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Tautology (logic)

Definition

In mathematical logic, a tautology is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. It is a logical truth. For example, a formula that states "the ball is green or the ball is not green" is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic.

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AND gate Abductive reasoning Abstract logic Academic Press Ackermann set theory Aleph number Alfred Tarski Algebraic logic Algebraic normal form Alphabet (formal languages)Ampersand Analytic–synthetic distinction Ancient Greek language Antecedent (logic)Antinomy Argument Argumentation theory Arity Arrow (symbol)Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bertrand Russell Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean domain Boolean function Boolean satisfiability problem Boolean valuation Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Class (set theory)Classical logic Co-NP-complete Codomain Compactness theorem Complement (set theory)Complete theory Completeness (logic)Computability theory Computable function Computable set Computably enumerable set Concrete category Conjunctive normal form Consequent Conservative extension Consistency Constructible universe Construction of the real numbers Constructive set theory Contingency (philosophy)Continuum hypothesis Contradiction Contraposition Converse (logic)Converse nonimplication Countable set Critical thinking D. 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(natural language)Formal system Formation rule Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Functional predicate Fuzzy set General set theory Gottlob Frege Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hans Reichenbach Harcourt (publisher)Henri Poincaré Herbert Enderton Hereditary set Higher-order logic Hilbert's axioms Hilbert system History of logic History of mathematical logic Horseshoe (symbol)Hypothetical syllogism IMPLY gate If and only if Image (mathematics)Immanuel Kant Inaccessible cardinal Independence (mathematical logic)Index of logic articles Inductive reasoning Inference Infinite-valued logic Infinite set Informal logic Information theory Inhabited set Injective function Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionistic logic Inverter (logic gate)Isomorphism Józef Maria Bocheński Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Law of excluded middle Law of identity Lemma (mathematics)Lindström's theorem Linguistic entailment List of Boolean algebra topics List of Hilbert systems List of axioms List of fallacies List of first-order theories List of formal systems List of logic symbols List of logicians List of mathematical logic topics List of mathematical theories List of paradoxes List of rules of inference List of set identities and relations List of set theory topics List of statements independent of ZFC Logic Logic in computer science Logic optimization Logic synthesis Logical NOR Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical form Logical graph Logical truth Logicism London Ludwig Wittgenstein Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Material conditional Material nonimplication Mathematical logic Mathematical object Metalanguage Metalogic Metamathematics Minimal axioms for Boolean algebra Model complete theory Model theory Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory NAND gate NIMPLY gate NOR gate NP-complete NP (complexity)Naive set theory Name Natural deduction Necessity and sufficiency Negation New Foundations New York City Non-Euclidean geometry Non-classical logic Non-logical symbol Non-standard model Non-standard model of arithmetic OR gate Open formula Operation (mathematics)Ordinal analysis Ordinal number Outline of logic P (complexity)P versus NP problem Paradox Paradoxes of set theory Partition of a set Peano axioms Pejorative Philosophical logic Philosophy of logic Philosophy of mathematics Polynomial-time algorithm Power set Predicate (mathematical logic)Predicate logic Predicate variable Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Probability Proof by cases Proof of impossibility Proof theory Proper subset Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Quantification (logic)Quantifier (logic)Quantifier rank Reason Recursion Recursive set Reductio ad absurdum Reference Relation (mathematics)Reverse mathematics Rhetoric Robinson arithmetic Rule of inference Russell's paradox SAT solver Satisfiability Satisfiability and validity Saturated model Schröder–Bernstein theorem Science and Hypothesis Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent calculus Set (mathematics)Set theory Sheffer stroke Signature (logic)Singleton (mathematics)Skolem arithmetic Soundness South Holland Spectrum of a sentence Spectrum of a theory Square of opposition Statement (logic)Stephen Kleene Strength (mathematical logic)String (formal languages)Structure (mathematical logic)Subset Substitution (logic)Substitution instance Substructure (mathematics)Supertask Surjective function Syllogism Symbol (formal)Syntax (logic)Synthetic truth T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautological consequence Tautology (disambiguation)Tautology (rhetoric)Tee (symbol)Term (logic)Term logic Theorem Theories of truth Theory (mathematical logic)Therefore sign Three-valued logic Tilde Timeline of mathematical logic Tractatus Logico-Philosophicus Transfer principle Transitive set Triple bar True (logic)True arithmetic Truth Truth function Truth predicate Truth table Truth tables Truth value Turing machine Turned A Turnstile (symbol)Type (model theory)Type theory Ultrafilter (set theory)Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Up tack Urelement Vacuous truth Validity (logic)Variable (mathematics)Vel (symbol)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Wedge (symbol)Well-formed formula XNOR gate XOR gate Zermelo–Fraenkel set theory

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