It is always a good idea to be precise in mathematics. Sometimes though, we can relax a little bit, as long as we all agree on a convention. An example of such a convention is to assume that sentences containing predicates with free variables are intended as statements, where the variables are universally quantified.
For example, do you believe that if a shape is a square, then it is a rectangle? But how can that be true if it is not a statement? To be a little more precise, we have two predicates: \(S(x)\) standing for “\(x\) is a square” and \(R(x)\) standing for “\(x\) is a rectangle”. The sentence we are looking at is,
This is neither true nor false, as it is not a statement. But come on! We all know that we meant to consider the statement,
and this is what our convention tells us to consider.
Similarly, we will often be a bit sloppy about the distinction between a predicate and a statement. For example, we might write, let \(P(n)\) be the statement, “\(n\) is prime,” which is technically incorrect. It is implicit that we mean that we are defining \(P(n)\) to be a predicate, which for each \(n\) becomes the statement, \(n\) is prime.