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Restricted quantifier

SEMANTICS: a quantifier which ranges over a subset of the universe of discourse selected by means of a predicate. Restricted quantification is sometimes represented as in (i), with the restricted quantifier between brackets and the predicate P indicating the subset:

(i)  [ All(x) : P(x) ] Q(x)
     [ ThereIs(x) : P(x) ] Q(x)
It can also be represented in standard predicate logic by means of connectives:
(ii) All(x) [ P(x) -> Q(x) ]
     ThereIs(x) [ P(x) & Q(x) ] 
In (ii) quantification is restricted to P: all or some entities that are P have property Q. In natural language, quantifiers are always restricted; either by the common noun following the quantifying determiner (every man, some woman) or by an inherent meaning element (everyone, something).
LIT. Gamut, L.T.F. (1991)