proofdetermining

Proofdetermining is the practice or study of establishing, for a given proposition, whether there exists a formal proof from the axioms of a specified theory, and, when possible, producing such a proof. It encompasses both theoretical questions about provability and practical methods to find proofs.

In formal logic, proofdetermining distinguishes syntactic provability from semantic truth. Some logics have decision procedures that determine whether a given statement is provable; in general first-order logic provability is not decidable, though semantic validity is linked to provability via completeness theorems. Gödel's incompleteness theorem shows that in any sufficiently strong, consistent theory, there are true statements that are unprovable within the theory, limiting the reach of proofdetermination. Conversely, in logics where the system is complete and constructive, provability and truth align for the statements in question.

Practices and methods include deductive calculi such as natural deduction, sequent calculus, and Hilbert systems, as well as automated techniques like resolution and tableaus. Automated theorem proving aims to implement proofsearch strategies, often with heuristics to prune search spaces. Decidability and complexity vary by logic: propositional logic is decidable; first-order logic is semi-decidable (a proof may be found if one exists, but proof search may run forever); higher-order logics are generally more complex and less tractable.

Applications include formal verification of software and hardware, formalized mathematics, and reasoning in artificial intelligence. Proof assistants such as Coq, Isabelle/HOL, and HOL Light assist in constructing and checking proofs within specific logical frameworks, while symbolic reasoning tools support automated proof discovery.

Related topics include proof theory, decidability, automated theorem proving, and proof assistants.