Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “ aboutness ” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be (...) any prime.” In this context, I will be using the term ‘p’ to reason about the primes. Although ‘p’ helps me secure the aboutness of my discourse, it may seem wrong to say that ‘p’ refers to anything. Be that as it may, this paper explores what mathematical discourse would be like if mathematicians were able to borrow freely from one another not just the reference of terms that clearly refer, but, more generally, the sort of aboutness present in a line of reasoning leading up to a universal generalization. The paper also gives reasons for believing that aboutness of this sort really is freely transferable. A key implication will be that the concept “set of natural numbers” suffers from no mathematically significant indeterminacy that can be coherently discussed. (shrink)

This paper investigates different readings of plural and reciprocal sentences and how they can be derived from syntactic surface structures in a systematic way. The main thesis is that these readings result from different ways of inserting logical operators at the level of Logical Form. The basic operator considered here is a cumulative mapping from predicates that apply to singularities onto the corresponding predicates that apply to pluralities. Given a theory which allows for free insertion of such operators, it can (...) then be shown that the lexical semantics of the reciprocal expressions each other/one another consists of exactly two components, namely an anaphoric variable and a non-identity statement. This receives further support from the observation that it is exactly these two components that can be focused by only; all that remains to be done is to correctly manipulate these components at the level of LF. (shrink)

When viewed as the most comprehensive theory of collections, set theory leaves no room for classes. But the vocabulary of classes, it is argued, provides us with compact and, sometimes, irreplaceable formulations of largecardinal hypotheses that are prominent in much very important and very interesting work in set theory. Fortunately, George Boolos has persuasively argued that plural quantification over the universe of all sets need not commit us to classes. This paper suggests that we retain the vocabulary of classes, but (...) explain that what appears to be singular reference to classes is, in fact, covert plural reference to sets. (shrink)

Gareth Evans'' proposal, as amended by Steven Neale –that a definite pronoun with a quantifiedantecedent that does not bind it has the sense ofa definite description – has been challenged inthe singular case by appeal to counter-examplesinvolving failure of the uniqueness condition forthe legitimacy of a singular description. Thischallenge is here extended to the plural.Counter-examples are provided by cases in which aplural description `the Fs'' does not denote,despite the propriety of the use of `they'' or`them'' it is to replace, because (...) of failure ofcumulativeness. A noncumulative predicate isnot distributive, and conditions for thelegitimacy of `the Fs'' designed to accommodatenondistributive `Fs'' are given in the context ofa two sorted theory with generalized quantifiers.Failure of cumulativeness is not to be confusedwith failure of maximality as Neale and othersdefine it. If not all Fs are Gs, `The Fs are Gs''is false; but it does not follow that `the Fs'' isillegitimate; and if `Fs'' is distributive. it isso only if there are no Fs. These differencesgranted, I give a partial defense of theEvans–Neale proposal from deficiencies in analternative based on the views of P. T.Geach. (shrink)