Journal of Philosophical Logic 22 (5):545 - 561 (1993)
|Abstract||The main objective of this paper is to give a characterization of those quantiiiers and operators that are freely interchangeable with all other quantiiiers or operators on a possibly different domain. The starting point is a slight generalization of an earlier result on unary extensional quantifiers. These are shown to be scopeless just in case they are ultraiilters with certain strong completeness properties: in many, though not all cases, a quantilier must be trivial or name-like (i.e. principal) in order to be scopeless. Which cases depends on the relative sizes of the domains of quantification. Operators other than unary extensional quantiliers for which the notion of scopelessness also makes sense include quantifiers in threevalued logic, intensional quantiliers and propositional operators in possible worlds semantics, as well as modifiers in natural language. These operators, about which the above results have nothing to say, are the subject of Section 2. The characterization result can in a sense, however, be extended to them. This is done in Section 3. Although the notion of a complete ultrafilter is as central in their case as it is in the case of unary extensional quantifiers, the results for the latter do not generalize as directly as one might think: scopelessness turns out to be even rarer in the more general setting. Finally, in Section 4, we brieiiy turn to notions of scoplessness that do not involve variable-binding, where it can be shown that completeness, but not ultraiilterhood, is irrelevant.|
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