Long motivation. I'll go over the technical bits quickly. First our logic is a formal

language which we're going to define inductively on a bunch of ingredients.

What we have, we have individuals we can talk about, that's the new stuff, right?

Every number is even or so things we can say. We have properties of individuals

being even, we have relations between individuals, five is greater

than two, and we have functions on individuals sometimes which could be the

father function or the successor function or addition as a function, those

kind of things. Those are the ingredients of first order predicate logic. You can

do a surprising amount of math in it. There are certain things we cannot say.

We cannot say there's a subjective function from the natural numbers into

the reals, which you need even stronger logics for, but this logic has very good

properties. It has the property that we have complete and sound calculi and that

a procedure called unification which we're going to see presently, which is

kind of the motor behind prolog and Vesuvius theorem proving, has good

properties and so on. If you go higher in the expressivity scale, then

first of all logic, all those vanish, all those good properties vanish, which is a

pity. So we're going to stay with first of all logic. Another thing you can't say,

you can't quantify, we say, over functions and predicates and sets, which

is something you sometimes want to do in math, but you can't also give

the meaning of words like most, some, few, those kind of things. This is

wrong. At least three you can say. In first order logic, homework, think about how you

would do that. But what you can say, more of this than of that, which is really

what most does. If I say most students sleep, then that means that the

number of objects that are both students and are sleeping is greater than the

number of objects that are students but are awake. And for that you need to

count. And counting is not something you can do in first order logic.