Predicate logic. Propositional logic is about simple statements, like the statement, Socrates, Is a man. Predicate logic is logic involving statements like for all or they exist. So the statement all men are mortal. Now, these two statements are different. The first statement, assuming that everyone knows who Socrates is, just says something about this individual Socrates. The second statement says something about all men. So the second statement is about a universe of discourse, in this case, omega, all men. And it says something about every instance of this universe of discourse. In this case, all men. So every member of this universe has a certain property. So this is the universe of discourse. We find several universes of discourse. In mathematics, it's often the sets like the set of real numbers, the set of integer numbers; collection of object, like all triangles. In the real world we talk about men, about mountains, about seas, about birds, for instance. Now, very famous proposition about birds was the following. And was the proposition that all swans are white. So there's the universal discourse, in this case comprising of all the swans. And the statement which was widely believed is that all swans are white. Until of course in the 17th century, someone went to Australia and found a black swan. So the negation of this statement would be there is a swan that is not white. So you see, in the first sentence we've got a quantifier saying something about all the members of the universe of discourse. And in the second sentence there's a different quantifier, saying that in the universe there exist a certain member with a certain property, in this case a swan that is not white. This is typical for predicate logic. The existence of these quantifiers and the, The duality between the quantify for all and varies. Now let's have a look at the details. So let's look at some more examples with quantifiers. I'll take as my universe of discourse the set of numbers 2, 3, 4, 5, etc. And we're excluding one for a reason. And I'm looking at a statement P(x) which now has a variable, Which says that, x is prime. Now remember, a number is prime if it cannot be written as the product of two smaller numbers and omega. So just to remind you, four can be written as two times two, is not prime, While five is prime. And so is two,.and so is three, etc. You see we're now looking at propositions, which are actually have a variable, and which can be true or false according to this variable. Now, for instance we could say that, for all x, px is true. All equivalently and a bit more shortly, for all x we have that px results. Well, this would be true if indeed every member of my universe of discourse omega would satisfy px so that if every number would hold that axis prime. But we already know that it's not true. This is not true for P(4) is false. Which means, There exists a number for at least but are more, such that Px is not true. Now we have one rule about the two quantifiers, and that's the following. For all x, P(x) is equivalent, it is not the case that there exists an x such that not P(x). And of course, equivalently, if there exists in x such that P(x) holds that is equivalent to it is not true that for all x, Not P(x) all. So this is the whole thing written out in symbols.
