 I'm Zor. Welcome to Unisor Education. Today's topic is logic, mathematical logic. Numbers, sets, equations, geometrical figures. There are many subjects mathematicians are talking about, but logic is the language which they're talking. This is how they're explaining their theories one to another, or published in the magazine. What is mathematical logic? Let's just examine certain concepts. It's just like learning another language basically. So the first concept which logic actually introduces to mathematics is a concept of true or false. We are saying that certain statements are true or false. That's basically the purpose of any scientific research. They take something and they prove that this is true or something which is actually false, doesn't really matter. So these two words are extremely important in this mathematical language which we call logic. Actually the purpose of any mathematical research is to prove that certain statement is true. Then somebody else takes this statement which has already been proven as true and derives some other statement a little bit more complex, derives using certain logical principles of derivation. We will talk about this principle a little later. But what's important is that from a true statement using a logical derivation mechanism we can come to another statement which by this derivation is proven to be true. And that's how the whole building is built. A couple of statements then somebody else takes them, comes to a conclusion using logical derivation to some other statements, etc. So that's how the whole building of mathematics is built. Alright, we understand how it grows but let's go down to the roots of this. How do I know that certain statement is true? Well as I was explaining most likely I derived this statement from previous statement which has already been proven true. But how that statement was proven to be true? Well there was something even more elementary. Alright, etc, etc. We don't want this process to be infinite. Well because if I want to explain that this statement is true I have to basically explain how it's derived. But if this process is infinite there is no way I can finish my explanation and find that number of hours. So logically we are coming to a certain conclusion that this derivation if we go backwards cannot really go indefinitely. It should stop somewhere and there are certain statements which we really cannot prove. Well, you can say that there are certain statements which are so obvious we don't have to prove them. Well, not so fast. There is nothing obvious in this world especially in mathematics. So let's just define it slightly differently. There are certain statements which we unfortunately have to accept for granted as a true statement just because we don't know how to prove them. There is nothing more elementary which we already know is true to derive from. So these statements which we unfortunately, unwillingly, kicking and screaming have to accept as true statements. And then having this as a foundation of our theory we can start building and derive other statements from them. So these first statements which we accept for granted as true statements are called axioms. Now having these axioms, maybe it's 1, maybe it's 10, maybe it's 1000, doesn't matter, having these statements in the foundation of our theory we can use logical derivation mechanism to come up with some other statements much more complex etc. So the geometry for instance was built in certain statements. The first person who really tried to put it in order was Greek mathematician Euclides and he had certain number of postulates as he said. But the axiom and postulate is synonyms. You can use them interchangeably. So he called them postulates and he basically derived the whole geometry, well he started at least to derive the whole geometry from these five initial postulates. It was the first attempt. Very successful by the way and it's not like 100% mathematically correct but there are very very important significant successes built on this. Alright so we have certain number of axioms which we unfortunately unwillingly accept as true statements and then we use logical derivation to prove other statements which we call theorems. Well obviously the number of axioms is usually very small, very limited. Like in case of Euclides he just came up with five absence and the number of theorems is extremely much more greater, it's extremely large and obviously everybody who is involved in mathematics, all they know is they're proving theorems one after another after another. Alright so basically the first statements are axioms, the next generation are theorems and we are using logical derivation. Now there is a term which mathematicians are using about logical derivation. It's called implication. Implication means basically that one particular statement is used to derive the other statement so we are saying that the first statement implies the second or this is the process of implication from the first statement to the second. Alright so these are just terms which we are using and let's try to do something with these true and false statements. There is an interesting concept of a function in mathematics. Function is something which you take something like an argument, you do something and you derive with some kind of a value which is a value of this function based on this particular argument. What's interesting is you can consider a set of all statements in the world, whatever anybody can say about anything and use these individual statements as arguments and the function which we actually take from it is the function which takes only two values, true or false. So for any statement we derive with a function value which is true or false. So you can say that logic is dealing with function which takes only two values, true or false, on the set of all the different statements. Actually it's not for every statement which this function is defined. There are certain statements which we cannot say whether it's true or false but let's not consider them right now maybe later. Right now we are considering only the statements which we can say about those statements whether it's true or false. Okay so let's say we have a statement S and let's say our function takes the value of true on this statement. For instance I'm standing here and talking right now. This is obviously a true statement and so what can we do with this function? Well first of all a function is defined on the statement S because always defined statement which is a negation of S. It's basically an applying of the word not. I am not standing here and talking to you. Well if I am standing and talking is true then the negation is false. This is just one of the rules of logic. If something is true its negation is false. That's how true and false are interconnected. Okay so we can apply one particular operation if you wish. It's an operation on a set of statements. From one statement we take an operation and take another statement. And by the way there is a great analogy. If you have a number 5 you can multiply it by minus 1 and get minus 5 which is negative 5. So operation of negation in logic is very much like multiplication by minus 1 in arithmetic. Great fine let's continue. What other operations we can do? With numbers for instance we can do addition or multiplication. Well what's interesting is we have a very good analogy here in logic. What is addition in numbers? We have two numbers and we get the third one. And we know it's value based on the values of these two. Same thing with logic. If you have two different statements S and G and whatever their value is. Let's say it's true and false. You can always come up with another statement which can be symbolically represented as S or G. I'm using the vertical bar as 4. So this is negation and this is logical or. These are operations. This is operation on one statement. This is operation on two statements. Well what's the value of a new statement which is a result of the logical or between S and G? Well let's think about. First of all let's talk about a concrete example. One statement which we have already spoken about is I'm standing here and talking to you. Another statement. Let's say the camera is taking my video. Those are true statements. If I'm connecting them with the word or I'm either standing here or the camera is taking the video. Well obviously if I'm standing and the camera is taking video then connecting by or will result in another statement which is also true. Let's just try to build a table actually. Of all the different values the operation or can present. So we are talking about operation or. If our arguments are both true the operation or applied to them will also result in a true statement. Correct? Either I'm standing here or the camera is taking picture. So the or is actually true as well. Now let's consider true and false connected with operation or. I'm either standing here or right now is two o'clock in the morning. And by the way right now is about ten o'clock in New York. So obviously the first statement is true. I'm standing here and talking and another statement is false that right now is two o'clock in the morning in New York. Well if I'm connecting them with an operator or either I'm standing here or this is two o'clock in the morning. Then since it's or and one of them is already a true statement obviously that the result will be a true statement. So the result of true or false is also true. Well similarly false true also true. And finally both statements are false like I'm not standing here right now and the camera is not really taking my picture. Now if I connect them with with operator or either I'm not standing here or the camera is not taking picture. Neither of these is true both are false so the result will be false. So this is our multiplication table if you wish. Remember we all had something like a multiplication table in school like 25 times 3 75 5 times 7 35 etc. So true or true is true. False or false is false. So basically this is our equivalent of a logical multiplication table. Alright so we have come up with this operator or any other operators yes. Another operator is I'm sure many of you have already figured out that there is an operator and in logic. I will use ampersand to signify. Alright now this is end and from the logical standpoint means that actually we are combining together those statements. Well it's quite obvious that if you combine true and true you will get the true statement. I'm standing here and right now is 10 o'clock in New York which is true. Then obviously this is a correct statement this is a true statement. Now obviously if you combine false together I'm not standing here and it's not 10 o'clock in New York then obviously you will get false statement. Maybe a little bit more complicated are the combination of true and false. Well let's just consider I am standing here and right now is 2 o'clock in the morning in New York and 2 o'clock in the morning. Because it's end it means that both have to be actually true. And right now we have only one of them so the result is false. Okay so we have introduced certain operations or operators in the set of all the different statements. And these operators help us to basically take two different statements and come up with the result of this operator. And we know what will be the value of the truthfulness for that other statement. That's very good that's what logic actually is using to derive the theorems. They have one particular statement which is true and then another statement they combine them together and they say hey this is true and this is true. Therefore this must be our result the end operator of these two statements should also be true. This is the process of derivation. Okay fine so we finished that we introduced our axioms and theorems and operators among statements. Alright so next thing is let's talk a little bit about implication and use some symbolics to basically write it down. Let's just consider some example of implication. Let's say I'm a man and all men are living beings so basically I am a living being. So the first is I am a man then all men are living beings. So this is statement X. This is statement Y. Obviously the new statement is basically the result of derivation from these statements. New statement is I am a living being. This is statement Z. Obviously statement Z is derived from X and Y. So how can it be described symbolically? Very simply I am a man and all men are living being. Let's start with this and this is the logical end. I am a man and all men are living being. It implies that I am a living being. It implies I will use this type of error as an implication sign. And that's basically a description of the fact that I am a living being is derived from logical ending between I am a man and all men are being. So this is the logical formula which basically represents a theory if you wish. That's the language. That's how people are describing what exactly they mean in mathematical logic. So let's talk about this operator if you wish or a relationship which we call implication. There are actually many different variations of this and I will consider four most important ones. Let's have this derivation. If I prove it, it means I have proved a theory. Whatever logical statements A and D are, if I prove this derivation, if I prove that A implies B, it means I have proven some theory. Now together with this kind of a theory there are some other theorems. Let's consider the following. Can it be proven? From A we derive B, but from B we can derive A as well. This by the way is called converse. This is a converse theorem to this one. Well, are there statements which are basically arranged like this? Yes, absolutely. They are equivalent statements. For instance, in geometry you can say that for instance there is a geometrical figure which consists of points and segments and they are connected to each other and there are only three points. And then there is another statement, the risk-particular figure which contains the same thing, but it has three sides. Well, three points or three sides, that seems to be kind of equivalent, right? And basically from one we can derive another. If there are three points and they are consecutively connected then there are three sides and vice versa in the loop I mean. So there are basically equivalent statements. And from one we can derive another, from another we can derive the other. But that's not necessarily true. There are some statements which you can prove the direct theorem but the converse theorem is just definitely not correct. Example, all men are living beings. The converse statement, all living beings are men. That's definitely incorrect. You have flowers, you have butterflies or whatever, they are not men and therefore the converse statement is not really true. Okay, another example is, and that's called inverse. I am not a man, therefore I am not a living being. Well, is it right? No, this is not correct. This is not correct. Again, flowers, butterflies, they are not men, but they are living being. So you cannot derive the statement that this is not a living being, you cannot derive it from a statement this is not a man. Because there are definitely other non-men but living beings. So again, if this is correct, the direct theorem is correct, the inverse theorem is not necessarily correct. And then the fourth variation is this. It's called, for the very long word, contra-positive. So this theorem which says in our example, if this is not a living being, it's not a man. Well, this seems to be a correct statement. So basically what I am saying is that this statement and this statement are basically related that if this is true, this is true as well. And by the way, these statements are also related in exactly the same fashion. And we can prove that. Alright, so we have introduced together with the direct theorem, a converse theorem, when we basically reverse the places of from and to. Inverse, when we are using negation on both sides. And contra-positive when we do both, we reverse the direction of derivation and we use the negation. And the last but not least, I would like to talk about connection between logic and set theory. In the set theory, if you remember, we were using something like unions and intersections between the different sets. Well, let me just say that there is a direct analogy between logical operation or, which is bar, and union in the set operation. Between logical and and intersection and intercept. And between logical negation and complement. Well, you are complementing one object, one particular set to a bigger set. Okay, let's consider each one of these in succession. So I will basically go into the details and explain what. And, and intersection. Intersection, if you remember, between two sets is this. Well, if I'm telling that a point belongs to a set A, and then I'm saying that the point belongs to the set B. What is intersection? It's a statement which says the point belongs to the set A and, logical and, the point belongs to the set D. Similarly, what is a union? Union is the combination of this. Obviously, from the logical standpoint, a statement, a point, an element belongs to the A, or the element belongs to the B. This is logical, or is equivalent to the union. And finally, negation, if you have a bigger set and a smaller set. So this is A, and this is some kind of a big set, which we can use Z, let's say, which includes A. If I'm saying an element does not belong to A, it means it belongs to everything around. And that's what complement actually is. So this is an exact correlation, exact connection between these two series. And finally, just to make a little bit, maybe to introduce some fun in this dry mathematical lecture. Let me just tell you that, if you remember in the beginning I told that not every statement can be qualified as true or false. And it means that this function, logical function which has values true or false, is not really defined on any statement whatsoever. There are statements for which we cannot really say whether they're true or false. Let me give you an example. What I am saying now is false. So whatever I am saying right now is false. Think about this statement. It cannot be true because the statement says that I am telling false, so it cannot be true. But it cannot be false because it means that the opposite statement will be true. Think about this yourself, and basically you'll come up to this conclusion that we cannot really say anything about this particular statement, whether it's true or false. Well, that's it for today. That's it for mathematical logic. Thank you very much.