Generalized quantifier: Difference between revisions

Revision as of 05:08, 5 April 2022

Revision as of 05:10, 5 April 2022

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====Monotone increasing GQs====

A ''generalized quantifier'' GQ is said to be [[monotone increasing]] (also called [[upward entailing]]) if, for every pair of sets ''X'' and ''Y'', the following holds:

::if <math>X\subseteq Y</math>, then GQ(''X'') [[Entailment|entail]]s GQ(''Y'').

The GQ ''every boy'' is monotone increasing. For example, the set of things that ''run fast'' is a subset of the set of things that ''run''. Therefore, the first sentence below [[Entailment|entail]]s the second:

#Every boy runs fast.

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====Monotone decreasing GQs====

A GQ is said to be [[monotone decreasing]] (also called [[downward entailing]]) if, for every pair of sets ''X'' and ''Y'', the following holds:

::If <math>X\subseteq Y</math>, then GQ(''Y'') entails GQ(''X'').

An example of a monotone decreasing GQ is ''no boy''. For this GQ we have that the first sentence below entails the second.

#No boy runs.

#No boy runs fast.

The lambda term for the [[determiner (linguistics)|determiner]] ''no'' is the following. It says that the two sets have an empty [[Intersection (set theory)|intersection]].

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

Monotone decreasing GQs are among the expressions that can license a [[negative polarity item]], such as ''any''. Monotone increasing GQs do not license negative polarity items.

#Good: No boy has '''any''' money.

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The lambda term for the (complex) [[determiner (linguistics)|determiner]] ''exactly three'' is the following. It says that the [[cardinality]] of the [[Intersection (set theory)|intersection]] between the two sets equals 3.

::<math display="block">\lambda X.\lambda Y. |X\cap Y|=3</math>

===Conservativity===

{{Further|Conservativity}}

A determiner D is said to be ''conservative'' if the following equivalence holds:

::<math display="block">D(A)(B) \leftrightarrow D(A)(A\cap B)</math>

For example, the following two sentences are equivalent.

#Every boy sleeps.