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Computer & Information Science
CIS 1166

Models 1 Models 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
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