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Truth, Vagueness, and Paradox: An Essay on the Logic of Truth

By Vann McGee

Description

Until the early 1970s there was really only one fully articulated technical solution to the semantic paradoxes: Tarski's. Most logicians--and philosophers who were interested--subscribed to it, faute de mieux. Since the seventies, however, satisfaction with this solution has waned and, starting with the work of Martin, Woodruff and Kripke, we have seen numerous articulated technical solutions. These have been accompanied by various degrees of philosophical sophistication. McGee's book provides the most recent work of this kind. It is, perhaps, the most sophisticated technically, drawing on the theory of inductive definitions, various results in recursion theory, and techniques involving possible world semantics. Only specialist logicians will be able to understand all that is going on and appreciate the ingenuity of many of the details. (The non-specialist may also be thrown by some of the typos. For example, formulae of the form II[MATHEMATICAL EXPRESSION OMITTED] are pretty uniformly printed simply as

[MATHEMATICAL EXPRESSION OMITTED] But the book also shows a high degree of philosophical sophistication: McGee is sensitive to the philosophical problems that beset technical solutions to the paradoxes, and which many others tend to sweep under the carpet.

The task that McGee sets himself is to show how to construct an adequate theory of truth for a formal language, the specification of which does not require the essential use of a more expressive metalanguage. This constraint is important, since the ultimate aim of his project is to show how, by using these techniques, a natural language can formulate its own theory of truth; and the theorist of truth for a natural language has no other language in which to work. The early chapters of the book lay the ground for the material. Chapters three to six explain and give a critique of the solutions of Tarski, Kripke, Gupta and Herzberger. In the process the basic formal techniques and results on which McGee's own solution draws are developed. The next three chapters give McGee's own solution, with the longish chapter eight being the heart of the matter. In the final chapter McGee reviews what has been achieved, discusses how he envisages the techniques being applied to a natural language, and points out problems, some highly non-trivial, that need to be tackled before this can be done.

McGee's central idea is that truth is a vague notion, like baldness, and that a suitable account of vagueness will solve the semantic paradoxes. According to McGee, sentences with vague predicates fall into three classes: "sentences that the rules of our language, together with the empirical facts, determine to be definitely true; sentences that the rules of our language, together with the empirical facts, determine to be definitely not true; and sentences that are left unsettled" (p. 6). Formally, he cashes out this view as follows. We suppose that we are given a language all of whose terms are precise, with its intended interpretation, 21. Consider some vague predicate, P, and a set of axioms concerning P, C, to be thought of as meaning postulates. The notion of definiteness can now be specified in a number of ways. McGee's preferred way is that a sentence is definitely true (false) iff it (its negation) is provable from C and the atomic facts of 21 in a certain infinitary (classical) logic. The set of meaning postulates that McGee envisages for the truth predicate, T, are essentially such as to determine as true/false those sentences that are true/false at a certain Kripke fixed-point. Showing this to be a recursive set (and so a set that can be specified without reference to any stronger metalanguage) that will do the job is the formal heart of the book.

There is much more to the book than this, however; its content is exceptionally rich, and there is much that anyone can learn from it whether or not they accept its main contentions; there are clear expositions and sharp appraisals of others' views, elegant theorems of quite general interest, and shrewd philosophical observations on numerous issues. The book won the Johnsonian Prize in Philosophy in 1988; and deservedly so. It is required reading for anyone with a serious interest in the area.

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