Predicate Logic

Abstract

In propositional logic we used large chuncks of mathematical language, namely those parts that can have a truth value. Unfortunately this use of language is patently insufficient for mathematical practice. A simple argument, such as “all squares are positive, 9 is a square, therefore 9 is positive” cannot be dealt with. From the propositional point of view the above sentence is of the form φ Λ ψ → σ, and there is no reason why this sentence should be true, although we obviously accept it as true. The moral is that we have to extend the language, in such a way as to be able to discuss objects and relations. In particular we wish to introduce means to talk about all objects of the domain of discourse, e.g. we want to allow statements of the form “all even numbers are a sum of two odd primes”. Dually, we want a means of expressing “there exists an object such that ....”, e.g. in “there exists a real number whose square is 2”.