Generalized quantifier: Difference between revisions

Revision as of 17:33, 3 March 2021

Revision as of 15:41, 22 May 2021

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*Expressions of type ''e'' denote elements of the [[universe of discourse]], the set of entities the discourse is about. This set is usually written as <math>D_e</math>. Examples of type ''e'' expressions include ''John'' and ''he''.

*Expressions of type ''t'' denote a [[truth value]], usually rendered as the set <math>\{0,1\}</math>, where 0 stands for "false" and 1 stands for "true". Examples of expressions that are sometimes said to be of type ''t'' are ''sentences'' or ''propositions''.

*Expressions of type <math>\langle e,t\rangle</math> denote [[Function (mathematics)|functions]] from the set of entities to the set of truth values. This set of functions is rendered as <math>D_t^{D_e}</math>. Such functions are [[Indicator function|characteristic functions]] of [[Set (mathematics)|sets]]. They map every individual that is an element of the set to "true", and everything else to "false." It is common to say that they denote ''sets'' rather than characteristic functions, although, strictly speaking, the latter is more accurate. Examples of expressions of this type are [[predicate (grammar)|predicates]], [[noun]]s and some kinds of [[adjective]]s.

*In general, expressions of complex types <math>\langle a,b\rangle</math> denote functions from the set of entities of type <math>a</math> to the set of entities of type <math>b</math>, a construct we can write as follows: <math>D_b^{D_a}</math>.

We can now assign types to the words in our sentence above (Every boy sleeps) as follows.

**Type(boy) = <math>\langle e,t\rangle</math>

**Type(sleeps) = <math>\langle e,t\rangle</math>

**Type(every) = <math>\langle\langle e,t\rangle,\langle \langle e, t\rangle, t\rangle\rangle</math>

Thus, every denotes a function from a ''set'' to a function from a set to a truth value. Put differently, it denotes a function from a set to a set of sets. It is that function which for any two sets ''A,B'', ''every''(''A'')(''B'')= 1 if and only if <math>A\subseteq B</math>.

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If we abbreviate the meaning of ''boy'' and ''sleeps'' as "''B''" and "''S''", respectively, we have that the sentence ''every boy sleeps'' now means the following:

::<math>(\lambda X.\lambda Y. X\subseteq Y)(B)(S)</math>
::<math>(\lambda Y. B \subseteq Y)(S)</math> — [[Lambda calculus#β-reduction|β-reduction]]

::<math>(\lambda Y. B \subseteq Y)(S)</math> — β-reduction

::<math>B\subseteq S</math>

The expression ''every'' is a [[determiner (linguistics)|determiner]]. Combined with a [[noun]], it yields a ''generalized quantifier'' of type <math>\langle\langle e,t\rangle,t\rangle</math>.