So, let's talk about propositions. A proposition, is a statement, that can be either true, or, false. Let's have an example. Let's assume that you face a door, and on the door there's a sign. And the sign says "In this room, there is a tiger". And it could be a matter of some interest to know whether this statement would be true or false. If it is true you probably would refrain from opening the door. If it's false, you might be interested to have a look what is behind the door. Now, in general propositions are often denoted by capital letters. And this is what we shall be doing. P.Q, etc. So, our example proposition "In this room, there is a tiger" we could denote P, and then, P can be either true, which we also write as P having the value 1 is true. And if P is false, then P has the value 0. This is often done in computer languages, but you could call one T for truth and zero, F, for false. Now the negation, we write it not P, of a proposition P, is obtained by prepending the phrase "it is not true that" to P. So, let's go to our example, P was the phrase "In this room there is a tiger", so "not P" would be the phrase "It is not true that in this room there is a tiger." And that is a bit of a more reassuring notice, isn't it? So if P is true, then "not P" is false, and vice versa. If P is false then "not P" would be true. Having these arrangement we have our first logical law. It's a very simple one but it says that, "not not P" is the same as P, so if you negate twice then you get P back again. And to prove this law we draw a little truth table. So here is P, P can have the values 1 for true, and 0 for false. "Not P" would be 0, because if P is 1, then "not P" is 0. And if P is 0, then "not P" is 1. "Not not P" reverses that again. And we see that the true values of P and the true values of "not not P" are the same. So these two propositions are equivalent but we haven't talked about equivalence yet. So two propositions, are equivalent, so the notation would be P is equivalent to Q if their truth values are same. This, again, we express by a truth table. And since we've got now two propositions, we write here the relation we want to define. And we've got here the proposition P and here the proposition Q. Now, P can be either true or false. And Q can be either true or false. And, over here, we will denote the value of the proposition P is equivalent to Q. So P is equivalent to Q if both P and Q are true or both P and Q are false. But it is not true if P is true and Q is false. And it's also not true if P is false and Q is true. So this is an exact way of saying what it means that P and Q are equivalent. If we have two propositions, let's say P and Q we can join them by two logical operations. And which are "and", and "or". As you see by the way and, and, or and most logical articles are so engrained in our language that it's rather hard to talk about them. This is why we need truth tables. So, "and" is in logical formalism often denoted by this small hat between P and Q. And the truth table of and is looking as follows, P is either true or false. Q is either true or false, and "P and Q" is only true if both Q and P are both true, otherwise, it is false. And the same way we have the truth table of "or", so that is denoted by this little wedge. And "P or Q", drawing the truth table, is true if either P or Q is true. So P is true or Q is true. But here, there's danger of confusion. For, there is also another way of saying "or", what logicians call the "exclusive or", and it turns out that the general population, when they use the word "or" some people use the "or" like I just explained, the logical "or", and others mean always the "exclusive or". So the "exclusive or" is not that often used, but if you use it, you should denote it like this, and it's written in the truth tables as follows. P, Q exclusive or, So it is true if P is true and Q is not, or Q is true and P is not, but not both. So let's write this down in words, "P exclusive or Q": P is true or Q is true, but not both, whereas the common meaning, the meaning of the logical "or" is P is true or Q is true or both are true. As I said, in general, you don't use the exclusive or, only use the logical or. Finally, we have by far the most important logical connection and that is the implication. And it's best also to start there with truth table. So P is either true or false, and Q is either true or false. And "P implies Q" means either P is false or P is true and Q is true. That is a bit odd, so let's try to rephrase that: "P implies Q" means that either P is false and the implication is always true or P is true and Q is true. That's what the truth table tells us. The implication is true if P is false, and if P is true, then the implication is true if Q is true as well. Now, let's look a real world example assume again that you face a door, and then we can discuss about the implication P: "you open the door", and Q: "you get eaten". Question is, assume that P implies Q, what does that mean? Then either, P is false, that is you do not open the door. Or P and Q are both true. That is, you open the door and you get eaten. So the implication is true in the following two cases. Either you leave the door alone, because then we can't find out whether the implication is true, or you open the door and you get eaten. The implication is false, if you open the door and you don't get eaten, by a tiger or whatever is behind the door. Now the implication is of course, fundamental to science. And what you do as a scientist, you want to establish that P implies Q, and if you then establish that P is true and this together should establish that Q is true as well. So you have to establish the what used to be called the antecedent and the implication and then you can have established the conclusion. But as you see, this is again a logical formula which would be of the form "(P implies Q) and P" implies Q. So we have to break that up in all kinds of little pieces, in order to be able to evaluate it. Now one of, we are going to treat these brackets in next lecture. But what we first can do is to see whether this implication P implies Q can be written differently. Fortunately it can because it's the same thing as writing "not P or Q". Why? Well, again the truth table. P implies Q has a truth table we just saw: if P's false just ones, and if Q is true and P is true 1, and otherwise 0. Not P or Q has a truth table Well if not P is true then it's true, and if Q is true then it's true as well, and you see that the truth tables are the same. So we have now established that P implies Q is equivalent to not P or Q.
