This short note is about the predicate calculus with equality. Equality is a
special predicate. I assume that it always make sense to ask of two things
whether or not they are equal. For any other predicate P, we have to specify
the domains of the various inputs to the predicate.
If we have a proposition in the predicate calculus, we might want to know
whether our proposition is true or not. To know whether proposition P is
true, it su▯ces (in principle) to know all about all the predicates that occur
in the proposition. (Actually in addition to predicates, our proposition might
contain function such as +, which given two reals as inputs outputs not a
truth value but a real number but strictly speaking we have not discussed
how such functions work and we can always replace functions by predicates.
Instead of writing x+y = z or x+y < z, we write L(x;y;z) to mean x+y = z
and 9w(L(x;y;w) ^ w < z) to mean x + y < z. So I ignore such functions.)
To say I know about all about all predicates is to say that I know which legal
combinations of inputs to a predicate result in output true and which result
in output false. Once I know this I can in principle (in practice it might
take be forever to actually do the computation ) the truth value of any ▯rst
order predicate calculus proposition. (If I also modelled functions, I would
have to know everything about the input-output behavior of any function I
use.) What I know when I know all about all the predicates, I call a model.
And given a proposition p there might be some models that make p true
and others that make it false. If I just write some models that make p true
and others that make p false, I have not really fully speci▯ed in general the
meaning of p but