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Logical equivalence

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	Topics Logical equivalence Topic Home In logicLogic Logic, from Classical Greek ?????, originally meaning the word, or what is spoken, is most often said to be the stud... , statements p and q are logically equivalent if they have the same logical content. SyntacticallySyntax (logic) In logic, syntax is a systematic statement of the rules governing the properly formed formulas of a logical system.... , p and q are equivalent if each can be proved from the other. Semantically, p and q are equivalent if they have the same truth value in every model. Logical equivalence is often confused with material equivalence. The former is a statement in the metalanguageMetalanguage In logic and linguistics, a metalanguage is a language used to make statements about other languages.... , claiming something about statements p and q in the object languageFormal system In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or tr... . But the material equivalence of p and q (often written "p ? q") is itself another statement in the object language. There is a relationship, however; p and q are syntactically equivalent if and only if p ? q is a theoremTheorem Overview A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions.... , while p and q are semantically equivalent if and only ifIf and only if In logic and fields that rely on it, such as mathematics and philosophy, "if and only if" is a logical connective between s... p ? q is a tautologyTautology (logic) In logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of ... . The logical equivalence of p and q is sometimes expressed as p = q or p ? q. However, these symbols are also used for material equivalence; the proper interpretation depends on the context. Example The following statements are logically equivalent: If Lisa is in FranceFrance France, officially the French Republic, is a country whose metropolitan territory is located in Western Europe and whi... , then she is in EuropeEurope Europe is one of the seven traditional continents of the Earth.... . (In symbols, f ? e.) If Lisa is not in Europe, then she is not in France. (In symbols, ~e ? ~f.) Syntactically, (1) and (2) are co-derivable via the rules of contrapositionContraposition In logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for i... and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in France is false or Lisa is in Europe is true. (Note that in this example classical logicClassical logic Classical logic identifies a class of formal logics that have been most intensively studied and most widely used.... is assumed. Some non-classical logics do not deem (1) and (2) logically equivalent.) See also Logical biconditionalLogical biconditional Summary In logic and mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only... Logical equalityLogical equality Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in p... If and only ifIf and only if In logic and fields that rely on it, such as mathematics and philosophy, "if and only if" is a logical connective between s... EquisatisfiabilityEquisatisfiability In logic, two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in oth...