A.4 Variables and predicates

So, mathematics is about propositions, implications, and their truth values. We have seen that we can formulate propositions and reason about pretty complicated expressions using our propositional calculus. However, the language we have built up so far is not suitable for propositions with variables.

For instance, think about the sentence

Because the truth value depends on x, this is not a well-formed proposition.

Sentences with variables are called predicates, and we denote them by emphasizing the dependence on their variables; for instance,

P (x) : x ≥ 0,

or

Each predicate has a domain from which its variables can be taken. You can think about a predicate P(x) as a function that maps its domain to the set {0,1}, representing its truth value. (Although, strictly speaking, we don’t have functions available as tools when defining the very foundation of our formal language. However, we are not philosophers or set theorists, so we don&...