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In logic, statements p and q are logically equivalent if they have the same logical content. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Syntactically, 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. Syntax in logic is a systematic statement of the rules governing the properly formed formulas (WFFs) of a logical system. ...
Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...
In general, semantics (from the Greek semantikos, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ...
In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...
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. 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, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...
Metalanguage in linguistics is a language used to make statements about language (the object language). ...
In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
It has been suggested that this article or section be merged with Logical biconditional. ...
Tautology refers to a use of redundant language in speech or writing, or, put simply, saying the same thing twice. Within the study of logic, a tautology is a statement that is true by its own definition. ...
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. (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. 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. Europe is conventionally considered one of the seven continents of Earth which, in this case, is more a cultural and political distinction than a physiographic one, leading to some dispute as to Europes actual borders. ...
In traditional logic, contraposition is a form of immediate inference in which from a given categorical proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality (affirmation or negation). ...
A double negative occurs when two or more ways to express negation are used in the same sentence. ...
(Note that in this example classical logic is assumed. 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. ...
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...
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