In any of several fields of study that treat the use of signs — for example, in

linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy Philosophy (f ...

semiotics Semiotics (also called semiotic studies) is the systematic study of sign processes ( semiosis) and meaning making. Semiosis is any activity, conduct, or process that involves signs, where a sign is defined as anything that communicates something ...

, and

philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...

— an extensional context (or transparent context) is a syntactic environment in which a sub-sentential expression ''e'' can be replaced by an expression with the same

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...

and without affecting the

truth-value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progra ...

of the sentence as a whole. Extensional contexts are contrasted with

opaque context An opaque context or referentially opaque context is a linguistic context in which it is not always possible to substitute "co-referential" expressions (expressions referring to the same object) without altering the truth of sentences. The expres ...

s where truth-preserving substitutions are not possible. Take the case of

Clark Kent Superman is a superhero who appears in American comic books published by DC Comics. The character was created by writer Jerry Siegel and artist Joe Shuster, and debuted in the comic book ''Action Comics'' #1 (cover-dated June 1938 and publish ...

, who is secretly

Superman Superman is a superhero who appears in American comic books published by DC Comics. The character was created by writer Jerry Siegel and artist Joe Shuster, and debuted in the comic book ''Action Comics'' #1 (cover-dated June 1938 and publi ...

. Suppose that

Lois Lane Lois Lane is a fictional character appearing in American comic books published by DC Comics. Created by writer Jerry Siegel and artist Joe Shuster, she first appeared in ''Action Comics'' #1 (June 1938). Lois is an award-winning journalist for ...

fell out of a window and Superman caught her. Thus the sentence "Superman caught Lois Lane" is true. Because this sentence is an extensional context, the sentence "Clark Kent caught Lois Lane" is also true. Anybody that Superman caught, Clark Kent caught. In opposition to extensional contexts are intensional contexts (which can involve modal operators and

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

), where terms cannot be substituted without potentially compromising the truth-value. Suppose that Lois Lane believes that Clark Kent will investigate a news story with her. Thus, the sentence "Lois Lane believes that Clark Kent will investigate a news story with her" is true. However, the statement, "Lois Lane believes that Superman will investigate a news story with her," is false. This is because 'believes' typically induces an intensional context. Lois Lane doesn't believe that Superman is Clark Kent and the

propositional attitude A propositional attitude is a mental state held by an agent toward a proposition. Linguistically, propositional attitudes are denoted by a verb (e.g. "believed") governing an embedded "that" clause, for example, 'Sally believed that she had won ...

"believe" induces an intensional context, so the substitution alters the meaning of the original sentence.

See also

Further reading