Overview of Logic and Computation: Notes

We study formal logic as a mathematical tool for reasoning and as a medium for knowledge representation. The central notion is that of a consequence relation defined over a formal language of some kind. This is intended to capture the intuitive concept of valid inference or entailment, at least in that it provides a catalogue of valid argument forms. 1.1 Consequence relations At the most abstract level, we do not say what is in the language, except that it is a set of objects which we may call sentences or, more neutrally, formulae. A set of formulae (assumptions or premises) entails a single formula (conclusion) if there is no way all of the premises could hold without the conclusion holding as well. The relation of entailment, which we shall symbolise with a “turnstile” (`), is a consequence relation in the technical sense if and only if it satisfies three conditions. For all sets Γ, ∆ and for all formulae A, B: 1. If A ∈ Γ then Γ ` A