Generalized quantifier: Difference between revisions

Revision as of 04:14, 27 November 2020

Revision as of 03:13, 5 January 2021

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===Monotonicity===

====Monotone increasing GQs====

A ''generalized quantifier'' GQ is said to be [[monotone increasing]], (also called [[upward entailing]], just in) caseif, for anyevery twopair 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:

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====Monotone decreasing GQs====

A GQ is said to be [[monotone decreasing]], (also called [[downward entailing]]) just in caseif, for anyevery pair twoof 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.

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====Non-monotone GQs====

A GQ is said to be ''non-monotone'' if it is neither monotone increasing nor monotone decreasing. An example of such a GQ is ''exactly three boys''. Neither of the following two sentences entailentails the other.

#Exactly three students ran.

#Exactly three students ran fast.

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#Every boy is a boy who sleeps.

It has been proposed that ''all'' natural language determiners (i.e. {{emdash}}in every natural language) {{emdash}}are conservative ([[Jon Barwise|Barwise]] and Cooper 1981). The expression ''only'' is not conservative. The following two sentences are not equivalent. But it is, in fact not common to analyze ''only'' as a [[determiner (linguistics)|determiner]]. Rather, it is standardly treated as a [[focus-sensitive]] [[adverb]].

#Only boys sleep.

#Only boys are boys who sleep.