Abstract
Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance.
Highlights
Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects
In order to set up a general semantic theory satisfying constraint (C), we need to be able to generalise on the syntactic position of every formula of our metalanguage
Let us reconsider our constraint (C) on a satisfactory general semantic theory, which we introduced in the previous section: (C) For all formulas D; F1; . . .; Fk of the metalanguage ML, there is an interpretation of the object language L according to which Pi means Fi and the quantifiers of L range over the Ds
Summary
Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. In ordinary discourse, our quantifiers are often restricted to some contextually salient domain. Some philosophers have questioned the intelligibility of (the required Fregean interpretation of) type theory, while others have criticised it for various expressive limitations (Godel 1983; Bealer 1982; Chierchia 1985; Menzel 1986; Chierchia and Turner 1988; Linnebo 2006; Weir 2006) It is for these reasons that it is desirable to develop some alternative solution. Even if the logical problem that we will deal with in the present paper is resolved, the most that we can hope for is quantification over domains that are unrestricted relative to some conceptual framework.
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