply each other just because they have a common implication. Try this example on for size. If Bill Gates is a man, that he's a human being, and if he's a woman, then he's a human being. Two propositions that both imply that he's a human being. Does it follow that Bill Gates is a Manning? Bill Gates is a woman, not even by a long shot. So I don't think Bill Gates would fall for this sort of reasoning, and neither should you. So let's move ahead and study motives. Opponents and Motus Tolins two very important forms of argument to very famous forms of argument in their Latin names. Don't let them intimidate you. Motus opponents literally means a mode or way of building up these air asserting mode. Take this. For example. 12 million Children die at yearly from starvation. Something is wrong with food distribution, and they are dying yearly from starvation moved to the conclusion something is wrong with through distribution. The way this argument worked is I took a premise that involved a horseshoe, and I affirmed the antecedent of that horseshoe, thus allowing me to move to the consequent thereof for my conclusion. Basically that simple. They're very with various ways of doing. This is well, for example, I could bring in Toto f implying by conditional again, the idea is to affirm the antecedent and then move straight down to the consequent as your conclusion. As long as you do that, you hold a valid form of inference. And it doesn't matter how complex the argument is either notice. Once again, I have a horseshoe claim of an implication claim is not first premise, and I have something that affirms the antis Eden thereof again, you have to remember, take the entire consequent of that conditional and move it straight down into your conclusion section or try this on. For example, uh k dot l being our first premise that the order of the premises don't matter. What matters is the second premise has the horseshoe is the main connective again, a firm, the antecedent, and moved to the entire consequent for your conclusion. As long as you do that, it's always a valid inference. So in short motives, opponents means affirming the antecedent. Now be careful for an invalid argument. Looks like Motus opponents, But it's not. If Napoleon was killed in a plane crash, that Napoleon is dead and Napoleon is, in fact dead. Does it follow that Napoleon was killed in a plane crash, not by a long shot. The problem is, I affirmed the consequent rather than the antis seed int of my conditional sentence affirming the consequent is a formal fallacy and very famous one in the example that we just looked at Does it does exactly that when you spot these formal fallacies, you know, in arguments always invalid just to give you another example. If bats are birds than bats would have wings because birds always do. And bats do have wings. So do you want to conclude that bats are birds? Well, we know they're not. They're mammals, of course, Noticed the problem here is not the truth or false. The truth of the premises, both of the premises air. True, it's that we made a bad inference by affirming the consequence. We did Motus opponents incorrectly. I like to do a little, uh, exercise called proving the ridiculous. This is a lot of fun. There is nothing so ridiculous that you cannot derive it from all true premises. If you're allowed to get away with affirming the consequent, that's a pretty good demonstration of the fact that affirming the consequent is a very bad form of inference. So I challenge you to find a ludicrous proposition and then prove them quote unquote to be true, using all true premises and by affirming the consequent. It's amazing how often you could do this. Well, just to prove my point, let's use a few examples. How about proving that Tom Cruise is God Almighty? Or how about that? I'm over 10 feet tall. Or how about that President Obama was born on Mars? You think we can't prove these? If you let people affirm, the consequent they'll uproot will prove it every time from all true premises. How about this? Tom Cruise is God Almighty. Don't believe it. I'll prove it. If Tom Cruise is God Almighty, he can certainly tie his own shoes strings, right? Well, if God compartment ocean, then he can certainly tie his own shoes strings. And we know that Tom Cruise can tie his shoes. So what does that prove? Proves absolutely nothing unless you try to get away with affirming the consequent. How about I'm over 10 feet tall? Think we can prove that? Well, if I was over 10 feet tall, wouldn't you have to agree that I'd be over four feet tall. 10 is bigger than four, and I'll let you know this much about me. I am over four feet tall. Does that prove anything? If you affirm the consequent, you'll get a ludicrous conclusion out of these two premises. Or how about President Obama was born on Mars? Don't believe it. I'll prove it by affirming the consequent. If a President Obama was born on Mars than he was born within a 1,000,000 light years of New York, that has to be true. And and Obama was, we know born within a 1,000,000 light years in New York were ever on earth. He was born. He would be born within that distance in New York. So what does that prove? Absolutely nothing. Burning the consequent is a bad form of inference. Now let's talk a little bit about Motus. Tolins Motus Toland's is the way of denying the way of tearing down. Just like motors opponents is the way of building up. If Japan cares about its endangered species than it stopped killing whales, it has not stopped killing whales, so it apparently doesn't care about endangered species. So what I did here is I took a horseshoe claim and I put a stopper on the consequent. I denied the consequent with my second premise. Consequently, I was able to also deny the conditions sufficient to get us to that particular consequence . What matters here is that any time you have a conditional sentence and another premise that denies the consequence of that conditional sentence, then you're entitled to say the sufficient conditions to get us to that consequence could not have been met. That happens even in cases such as the example over to your right. What, you're dealing with some pretty complex stuff. I'm dealing with double negation in this particular case. But I did deny and the second premise the consequence of the conditional in the first premise. What does that mean? I can move to the negation of whatever was sufficient to get us to that consequence Notice . I'm putting a tota in here. That is a very important step. Even if I have to do double negation to get to it, we'll take a look at the example on the bottom. The second premise is a very long conditional sentence, but there is the first premise that denies the consequent there. So if you deny the consequent, take the entire and decedent good as a conclusion, making sure that you put a negation on it to show that it cannot be the case. So in short motives, Toland's basically means denying the consequence. That's another way you can think of it or the way that you can name it now, just like there's an impostor version of Motus opponents. There is an imposter version of motives. Toland's Let me illustrate if Napoleon was killed in a plane crash that Napoleon would be dead for sure, but he was not killed in a plane crash. Well, does it follow from that that Napoleon is not dead? Not by a long shot. That's just one of many ways that Napoleon could have died. Basically, what we have going on here is denying the antecedent. Instead of denying the consequent that's bad. It's another formal fallacy. So watch out for situations in which a person tries to do what you see over to your right here that is putting the stopper on the antecedent rather than the consequent the consequent lis. Necessary conditions for the antecedent. So if there's necessary conditions aren't met, then the antis Eden can't hold. But the reverse is not true. Well, this might be another good point to play a game. Approving the ridiculous. You remember the rules of the game, right? We tried to show how bad certain bad argument forms are by finding the most ludicrous propositions we can and then quote unquote proving them to be true, using Onley two premises and showing that we can get to that conclusion if we're allowed a bad argument form. How about this for a bad conclusion? Bullets cannot harm me. Well, I bet you money that they can. Let's prove it by denying the incident. And while we're at it, how about this logic? Education is not a worthwhile endeavor. I think that's a pretty ludicrous proposition. And I hope by this point in Syria's you agree. Well, all I need is this conditional premise. If bullets could give me lung cancer than bullets can harm me, L. A. And all of you know that's not the way that bullets go around harming people. But if I point out the fact that bullets can't give me lung cancer and I allowed to move to the conclusion that bullets can't harm me, not by using good rules of inference. Well, consider this logic. Education is not a worthwhile endeavor. Why no One way it could be a worthwhile endeavor if logic education could help me when the lottery would be a worthwhile endeavor. Unfortunately, it can't do that. But if it can't do that, does it follow that it's not a worthwhile endeavor? No, that's the fallacy of denying the antecedent. So notice. What makes this all confusing is that all four argument types both motors opponents motus, Tolins and they're imposters. Start with the premise P implies. Q. Both the good and the bad argument form start that way, so it's very important to distinguish the argument types in the green boxes here from the ones in the red boxes. And the difference is, ah, the good types of arguments in the bad type. And while very different forms of argument take modus opponents, it starts from a premise P implies. Q. But it tries to reach the conclusion that Q. You gonna be perfectly fine if you've got P to carry you along across that horseshoe headed the right direction. No problems with that. The problem is when you try to get to a different conclusion, using this exact same premise that P implies. Q. By way of contrast, if you tried to get to the premise that P by affirming the consequent que then you headed in the wrong direction and that's a recipe I'm afraid for, well, logical disaster. A similar thing happens when you're dealing with Motus Toland's. Except in that case, you're dealing with negations. What I mean by that is that is long. You're trying to get to the conclusion that told API by denying the consequent of your conditional sentence. Well, then, that's a safe inference. You're going to travel just fine with that. But if you try to go the opposite direction, so you tried to reach the conclusion instead of till two p that tilted Q is true. Instead. But well, you're not gonna be ableto do that with Tilda P. You're headed in the wrong direction, and you're gonna get the same result. Bad argument and bad logic. So these arguments are very distinct. You need to memorize the four types and how to use them and how to avoid the incorrect types. And we said that natural deduction works when you use premises, reach intermediate conclusions and then use those intermediate conclusions to deduce your final conclusion. Let's use our four rules of inference that we studied in order to do that. Let's go ahead and try this derivation for practice. How about we have a premise that says A or B is true until to see implies till today See implies D until two D is true. Let's try to reach the conclusion that be you notice that be only occurs in the premises up their on premise Number one. It sure would be helpful to have Tilda A. To go use a disjunctive syllogism to reach be. But in order to get to tilt A, we have to somehow affirm till to see weaken. Get to till to see if we can use three and four to do motives. Toland's. Once you lay out a strategy, start laying out your proof still to see follows from three and four by motives Tolins Tilt A with them, follow from two and five by Motus opponents, and finally we get be by using mine one and lined six by disjunctive syllogism. You just take it step by step and you get your proofs rolling like this. Let's try another proof. How about E implies that K implies l f implies that l implies em. G or E is true. Total G and F Therefore, K implies em Gonna have to lay out a strategy here. Look at all these propositions. Aren't you glad you're not doing truth tables for this argument? Well, first step is we need to get if we want to get to K implies em. We got to get to the consequence of both premise one and premise to so we have to find a way to affirm E and a firm F Well, f is already given to us. So that's not gonna be a problem. To get the e will have to use three and four by disjunctive syllogism. So let's go ahead and get even by three and four disjunctive syllogism, and then we'll get to K implies l using one and six motors opponents. Once you got into K implies l we can get l implies him. Since we already have f in premise five that affirms the antecedent of two. Once we have those two simple, hypothetical syllogism is going to get us to K implies, m It's really easy when you just lay out a plan. Well, I've got three more rules of inference for you. This is gonna be a longest lesson. You're probably thinking that logic is gonna be quite a drain. Trust me, the point is I'm gonna obey the old acronym kiss. Uh, kiss just means keep it simple, Stupid. The remaining three rules are extremely intuitive. You're not gonna have any trouble memorizing them. The three rules. We're going to study our conjunctions simplification. In addition, they are very easy and intuitive rules. You're not gonna have any trouble remembering these. So start with the conjunction rule. If you have a P and a Q, can't you put a dot between him and reach? That is Ah, by way of inference. That seems pretty obvious. You can you can do with more complex propositions as well. For example, of e and totally in total gear. True. Just go ahead and put a dot between him. You're allowed to reach that by rule of inference, Conjoined ing Any two previous premises that you had basically or take these two Now you can conjoined them. But you got to make sure and put parentheses in their proper place. Otherwise, you're not gonna end up with well formed form. Sometimes you're gonna have to get even more complex and just put brackets around the two propositions in question in order to keep your conclusion or that intermediate conclusion that you reached as a well formed form. So remember, the only thing complex about the conjunction rule is you have to remember where to put parentheses and brackets to make sure you end up with wealth, warm forms. Otherwise, it's perfectly intuitive. And simplification is another rule that applies to the DOT. Um, if you have ah conjunctivitis claim you could move to the truth of one of the con junks and it works with more complex propositions as well. Just eliminate one of the con junks and moved to the other by way of inference. It just works that easy. After all. If both con junks are true, even in a complex proposition like this, you should be able to eliminate one and reach the other is a conclusion up. Remember, when you're eliminating one, you're not calling it false. It's not false, but you don't want to have that. In your argument, you want to have a way of simplifying things, and consequently, this rule comes in really handy now. Remember, in this case, as opposed to the last rule, you sometimes have to remember when to eliminate brackets that are unnecessary Once you've simplified a proposition and watch out for one more thing, there are some textbooks that are very strict and say that you can Onley simplify to the first con jumped. Obviously, we know it be equally valid to simplify to the second. Some textbooks just don't allow it, so check whichever textbook you're using. So if you do have a textbook that tells you that with it, when it comes to simplification, you got to take the first and leave the rest. Then I suggest following your textbooks advice. Now we have rules for introducing and eliminating both dots and wedges at this point to eliminate a dot, we use simplification to introduce it. We use conjunction, and to eliminate wedges, we use the disjunctive syllogism that we studied earlier. It's all that's gonna do it, but to introduce a new wedge claim in the course of an argument What do we use for that? We'll use something called addition because the roof with wedges so generous, really, if you have any premise that's true, you can really wedge off from that premise to anything, whether it's true or false. And you end up with the correct inference, for example, of tilt F is true. Well, then it should be the case that tilde f wedge. I don't know any old thing this by conditional. For example, it is true. The one thing you have to remember with addition, Sometimes when you use it, you have to take whatever your wedging off from, and you have to put parentheses around it. That's just a way of maintaining well formed forms. Sometimes you even have to use brackets like in the case of this last example. But as long as you remember to use, your brackets and wedges are about their parentheses. To maintain well formed forms, you can construct all sorts of disjunctive claims you need in the course of a proof. So just remember your parentheses in your brackets to maintain well form forms. And then what can you do? Well, you can dive or wedge off without fear to any old proposition that you want. The inference is gonna be a valid one. So I promised you three new rules. All explained simply, they don't get any more simple than the last three rules we study. Conjunction, simplification. In addition. And first of all, let's start off the premise that says H er til to be implies are and HR told an implies P. We have h Can you get toe R and P? Obviously, you're gonna want to use motives opponents combined with your new rule of addition for the wedge. So I'll just introduce what I need a firm. They antis seed into premise one, and I'll introduce what I need to affirm. The antecedent of Premise two Edition will give me both of those straight out of premise. Three. Once I have those, I get R and P by motives opponents. Now we'll use our conjunction rule. It's a pretty simple proof, isn't it? Let's try another argument. How about C implies n Andy de wedges to an implying D till the D. Therefore try to each toe to see Wedge P. Now, if we start that, I can see where we can get a still to see out of this argument at some points, see, is hiding up there in the first premise for us to get it out. But I don't see a P in this argument. I think what we're gonna have to do with the end of this proof is use our addition, rule with it for the wedge in order to get from till to see to the claim till to see wedges to pee. So let's just concentrate on getting to tilt A C. One step is going to be to use our disjunctive syllogism rule to get to end implying D. Now, once we have that, we can get to till the end through Motus Tolins because we already have a premise that says Tilde de. Once we get to Toto end, look up there by simplifying ah, premise Number one. Using our conjunction rules, we can get C implying n and do another motus Toland's. Once we do that, we've got our till to see, and that's all we need to get to the conclusion of this argument. By addition, let's try another proof. Let's try one more example. This one's gonna be a little daunting This is a pre complex premise. And so is this one to horseshoe claims and some conjunctions, horseshoes and wedges embedded in them. And till this up sorry C and Tilda are will be our third premise. The simplest one. We're gonna try to get to the collusion that D wedges toe are now, this is gonna be difficult. We need a strategy. First of all, what I'm going to suggest is that our does not seem to be involved in these premises at all . In fact, the third premise pretty go openly. Ah, firms that Tota are is the case. What that tells me is I think we're gonna have to prove that d is the case. And then we'll just use our addition rule to get to the wedge claim in the conclusion. So we need to prove that d is true. To get this argument going, I'm gonna use, See, First of all, I want to get that out by simplification, cause I'm gonna use it on Premise one eventually. But in order to use it on premise one, I need to build it up a little bit. I'm gonna build it up through addition. So that we have something here that actually affirms the antis seed into premise one. Having done that, I can move to the consequent thereof. Now I've got a lot of dot claims to work with. Notice. I can take the C that I got four and I can take out that tilde p that I have in six. That puts me in another position where I can put them both together through conjunction and affirm the antecedent of premise number two. Once I do that, I have C implies d almost all the way home. I can use premise number or line number four to affirm the antecedent of Line number nine and that will get me D. And that's all I really need to get to my conclusion that D wedges to our This is some pretty stuff stuff. I hope you guys have enjoyed my longest logic lessons so far. There's a lot here to digest, so I hope you guys spend a lot of time working on this particular lesson. It's still a crash course and uncovering proposition a logic and natural deduction. We're just gonna have three easy lessons. So digest as much as you can in each one. They're gonna be the longest lessons I have online. We'll see 19. Natural Deduction in Propositional Logic Continued: with everybody and welcome back to my crash course in formal logic. This time we're gonna study Net mawr on natural deduction in Proposition A logic, basically more rules of inference. So let's get started. It's gonna be a longish lesson. Take your time. Now. What we said about truth tables is that they haven't inconvenient feature that they can get very long. If there's too many simple sentences being involved and indirect truth tables can get very long and hardly able to be done. If the conclusion is too complex, we needed a more convenient and powerful system. The solution came up with was a natural deduction system, which is a system for establishing validity and a proposition all argument by moving from premises to conclusions to intermediate steps. The key thing is that these steps that we making the argument have to be justified by rules of inference, whether how to prove in validity. Well, we still need a truth table for that. I prefer the indirect truth table method. Remember, we have already studied seven rules of inference so far. I hope you've committed east to memory. There's no other way to handle them. But four were kind of difficult, and three were extremely simple and very intuitive. Now these rules of inference still hold. We studied rules of implication in the last lesson. This time we're gonna move forward. Remember, implication is a one way street equivalents is kind of like a two way street. For example. In the last lesson, I said A and B certainly by simplification implies a, but it doesn't go back. Vice versa. A does not imply both A and B. However, by way of contrast, a M B implies the proposition B and A and vice versa. That sort of move is an equivalence, and same thing goes forward things like a being equivalent to be until toe A. Those two propositions combined imply till to be. But you can't move from toe to be to the claim that tilde a or to the claim that a is equivalent to be. It's not a two way street. So in this lesson, we're gonna start studying rules of replacement. The equivalents claims the difference is the rules of implication are derivations of a line from one or two emphasis on or two previous lines in a proof they are valid argument forms like hypothetical syllogism and motives opponents, and they are applicable only to entire lines and a proof. What I mean by that is, if you have a claim like a implies that be imply. See, you can't take B and take a part of premise one and say, I'm gonna move to see by motives opponents that is an incorrect move. You have to apply modus opponents to the entire proposition of number one, and just not. You can't just take a part of the proposition and use it to do motives opponents. That's invalid now. The rules of replacement, by way of contrast, are derivations from Onley one previous line, because the line you derive is logically equivalent to the line you derived from, and they are applicable to entire lines and a proof or important difference. Two parts of a line so you could take something like a implies B and moved to the claim by double negation that a implies Tilda till to be cause Tilda till to B and B say about the same thing so you can apply these rules going to study today two parts of lines in a proof big difference as Kenneth Conan Dyke puts it in his introduction. Motile Mota logic book Unlike the rules of replacement of equivalence, is that's the stuff going to study today, he says. Rules of inference stuff we studied last time may not be applied within formulas or two parts of formulas. They cannot be applied to parts of premises or two parts of lines of approved. The rules we study previously have to apply the lines as a whole. Now, the good news is, most textbook used the same rules. And this distinction between the rules of replacement rules of equivalence is gonna hold throughout any textbook that you study in a college level course. So let's get underway. Well, last time when we studied our rules, we took it in this order. We said, We're going to study the hard stuff first, and then we'll go ahead and move to some simpler rules, like simplification and things of that nature. This time, we're gonna do things differently. This time, it's gonna be an uphill battle. When we study rules of equivalents, I'm gonna give you the easy stuff, first stuff that's intuitive, and then we'll work up to the difficult rules of equivalence later on. When we talk about equivalence is I'm gonna use a double colon as the symbol for logical equivalence with what that means is that stuff on the left could be exchanged for stuff on the right and vice versa that basically, they say logically, the exact same thing. Two way street. So let's study the first to really easy rules of equivalents double negation and tautology and get those under our belts. Now, when we talk about the truth function for the negation, what we said is it's basically the flip flop function. It just takes whatever truth value you put in of a proposition, and it spits out the opposite value. And that's because of the truth table for the negation, its truth functional definition. But the negation doubled. Then. If we take these truth values and run them through a second time, you should return back to your original value right? That makes perfectly good sense. And that just means that basically a double negative is equal to the original value of the proposition. So basically, P is equivalent to Tilda Tilda P, and vice versa makes perfect sense, right? So we, even when you're dealing with complex propositions like W and F. You could double negate that entire conjunction, and you've said the exact same thing. Of course, if you take a simple proposition like you, you could move from the double negation straight down to the UN negated version and vice versa. And look at a proposition as long as the one I gave you at the bottom. Make sure you put brackets around the entire thing before you add double negatives. As long as you do that you've ended up with an equivalence. You can move from one to the other, and importantly, you could go vice versa. But also you can apply these sorts of equivalence is two parts of proofs or parts of lines . I should say What if instead, I wanted to take the latter part of the original premise s implies tea and double negate that that's perfectly acceptable. Or if I wanted to take some simple proposition embedded deeply in the long complex one, I'll take em and double negate that. That's okay, because Tilda Tilda M. Says the exact same thing is M. Or if I wanted to take s and double negate that or vice versa, not a problem there equivalent. Now there are forms of logic that involved three truth values. If there was a truth value that was neither true or false will, then our double negation function would not work. But we're doing classical logic, the traditional forms of logic, and we recognized two truth values. And true is the opposite of F and vice versa. No middle ground. That's why this negation function works the way it does. Let's talk about tautology. That's another simple one. P is equivalent to P or P, and the same thing happens with the dot P is the same thing as PNP. Now we already have studied something like thes rules. We had a rule called addition, which said that if you have started with a proposition P, you could wedge it to anything. The tautology rule says if you have exact same proposition on each side of the wedge, you can eliminate one side. That's something new. And we did have a rule that said, If you have PNP, you could simplify down to pee. That was a rule for the conjunction. But now this rule says, if this proposition on each side of the dot is the exact same proposition. Then it becomes a two way street. So very often this rule is just a technical help. It may seem autologous, as the name implies, but but, for example, you're not gonna get through an argument. It says a or B in the A imply C or C to the claim that see, unless you have this technical rule in your pocket, you can reach a by simplification on one and by motives, opponents. Then you can reach C or C. That's not a problem. But then, to get to the conclusion that see, you gotta have something that allows you to eliminate the wedge and that only happens with both sides of the wedge, say exactly the same thing. Then the movies autologous or, for example, P implies that, not Q and Q. Well, if Q is true, then does that mean Pius false? Seems like we should be able to reach that by motives. Toland's, but only if we have a rule that lets us move from the claim that Q two Q Doc, you tautology will let us do that. Then, by double negation, we have the negation of the consequent on premise one. Once we have that, we can move to Tilda P by motives. Toland's See how it works. Now this roommates seem to be just a technicality, but it will be useful, and we'll get more more uses as we go on Scout's honor. Well, let's talk about more conferences. How about communicative ITI and associative ity? Very soft, simple rules. As the name implies, communicative ity involves some sort of commuting. Have you ever been a commuter? You know, that means traveling. We're gonna let propositions travel a bit. That's a good pneumonic device for remembering this rule. Communicative ity means if you have a dot or wedge, you can let the propositions travel. P Wedge Que is just Q wedge p p dot q means que dot p around the wedge in the dot propositions. However complex conduce um, traveling fact. It's similar to situation in math, where you can flip flop around. The addition symbol three and five, means five and 32 times three is three and two. There are other connective is just like there are things like division in which it doesn't hold. For example, communicative ity won't hold for the horseshoe. Just the wedge in the dot. Remember that associative ity? Well, here's another pneumonic device for you. If you have three propositions that involved and all the propositions air linked up with, like a wedge or a dot well, then you can start moving the parentheses within them so that propositions associate with other propositions differently. Kind of like you can associate with some people who are. You may choose to associate with others. It's your choice, or vice versa when you can move the brackets or parentheses around in this fashion, that's called associative ity, kind of like associations you keep with people now notice. No one commuted. Nobody traveled from spot to spot here. The peas cues and ours, and our propositions similarly did not move. This is not the same rules. Commune Tiv ity. It's a rule for moving parentheses and brackets, and the same thing holds for associative ity as communicative ity. Namely, that it only applies to dots and wedges. For example, in mathematics three plus five plus two. As long as the plus symbols moved throughout, you can move the parentheses around. It's the same as three plus +25 plus two with the brackets or parentheses move slightly to the right. Same thing holds with multiplication. And again, it doesn't hold for the division symbol in math. And similarly, our rule for associative ity does not hold for every single connective either. It doesn't hold for the horseshoe. So when you have this sort of rule in your pocket, you can start moving propositions into place in order to get your proofs toe work. That's really what these two rules help us out with. Let's do a little practice with these new rules that we've just gotten under our belts so we can see how they help us solve proofs that we could not have solved in ah, with just the rules that we had in the previous lesson. So how about this one? Suppose I have till the J gives me Tilda K. And a premise that says Tilda Que Well, obviously I could do something with these two premises like motives. Toland's and I should be able to get to a conclusion out of them like Jay, very simply. But look at the long conclusion I have here J wedges to L wedge J. Well, I after I get to J. I should be able to get to something like that. But I'm gonna need a lot of my new rules in order to get that long conclusion from these two premises. So let's just use the obvious. My second premise is pretty much equivalent to double negation K. And that's gonna give me by Motus Toland's The Tilda Tilda J. Now, where do I go from here? Double negation will help again. Look, I finally got my J. What do I do now? Well, what I'm gonna do is I'm gonna reach J or J by tautology because my conclusion has a second j in it. Then I'm gonna wedge that off to L. Because of the rule Edition says I can wedge off to anything I want. And after that, I'm gonna have to use associative iti to move my parentheses down towards the right where I need them. In my conclusion, by commuted ity, I get to my conclusion. I could just move jr el into each other spot. They do a little traveling or commuting around the wedge. My proof is done. For example. Here Zamora were rather more complicated. Example p dot que dot p and with some brackets don't in there. This asserts both Q and P. But the propositions air locked up pretty tight. I need Q impede work with premise to if it says that Q is false, which, of course, premise one says it's not, then our implies. Tilda P. But of course, premise. One says that Tilda P can't be the case. Can I get to tilt? Are I need to start unlocking my P's and Q's in orderto work with premise to. So I'm gonna start by moving the parentheses and premise one by associative itI, and then flip flopping the positions of Q and P. Why do that so I can move the parentheses back by associative ity. Once again, an unlocked Q. I need Q stock and double negated. And do you use premise to by disjunctive syllogism to get to the horseshoe claim Now, once I have that horseshoe claim, looking back, it premise one. I have a rule called simplification that allows me to get straight to pee once I get P. Aiken double negated Do Motus Toland's on premise number eight, and then I'm home free. So associative itI and community are very helpful in getting back through these sorts of proofs, but notice we never swapped out a wedge claim for a dot claim or a dot claim for a wedge claim. As we struggle uphill towards more complex rules, we're gonna discuss rules that allow us to exchange one type of these claims a disjunction for a conjunction or vice versa. Why do that? Well, sometimes you just need something that's more useful in your proofs. For example, dot claims tend to be a lot more help in a proof than wedge claims, because dot claims give you more information. So our rules for swapping out one of these operators for the other are the rules of distribution and dim organs rule. As we struggle uphill, let's try and wrestle with these more complicated rules for logical operators. Now, the rule of distribution. There's no getting around it. This is a complicated rule. As you try to memorize these principles, I want you to notice two things. Number one. The key thing is that the main operator changes as you move from left to right and vice versa. And that's the most helpful feature of this rule. And secondly, it's called distribution because of the move from the left to the right, P gets distributed in the way that it's forced to associate with the Q and R respectively. Hence the term There's no getting around it. This is a hard rule, and you're gonna have to spend some time committing it to memory and even more time learning how to use it. But it is very helpful for getting some. I find it especially helpful when I exchange a wedge claim, which doesn't have a whole lot of information in it for a DOT claim, which Aiken simplify to one of the other con junks. And that's very helpful in a proof. Another interesting feature of this particular rule is that, unlike the rules that we study previously, it's one of those rules where we also have a situation where the wedge claims and dot claims are mixed. Associative, iti and commuted. ITI just involved one of thes operators, wedges or dots, respectively, throughout the entire proposition. I don't know why, but this always helps me remember the rule it slices, dices, mixes and matches it slices up Thea Proposition P and distributes it, and it mixes up the connective and it even works in reverse. I don't know why, but that helps me remember it. Another way to remember this rule is just to commit the first form of it, that the implication goes one way. Commit that principle to memory. And then the other principles fought well. The other versions followed. The same general principle used these sorts of tricks to help get your mind around the distribution rule. It can be very helpful. Well, in a previous lecture, we already discussed Day Morgan's rule, so hopefully you're already familiar with that. It's named after Augustus de Morgan, the 19th century logician who discovered that told us could distribute in a neat, unique way that allowed you to change wedge claims to dot claims or dot claims to wedge claims. Again, you're gonna have to do some memorization here, but the principles should be pretty clear now. The reason why this works after you've memorized those two formulas, because the disjunction function is basically one that allows you to put in inputs of a certain sort and get out truths of a certain sort of known gives you an F When you get F sport, both dis chunks the bottom end of this table. The conjunction function, by way of contrast, is only going to give you an F. Well, it's gonna give you an F on every occasion except for the situation where you put in truth values t for both propositions. That's just the way our table worked. Now take a look at these two tables side by side. What you notice is the only time that you get an F is pretty much parallel to the only time you get a t for the conjunction. Um, unique parallel here that allows Derrick Morgan's rule toe work the way it does. So here's a little pneumonic device to remember. Mind you how to distribute the tilde Z across conjunctions or wedges appropriately, What you do with them organs rule is you get a different main connective when you move the negation. That's the way you work it. Hopefully you guys can see how negations distribute or un distribute, depending on how you go left to right or right to left. And the main issue here is you get a different main operator and that when you do that, that is very helpful in doing your proofs. Well, Here's a fun little example from Saturday Night Live to help you get your mind around a Morgan's room what it can and can't prove supposed. But he says that Pat, the SNL character known for his or her androgyny, is not a bachelor. Well, does that little bit of information help you solve the mystery as to whether Pat is male or female? Ah, bachelors by definition, a not married male. So basically, if Pat is not a bachelor, you get Tilda parentheses till to m dot a. So what does that tell us? By De Morgan's rule? Well, I'd shift the tilde in across the tota m and to the tilt A. And I changed the main connected to a wedge. So we find out is that either Pat is married or not a male. Does that tell you enough? Well, unfortunately, no, it doesn't unless you know the marital status of Pat. So I guess we can't solve a whole every mystery with the Morgans rule, but it gets us a little closer in some cases, just a little fun for you. Let's take a look at a few examples of how to use them. Organs rule how about the first up in Europe are left? I'm gonna use the Morgans rule by first moving the total to the insides and changing the main operator. It's a two step process. Now, over in the next example, I'm gonna used a Morgan's in this spot. First thing I'm gonna do, I'll change the main operator. And secondly, move. The tilde is to the outside of the parentheses. So it governs the operator that we just changed from a dot to a wedge, but I could have used not well. Now I'm in a position used a Morgan's in this position again. It's a two step process changed the main operator. Move the totals to the outside of a set of brackets. That way that until it a once it's moved, governs the operator that you just switched in. This case switched from a wedge to a dot. In this situation, I'm gonna go ahead and you distribute the tilde to the inside, so I'm gonna change the main operator and distribute the Tota C two step process every single time. We're not that, you know, Day Morgan's ruling distribution. Let's find an example of a proof where you have to use both of these rules in order to get to the conclusion. In this premise, we have a negation governing conjunction. And in this one, you notice curiously enough that to elope occurs twice The second premise. That's a premise that we're gonna use distribution on. We're gonna try and reach the conclusion, Toto. Oh, and I want you to pay attention. The fact that oh is up there on premise number one. Well, first thing, I'm gonna try and get O out into the open by using Day Morgan's rule on premise number one Now notice I'm set to get into my conclusion. Tota Oh, if only I could do disjunctive syllogism of some type on premise three or rather, step three. So what I need to do is try and get to something like Tilda P now, till the P is locked up their on premise to I use commutation to put it in the right position and distribution to get P outside of the parentheses and by itself. Then I can get Tilda P by itself through simplification. Now I'm almost set to use Ah, my disjunction destructive syllogism. I just need to use double negation in order to get the Tilda Tilda p turned into a P and then switch by commutation to set up P in a position where premise seven can be used to help us obtain tota Oh, which is our conclusion. Let's try another example. How about this, um, long disjunctive premise followed by a simple one p p. Ah. Implies Oh, but three conclusions here is gonna be in notice. End is locked up there. In that first premise, we need toe take that first premise apart. First thing I always do is Day Morgan's when I can, in order to get the tilde within the parentheses. Next things simpler and then by double negation. Well, that's pretty simple. Move on three. Now what the reason why I did that? You can see I was setting myself up for distribution. Any time that I can simplify a wedge claim like four down toe dot claim like five, I'm in a good position. Then it get out of hand and had a simple proposition. P I use that word along with premise to to get toe Oh, now notice. I'm setting myself up for a disjunctive syllogism in order to get to end. Basically, what I need to do is use calm utility to get to that. Ah, Tilda. Oh, wedge, end claim Once I do that and unlock it with simplification on line nine, then I can set myself up through double negation to do a disjunctive syllogism and I'm home free. So far, we haven't had any rules to help us with conditional claims. So now I'm gonna introduce to because horseshoes are very important and logic, we're gonna introduce two rules, transposition and material implication. You're going to use these an awful lot when it comes to dealing with conditional sentences . And again, we're dealing with equivalents Claims. Transposition, I think, is a bit of an easy rule to grasp material implication a bit harder now regarding transposition. We said that when it comes to dots and wedges, you can switch the position of dis chunks and con junks. But you can't switch the positions of P's and Q's in this case around a horseshoe. It just doesn't work. But you can do that if you add another step where you negate each of the propositions involved. Why does that work? Take a look at the bottom examples where I have two instances of motus Tolins we began in the first premise around. The first argument to your left with P implies Q Tilda que, therefore will take you to pee notice. That's pretty much just transposition moving from the left formula to the right formula thereof. We're look at the example to your right, and the gray toda que implies told API, was not enough to say that Double Tilda P will take you to double Q, which is equivalent to claim that P takes you to queue. That's why transposition works. It's basically an application of motives. Toland's simplified Compare this to Contra position. If all apples are fruit, we said, then if it's not a fruit, it's not Apple. Well, if we said all bulls or cows them. Basically, what that means is that all the bulls are in the cow class and consequently that if it's not a cow than it's not a bull. Well, if you recall how Contra position works transposition to be easier, remember, basically A. In the case of propositions, the proposition it's an apple implies it's a fruit. True, we can't just, uh, commute the two propositions. It's a fruit does not imply, of course, that it's an apple, but that if it's not a fruit, that's certainly implies that it's not an apple again. Switch the positions and negate. Now you can use this in a variety of ways. For example, just put flop the too complex propositions around. Ah, horseshoe. Add your negatives. We just did it with more complex propositions or, in this case, flip negate. The move works again, or in this case, we can use it on part of a prop proposition, since it rule is a rule of equivalents. So we'll just take the two propositions, move them into different positions on the horseshoes and negate. Remember, do all the spinning of your propositions around the horse you want. Just remember to bring those negatives with you now through dots, wedges and in a moment we'll see by conditional is transposing is easy. Just flip flop. The two propositions. The horseshoe is extra special. Bring along your to TODAS and you'll do okay. The second rule that we have for dealing of horseshoe claims is the material implication rule dealing with the material conditional. We have to remember that what goes in the anti seed on consequent columns makes a big difference. And we only get output of false right there where the ants seat was true in the consequent false. Now, if you can remember this truth table for the horseshoe, then you should be able remember the material implication rule. This table should remind you that all the horseshoe claims is that you're not going to get or rather, that you're going to get either a false and seeding or else a true consequence up for those who like a ah, actually, a formal demonstration of the equivalence of these two claims. Let's go ahead and take a look. Now we know that P implies Q is pretty well equivalent to the claim that it's not the case that P and Tilda Que not the case that true incident and false consequent. That's what the table told us right now. If that's the case, then that claim is also equivalent to the claim that by de Morgan's rule Tilda P or else Tilda Tilda Q. And by double negation. Yep, you got it straight to the bottom. Okay, enough with the formal demonstrations is exactly why material implication works by and large. I guess you're just gonna have to memorize the rule. But if you know your truth, Table and, ah, formal proofs of how material implication eyes derived. I think you're a long way towards understanding and using the rule. Let's try to do approve, from which we have to deal with a lot of horseshoes till the H implies K. It also implies J. And it can't be the case that both K and J. Now that third premise should tell you that if it can't be the case that both of them, then we should be able to do motives Toland's or something like that on either one or two and get to our conclusion H somehow we're gonna have to do a lot of work with horseshoes to get their first things first. I want to do Day Morgan's to get that Tilda distributed. I don't like Tilda is in front of parentheses as a general rule. Now, I'm gonna do implication to get a horse. You claim out of this by line eight. In this proof, you're going to see why I just did that. What I'm going to try and do is say that tilde h implies told a J by hypothetical syllogism . Look, it lines one through five, where K does the linking up work. And then j implies age by transposition. I just took line six flip flopped and dropped the negatives. Now, doing that, I could do hypothetical syllogism one more time. This time using lines two and seven. I told you buy online eight. You'd see what I'm up to. Line eight shows you a little technique that's very important. Improves if anything implies its opposite than that opposite is gonna be true. How do you know that this trick always works? It will get me to H or the consequent every time. All you gotta do is use implications. See the importance of the rule there. Once I dropped that double negative that I haven't nine, I get H or H, and I told you this rule would be useful by tautology. Now you can tell that why I tried to get to line eight. Any time I get a claim like that, there's a little trick like lines nine through 11 that will always get you to the truth of the consequent like in eight. Okay, I know you guys must be getting tired. Please don't give up. I've got one simple rule for you to learn material equivalence, and one that's a little difficult. Let's take the easy one first. Both the first one's gonna deal with by condition. ALS material equivalence is the only will we have for dealing with triple bars, and it's the only rule for triple bars you're ever gonna need. The first version says that if P triple bars to Q then P implies Q and Q implies P. The second version says that P triple bars to Q than either P and Q are both true or P and Q are both false noticed. These rules make per better sense when we put it in plain English. It's what you've known about the triple bar all along. Onley here. We haven't laid out in explicit formal terms, and it may be a little difficult to read. Read it a couple times and you'll get the basic concept. The basic concept basically will goes like this. The triple bar rules State rule number one implication goes both ways, And if a triple bar holds either both propositions are true or both. The false, just that simple. Now the triple bar rule or material equivalence is really helpful when you need to get something more useful. Let me give you a couple instances of this. Here's a couple of useful tricks. When you're using this rule, look over to your left. If P triple bars to Q disk, you double triple bar to pee. We didn't have a rule of commuted ity for, Ah, the Triple bar, but the world that we have will give you that, because what our rule says is that what this triple bar amounts to is P implies Q. And Q implies P. Now you can commute around the dot and therefore arrive at the claim that Q. Dust Ripple, Bart Api. So in this extended sense, yes, we do have a calm utility rule for the triple bar, thanks to material equivalents. Now look over to your right. Suppose you find out that P and Q have the same truth value, namely Tilda P. Until the Q. Can you get a triple bar out of that? Here's a trick. Just go ahead and conjoined the two lines that you have informed one conjunction through conjunction and then because you can conjoined and then wedge off to any claim you want. Just wedge off. Two p dot Cure anything else? For that matter, you can commute around a wedge. So do our triple bar rule. This is equivalent to P implies. Q. This is a trick that you can use any time that you know propositions have the same truth value to get a triple bar out of it. Useful in many proofs. So let's use a proof where we have to use our triple bar rule. Thanks toa premise Number one here and who have a disjunctive premise of tilde or till the K and R implies a Does it follow that till decay or are now here? I want to show you another trick with respect approves. Look at that conclusion that's a little bit too hard. Derive all by itself going to give away a hint. If we want to derive by line 15 of our proof, the conclusion in question. We need to reach it through double negate or rather, day Morgan's rule. What we need is to get the line 14 from our premises. I suggest we try to prove till decay and then we try to prove Total are then we'll conjoined them by lying 14 and then used a Morgan's rule to get to our conclusion. Line 15. That's a good strategy, I think. First step we're gonna do is use the equivalents claim that we just studied to dismantle premise number one. And since I think I want to deal with K horse used a first, I'm gonna just flip flop those to buy commuted ity so that I can simplify down to the first con junked in five. Once I've done that, I'm gonna use implication online, too, because I don't like to deal with wedges. I like dealing with horseshoes better. I noticed that sets me up for hypothetical syllogism. Now, if you remember our last proof, you can tell right away why I wanted to get to line eight in a few simple steps, I should be able to prove from this sort of line that till decay is the case, I do that by the same moves is before implication to tautology. There we go. We're halfway there. The next step of men don't do a simplification on line number four and get to a implies K sort of like I simplified five before I'm gonna get Teoh mean 12 by Motus Toland's My intent is helpful there and then, ah, three and 12 motus Toland's is going to give us the second con junked that we need. My M 14 follows by a simple conjunction and a Morgan's rule gives us our conclusion. End of proof and it's almost the end of our lectures on these. Ah, new rules. Exportation I promised you would be a hard ruled learn P and Q Together imply are is basically the equivalent of saying P implies that Q is enough to get you to our well. How does that work? I call it the You're almost There rule. Here's how you explain the rule to try and make it more intuitive. Basically, what this rule says is that if P and Q together will get me are, then that means that if you have P already then accused the last thing you need and you will get you there. Here's an illustration suppose somebody says, If you have three tens and two fours together, that's enough to get you a full house in a card game. Well, what that is equivalent saying is that if you already have three tens, well, then that implies that you're almost there. All you have left overs to get those two fours and then that will get you a full house. It's basically you're almost there. There's a short illustration of the how to use the rule. If a and B together imply See, and you've already got be Well, it looks like you're almost there. All you need leftover is a to get to see As the conclusion says this conclusion should follow. We need our exportation will show us why First I'm gonna commute be and a so that we get be right out there in the front and then I'm gonna use exportation by exportation. Be implies that a is enough to get you to see notice what I did. I set us up for motus opponents with line to a really is enough to get you to see given line to that. One of the conditions for C has already been met. You just have another one left over now. You might feel like you're drowning in all these rules. It's going to take a little while to get your mind around Hall of them. No doubt they are kind of difficult. This is gonna be one of the most difficult, Ah, lessons that you've had. I'm gonna provide a lot to help you later to get your mind around these concepts. Even better. So plenty of exercises to come. They're gonna be necessary in order to wrap your mind around the concept you just studied. In the meantime, watch this video a few times and hopefully you'll have a lot of fun with your logic textbooks. Take care, and we'll see you next time for our very last lesson in this first lecture series on logic . 20. Conditional Proof, Indirect Proof and Proof of Logical Truths: Well, welcome back, everybody to my crash course. In formal logic, this is the last lesson of my first Siri's. So congratulations for making it this far. I hope you guys had fun with last lesson. That was definitely the hardest lesson I presented online. Now, in this lesson, we're gonna study three things. The conditional proof, the indirect proof thes two are very closely related to one another as we're gonna find out , and lastly, approves of logical truths. And if you master conditional and indirect proofs, you should be able to prove logical truths very easily. Let's begin with the conditional proof of very common type of technique used in a lot of logic textbooks. Now, earlier, we saw too inconvenient, inconvenient truths. Truth tables can get varied long need hopelessly long if too many simple sentences are involved. But if you try to go toe indirect truth tables that can be very confusing and complex. If you're conclusion is complex, they're not very efficient under those conditions. So here's how you do it the hard way, - so these can proofs can become needlessly long and tedious. I mean, actually, the reasoning from one to the conclusion is pretty simple. Really? What we need to think to ourselves. His mam wouldn't have been convenient if just for a minute we'd had an A. We need a little horse. You help here and I'm gonna show you with conditional proof how conditional proofs offer us to help with horseshoes that we really need in plain talk. We would never have walked through a proof like that. It's just too complicated. We would just use common sense. And here's how we would have done it. We have taken the proof or rather the argument listed on the blackboard above. And we just said number one. Suppose importantly, hypothetically that we had had in a then according to premise one we'd get be dot c and be dot c Note line number three here says, Hence we get be by simplification and four, we just say, And if b is true, then beardy is true. And by that B or D line to would take us by motus opponents to e so a does get us tea after all. Now what we're doing here in plain talk is a few steps. First offer saying hypothetically assumed that I did have in a after that, Uh, for conditional proof, we're just going to say what would follow This is the scope of our assumption of a once we're done with a we show that we finally did get to E under that assumption. So we concluded, discharging the assumption they would have gotten us to e after all. See how easy that is. But all is not lost. We can actually prove the proof. There was argument that was on the Blackwood earlier. The easy way in our formal logical system are formal logic doesn't need toe, be devoid of common sense, approaches proofs. So rather than this longest proof that we just did a second ago, Let's go ahead and shorten it up a little bit instead of doing that, that we're going to say, Suppose hypothetically notice I'm in denting a little bit. Suppose hypothetically had a I'm gonna introduce a scope line to keep track of how long I'm using my assumption of a Well, A is true. Then we're gonna get, uh, to be dot c through motus opponents on one and three. And then we're going to simplify down to be after that weaken wedge off. Two D Once we have bi wedge de weaken doom Otis opponents on to and get to E. And that shows that they would have gotten us all the way to e notice I un in Dent back out of the scope line. Here's how it worked. I introduced on assumption for conditional proof than I did Motus opponents using the Assumption Line three. And then I simplified used addition and did modus opponents one more time. Now all of those justifications occur within the scope of the assumption of a Finally I got via 337 by a conditional proof that a really did get us to e. Now remember, we're gonna be in denting here when we ever use an assumption. Because E does not follow. Line seven does not follow from lines one into it followed from 12 and three are assumption for conditional proof. And we need an indentation to note that. So basically any conditional proof you ever have is gonna have three steps basically gonna introduce us some. I hope this is a good pneumonic device, usually or some for your assumption for conditional proof. After that, you're gonna have to make some sort of jump to whatever it is you want in the consequent of your conditional. And once you make that jump along your scope line, well, then you gotta dump your assumption. So make Theus sump, make your introduce your some rather make your jump, then dump. Dump your assumption for conditional proof. It's that easy every time. I guess we should put this, um, or technical terms because introduced Theus. Some get over the jump and dump really just doesn't do it well, does it? The key point is you need to explicitly introduce your assumption for conditional proof Once you've done that like we did on line three of the last proof keep track of how long you're using that assumption, how long you're using it is called the scope of the assumption. And then when you're done using the original assumption, discharge it when you're done with it and mentioned that the result the horseshoe claim that you finally reached follows by conditional proof or C p for short. So if you remember these three points, then you've pretty well mastered the idea of a conditional proof. It's a great horse. You help get lets you reach horseshoes quickly and efficiently that in ways that would otherwise be long and tedious. But still, no matter how Ah, technically, you introducing the idea of a conditional proof or explain it is still going to come down to three basic things. You know how I need to know how to introduce your assumption, Make your jump and then discharge the assumption when you're done, introduce your sump, make your jump, then don't and then you're done. Now there's two things that you're allowed to do with conditional proofs that we need to cover. And there's three things that you're not. These are going to seem kind of intuitive. So don't be too intimidated by the fact that you need to memorize five rules you first permission as you may use conditional proof more than once. Here's an example. So suppose that G implies h dot I and J implies K and Premise three G or J. Therefore, can we reach this long conjunctivitis claim that you find for our conclusion now to get this proof to work, I'm gonna try to prove the second conditional of our conclusion. In other words, I'm gonna try to get from Tilda K all the way down to H. And, of course, I'm gonna have to fill in the blanks along this proof. Once I do that, I can discharge the assumption Toda que would imply age. If I can get this whole proof inside the green scope line to work after that, I'll just go ahead and try and prove the first con junk of the conclusion that total H implies K. I'll just introduce total H is ah, assumption for conditional proof. Try to get to K, and if I can fill in those blanks, I'll have total h implies K notice. If I get Line 10 and line 14 I pretty well have my conclusion. All I have to do is add the two together, right? Well, rather in that case, conjoined them. So obviously I could get the line five once under the scope of four. I just need to him four motors Tolins. After that, I could get to J or G. All I have to do is take premise number three and flip flop around the wedge and premise. Three. I'm only doing that because I want to do disjunctive syllogism to get G and from G I want to get to h dot i. Why? Because after that, I can simplify down to h. So Tilda K did get us to H. Now it's fill in the second scope line under the assumption of Tilda H line 11. I can demonstrate that. Tilda Tilda K. How'd I do that? I just used the conditional that I just proven before that toda que does imply age and a double negative amounts to a positive. So 13 follows from 12. So I did get all the way from total H two K as line 14 indicates all I have to do to get my conclusion. I conjoined the two things that I demonstrated from my conditional proofs. You can use more than one conditional proof in a proof of any particular argument. Well, now that I've given you a Commons Ah, very common sense permission. I'm gonna give you one very common sense restriction on conditional proofs. That restriction is you may never use lines from a discharged conditional proof. For example, let's take a look at the proof that we did just a moment ago. I shorten it up a little bit now suppose instead I tried to reach line 11 line that says, H and I made my justification. This I'm going to use lines four and 10 and do Motives opponents. Line number four, though, shouldn't be able to be used. We discharge the assumption lot on number four. Let's take a look at what a case in which this logical maneuvering clearly commits a fallacious inference. Now line number one of our proof. Where Premise one says P line to promise to says Q. Implies are does it follow that? Q. Not at all. But what if we used this technique? Will say till two p is an inception for conditional proof and will use line number one toe wedge off to que. Having done that, we can use lines three and four to get to queue. Now what is this proof till the P does take us to Q under the assumptions just given No problem with the argument thus far. But what if we try to say, therefore Que, after all, doesn't que follow from lines three and six? Well, here's the problem. What was going on within the scope of the assumption should have stayed within the green scope line of the assumption Number three should not have been able to be used after Line six, where the assumption was discharged. It's kind of like what happens in Vegas stays in Vegas, except, of course, when what happens in Vegas stays in God's mind for all eternity. Well, in this case, what happened within the scope line of the assumption should have stayed there. What happens in the box, I tell students, stays in the box. Once you discharge your assumption, you're out of the box. Can't use what's in it anymore. Well, let's return to our topic of what's permitted and what's restricted when it comes to conditional proofs. Now, one thing you have permission to do is to use conditional proofs within the scope of other conditional proofs, and that can come in handy. Try this, for example. Suppose we have a promise. L implies that implies and wedge Oh, and a second premise that says M implies till the end Does it follow that l horseshoes toe till toe M wedge. Oh, take a look at the conclusion there's obviously one horseshoe, but look inside the parentheses till two AM Wedge. Oh, is another horse you claim If you use implication, you can see that That means that m implies. Oh, so let's just try to find to prove that l implies the what's in the parentheses, and then we'll try to see if m really does imply. Oh, So first I introduced L and by motives opponents on one that's going to get us a little bit of our distance. Now what I'm gonna do next is I'm gonna assume M Why am I doing that? Because I want to get to the final ah conclusion when I jettison this conditional proof that m implies. Oh, so here's what I'm gonna do once I get to em implies, Oh, by implication, that is till toe M wedge. Oh, see how that works. So fill in the blanks after five line number six is gonna be in wedge Oh, by motus opponents and then till the end by two and five motus opponents. Now, once I get to that point, I can prove that Oh, by disjunctive syllogism on six and seven. So I m really does imply Oh, as line nine says not line nine that rhymes notice. I've just jettisoned one condition. Just conditional proof. I've discharged a conditional proof, rather lines five through eight have been discharged. But the conditional proof on three through 10 is still going strong. Uh, jettison that or discharged that conditional proof. We need to just say Line three really did imply Line 10. And that, really is all of our conclusion now, long the outside of any proof. There's always an imaginary final scope line, if you want to call it that. Basically, what we've shown is that lines one into kind of like assumptions prove Line 11 that scope lines always there, but kind of imaginary. Well, that mentioning of, ah, imaginary scope line that's always there brings us back to our next issue about restrictions. Don't end approve on an indented line. Always get back out to that original scope of the original premises. So in the last proof we had this sort of thing going on that we just illustrate the importance of this particular rule. Suppose that instead of going back out to the scope of the original assumptions on our conclusion, which was Conclusion Line 11 suppose instead is a conclusion we tried to introduce. Oh, now one and two definitely do not prove Oh, but how? What if I try to say this, I'll just end my proof Online eight and therefore premises wanted to really do prove. Oh, is this a problem? Let's take a look at a case in which this Keiper type of ah logical thinking clearly leads to bad inferences. A premise number one says p. And the conclusion is Q implies are now. This argument shouldn't work. But what if I said Tilda Que by assumption for conditional proof and there's no room what you can introduces assumption, and then I wedge that off to our By addition now from their by line number three is equivalent to Q. Implying are by implication. Now, does this argument or proof rather show that P really did imply that Q Horseshoes toe are no. It shows that line number one and line number to lead to the line number four, but it does not show that the original argument we were analyzing is valid. So let's return to our two permissions. Three restrictions talk and talk about the third restriction. You cannot discharge two assumptions at a time. You gotta go one assumption discharge at a time. If you do it at all. They're going back to the argument that we just did. I'll erase this final scope line just so that you can see the argument more clearly and notice I put a new conclusion in l horseshoes. Oh, now does that conclusion follow from one and to take could look at one and two By now, you should be convinced it doesn't. But here's what I'll do. I'm gonna shorten up our entire proof and say, I'll just end it, line eight And then I'll jump over to scope lines and say, Look, line number three on that scope line ended on Oh, therefore l horseshoe toe by five through eight. Or, if you prefer three through eight conditional proof something's gone wrong here. Let's take a look in another example where this sort of logical maneuvering commits fallacious inferences. Now line number one or premise says Q implies are it should be obvious that the conclusion here shouldn't follow. That, P implies are. But here's my fallacious proof. How about I say P as an assumption for a conditional proof and introduce my scope line, and I'll introduce an assumption for conditional proof within that original assumption That's okay So far now, by lines one and three Motus opponents that will get you toe are have I shown that two takes you all the way to four? Well, no, I haven't. I made a mistake here. What happened is the conditional proof that ran from Lines three and four was not discharged before the line two through four was discharged. You can't discharge two assumptions at once. You got to take him one of the time. So to review any time that you're going to do a conditional proof, introduce your assumption for conditional proof that is introduced the assumption. Then make your jump within your scope of your assumption. Once you make your jump, you will have to learn to dump. If you remember to do all those correctly and remember our three permissions. I'm sorry to permissions three restrictions. You're gonna do really well with conditional proofs. Well, now I'm ready to now we're ready to move on and study the indirect proof. A close cousin of the conditional proof. Now we said earlier that proves the natural deduction can become needlessly long and tedious after also have to mention they becoming become very difficult to figure out brain teasers. Take a look at this one A or B implies CND premise to see implies the negation of D. Therefore, tilt A now to prove this the hard way, you might have to take the following route. One thing you could do is just point out. That, too, is equivalent to three by implication. And if that's the case, this is one of those few times I'm gonna used a Morgan's rule to put the tilde outside a set of parentheses. Why did I do that? Because I'm setting myself up for Motus Toland's. I want to say that no line number four is the negation of the consequent in line number one . Now when I have that are used, a Morgan's toe unpack Line five. And that's why I'm gonna wind up with my Tilda A by simplification on the conjunction in six. Well, it's pretty straightforward when I lay it out there for you like this, but it's a very tough puzzle. It's a brain teaser, and you could be ah, you could do this more simply in plain speech, which might want to look at is what would have happened if they were true. Notice these two premises and work your way along them. If they were the case, you should be able to prove D online one. And since you did get see out of line one as well, you should be able to prove till two. D Simply put a seems to lead to contradictions. So you could say suppose hypothetically that we had at a If we had a assumption for indirect proof, then we'd get to a or B because they can wedge off to anything then by Line one and motus opponents that gets you to C and D so you'd have both c and D in your pocket and see online to would get you the negation of D. Hence you're gonna get Teoh both d and its negation. A proves contradictions on these premises. So if they can't be true, there's only one option. It's opposite Tilda. A must be basic principle behind an indirect proof is that anything that proves contradictions has to be wrong. You can't have contradictions turning out true. So if an assumption leads to contradictions than that assumption must be wrong. Now, this proof is gonna be a little bit longer, but it's much more intuitive If you want to, uh, do this argument, he may just prove Start out by assuming the opposite of your conclusion. A. Now, if a is an assumption for your indirect proof A i p, we're gonna run a scope line here and along that assumption, we're line. We're going to say a or B must be the case by addition, of course. And by that, C or D must be the case because of line one by motus opponents. Now that gives us see which I'm gonna use in just a minute, because I need C to get to tilde de by using line number two. Now, if I have tilde de look back at Line five, I'm gonna flip flop C and D in line five by commuting them. And then I'm gonna wind up with D by simplification. Now, when you put seven and nine together, you wind up with the contradiction. Therefore, since whatever proves contradictions must be false. Line three a. While Assumption for indirect proof proved a contradiction. Seven and nine were just conjoined in line 10 and tilt a follows Do two lines three through 10 by indirect proof basic idea here is if Line three proves contradictions. Line three has to be rejected. So again, our basic principle here is the same as the conditional assumption her quick, rather conditional proof. Introduce your sump, make your jump. And once you've made your jump along the scope line, then you discharge the assumption in this case you discharge the assumption for indirect proof. But of course, we want to put all this in technical terms. The key points are explicitly introduced. Your assumption for indirect proof. Keep track of how long you're using this assumption and notice. I added a new point here point. See some systems not all require you. Do you have a line where you state explicitly that some statement of form P and Tilda P has been proven? I put that in the blue on the last slide. Be sure to discharge the assumption when you're done with it, mentioning that your result followed by indirect proof some systems don't require points. See, Sometimes they just require that a formula and its negation occur on the scope line. But some systems want you to conjoined both the formula and its negation that way everything's explicit. It's important to note that all the restrictions that we had wrong conditional proof supply to indirect proofs. But the permissions are extended as well. Well, pause the video for a second and familiarize yourself with these premises and the conclusions. Now when you do that, make sure you take special note of the conclusion because it is a little bit complex, really, which you should think about that conclusion in terms of Day Morgan's law, indirect proofs can be used repeatedly. Maya strategy is gonna be this assumed G and prove it leads to contradictions, then assume K and prove que leads to contradictions that's going to lead to the conclusion that eight and 14 congee conjoined. And once I conjoined them, Day Morgan's law will give me my conclusion. So let's get started back up at Lime for assuming G obviously weaken doom. Otis opponents on one and then that's gonna get us Thanks. Tow line number three to tilt L, but also l and Tilda L. By three and line six. You can use lines to the left inside your scope line. You just can't take things outside the box or the scope line right so, uh, till the G follows because G lead to contradictions. Now let's go down the line nine. Assuming K Then we get l dot tilda l by one and two. Now give it our and hypothetical syllogism, of course. And if that's the case than l leads to Tilda, l now line three says that l is true. So till the l follows my modus opponents and we're just gonna conjoined a couple lines were minute conjoined line three, which were allowed to use within the scope of this proof. Ah, with the line number 12 that gives us a contradiction online 14 the rest of the conclusion , Or rather, the conclusion follows through the rest of this proof. In fact, not only can you use indirect proof within the scope of, ah, another indirect proof, you can use conditional and indirect proofs within the scope of one another. That's kind of neat trick. So look at this argument. We've done it before. We have introducing an assumption for conditional proof here because our conclusion is a conditional sentence that seems helpful. And I could get a little ways towards that conclusion by using line one. Now what? I'm gonna do is since the conclusion involves a negation until 2 a.m. It looks like I should just assume if I want to get toe l implies till toe M Maybe I should just assume m for indirect proof and show that contradictions follow. If so, then I'll reach Toto Em eventually and my conclusion just like so. So all I have to do is still outlines five through 11 and I'll be home free. Because the last two lines in the proof number 12 and 13 incidentally, are going to jettison or discharge assumptions one at a time, according to one of our restrictions Now given line five, I'm gonna use in line four to get to end and oh, and simplified it in. And since I have oh available to me in line six, I just have to commute and simplify. Then I can use that in line number two to get to till the end. Now, once I've done that, I've got a contradiction between lines seven and 10. Given a contradiction, I moved the line 12 by indirect proof, discharging one assumption. And then I moved from 2 13 by conditional proof, discharging a second assumption. There's plenty of tricks that you can use. Ah, when you're dealing with indirect and conditional proofs. One of the tricks is indirect. Proves compel. Prove conclusions that aren't negations. Let's take a look at how you can do that. Now here's a Here's a proof that we did earlier in order to demonstrate how indirect proofs work. I hope you're still familiar with it, but notice we didn't have to prove the conclusion that tilde a we could have. If you swap a fertility A throughout this entire proof uniformly one for the other, you can see that we can actually prove a conclusion like a So I just swap the A's until the A's noticed that when you get to the bottom because Online three we introduced Tilda as an assumption for conditional proof, we're going to negate that down on line 11 and then one last move just gonna use double negation. So notice we proved a conclusion a using the exact same proof and set exact same form of argument before. But that just demonstrates that indirect proofs are well handy for all sorts of proofs, proofs of negations and proofs of positive claims. Here's another little trick you should remember. Technically speaking, indirect proof and conditional proof are redundant. Anything you can prove with one, you should be able to prove with the other. You really just need one technique. Well, if you haven't figured it out by now, anything that Platt implies its own opposite has to be false. Look at these two premises. Toto Athletes to g dot K and K leads toa f. Obviously, Tilda F is gonna take us toe f. And that's why this conclusion involving our is gonna follow. Because if f is gonna be proven true by the premises till the f word, it's opposite horseshoes. Anything? I'll show you of that technique if you haven't learned it yet. So I'm gonna assume till two f for indirect proof and then get to G and K by motives, opponents flip flop the gnk by commuting them and reduced decay. Once I get K by itself, I can use line to unlock F by motus opponents. Now, obviously we have a contradiction between line and 63 and seven. And that means our original assumption has to be false. So I double negate line three. And with that in mind. I can reduce that to f Remember, you can use indirect proofs to prove positive claims, which is what we really just did with F in my pocket. I can wedge off to our if I want to, and then I get by implication that tilde f implies are It's a tricky little proof, but it's interesting that you can prove in this particular context positive claims. Using an indirect proof. We just had that essential line involving double negation on 10 but we could have taken over the proof from this point onward and done it differently. Suppose instead, we had said Tilda F implies F as our conclusion. That is, when we discharged, we said We're gonna discharge lines three through seven by conditional proof till toe F proved that f we could have done that. If that's the case, then double Tilda F or F by implication, which reduces toe F wedge f by double negation and therefore f by tautology. So if F is true, we can wedge off again are just like we did on Line 11 in the previous version of the Proof , and we get the same result till the F implies are this is just one example of how conditional and indirect proofs are really interchangeable. In an earlier lesson, I told you that tautology would be a useful ruling we just used for help us through technicalities previously. Now you can see how tautology was a help. It was essential. Net. Uh, little sub proved to the right that I did on the previous slide. Essentially, what happens is that it reduces indirect proved to conditional proof. If anything implies its own negation conditionally than the negation is true. So if you get f implying that tilde f well, then that means Tilda F. Or till the F implies f means f. So this is a rule tautology that's gonna help us reduce in director proofs to conditional proofs. Well, let's move on to our next topic. Proving logical truce. What are logical truths? Well, logical truths are tautology ease that occur that are due to form tautology, zehr sentences that cannot possibly be false like all bachelors are men. But that is a tautology. Due to semantics. A tautology due to form like if it's raining, it's raining, are implies are that is necessarily true. Cannot possibly be false. But that's due to its logical shape or form, not just semantics. So recall when we were classifying statements in earlier lessons, we said There's tautology is self contradictions and contingent sentences. And when we were doing truth tables, we pointed out that every sentence is gonna be in one and only one of these categories. Tautology is we proved by truth tables because they had all truths in their columns, like in the column to the right on this table, this is, ah, truth table that we did in an earlier lesson to try and demonstrate that a compound sentence was actually a tautology us, which means it could not possibly be false. Statements that are tautology is and proposition all logic should be provable from no premises whatsoever. In other words, the tautology is truth should follow not from any particular set of premises, but just due to logic itself, no premises needed. So I should be able to prove the sentence that we compound sentence. We were looking at on the last slide from no premises at all. How can I do that? Will notice that line number six down here. The sentence we were looking at is a whole shoe claim. I'm gonna start out by assuming the antecedent, and I'm gonna try to reach by line number five in this proof the consequent and then discharge the proof. So I'm just gonna flip flop the two elements in line number one And then I'm gonna pull out G horseshoes th by simplification on one and g by simplification on 23 and four by motus Opponents will get me to H. So I introduced an assumption for conditional proof. I did not need premise season this argument And then I commute on one and I drag out the first con junked. And then secondly, I drag out the 1st 1st con jumped on the first line, put them together by Motus opponents, and voila! I discharge the proof. I discharge the assumption one, But notice this proof did not require any premises. Not every argument does. The importance of logical truths is that they are necessity claims they are necessary and virtue of logic alone. They are not necessary and virtue of any particular sets of premises. That's gonna be important when we do motile logic later on. So keep your eyes on this reason is because necessity claims or necessary necessary truths can also be treated in Proposition logic. As Theorems and Kenneth Conan Dykes book, he defines AH, Formula Pia's a deer. Um, if there is a proof of it which uses no premises, prove ability without premises is sufficient for theorem hood, and it is sufficient for necessity claims things that cannot possibly be false or, as we put it, proposition, logic, pathologies. Let's give an example, um, or a challenge. Rather, let's prove motive. Opponents is a the're um instead of taking it is one of the rules of our systems using only the other rules of inference. In other words, let's make believe that Motus opponents is not one of our rules, and we have to derive it from the other rules that we have now. The last proof that I did, which we proved ah, long sentence in line line number six from no premises whatsoever involved modus opponents . I guess the real challenge here is could we have done proofs without motive opponents? Does Motus opponents or something like it follow just by using other rules? Here's a rather long Bishan boring way of doing it. My strategy is going to be to prove Line seven, The general form that p implies. Q. Along with the unseeded P would imply. Q. I can get there if I can get to line six. What I'm basically doing is on exporting. Here's how the rest of the proof is gonna look. Assume the antecedent of line number six and try to prove the consequent. Now, you might stop to think here a moment, assume the antecedent of line number six and approve the consequent. They're the exact same thing. Yes, but we have to go through a few steps in this particular system that I have laid out. Actually, we're gonna have to assume ah, the opposite of number one for indirect proof conjoined the to and get an explicit contradiction. Once we have that, we can move to the double negation of that and by double negation elimination P implies. Q. I know this seems like a longest way of getting from line one to line five. There are certain systems that have something called the reiteration rule, which allows you to repeat any line that you previously had. Since our system doesn't have it, I had to lay out this proof. The point is that by the time you get to line six by conditional proof, then all you have to do is export. And line number seven is basically the rule modus opponents. Therefore, it is provable is a dear um, and notice. I proved it on my booking. At my justifications, the right approved it without assuming it. This may lead you to ask if we didn't need motive opponents. Are there systems which use rules other than the ones we've studied? Yes. Oh, exactly which rules or using a particular system of logic is really It's not altogether arbitrary. But there is some leeway a system could have treated Motus opponents instead of as a rule, as derive herbal, the Rome derive herbal from rules of a system. And that's true for other rules as well. Consider something like the rule of absorption, which we have not studied. P implies. Q is equivalent to the claim that P implies well, both itself and Q. And consider the rule exportation, which we do have in our system. For Patrick Hurley's textbook, which I've used, exportation is a rule and absorption is provable from exportation and other rules For Conan Dyke, absorption is a rule in his system. Exportation is not. Is there some sort of leeway here? And generally there is. So consider the rule absorption, which we do not have have in our system. It's not in our system as an axiom or as a rule, but we could have proved it from the rules that we do have. All we have to do is use the proofs of logical truths that we've studied earlier. How do we do that? Well, first we need to get a game plan. Always lay out exactly what it is we want to do in order to demonstrate a claim. Now look at the in order to get your proof going. Noticed that they are absorption rule. At the bottom of your screen is a triple bar. I'm gonna give you just a few hints on how to fill out this proof and let you fill in the blanks. It's actually pretty straight forward. Once you get a game plan. First you need to demonstrate that the implication goes one direction, and you need to prove the implication goes the opposite direction as well, which I'll demonstrate earlier in the proof. Now notice what you're trying to demonstrate when you discharge the two assumption lines that I have. You're trying to prove horseshoe claims. So when that happens, you assume the antecedent and try to get to the consequent and again assume the antecedent and try to get to the consequent in the second half of the proof. Now, from here, it should seem pretty easy how to fill in the blanks. And so I'm gonna leave that for you as an exercise. But this is the general form of the proof by which you could prove an hour system that absorption is provable from our rules. Just make sure that when you fill out the proof you recognize you're gonna need another conditional proof as well, because you're trying to get from well, the antecedent listed as your assumption to the consequent See how the proof is gonna fill out. Go ahead and fill it in. So is any theorem a candidate for a rule of inference? Yes, logician is who construct their derivation systems have some leeway here, but the general rule is don't put too many rules in your system, which is why we don't have absorption in our system. As a rule, we had exportation. But give enough of the rules for the system to be easy to use. You don't want to leave out something basic like motives, opponents or hypothetical syllogism. That's why all textbooks tend tohave almost the exact same set of rules in them. And the result is that any useful system at least offers a rule for introducing a connective like a horseshoe, introducing a wedge and for eliminating it so bare minimum. If you have five connective Zina system, you're gonna have at least 10 rules. All right, well, that's enough theoretical stuff to get us set up for our next course in logic. But for now, you just finished your first course. Congratulations. You should be proud. The second course is still to come, and I look forward to seeing you when I put that together. I hope these videos have been helpful and have gotten you through plenty of logic classes at your college or university where you're studying, take care and have a nice ah, nice time with logic.