Soundness

A deductive system is sound when every statement it proves is true in every model of its premises. Soundness ensures that proof rules do not derive conclusions that fail to follow semantically.

Deductive system: A formal language together with axioms and inference rules for deriving statements. Soundness evaluates whether this system's derivations preserve truth.
Soundness theorem: A theorem stating that every formula derivable in a specified logical system is valid under its semantics. It formalizes the exact guarantee captured by soundness.
Completeness theorem: A theorem stating that every semantically valid formula is derivable in a specified logical system. Completeness reverses soundness's direction, from semantic validity to provability.
Type soundness: A property of a programming language's type system ensuring well-typed programs avoid specified runtime errors. It applies soundness-style reasoning to typing judgments and program behavior.
Formal proof: A finite sequence of statements, each justified by an axiom or an inference rule. Soundness constrains what conclusions the system's proofs can establish.
Induction on proofs: A proof technique that establishes a property for all derivations by treating axioms and inference steps. It carries truth preservation from individual proof steps to complete derivations.
Gödel's completeness theorem: The theorem that first-order logic proves every sentence true in all models. It pairs first-order logic's completeness with its separate soundness guarantee.
SAT solver: A program that decides whether a propositional formula has a satisfying truth assignment. A sound solver's reported conclusions must be justified by its formal reasoning.
Model theory: The study of mathematical structures that interpret formal languages and satisfy theories. It supplies the models against which proved statements are checked.
Truth preservation: The property that an inference step yields a true conclusion whenever its premises are true. Soundness follows when every rule preserves truth in every interpretation.

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