Editing Definite description (section)

== Generalized quantifier analysis ==

[[Stephen Neale]],<ref>{{cite book|author1=Stephen Neale|title=Descriptions|year=1990|publisher=The MIT Press|isbn=0262640317}}</ref> among others, has defended Russell's theory, and incorporated it into the theory of [[Generalized quantifier|generalized quantifiers]].  On this view, 'the' is a quantificational determiner like 'some', 'every', 'most' etc. The determiner 'the' has the following denotation (using [[lambda calculus|lambda]] notation):

{{block indent|<math>\lambda f. \lambda g.\exists x(f(x)=1 \land \forall y(f(y)=1 \rightarrow y=x) \land g(x) = 1)</math>}}

(That is, the definite article 'the' denotes a function which takes a pair of [[property|properties]] {{var|f}} and {{var|g}} to truth [[if and only if|if, and only if]], there exists something that has the property {{var|f}}, only one thing has the property {{var|f}}, and that thing also has the property {{var|g}}.) Given the denotation of the [[Predicate (mathematical logic)|predicates]] 'present King of France' (again {{var|K}} for short) and 'bald' ({{var|B}} for short)

{{block indent|<math>\lambda x.Kx</math>}}
{{block indent|<math>\lambda x.Bx</math>}}

we then get the Russellian truth conditions via two steps of [[function application]]: 'The present King of France is bald' is true if, and only if, <math>\exists x((Kx \land \forall y(Ky \rightarrow y =x)) \land Bx)</math>.  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]], or "incomplete symbols"); but the view retains the essentials of the Russellian analysis, yielding exactly the truth conditions Russell argued for.