Generalized quantifier: Difference between revisions

Revision as of 05:06, 5 April 2022

Revision as of 05:08, 5 April 2022

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==Typed lambda calculus==

A useful way to write complex functions is the [[lambda calculus]]. For example, one can write the meaning of ''sleeps'' as the following lambda expression, which is a function from an individual ''x'' to the proposition that ''x sleeps''.

::<math display="block">\lambda x. \mathrm{sleep}'(x)</math>

Such lambda terms are functions whose ___domain is what precedes the period, and whose range are the type of thing that follows the period. If ''x'' is a variable that ranges over elements of <math>D_e</math>, then the following lambda term denotes the [[identity function]] on individuals:

::<math display="block">\lambda x.x</math>

We can now write the meaning of ''every'' with the following lambda term, where ''X,Y'' are variables of type <math>\langle e,t\rangle</math>:

::<math display="block">\lambda X.\lambda Y. X\subseteq Y</math>

::<math>\lambda X.\lambda Y. X\subseteq Y</math>

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 display="block">(\lambda X.\lambda Y. X\subseteq Y)(B)(S)</math>

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

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

and

<math display="block">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>.