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<title>Extension (semantics)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Extension (semantics)</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Extension (logic)" redirects here. For another use in mathematical logic, see <a href="Conservative_extension" title="Conservative extension">Conservative extension</a>.</div>

<p>In any of several fields of study that treat the use of signs — for example, in <a href="Linguistics" title="Linguistics">linguistics</a>, <a href="Logic" title="Logic">logic</a>, <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Semantics" title="Semantics">semantics</a>, <a href="Semiotics" title="Semiotics">semiotics</a>, and <a href="Philosophy_of_language" title="Philosophy of language">philosophy of language</a> — the <b>extension</b> of a concept, idea, or sign consists of the things to which it applies, in contrast with its <a href="Comprehension_(logic)" title="Comprehension (logic)">comprehension</a> or <a href="Intension" title="Intension">intension</a>, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.

</p><p>In philosophical <a href="Semantics" title="Semantics">semantics</a> or the <a href="Philosophy_of_language" title="Philosophy of language">philosophy of language</a>, the 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy. Concepts and expressions of this sort are <a href="Monad_(Greek_philosophy)" class="mw-redirect" title="Monad (Greek philosophy)">monadic</a> or "one-place" concepts and expressions.

</p><p>So the extension of the word "dog" is the set of all (past, present and future) dogs in the world: the set includes Fido, Rover, <a href="Lassie" title="Lassie">Lassie</a>, Rex, and so on. The extension of the phrase "Wikipedia reader" includes each person who has ever read Wikipedia, including <i>you</i>.

</p><p>The extension of a whole statement, as opposed to a word or phrase, is defined (since <a href="Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>'s "<a href="On_Sense_and_Reference" class="mw-redirect" title="On Sense and Reference">On Sense and Reference</a>") as its <a href="Truth_value" title="Truth value">truth value</a>. So the extension of "Lassie is famous" is the logical value 'true', since Lassie is famous.

</p><p>Some concepts and expressions are such that they don't apply to objects individually, but rather serve to relate objects to objects. For example, the words "before" and "after" do not apply to objects individually—it makes no sense to say "Jim is before" or "Jim is after"—but to one thing in relation to another, as in "The wedding is before the reception" and "The reception is after the wedding". Such "relational" or "polyadic" ("many-place") concepts and expressions have, for their extension, the set of all sequences of objects that satisfy the concept or expression in question. So the extension of "before" is the set of all (ordered) pairs of objects such that the first one is before the second one.

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<div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Extension_(predicate_logic)" title="Extension (predicate logic)">Extension (predicate logic)</a></div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the 'extension' of a mathematical concept <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> that is specified by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</p><p>For example, the extension of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> is a set of <a href="Ordered_pair" title="Ordered pair">ordered pairs</a> that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>, such as a <a href="Group_(mathematics)" title="Group (mathematics)">group</a>, is the <a href="Underlying_set" class="mw-redirect" title="Underlying set">underlying set</a> of the object. The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the <a href="Axiom_of_extensionality" title="Axiom of extensionality">axiom of extensionality</a> in <a href="Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>.

</p><p>This kind of extension is used so constantly in contemporary mathematics based on <a href="Set_theory" title="Set theory">set theory</a> that it can be called an implicit assumption. A typical effort in mathematics evolves out of an observed <a href="Mathematical_object" title="Mathematical object">mathematical object</a> requiring description, the challenge being to find a <a href="Characterization_(mathematics)" title="Characterization (mathematics)">characterization</a> for which the object becomes the extension.

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<div class="mw-heading mw-heading2"><h2 id="Computer_science">Computer science</h2></div>

<p>In <a href="Computer_science" title="Computer science">computer science</a>, some <a href="Database" title="Database">database</a> textbooks use the term 'intension' to refer to the <a href="Logical_schema" title="Logical schema">schema</a> of a database, and 'extension' to refer to particular <a href="Row_(database)" title="Row (database)">instances</a> of a database.

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<div class="mw-heading mw-heading2"><h2 id="Metaphysical_implications">Metaphysical implications</h2></div>

<p>There is an ongoing controversy in <a href="Metaphysics" title="Metaphysics">metaphysics</a> about whether or not there are, in addition to actual, existing things, non-actual or nonexistent things. If there are—if, for instance, there are possible but non-actual dogs (dogs of some non-actual but possible species, perhaps) or nonexistent beings (like Sherlock Holmes, perhaps)—then these things might also figure in the extensions of various concepts and expressions. If not, only existing, actual things can be in the extension of a concept or expression. Note that "actual" may not mean the same as "existing". Perhaps there exist things that are merely possible, but not actual. (Maybe they exist in other universes, and these universes are other "<a href="Possible_worlds" class="mw-redirect" title="Possible worlds">possible worlds</a>"—possible alternatives to the actual world.) Perhaps some actual things are nonexistent. (Sherlock Holmes seems to be an <i>actual</i> example of a fictional character; one might think there are many other characters <a href="Arthur_Conan_Doyle" title="Arthur Conan Doyle">Arthur Conan Doyle</a> <i>might</i> have invented, though he actually invented Holmes.)

</p><p>A similar problem arises for objects that no longer exist. The extension of the term "Socrates", for example, seems to be a (currently) non-existent object. <a href="Free_logic" title="Free logic">Free logic</a> is one attempt to avoid some of these problems.

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<div class="mw-heading mw-heading2"><h2 id="General_semantics">General semantics</h2></div>

<p>Some fundamental formulations in the field of <a href="General_semantics" title="General semantics">general semantics</a> rely heavily on a valuation of extension over <a href="Intension" title="Intension">intension</a>. See for example extension, and the <a href="#External_links">extensional devices</a>.

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<ul><li><a href="Enumerative_definition" title="Enumerative definition">Enumerative definition</a></li>

<li><a href="Extensional_definition" class="mw-redirect" title="Extensional definition">Extensional definition</a></li>

<li><a href="Extensional_logic" class="mw-redirect" title="Extensional logic">Extensional logic</a></li>

<li><a href="Generalization" title="Generalization">Generalization</a></li>

<li><a href="Sense_and_reference" title="Sense and reference">Sense and reference</a></li>

<li><a href="Semantic_property" title="Semantic property">Semantic property</a></li>

<li><a href="Type%E2%80%93token_distinction" title="Type–token distinction">Type–token distinction</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://ontology.buffalo.edu/bfo/Terminology_for_Ontologies.pdf">Towards a Reference Terminology for Ontology Research and Development</a></li>

<li><a rel="nofollow" class="external text" href="http://esgs.free.fr/de/ext.htm">Extensional processes</a></li></ul>

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</style><div id="Formal_semantics_(natural_language)78" style="font-size:114%;margin:0 4em"><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Central concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Principle_of_compositionality" title="Principle of compositionality">Compositionality</a></li>

<li><a href="Denotation" title="Denotation">Denotation</a></li>

<li><a href="Linguistic_entailment" title="Linguistic entailment">Entailment</a></li>

<li><a href="Generalized_quantifier" title="Generalized quantifier">Generalized quantifier</a></li>

<li><a href="Intension" title="Intension">Intension</a></li>

<li><a href="Logical_form_(linguistics)" title="Logical form (linguistics)">Logical form</a></li>

<li><a href="Presupposition" title="Presupposition">Presupposition</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Reference" title="Reference">Reference</a></li>

<li><a href="Scope_(formal_semantics)" title="Scope (formal semantics)">Scope</a></li>

<li><a href="Speech_act" title="Speech act">Speech act</a></li>

<li><a href="Syntax%E2%80%93semantics_interface" title="Syntax–semantics interface">Syntax–semantics interface</a></li>

<li><a href="Truth-conditional_semantics" title="Truth-conditional semantics">Truth conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Areas</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anaphora_(linguistics)" title="Anaphora (linguistics)">Anaphora</a></li>

<li><a href="Ambiguity" title="Ambiguity">Ambiguity</a></li>

<li><a href="Binding_(linguistics)" title="Binding (linguistics)">Binding</a></li>

<li><a href="Conditional_sentence" title="Conditional sentence">Conditionals</a></li>

<li><a href="Definiteness" title="Definiteness">Definiteness</a></li>

<li><a href="Disjunction" class="mw-redirect" title="Disjunction">Disjunction</a></li>

<li><a href="Evidentiality" title="Evidentiality">Evidentiality</a></li>

<li><a href="Focus_(linguistics)" title="Focus (linguistics)">Focus</a></li>

<li><a href="Indexicality" title="Indexicality">Indexicality</a></li>

<li><a href="Lexical_semantics" title="Lexical semantics">Lexical semantics</a></li>

<li><a href="Modality_(semantics)" title="Modality (semantics)">Modality</a></li>

<li><a href="Negation" title="Negation">Negation</a></li>

<li><a href="Propositional_attitudes" class="mw-redirect" title="Propositional attitudes">Propositional attitudes</a></li>

<li><a href="Tense%E2%80%93aspect%E2%80%93mood" title="Tense–aspect–mood">Tense–aspect–mood</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantification</a></li>

<li><a href="Vagueness" title="Vagueness">Vagueness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antecedent-contained_deletion" title="Antecedent-contained deletion">Antecedent-contained deletion</a></li>

<li><a href="Cataphora" title="Cataphora">Cataphora</a></li>

<li><a href="Coercion_(linguistics)" title="Coercion (linguistics)">Coercion</a></li>

<li><a href="Conservativity" title="Conservativity">Conservativity</a></li>

<li><a href="Counterfactuals" class="mw-redirect" title="Counterfactuals">Counterfactuals</a></li>

<li><a href="Crossover_effects" title="Crossover effects">Crossover effects</a></li>

<li><a href="Cumulativity_(linguistics)" title="Cumulativity (linguistics)">Cumulativity</a></li>

<li><a href="De_dicto_and_de_re" title="De dicto and de re">De dicto and de re</a></li>

<li><a href="Deontic_modality" title="Deontic modality">Deontic modality</a></li>

<li><a href="Discourse_relation" title="Discourse relation">Discourse relations</a></li>

<li><a href="Donkey_anaphora" class="mw-redirect" title="Donkey anaphora">Donkey anaphora</a></li>

<li><a href="Epistemic_modality" title="Epistemic modality">Epistemic modality</a></li>

<li><a href="Exhaustivity" title="Exhaustivity">Exhaustivity</a></li>

<li><a href="Faultless_disagreement" title="Faultless disagreement">Faultless disagreement</a></li>

<li><a href="Free_choice_inference" title="Free choice inference">Free choice inferences</a></li>

<li><a href="Givenness" title="Givenness">Givenness</a></li>

<li><a href="Homogeneity_(linguistics)" class="mw-redirect" title="Homogeneity (linguistics)">Homogeneity (linguistics)</a></li>

<li><a href="Hurford_disjunction" title="Hurford disjunction">Hurford disjunction</a></li>

<li><a href="Inalienable_possession" title="Inalienable possession">Inalienable possession</a></li>

<li><a href="Intersective_modifier" title="Intersective modifier">Intersective modification</a></li>

<li><a href="Logophoricity" title="Logophoricity">Logophoricity</a></li>

<li><a href="Mirativity" title="Mirativity">Mirativity</a></li>

<li><a href="Modal_subordination" title="Modal subordination">Modal subordination</a></li>

<li><a href="Opaque_context" title="Opaque context">Opaque contexts</a></li>

<li><a href="Performative_utterance" title="Performative utterance">Performatives</a></li>

<li><a href="Polarity_item" title="Polarity item">Polarity items</a></li>

<li><a href="Privative_adjective" title="Privative adjective">Privative adjectives</a></li>

<li><a href="Quantificational_variability_effect" title="Quantificational variability effect">Quantificational variability effect</a></li>

<li><a href="Responsive_predicate" title="Responsive predicate">Responsive predicate</a></li>

<li><a href="Rising_declarative" title="Rising declarative">Rising declaratives</a></li>

<li><a href="Scalar_implicature" title="Scalar implicature">Scalar implicature</a></li>

<li><a href="Sloppy_identity" title="Sloppy identity">Sloppy identity</a></li>

<li><a href="Subsective_modifier" title="Subsective modifier">Subsective modification</a></li>

<li><a href="Subtrigging" title="Subtrigging">Subtrigging</a></li>

<li><a href="Telicity" title="Telicity">Telicity</a></li>

<li><a href="Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li>

<li><a href="Veridicality" title="Veridicality">Veridicality</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formalism</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Formal systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternative_semantics" title="Alternative semantics">Alternative semantics</a></li>

<li><a href="Categorial_grammar" title="Categorial grammar">Categorial grammar</a></li>

<li><a href="Combinatory_categorial_grammar" title="Combinatory categorial grammar">Combinatory categorial grammar</a></li>

<li><a href="Discourse_representation_theory" title="Discourse representation theory">Discourse representation theory (DRT)</a></li>

<li><a href="Dynamic_semantics" title="Dynamic semantics">Dynamic semantics</a></li>

<li><a href="Generative_grammar" title="Generative grammar">Generative grammar</a></li>

<li><a href="Glue_semantics" title="Glue semantics">Glue semantics</a></li>

<li><a href="Inquisitive_semantics" title="Inquisitive semantics">Inquisitive semantics</a></li>

<li><a href="Intensional_logic" title="Intensional logic">Intensional logic</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Mereology" title="Mereology">Mereology</a></li>

<li><a href="Montague_grammar" title="Montague grammar">Montague grammar</a></li>

<li><a href="Segmented_discourse_representation_theory" class="mw-redirect" title="Segmented discourse representation theory">Segmented discourse representation theory (SDRT)</a></li>

<li><a href="Situation_semantics" title="Situation semantics">Situation semantics</a></li>

<li><a href="Supervaluationism" title="Supervaluationism">Supervaluationism</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autonomy_of_syntax" title="Autonomy of syntax">Autonomy of syntax</a></li>

<li><a href="Context_set" class="mw-redirect" title="Context set">Context set</a></li>

<li><a href="Continuation" title="Continuation">Continuation</a></li>

<li><a href="Conversational_scoreboard" title="Conversational scoreboard">Conversational scoreboard</a></li>

<li><a href="Downward_entailing" title="Downward entailing">Downward entailing</a></li>

<li><a href="Existential_closure" title="Existential closure">Existential closure</a></li>

<li><a href="Function_application" title="Function application">Function application</a></li>

<li><a href="Meaning_postulate" title="Meaning postulate">Meaning postulate</a></li>

<li><a href="Monad_(functional_programming)" title="Monad (functional programming)">Monads</a></li>

<li><a href="Plural_quantification" title="Plural quantification">Plural quantification</a></li>

<li><a href="Possible_world" title="Possible world">Possible world</a></li>

<li><a href="Quantifier_raising" class="mw-redirect" title="Quantifier raising">Quantifier raising</a></li>

<li><a href="Quantization_(linguistics)" title="Quantization (linguistics)">Quantization</a></li>

<li><a href="Question_under_discussion" title="Question under discussion">Question under discussion</a></li>

<li><a href="Semantic_parsing" title="Semantic parsing">Semantic parsing</a></li>

<li><a href="Squiggle_operator" title="Squiggle operator">Squiggle operator</a></li>

<li><a href="Strawson_entailment" title="Strawson entailment">Strawson entailment</a></li>

<li><a href="Strict_conditional" title="Strict conditional">Strict conditional</a></li>

<li><a href="Type_shifter" title="Type shifter">Type shifter</a></li>

<li><a href="Universal_grinder" title="Universal grinder">Universal grinder</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cognitive_semantics" title="Cognitive semantics">Cognitive semantics</a></li>

<li><a href="Computational_semantics" title="Computational semantics">Computational semantics</a></li>

<li><a href="Distributional_semantics" title="Distributional semantics">Distributional semantics</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Formal grammar</a></li>

<li><a href="Inferentialism" class="mw-redirect" title="Inferentialism">Inferentialism</a></li>

<li><a href="Logic_translation" title="Logic translation">Logic translation</a></li>

<li><a href="Linguistics_wars" title="Linguistics wars">Linguistics wars</a></li>

<li><a href="Philosophy_of_language" title="Philosophy of language">Philosophy of language</a></li>

<li><a href="Pragmatics" title="Pragmatics">Pragmatics</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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