## Transcripts

1. What Is Logic: So what is logic? Well, logic is a science that is concerned by truth, improve nous off statements and by the relationship logic. The day is more about idealization and formalization. What do I mean by that? Well, let me show you Example, if I have your these two statements 1st 1 is if it rains, then it is cloudy and second oneness. It drains. So from these two statements, we can assume that it's cloudy. Right? Let's look at another one first, say that if Peter is sired than Peter Steeps and the second say, Peter is tired. So from these Sue, we can assume that beater steeps. So in these two examples, you may notice that even though these expressions had completely different content, 1st 1 was about raining and and 2nd 1 about some Peter. They both had the same form, and the form simply looks like this. The first statement looked like this where the sign between A and B means implication. You better remember that because we will use that lay their own and one this implication means in real world, it means, if a then b where instead, off A and B can be some any bullion statement and bullion means that it can be true or false. So if I go back to our examples, I can ride that like this raining implies cloudy. So instead of fe we have raining and instead of B, we have cloudy. The second statement can transfer to reigning because there is no logical connective and I will talk more about this in next. Lecturers, it is okay if you don't understand something from this lecture, I will talk more about these things in following lecturers. So the assumption dead we used was from a implies B and a we can deduct be so if I transfer previous example using this form, it will look like this raining implies cloudy and raining. And we deducted from these two statements that it is cloudy and the second example looks similar. Tired implies sleep and start and we deducted sleep. This might still be a bit confusing, but don't worry. I will go to details in following electricity. So the important thing to take from this lecture is that we use something called form in logic and that is just this a implies B. We use something called logical connective. And that is, for example, implication. And we are trying to deduct from some expressions other expressions where expression is, For example, it drains or Peter's start. Okay, that is all for his lecture, and I will soon excited.
2. Expressions and Logical Connectives: let's start with expressions. More precisely, logical expressions. So logical expression is just something that can be true or false. So, for example, five is smaller than free. You can obviously see that it is complete nonsense. Bad. It is still logical statement That is false, right? Another expression could be. For example, my father is 50 years sold because it can either be true or false. There is nothing between no third option, right? And since we care about the form and not countin, we will call all statements by letters from A to Z. And since in our natural language we are talking using these statements and connective. So in order for us to be able to transfer sentence into logical formula, we need logical connective. And thankfully, there's only five of them. So the 1st 1 is negation. It looks like this, and it represents not in our spoken word, logical formula that looks like this negation off a where a represents statement. But since instead off a, we can put any statement remember, we care about the form, not about content bad. We know that it can either be true or false. And since we are lazy. We represent true by one and falls by zero. So if a is true negation of true is false And if a zero negation of false is true so that means one. So what negation does It kind of flips theological value in simple words. Okay, the next one is or and if you think about it, when you talk you use or all the time in logical formula, it looks like we so a or B looks like this. If a and B is zero, this logical formula will result into false that if a is false and B is true, this formula will result into true if a is true and bees force, we get true. And if a is true and bees true, we get true so or is true if at least one off its operate and is true. And if you think about it, it kind of makes sense and it curse bones with or in our natural language. And since we have or we also have end and looks like this. So it is kind of a without the line in middle, so a and B looks like this. And when a it's false and be its force A and B will result into force when a is false and bees true, we get falls again when a is true and B it's false. We get still false that when a is true and B is true, we get true. So in order for end to be true, we need A and B two b both true. You can think of it as multiplication One times one gives you one. Okay, let's look at another one. The next one is pretty simple and can a straightforward it ISS equivalents sometimes known as logical big conditioner. And it is true if all friends have same value. So if a and B this false we get true because they are equal. If a and B is true, we get true. Otherwise we get false so you can think about it as an equal sign. Zero is equal to zero by zero is not equal to one, right. The last one is implication and I kind of show you implication in previous lecture. But I want to talk more about its purpose in this one. So it looks like this, as you already know, and if a is false. Then no matter what B is, we get true that if a is true and B is force, we get force. And if a is true and bees also true, we get true. So this one is not consistent and you might think it is kind of weird. And I think the best way for you to understand implication is fruit. This sentence. If it's raining, then it's cloudy. So a is raining and B it's cloudy and let's go through all the logical values they can have . So 1st 1 is that a is false and B is also false. So if it's not raining, then it's not cloudy and that is OK. That makes sense, right, and it is true. Next one says that if it's not raining, then it's cloudy and that also makes sense, right. You can have clouds without rain, but now it gets interesting. If it is raining, then it is not cloudy and this doesn't make sense, right, because in order to rein, you need clouds. So this is walls and last one says if it is raining, it is cloudy and didn't make sense. So it is true. Okay, That's pretty much it for dis lecture. We'll play more with these in next lecturers. So if you have any questions, feel free to ask and I will see you next time.
3. Logical Formula: logical formula is something that is made up from statements that means from under se two Z and from logical connective. Yeah, and also parentheses. So, for example, logical formula can look like this negation. A implies B is the correval and with a or B, and every formula is either tautology. That means no matter what logical value A and B half for every logical value the formula will result into true or former can be, satisfy a ble and formalize, satisfy a ble. If it is true, under at least one interpretation off A and B and last one is contradiction and that means formalize false, no matter world, logical value A and B had. So how do you find out what kind of formula we have here? We will use table. So the arguments are A and B, and we will consider every possible scenario. Now let's look at our formula and you can kind of split it into smaller parts, right? Compute these smaller parts and then put the result together. So on right side we have A or B and on left side we have negation. A implies B, so let's compute left side first I recommend compute negation a first and then compute implication off negation A and B. So let's do this. As you already know, negation flips value. So where a is zero negation of a will be one and where a is one negation of a will be zero like this. So now let's compute implication off negation A and B and we know that implication results into false Onley. A first operandi is true and 2nd 1 is false. So if we look at our table, we can see that this line will be false. Right? Because negation of a is true and be this force bad all the other lines will be true So we have compute are left side Now let's compute the right side this oneness easier We just do for over a and B so or it's false when both oh Prince are false and that happens only in the first row all the arrows will be true. So now So now let's boot left side and right side Together we have equivalents between them . So that means if value of left side this same as right side, we get true Otherwise force that if you look at a stable. You can see that negation a implies B is the same as say or be. So this formula will be true no matter what a and B are. So this formula is a tautology. Now I want you to try this on your own. With these two formulas, the 1st 1 have only negation in front off. Or so that means you do negation off the result off this or operation. And this formula should be contradiction. That means balls everywhere. And the 2nd 1 looks like this. So instead off or Derris end. And this one should be satisfied. Herbal. And it should be one zeros. You're one So tried these on your own. And if you get stuck, just posed the question. Now, from this first example, you might notice that it tells us something. This formula is tautology, and that means left side is equal to right side. And that means we can use negation and implication instead of our. And now I'm going to tell you a secret. We can use negation and implication instead off and to and instead of AC woolens to. So in order for us to write any formula. We need Onley, negation and implication. As you saw in the first example, negation A implies B is equivalent to A or B negation of a implies negation. Off B is equivalent to A and B and equivalents is a bit complicated. Is negation of a implies B that implies negation Off be imp, I say. And that is the same as if you write a is equivalent to be. You can try to prove it like I did. And first example it should always be tautology. Okay, that is pretty much it for this lecture. If you have any questions, feel free to ask. I suggest you play with these and if you get stuck, feel free to pose the question. I will be there to help you and I will soon exam.
4. Quantifiers: we have to quantification hours 1st 1 looks like this and it says that for every X is true , that's something where instead of something, you have some statement, obviously. And the 2nd 1 looks like this and it says, Dad, there exists a for which is true That's something. So if you think about it, the first quantification er is basically kind of total aji quantify gator for every X must be true and the 2nd 1 is basically set us viable quantification er, that means for at least one is true. So now let's take a look at some examples. 1st 1 says that for every X, that is really number. So this are represents set off all riel numbers like for example, 05 minus su and so on. And this euro kind of sign says that X is from this set. So no matter what number we pick from our for every number should be true that this number to the power of to is greater than zero. So feel free to think about it. But keep in mind that it must be true for every number that you can think off. And if you really think about it and Big zero SX, you will find out that zero to the power of to is still zero. And that means this is not true for every X. So, overall, this statement is false. Let's do another one. This one says that there exists X that is from riel numbers and it is greater than 1000. And there is because our represents all the numbers in our world. So if you pick for example 1,000,000,000 for X, you will get true. And since it's going to fire, says that it is true for at least one we are done. And this statement is true. And last one says that for every X is true that there exists, I data smaller and blow. These numbers are from our So if you think about it, if you pick any number in this world, you can always find smaller run. And if you think about it, no matter what X you pick. If you choose I to be equal to X minus one, it will always be smaller than X, right. Okay, that is bringing much it for dis lecture. Feel free to ask if you don't understand something. We will use these Quentin fires later on, and with that being said, I will soon exam