Online research in philosophy

The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward L?wenheim-Skolem theorem to proper classes is used to argue that a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus.

There are certain metaphysically interesting arguments ‘from vagueness’, for unrestricted mereological composition and for four-dimensionalism, which involve a claim to the effect that idioms for unrestricted quantification are precise. An elaboration of Lewis’ argument for this claim, which assumes the view of vagueness as semantic indecision, is presented. It is argued that the argument also works according to other views on the nature of vagueness, which also require for an expression to be vague that there are different admissible alternatives of the relevant sort, such as epistemicism, as defended by Williamson. Recent attempts to resist the argument are discussed and rejected.

Many of those who accept the universalist thesis that mereological composition is unrestricted also maintain that the folk typically restrict their quantifiers in such a way as to exclude strange fusions when they say things that appear to conflict with universalism. Despite its prima facie implausibility, there are powerful arguments for universalism. By contrast, there is remarkably little evidence for the thesis that strange fusions are excluded from the ordinary domain of quantification. Furthermore, this reconciliatory strategy seems hopeless when applied to the more fundamental conflict between universalism and the intuitions that tell against it.

There are four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned—one based upon the existence of semantic indeterminacy, another on the relativity of ontology to a conceptual scheme, a third upon the necessity of sortal restriction, and the last upon the possibility of indefinite extendibility. The argument from semantic indeterminacy derives from general philosophical considerations concerning our understanding of language. For the Skolem–Lowenheim Theorem appears to show that an understanding of quanti- fication over absolutely everything (assuming a suitably infinite domain) is semantically indistinguishable from the understanding of quantification over something less than absolutely everything; the same first-order sentences are true and even the same first-order conditions will be satisfied by objects from the narrower domain. From this it is then argued that the two kinds of understanding are indistinguishable tout court and that nothing could count as having the one kind of understanding as opposed to the other.

Quantification is haunted by the specter of paradoxes. Since Russell, it has been a persistent idea that the paradoxes show what might have appeared to be absolutely unrestricted quantification to be somehow restricted. In the contemporary literature, this theme is taken up by Dummett (1973, 1993) and Parsons (1974a,b). Parsons, in particular, argues that both the Liar and Russell’s paradoxes are to be resolved by construing apparently absolutely unrestricted quantifiers as appropriately restricted.

In my book, Worlds and Individuals, Possible and Otherwise , I use the novel idea of modal tense to respond to a number of arguments against modal realism. Peter van Inwagen’s million-carat-diamond objection is one of them. It targets the version of modal realism by David Lewis and exploits the fact that Lewis accepts absolutely unrestricted quantification. The crux of my response is to use modal tense to neutralize absolutely unrestricted quantification. Seahwa Kim says that even when equipped with modal tense, I am unsuccessful, given my view of reality and the proper use of modal tense in speaking of reality. I counter her attempt at resurrecting van Inwagen’s objection and clarify how we should use modal tense and how we should talk about reality.

In this paper, I examine Takashi Yagisawa’s response to van Inwagen’s ontic objection against David Lewis. Van Inwagen criticizes Lewis’s commitment to the absolutely unrestricted sense of ‘there is,’ and Yagisawa claims that by adopting modal tenses he avoids commitment to absolutely unrestricted quantification. I argue that Yagisawa faces a problem parallel to the one Lewis faces. Although Yagisawa officially rejects the absolutely unrestricted sense of a quantifying expression, he is still committed to the absolutely unrestricted sense of ‘is a real.’.

I present a puzzle for absolutely unrestricted quantification. One important advantage of absolutely unrestricted quantification is that it allows us to entertain perfectly general theories. Whereas most of our theories restrict attention to one or another parcel of reality, other theories are genuinely comprehensive taking absolutely all objects into their domain. The puzzle arises when we notice that absolutely unrestricted theories sometimes impose incompatible constraints on the size of the universe.

I have two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higherorder quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language in a language of the same logical type. I claim that this leads to a trilemma: one must choose between giving up absolutely general quantification, settling for the view that adequate semantic theorizing about certain languages is essentially beyond our reach, and countenancing an open-ended hierarchy of languages of ever ascending logical type. I conclude by suggesting that the hierarchy may be the least unattractive of the options on the table.