Chapter 6: Proof Theory

 We've worked through a substantial bit of logic, and we're finally just getting to proofs.  In the material covered so far, we've emphasized the idea that logic is about classifying arguments and evaluating the semantic status of propositions, and we've done that without saying anything about proofs or about proving anything. We'll see in the next section that there's a good reason for this procedure. 

The syntax of propositional logic specifies the sentences of the logical language. Semantics enables us evaluate arguments as valid or invalid.  Proof theory is a third branch of logic, and it provides rules for inferring sentences from other sentences.  It may not be immediately obvious how proof theory differs from semantics, so consider this: When we do semantics, we already have arguments with premises and conclusions. Our truth table rules, together with our definition of validity, enable us to evaluate those arguments and classify them as valid or invalid.  By contrast, In proof theory, we are given just a set of initial premises, and we ask: "What follows from those premises?"  or alternatively, we're given a set of premises, and a conclusion, and we ask: "What set of inferences enable us to reach the conclusion from those premises?"  Proof theory is a set of rules, called "rules of inference" which specify the allowable inferences we can make to reach a conclusion from a set of premises.