Online research in philosophy

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably.

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably.

The notion of existence is a central philosophical notion and at the heart of discussions about ontological commitment. The linguistic form of statements of ontological commitment or existence plays a not unimportant role in that discussion as well, and the semantics of statements of existence or nonexistence has often been taken as revealing as to the ontological issues themselves. While there are also views according to which questions of ontological commitment can and perhaps should be pursued independently of the linguistic form of the relevant sentences, it is a worthwhile project in itself to see what the actual semantics of sentences expressing ontological commitment and existence in fact is. The verb exist is of course a central expression for making statements about existence, and while many philosophers have expressed particular views concerning that expression (or occurrences of it), exist has hardly been a subject of study in linguistic semantics, possibly because of its apparent ‘technical’, and thus marginal status. In this paper, I will take a closer and systematic look at statements of existence from the point of view of natural language semantics. I will focus on the verb exist when it occurs predicatively, as in (1) and (2).

There are two doctrines for which Quine is particularly well known: the doctrine of ontological commitment and the inscrutability thesis—the thesis that reference and quantification are inscrutable. At first glance, the two doctrines are squarely at odds. If there is no fact of the matter as to what our expressions refer to, then it would appear that no determinate commitments can be read off of our best theories. We argue here that the appearance of a clash between the two doctrines is illusory. The reason that there is no real conflict is not simply that in determining our theories’ ontological commitments we naturally rely on our home language but also (and more importantly) that ontological commitment is not intimately tied to objectual quantification and a reference-first approach to language. Or so we will argue. We conclude with a new inscrutability argument which rests on the observation that the notion of objectual quantification, when properly cashed out, deflates.

Truthmaking and ontological commitment How do we determine the ontological commitments of a theory? Quine told us to look to the quantifier.1 What must be in the domain of the quantifiers if the (regimented) sentences of the theory are all to be true? Those are the ontological commitments of the theory.

According to the familiar Quinean understanding of ontological commitment, (1) one undertakes ontological commitments only via theoretical regimentations, and (2) ontological commitments are to be identified with the domain of a theory’s quantifiers. Jody Azzouni accepts (1), but rejects (2). Azzouni accepts (1) because he believes that no vernacular expression carries ontological commitments. He rejects (2) by locating a theory’s commitments with the extension of an existence predicate. I argue that Azzouni’s two theses undermine each other. If ontological commitments follow from predications of existence, then ontological commitments can be expressed in the vernacular via negative existential sentences.

Discourse carries thin commitment to objects of a certain sort iff it says or implies that there are such objects. It carries a thick commitment to such objects iff an account of what determines truth-values for its sentences say or implies that there are such objects. This paper presents two model-theoretic semantics for mathematical discourse, one reflecting thick commitment to mathematical objects, the other reflecting only a thin commitment to them. According to the latter view, for example, the semantic role of number-words is not designation but rather the encoding of cardinality-quantifiers. I also present some reasons for preferring this view.

There are two different ways of understanding the notion of ‘ontological commitment’. A question about ‘what is said to be’ by a theory or ‘what a theory says there is’ deals with ‘explicit’ commitment; a question about the ontological costs or preconditions of the truth of a theory concerns ‘implicit’ commitment. I defend a conception of ontological commitment as implicit commitment, and argue that existentially quantified idioms in natural language are implicitly, but not explicitly, committing. I use the distinction between the two kinds of ontological commitment to diagnose a flaw in a widely used argument to the effect that existential quantification is not ontologically committing.

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives?

Byeong-Uk Yi has argued that number words like ‘two’ primarily function as numerical predicates as in ‘Socrates and Hippias are two (in number)’, and other grammatical uses of number words can be paraphrased in terms of the predicative use. This paper critically examines Yi’s paraphrase scheme and also some other alternative schemes, and argues that the adjectival use of number words as in ‘The Scots and the Irish are two peoples’ cannot be paraphrased in terms of the predicative use.