Proof Strategies

Abstract

This chapter introduces some elementary proof strategies for deriving theorems in propositional and predicate logic from assumptions, and these strategies are illustrated on a number of example proofs. A general distinction is made between strategies that introduce a certain logical connective into a proof and those that eliminate a connective from a proof; the former is used to derive a proof goal in a compositional way from simpler ones, while the latter is used to decompose an assumption or statement into simper ones on the way to deriving a goal. The strategies include well known principles such as modus ponens, modus tollens, proof by contradiction, contraposition and case analysis. Proof examples examined include Pythagoras’ proof that the square root of two is irrational, Euclid’s fundamental theorem of arithmetic, a proof that there are infinitely many prime numbers, and a non-constructive argument that one irrational number raised to the power of another can be rational.