The following is not intended as an introduction to classical logic, but rather as a review of the concepts and a setting of notation. We start with propositional calculus and then move to first-order logic. (We do the latter for completeness, but in fact first-order logic plays almost no role in this book.)

Propositional calculus

Syntax

Given a set P of propositional symbols, the set of sentences in the propositional calculus is the smallest set ℒ containing P such that if φ, ψ ∈ ℒ then also ¬φ ∈ ℒ and ∈ ∧ ψ ℒ. Other connectives such as ∨, →, and ≡ can be defined in terms of ∧ and ¬.

Semantics

A propositional interpretation (or a model) is a set M ⊂ P, the subset of true primitive propositions. The satisfaction relation ⊧ between models and sentences is defined recursively as follows.

• For any p ∈ P, M ⊧ p iff p ∈ M.

• M ⊧ φ ∧ ψ iff M ⊧ φ and M ⊧ ψ.

• M ⊧ ¬φ iff it is not the case that M ⊧ φ.

We overload the ⊧ symbol. First, it is used to denote validity; ⊧ φ means that φ is true in all propositional models. Second, it is used to denote entailment; φ ⊧ ψ means that any model that satisfies φ also satisfies ψ.