In this chapter we present the logical foundations for theories of bounded arithmetic. We introduce Gentzen's proof system LK for the predicate calculus, and prove that it is sound, and complete even when proofs have a restricted form called “anchored”. We augment the system LK by adding equality axioms. We prove the Compactness Theorem for predicate calculus, and the Herbrand Theorem.

In general we distinguish between syntactic notions and semantic notions. Examples of syntactic notions are variables, connectives, formulas, and formal proofs. The semantic notions relate to meaning; for example truth assignments, structures, validity, and logical consequence.

The first section treats the simple case of propositional calculus.

Propositional Calculus

Propositional formulas (called simply formulas in this section) are built from the logical constants ⊥, ⊤ (for False, True), propositional variables (or atoms) P1, P2, …, connectives ¬, ∨, ∧, and parentheses(,). We use P, Q, R, … to stand for propositional variables, A, B, C, … to stand for formulas, and Φ, Ψ, … to stand for sets of formulas. When writing formulas such as (P ∨ (Q ∧ R)), our convention is that P, Q, R, … stand for distinct variables.