Just Supposing: Proof and Consequence

Abstract

In the last two chapters, we learned quite a lot about propositional and quantificational logic and in particular their relations of logical implication. In this chapter, we look at how simple implications may be put together to make a deductively valid argument or proof. At first glance, this may seem trivial: just string them together! But although it starts like that, it goes well beyond, and is indeed quite subtle.

We begin by looking at the easy process of chaining, which creates elementary derivations, and show how its validity is linked with the Tarski conditions defining consequence relations/operations. We then review several higher-level proof strategies used in everyday mathematics and uncover the logic behind them. These include the strategies traditionally known as conditional proof, disjunctive proof and proof by cases, proof by contradiction and argument to and from an arbitrary instance. Their analysis leads us to distinguish second-level from split-level rules, articulate their recursive structures and explain the informal procedure of flattening a split-level proof into its familiar ‘suppositional’ form.