# Definite description

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Definite_description
> Markdown URL: https://mediated.wiki/source/Definite_description.md
> Source: https://en.wikipedia.org/wiki/Definite_description
> Source revision: 1354288853
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Denoting phrase in the form of "the X"

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (June 2016) (Learn how and when to remove this message)

In [formal semantics](/source/Formal_semantics_(natural_language)) and [philosophy of language](/source/Philosophy_of_language), a **definite description** is a [denoting](/source/Denotation) [phrase](/source/Phrase) in the form of "the X" where X is a noun-phrase or a singular common [noun](/source/Noun). The definite description is *proper* if X applies to a unique individual or object. For example: "[the first person in space](/source/Yuri_Gagarin)" and "[the 42nd President of the United States of America](/source/Bill_Clinton)" are proper. The definite descriptions "the person in space" and "the man from Ohio" are *improper* because the noun phrase X applies to more than one thing, and the definite descriptions "the first man on Mars" and "the man from [Jupiter](/source/Jupiter)." are *improper* because X applies to nothing. Improper descriptions raise some difficult questions about the [law of excluded middle](/source/Law_of_excluded_middle), [denotation](/source/Denotation), [modality](/source/Linguistic_modality), and [mental content](/source/Mental_content).

## Russell's analysis

Main article: [Theory of descriptions](/source/Theory_of_descriptions)

As [France](/source/France) is [currently a republic](/source/French_Fifth_Republic), it has no king. [Bertrand Russell](/source/Bertrand_Russell) pointed out that this raises a puzzle about the truth value of the sentence "The present King of France is bald."[1]

The sentence does not seem to be true: if we consider all the bald things, the present King of France is not among them, since there is [no present King of France](/source/List_of_French_monarchs). But if it is false, then one would expect that the [negation](/source/Negation) of this statement, that is, "It is not the case that the present King of France is bald", or its [logical equivalent](/source/Logical_equivalence), "The present King of France is not bald", is true. But this sentence does not seem to be true either: the present King of France is no more among the things that fail to be bald than among the things that are bald. We therefore seem to have a violation of the [law of excluded middle](/source/Law_of_excluded_middle).

Is it meaningless, then?[*[to whom?](https://en.wikipedia.org/wiki/Wikipedia:Avoid_weasel_words)*] One might suppose so (and some philosophers have)[*[who?](https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions)*] since "the present King of France" certainly does [fail to refer](/source/Failure_to_refer). But on the other hand, the sentence "The present King of France is bald" (as well as its negation) seem perfectly intelligible, suggesting that "the present King of France" cannot be meaningless.

Russell proposed to resolve this puzzle via his [theory of descriptions](/source/Theory_of_descriptions). A definite description like "the present King of France", he suggested, is not a [referring](/source/Reference) expression, as we might naively suppose, but rather an "incomplete symbol" that introduces [quantificational](/source/Quantifier_(logic)) structure into sentences in which it occurs. The sentence "the present King of France is bald", for example, is analyzed as a conjunction of the following three [quantified](/source/Quantifier_(logic)) statements:

1. there is an x such that x is currently King of France: ∃ x K x {\displaystyle \exists xKx} (using 'Kx' for 'x is currently King of France')

1. for any x and y, if x is currently King of France and y is currently King of France, then x=y (i.e. there is at most one thing which is currently King of France): ∀ x ∀ y ( ( K x ∧ K y ) → x = y ) {\displaystyle \forall x\forall y((Kx\land Ky)\rightarrow x=y)}

1. for every x that is currently King of France, x is bald: ∀ x ( K x → B x ) {\displaystyle \forall x(Kx\rightarrow Bx)} (using 'B' for 'bald')

More briefly put, the claim is that "The present King of France is bald" says that some x is such that x is currently King of France, and that any y is currently King of France only if y = x, and that x is bald:

        ∃
        x
        (
        (
        K
        x
        ∧
        ∀
        y
        (
        K
        y
        →
        y
        =
        x
        )
        )
        ∧
        B
        x
        )

    {\displaystyle \exists x((Kx\land \forall y(Ky\rightarrow y=x))\land Bx)}

This is *false*, since it is *not* the case that some x is currently King of France.

The negation of this sentence, i.e. "The present King of France is not bald", is ambiguous. It could mean one of two things, depending on where we place the negation 'not'. On one reading, it could mean that there is no one who is currently King of France and bald:

        ¬
        ∃
        x
        (
        (
        K
        x
        ∧
        ∀
        y
        (
        K
        y
        →
        y
        =
        x
        )
        )
        ∧
        B
        x
        )

    {\displaystyle \lnot \exists x((Kx\land \forall y(Ky\rightarrow y=x))\land Bx)}

On this disambiguation, the sentence is *true* (since there is indeed no x that is currently King of France).

On a second reading, the negation could be construed as attaching directly to 'bald', so that the sentence means that there is currently a King of France, but that this King fails to be bald:

        ∃
        x
        (
        (
        K
        x
        ∧
        ∀
        y
        (
        K
        y
        →
        y
        =
        x
        )
        )
        ∧
        ¬
        B
        x
        )

    {\displaystyle \exists x((Kx\land \forall y(Ky\rightarrow y=x))\land \lnot Bx)}

On this disambiguation, the sentence is *false* (since there is no x that is currently King of France).

Thus, whether "the present King of France is not bald" is true or false depends on how it is interpreted at the level of [logical form](/source/Logical_form): if the [negation](/source/Negation) is construed as taking wide scope (as in the first of the above), it is true, whereas if the negation is construed as taking narrow scope (as in the second of the above), it is false. In neither case does it lack a truth value.

So we do *not* have a failure of the [Law of Excluded Middle](/source/Law_of_Excluded_Middle): "the present King of France is bald" (i.e. ∃ x ( ( K x ∧ ∀ y ( K y → y = x ) ) ∧ B x ) {\displaystyle \exists x((Kx\land \forall y(Ky\rightarrow y=x))\land Bx)} ) is false, because there is no present King of France.

The negation of this statement is the one in which 'not' takes wide scope: ¬ ∃ x ( ( K x ∧ ∀ y ( K y → y = x ) ) ∧ B x ) {\displaystyle \lnot \exists x((Kx\land \forall y(Ky\rightarrow y=x))\land Bx)} . This statement is *true* because there does not exist anything which is currently King of France.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

## Generalized quantifier analysis

[Stephen Neale](/source/Stephen_Neale),[2] among others, has defended Russell's theory, and incorporated it into the theory of [generalized quantifiers](/source/Generalized_quantifier). On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The determiner 'the' has the following denotation (using [lambda](/source/Lambda_calculus) notation):

        λ
        f
        .
        λ
        g
        .
        ∃
        x
        (
        f
        (
        x
        )
        =
        1
        ∧
        ∀
        y
        (
        f
        (
        y
        )
        =
        1
        →
        y
        =
        x
        )
        ∧
        g
        (
        x
        )
        =
        1
        )

    {\displaystyle \lambda f.\lambda g.\exists x(f(x)=1\land \forall y(f(y)=1\rightarrow y=x)\land g(x)=1)}

(That is, the definite article 'the' denotes a function which takes a pair of [properties](/source/Property) f and g to truth [if, and only if](/source/If_and_only_if), there exists something that has the property f, only one thing has the property f, and that thing also has the property g.) Given the denotation of the [predicates](/source/Predicate_(mathematical_logic)) 'present King of France' (again K for short) and 'bald' (B for short)

        λ
        x
        .
        K
        x

    {\displaystyle \lambda x.Kx}

        λ
        x
        .
        B
        x

    {\displaystyle \lambda x.Bx}

we then get the Russellian truth conditions via two steps of [function application](/source/Function_application): 'The present King of France is bald' is true if, and only if, ∃ x ( ( K x ∧ ∀ y ( K y → y = x ) ) ∧ B x ) {\displaystyle \exists x((Kx\land \forall y(Ky\rightarrow y=x))\land Bx)} . On this view, definite descriptions like 'the present King of France' do have a denotation (specifically, definite descriptions denote a function from properties to truth values—they are in that sense not [syncategorematic](/source/Syncategorematic), or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.

## Fregean analysis

The Fregean analysis of definite descriptions, implicit in the work of [Frege](/source/Frege) and later defended by [Strawson](/source/P._F._Strawson)[3] among others, represents the primary alternative to the Russellian theory. On the Fregean analysis, definite descriptions are construed as [referring expressions](/source/Referring_expression) rather than [quantificational expressions](/source/Quantifier_(logic)). Existence and uniqueness are understood as a [presupposition](/source/Presupposition) of a sentence containing a definite description, rather than part of the content asserted by such a sentence. The sentence 'The present King of France is bald', for example, is not used to claim that there exists a unique present King of France who is bald; instead, that there is a unique present King of France is part of what this sentence *presupposes*, and what it *says* is that this individual is bald. If the presupposition fails, the definite description *fails to refer*, and the sentence as a whole fails to express a [proposition](/source/Proposition).

The Fregean view is thus committed to the kind of [truth value](/source/Truth_value) gaps (and failures of the [law of excluded middle](/source/Law_of_excluded_middle)) that the Russellian analysis is designed to avoid. Since there is currently no King of France, the sentence 'The present King of France is bald' fails to express a proposition, and therefore fails to have a truth value, as does its [negation](/source/Negation), 'The present King of France is not bald'. The Fregean will account for the fact that these sentences are nevertheless *meaningful* by relying on speakers' knowledge of the conditions under which either of these sentences *could* be used to express a true proposition. The Fregean can also hold on to a restricted version of the law of excluded middle: for any sentence whose presuppositions are met (and thus expresses a proposition), either that sentence or its negation is true.

On the Fregean view, the definite article 'the' has the following denotation (using [lambda](/source/Lambda_calculus) notation):

        λ
        f
        :
        ∃
        x
        (
        f
        (
        x
        )
        =
        1
        ∧
        ∀
        y
        (
        f
        (
        y
        )
        =
        1
        →
        y
        =
        x
        )
        )
        .

    {\displaystyle \lambda f:\exists x(f(x)=1\land \forall y(f(y)=1\rightarrow y=x)).}

 [The unique z such that

        f
        (
        z
        )
        =
        1

    {\displaystyle f(z)=1}

]

(That is, 'the' denotes a function which takes a property f and yields the unique object z that has property f, if there is such a z, and is undefined otherwise.) The presuppositional character of the existence and uniqueness conditions is here reflected in the fact that the definite article denotes a [partial function](/source/Partial_function) on the set of properties: it is only defined for those properties f which are true of exactly one object. It is thus undefined on the denotation of the predicate 'currently King of France', since the property of currently being King of France is true of no object; it is similarly undefined on the denotation of the predicate 'Senator of the US', since the property of being a US Senator is true of more than one object.

## Mathematical logic

Further information: [Uniqueness quantification](/source/Uniqueness_quantification) and [Epsilon calculus](/source/Epsilon_calculus)

Following the example of *[Principia Mathematica](/source/Principia_Mathematica)*, it is customary to use a definite description operator symbolized using the "turned" (rotated) Greek lower case iota character "℩". The notation ℩ x ( ϕ x ) {\displaystyle x(\phi x)} means "the unique x {\displaystyle x} such that ϕ x {\displaystyle \phi x} ", and

        ψ
        (

    {\displaystyle \psi (}

℩

        x
        (
        ϕ
        x
        )
        )

    {\displaystyle x(\phi x))}

is equivalent to "There is exactly one ϕ {\displaystyle \phi } and it has the property ψ {\displaystyle \psi } ":

        ∃
        x
        (
        ∀
        y
        (
        ϕ
        (
        y
        )

        ⟺

        y
        =
        x
        )
        ∧
        ψ
        (
        x
        )
        )

    {\displaystyle \exists x(\forall y(\phi (y)\iff y=x)\land \psi (x))}

## See also

- [Lambert's law (logic)](/source/Lambert's_law_(logic))

- [Philosophy of language](/source/Philosophy_of_language)

- [John Searle](/source/John_Searle)

- [Vacuous truth](/source/Vacuous_truth)

## References

1. **[^](#cite_ref-ondenoting_1-0)** Russell, Bertrand (1905). "On Denoting". *Mind*. **14** (4): 479–493. [doi](/source/Doi_(identifier)):[10.1093/mind/XIV.4.479](https://doi.org/10.1093%2Fmind%2FXIV.4.479).

1. **[^](#cite_ref-2)** Stephen Neale (1990). *Descriptions*. The MIT Press. [ISBN](/source/ISBN_(identifier)) [0262640317](https://en.wikipedia.org/wiki/Special:BookSources/0262640317).

1. **[^](#cite_ref-onreferring_3-0)** Strawson, Peter (1950). "On referring". *Mind*. **59** (235): 320–344. [doi](/source/Doi_(identifier)):[10.1093/mind/LIX.235.320](https://doi.org/10.1093%2Fmind%2FLIX.235.320).

## Bibliography

- [Donnellan, Keith](/source/Keith_Donnellan), "Reference and Definite Descriptions," in *[Philosophical Review](/source/The_Philosophical_Review)* 75 (1966): 281–304.

- Neale, Stephen, *Descriptions*, MIT Press, 1990.

- Ostertag, Gary (ed.). (1998) *Definite Descriptions: A Reader* Bradford, MIT Press. (Includes Donnellan (1966), Chapter 3 of Neale (1990), Russell (1905), and Strawson (1950).)

- Reimer, Marga and Bezuidenhout, Anne (eds.) (2004), *Descriptions and Beyond*, Clarendon Press, Oxford

- Russell, Bertrand, "[On Denoting](/source/On_Denoting)," in *[Mind](/source/Mind_(journal))* 14 (1905): 479–493. [Online text](http://www.fh-augsburg.de/~harsch/anglica/Chronology/20thC/Russell/rus_deno.html).

- Strawson, P. F., "On Referring," in *Mind* 59 (1950): 320–344.

## External links

- [Zalta, Edward N.](/source/Edward_N._Zalta) (ed.). ["Descriptions"](https://plato.stanford.edu/entries/descriptions/). *[Stanford Encyclopedia of Philosophy](/source/Stanford_Encyclopedia_of_Philosophy)*. [ISSN](/source/ISSN_(identifier)) [1095-5054](https://search.worldcat.org/issn/1095-5054). [OCLC](/source/OCLC_(identifier)) [429049174](https://search.worldcat.org/oclc/429049174).

v t e Analytic philosophy Related articles Areas of focus Metaphysics Epistemology Language Mathematics Science Turns Aretaic Historical Linguistic Performative Logic Classical Deviant Mathematical Non-classical Paraconsistent Philosophical Predicate Theories Anti-realism Causal theory of reference Descriptivism Emotivism Feminism Functionalism Logical atomism Logical positivism Marxism Neurophilosophy Ordinary language Pragmatism Quietism Scientific structuralism Sense data Analytic theology Analytical Thomism Concepts Analysis (paradox of analysis) Analytic–synthetic distinction Counterfactual Natural kind Reflective equilibrium Supervenience Modality Actualism Necessity Possibility Possible world Realism Rigid designator Philosophers Tyler Burge David Chalmers Noam Chomsky Keith Donnellan Gottlob Frege Edmund Gettier Jaakko Hintikka David Kaplan Ernst Mally Richard Montague Ernest Sosa Robert Stalnaker Barry Stroud Australian realism David Malet Armstrong J. L. Mackie Peter Singer J. J. C. Smart Cambridge C. D. Broad Casimir Lewy Norman Malcolm G. E. Moore Graham Priest Bertrand Russell Frank P. Ramsey C. L. Stevenson A. N. Whitehead Ludwig Wittgenstein Oxford G. E. M. Anscombe J. L. Austin Michael Dummett Antony Flew Philippa Foot Peter Geach Paul Grice R. M. Hare H. L. A. Hart Alasdair MacIntyre Derek Parfit W. D. Ross Gilbert Ryle John Searle P. F. Strawson Richard Swinburne Charles Taylor Bernard Williams Timothy Williamson Crispin Wright Logical positivists A. J. Ayer Ernest Nagel Berlin Circle Carl Gustav Hempel Hans Reichenbach Vienna Circle Rudolf Carnap Hans Hahn Otto Neurath Moritz Schlick Friedrich Waismann Harvard Roderick Chisholm Donald Davidson Daniel Dennett Nelson Goodman Christine Korsgaard Thomas Nagel Robert Nozick Hilary Putnam W. V. O. Quine John Rawls Notre Dame Robert Audi Peter van Inwagen Pittsburgh School Robert Brandom Patricia Churchland Paul Churchland Adolf Grünbaum John McDowell Ruth Millikan Alexander Pruss Wilfrid Sellars Bas van Fraassen Pragmatism Susan Haack Nicholas Rescher Morton White Princeton Alonzo Church Jerry Fodor Kurt Gödel David Lewis Jaegwon Kim Saul Kripke Richard Rorty Nathan Salmon Michael Walzer Quietism James F. Conant Alice Crary Cora Diamond Reformed William Lane Craig Alvin Plantinga Nicholas Wolterstorff Science Paul Feyerabend Thomas Kuhn Imre Lakatos Karl Popper Stanford School Nancy Cartwright John Dupré Peter Galison Ian Hacking Patrick Suppes Lwów–Warsaw Jan Łukasiewicz Alfred Tarski Category

v t e Philosophy of language Philosophers Confucius Gorgias Cratylus Plato Aristotle Eubulides Diodorus Chrysippus Zhuangzi Xunzi Averroes Ibn Khaldun Hobbes Wilkins Arnauld Lancelot Leibniz Berkeley Herder von Humboldt Mauthner Ricœur de Saussure Frege Boas Tillich Sapir Bloomfield Bergson Vygotsky Wittgenstein Russell Carnap Derrida Whorf Austin Chomsky Gadamer Kripke Ayer Anscombe Hintikka Dummett Davidson Grice Ryle Strawson Quine Putnam Lewis Searle Watzlawick Theories Causal theory of reference Contrast theory of meaning Contrastivism Conventionalism Cratylism Deconstruction Descriptivism Direct reference theory Dramatism Dynamic semantics Expressivism Inquisitive semantics Linguistic determinism Mediated reference theory Nominalism Non-cognitivism Phallogocentrism Relevance theory Semantic externalism Semantic holism Situation semantics Structuralism Supposition theory Symbiosism Theological noncognitivism Theory of descriptions Definite description Theory of language Verification theory Concepts Ambiguity Cant Class Concept Categories Family resemblance Intension Language Linguistic relativity Logical form Mental representation Metalanguage Modality (semantics) Presupposition Principle of compositionality Property Proposition Sense and reference Sentence Set Sign Speech act Statement Symbol Truth-bearer Use–mention distinction more... Works Cratylus (n.d.) Port-Royal Grammar (1660) De Arte Combinatoria (1666) An Essay Towards a Real Character, and a Philosophical Language (1668) Alciphron (1732) "On Denoting" (1905) Tractatus Logico-Philosophicus (1921) Language, Truth, and Logic (1936) Two Dogmas of Empiricism (1951) Philosophical Investigations (1953) Of Grammatology (1967) Naming and Necessity (1980) Wittgenstein on Rules and Private Language (1982) Limited Inc (1988) Related articles Analytic philosophy Philosophy of information Philosophical logic Linguistics Pragmatics Rhetoric Scholasticism School of Names Semantics Formal semantics General semantics Semiotics Category Task Force Discussion

v t e Formal semantics (natural language) Central concepts Compositionality Denotation Entailment Extension Generalized quantifier Intension Logical form Presupposition Proposition Reference Scope Speech act Syntax–semantics interface Truth conditions Topics Areas Anaphora Ambiguity Binding Conditionals Definiteness Disjunction Evidentiality Focus Indexicality Lexical semantics Modality Negation Propositional attitudes Tense–aspect–mood Quantification Vagueness Phenomena Antecedent-contained deletion Cataphora Coercion Conservativity Counterfactuals Crossover effects Cumulativity De dicto and de re De se Deontic modality Discourse relations Donkey anaphora Epistemic modality Exhaustivity Faultless disagreement Free choice inferences Givenness Homogeneity (linguistics) Hurford disjunction Inalienable possession Intersective modification Logophoricity Mirativity Modal subordination Opaque contexts Performatives Polarity items Privative adjectives Quantificational variability effect Responsive predicate Rising declaratives Scalar implicature Sloppy identity Subsective modification Subtrigging Telicity Temperature paradox Veridicality Formalism Formal systems Alternative semantics Categorial grammar Combinatory categorial grammar Discourse representation theory (DRT) Dynamic semantics Generative grammar Glue semantics Inquisitive semantics Intensional logic Lambda calculus Mereology Montague grammar Segmented discourse representation theory (SDRT) Situation semantics Supervaluationism Type theory TTR Concepts Autonomy of syntax Context set Continuation Conversational scoreboard Downward entailing Existential closure Function application Meaning postulate Monads Plural quantification Possible world Quantifier raising Quantization Question under discussion Semantic parsing Squiggle operator Strawson entailment Strict conditional Type shifter Universal grinder See also Cognitive semantics Computational semantics Distributional semantics Formal grammar Inferentialism Logic translation Linguistics wars Philosophy of language Pragmatics Semantics of logic

---
Adapted from the Wikipedia article [Definite description](https://en.wikipedia.org/wiki/Definite_description) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Definite_description?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.