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In propositional logic, tautology is either of two commonly used rules of replacement. [ 1 ] [ 2 ] [ 3 ] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs .
Many logicians in the early 20th century used the term 'tautology' for any formula that is universally valid, whether a formula of propositional logic or of predicate logic. In this broad sense, a tautology is a formula that is true under all interpretations, or that is logically equivalent to the negation of a contradiction.
In literary criticism and rhetoric, a tautology is a statement that repeats an idea using near-synonymous morphemes, words or phrases, effectively "saying the same thing twice". [ 1 ] [ 2 ] Tautology and pleonasm are not consistently differentiated in literature. [ 3 ]
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
A phonological rule is a formal way of expressing a systematic phonological or morphophonological process in linguistics.Phonological rules are commonly used in generative phonology as a notation to capture sound-related operations and computations the human brain performs when producing or comprehending spoken language.
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
Rules for the propositional sequent calculus LK, in Gentzen notation Syntactic proof systems, in contrast, focus on the formal manipulation of symbols according to specific rules. The notion of syntactic consequence, φ ⊢ ψ {\displaystyle \varphi \vdash \psi } , signifies that ψ {\displaystyle \psi } can be derived from φ {\displaystyle ...
A propositional proof system P is polynomially bounded (also called super) if every tautology has a short (i.e., polynomial-size) P-proof. If P is polynomially bounded and Q simulates P, then Q is also polynomially bounded. The set of propositional tautologies, TAUT, is a coNP-complete set. A propositional proof system is a certificate-verifier ...