Laurence Goldstein

(1947 - 2014)

Logicism is one of the great reductionist projects. Numbers and the relationships in which they stand may seem to possess suspect ontological credentials – to be entia non grata – and, further, to be beyond the reach of knowledge. In seeking to reduce mathematics to a small set of principles that form the logical basis of all reasoning, logicism holds out the prospect of ontological economy and epistemological security. This paper attempts to show that a fundamental logicist project, that of defining the number one in terms drawn only from logic and set theory, is a doomed enterprise. The starting point is Russell's Theory of Descriptions, which purports to supply a quantificational analysis of definite descriptions by adjoining a 'uniqueness clause' to the formal rendering of indefinite descriptions. That theory fails on at least two counts. First, the senses of statements containing indefinite descriptions are typically not preserved under the Russellian translation. Second (and independently), the 'uniqueness clause' fails to trim 'some' to 'one'. The Russell–Whitehead account in Principia Mathematica fares no better. Other attempts to define 'one' are covertly circular. An ontologically non-embarrassing alternative account of the number words is briefly sketched

A syntactically correct number-specification may fail to specify any number due to underspecification. For similar reasons, although each sentence in the Yablo sequence is syntactically perfect, none yields a statement with any truth-value. As is true of all members of the Liar family, the sentences in the Yablo sequence are so constructed that the specification of their truth-conditions is vacuous; the Yablo sentences fail to yield statements. The ‘revenge’ problem is easily defused. The solution to the semantical paradoxes offered here revives the mediaeval cassatio approach, one that largely disappeared due to its incomprehending rejection by influential contemporary writers such as William Shyreswood and Thomas Bradwardine. The diagnosis readily extends to the set-theoretic paradoxes

Romane Clark has complained of the dissimilarity between Sellars’s treatment of conceptual thought and his treatment of sense impressions. For sense impressions are intrinsic to perceptions and, on Sellars’s view, both conceptual thought and perception are species of judgment. In the first section of this paper I want to raise a converse sort of complaint: Sellars offers an ‘adverbial’ theory of sense impressions and a similar account of conceptual thought. But this similarity of treatment is not justified by what Sellars says in its defence. If the adverbial theory of conceptual thought is true, as I believe it is, then a number of consequences, important for the philosophy of action, follow. Sellars traces some of these in his paper “Metaphysics and the Concept of a Person.” It would be desirable, therefore, if the adverbial theory of conceptual thought could be made secure. This is attempted in the second section and, in conclusion, the power of the theory is exhibited by directing it briefly at a long-standing problem in the philosophy of logic.