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Weakening (logic)
In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgment (mathematical logic), judgment or sequents directly. Structural rules often mimic intended meta-theoretic properties of the logic. Logics that deny one or more of the structural rules are classified as substructural logics. Common structural rules Three common structural rules are: * Weakening, where the hypotheses or conclusion of a sequent may be extended with additional members. In symbolic form weakening rules can be written as \frac on the left of the Turnstile (symbol), turnstile, and \frac on the right. * Contraction, where two equal (or unifiable) members on the same side of a sequent may be replaced by a single member (or common instance). Symbolically: \frac and \frac. Also known as factoring in automated theorem proving systems using Resolution (logic), resolution. Known as idempotency of entailment in classical logic. * Exchange, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |