Relatively Unrestricted Quantification

Philosophical Studies, 2022

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.

Unrestricted Quantification: New Essays, 2007

"Substitutional and interpretational accounts of logical consequence explicate it as truth-preservation in all cases, cases being construed either as admissible substitutional variants or admissible semantic interpretations. This series of three connected studies aims to examine the merits and demerits of “quantificational accounts” of this sort, focusing on seminal contributions of Bolzano, Russell, Tarski, Carnap, Quine and the standard model-theoretic approach. Though extremely influential in the tradition, quantificational accounts have received much critical attention, especially in the wake of Etchemendy 1990. I reconstruct the main objections, arguing that the model-theoretic account appears as the most promising of quantificational accounts vis-à-vis them. Finally, by way of responding to one allegedly devastating objection due to Kneale and Etchemendy, I explain that quantificational slogans do not by themselves capture all there is to logical properties, if not backed up by a plausible account of semantic structure and the specific contribution of logical elements to it. But proponents of the interpretational approach in model-theoretic style need not be paralysed, since they have such a semantic story at their disposal. This, in a nutshell, is my overall agenda. In the present study, I focus on the beginning of systematic theorizing of consequence in Aristotle‘s work, which contains the rudiments of both modal and quantificational accounts of logical consequence. It is pointed out, inter alia, that (1) there is no evidence for the claim that Aristotle would have subscribed to the reductionist spirit of the latter, and that (2) for a full-fledged quantificational approach we need to turn to Bolzano’s substitutional approach, whose motivation, structure and problems are explained in the second part of this study. "

Aristotelian Metaphysics, 2012

Australasian Journal of Philosophy, 2004

A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The second is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences?

2017

Vagueness is a phenomenon whose manifestation occurs most clearly in linguistic contexts. And some scholars believe that the underlying cause of vagueness is to be traced to features of language. Such scholars typically look to formal techniques that are themselves embedded within language, such as supervaluation theory and semantic features of contexts of evaluation. However, when a theorist thinks that the ultimate cause of the linguistic vagueness is due to something other than language-for instance, due to a lack of knowledge or due to the world's being itself vague-then the formal techniques can no longer be restricted to those that look only at within-language phenomena. If, for example a theorist wonders whether the world itself might be vague, it is most natural to think of employing many-valued logics as the appropriate formal representation theory. I investigate whether the ontological presuppositions of metaphysical vagueness can accurately be represented by (finitely) many-valued logics, reaching a mixed bag of results.

It is widely believed that existential quantifiers can bring about the semantic effects of a scope which is wider than their actual syntactic scope (See Fodor & Sag (1982), Cresti (1995), Kratzer (1995), Reinhart (1995) and Winter (1995), among many others.) On the other hand, it is assumed that the syntactic scope of universal quantifiers can be determined unequivocally by the semantics. This paper shows that this second assumption is wrong; universal quantifiers can also bring about scope illusions, though in a very specific ...

Philosophical Perspectives, 2008