Logical consequence

Logical consequence is the relation that holds between a set of premises and a conclusion when the conclusion must be true in any model where the premises are true. It is the central concept of logic: logic is not the study of truth but the study of what follows from what. The definition is due to Alfred Tarski, who characterized logical consequence as truth preservation across all interpretations.

This model-theoretic definition — often called semantic consequence (written Γ ⊨ φ) — contrasts with syntactic consequence (Γ ⊢ φ), defined as derivability in a proof system. A fundamental goal of foundations of mathematics is proving that these notions coincide: that what can be derived is exactly what must be true. Kurt Gödel's completeness theorem for first-order logic establishes this coincidence for the most important case; his incompleteness theorems show it fails for stronger systems.

The concept of logical consequence is not merely technical. It underwrites the normativity of reasoning: if you accept the premises and the argument is valid, you are rationally compelled to accept the conclusion. Whether this compulsion is psychological, metaphysical, or purely conventional is one of the enduring disputes in philosophy of logic. The model-theoretic account grounds consequence in structural preservation; the proof-theoretic account grounds it in rule-following. That these accounts agree in first-order logic but diverge elsewhere suggests that logical consequence may not be a single, unified phenomenon.