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

linguistics Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...

semiotics Semiotics ( ) is the systematic study of sign processes and the communication of meaning. In semiotics, a sign is defined as anything that communicates intentional and unintentional meaning or feelings to the sign's interpreter. Semiosis is a ...

, and

philosophy of language Philosophy of language refers to the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy), me ...

— 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 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''). Truth values are used in c ...

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 created by writer Jerry Siegel and artist Joe Shuster, which first appeared in the comic book ''Action Comics'' Action Comics 1, #1, published in the United States on April 18, 1938.The copyright date of ''Action Comics ...

, who is secretly

Superman Superman is a superhero created by writer Jerry Siegel and artist Joe Shuster, which first appeared in the comic book ''Action Comics'' Action Comics 1, #1, published in the United States on April 18, 1938.The copyright date of ''Action Comics ...

. 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'' Action Comics 1, #1 (June 1938). Lois is an award-winning ...

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 kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...

), 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 or organism toward a proposition. In philosophy, propositional attitudes can be considered to be neurally realized, causally efficacious, content-bearing internal states (personal princip ...

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

See also

Further reading