Editing Definite description (section)

==Fregean analysis==

The Fregean analysis of definite descriptions, implicit in the work of [[Frege]] and later defended by [[P. F. Strawson|Strawson]]<ref name=onreferring>{{Cite journal|last=Strawson|first=Peter|date=1950|title=On referring|journal=Mind|language=en|volume=59|issue=235|pages=320–344|doi=10.1093/mind/LIX.235.320}}</ref> among others, represents the primary alternative to the Russellian theory. On the Fregean analysis, definite descriptions are construed as [[referring expression]]s rather than [[Quantifier (logic)|quantificational expressions]].  Existence and uniqueness are understood as a [[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]].

The Fregean view is thus committed to the kind of [[truth value]] gaps (and failures of the [[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]], '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 calculus|lambda]] notation):

{{block indent|<math>\lambda f: \exists x(f(x)=1 \land \forall y(f(y)=1 \rightarrow y=x)).</math> [The unique z such that <math>f(z)=1</math>]}}

(That is, 'the' denotes a function which takes a property {{var|f}} and yields the unique object {{var|z}} that has property {{var|f}}, if there is such a {{var|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]] on the set of properties: it is only defined for those properties {{var|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.