The diagonal argument and the Liar

Abstract

There are arguments found in various areas of mathematical logic that are taken to form a family: the family of diagonal arguments. Much of recursion theory may be described as a theory of diagonalization; diagonal arguments establish basic results of set theory; and they play a central role in the proofs of the limitative theorems of Godel and Tarski. Diagonal arguments also give rise to set-theoretical and semantical paradoxes. What do these arguments have in common what makes an argument a diagonal argument? And why do some diagonal arguments lead to theorems, while others lead to paradox? In this paper, I attempt to answer these questions. Cantor's first uses of the diagonal argument are presented in Section II. In Section III, I answer the first question by providing a general analysis of the diagonal argument. This analysis is then brought to bear on the second question. In Section IV, I give an account of the difference between good diagonal arguments (those leading to theorems) and bad diagonal arguments (those leading to paradox). The main philosophical interest of the diagonal argument, I believe, lies in its relation to the Liar paradox. The familiar Liar is generated by our ordinary semantical concepts of truth and falsity. Its proper setting is natural language, in which our ordinary semantic terms appear. As Tarski has made clear, this means that the Liar is inextricably linked with another vexed semantical problem, that of universality. Perhaps the central question here is this: Are natural languages universal? Roughly speaking, a language is universal in Tarski's sense if it can say everything there is to be said. If natural languages are universal in this sense, then they can say everying there is to be said about their own semantics. But then it would seem that natural languages fall foul of the Liar.