Laurence Goldstein

(1947 - 2014)

If nowadays categories seems to cover a multitude of different enquiries, we can see some continuity and coherence among them, and we can get some sense of what the subject is, by going back to the first treatise to receive the name, the Categories of Aristotle. The scheme of categories worked out by Aristotle in that book was used by him in subsequent works to solve a variety of problems. On one plausible hypothesis, Aristotle’s scheme was partly shaped by ontological considerations. However, one can construct categorial schemes that are free of ontological assumptions, and I call such schemes and the principles on which they are constructed “pure.” A purified Aristotelian scheme is one of a multiplicity of categorial schemes that can be generated by a sufficiently general pure categorial principle. In the penultimate section of this paper I consider a useful scheme, generated in this way, that categorizes the elements of our discourse about discourse. This links up with J. L. Austin’s discussion of the nature of illocutionary acts and throws light on the interconnections between such problematic notions as sentence, meaning, and proposition.

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

The Russell class does not exist because the conditions purporting to specify that class are contradictory, and hence fail to specify any class. Equally, the conditions purporting to specify the Liar statement are contradictory and hence, although the Liar sentence is grammatically in order, it fails to yield a statement. Thus the common source of these and related paradoxes is contradictory (or tautologous) specifying conditions-for such conditions fail to specify. This is the diagnosis. The cure consists of seeking and destroying the deep-seated preconceptions that make almost irresistible our belief in the existence of items which provably do not exist