Logical equivalence - Citizendia

In logic, statements p and q are logically equivalent if they have the same logical content. Logic is the study of the principles of valid demonstration and Inference.

Syntactically, p and q are equivalent if each can be proved from the other. In Logic, syntax comprises the rules governing the composition of texts in a Formal language that constitute the properly formed formulas (WFFs of a logical Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Semantically, p and q are equivalent if they have the same truth value in every model. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Logical equivalence is often confused with material equivalence. ↔ The former is a statement in the metalanguage, claiming something about statements p and q in the object language. In Logic and Linguistics, a metalanguage is a Language used to make statements about statements in another language which is called the Object In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist 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 theorem, while p and q are semantically equivalent if and only if p ↔ q is a tautology. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements ↔ In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation

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 France, then she is in Europe. This article is about the country For a topic outline on this subject see List of basic France topics. (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 contraposition and double negation. For contraposition in the field of traditional logic see Contraposition (traditional logic. A double negative occurs when two forms of Negation are used in the same sentence. 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 logic is assumed. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used Some non-classical logics do not deem (1) and (2) logically equivalent. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used )

Example

See also