Logical Inference

This book is about mathematical logic, but so far the discussion has focused only on first-order logic as a formal language in which properties of structures can be expressed. Finding a language that can serve this purpose was just the first and not an insignificant step towards realization of Hilbert’s program. This chapter is about the second step: formalizing the notion of logical consequence. We will describe the model-theoretic approach, which allows us to give short proofs of several major results by bypassing the necessary technicalities of proof theory, but first a few words must be said about the proof-theoretic account.