Costing Non-classical Solutions to the Paradoxes of Self-reference 1 Non-classical Solutions

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell's paradox and their cousins pose enormous diiculties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in the paradoxical inferences, and one or other of these problematic inferences are rejected, to give a consistent (or at least, a coherent) theory. I am not going to show that non-classical accounts of the paradoxes are misguided or wrong-headed. (On the contrary, I think that the general approach is quite sane, and have argued as much in print 16].) However, I am going to show that such approaches do come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics. Non-classical solutions are not cheap. In this section I will sketch the kinds of non-classical solutions I have in mind. They have the same general features: Firstly, we keep whatever semantic, or set-theoretic principles are in dispute. For example, if it is the liar paradox in question, we keep the na ve T-scheme, to the eeect that ThAi $ A where h|i is some name-forming functor, taking propositions to names, and where $ is some form of biconditional. This scheme says, in eeect, that ThAi is true under the same circumstances as A. To say that A is true, is saying no more and no less than saying A. 1 Not only do we keep the na ve T-scheme, but we ensure that our language has a degree of self-reference. As a result, we can express sentences such as the liar: \This very sentence is not true." If the language in question is a natural language, then indexicals will do the trick. If the language is a formal language 1 This is rather na ve, of course, for more must be said about propositions. If we take the bearers of truth to be sentences then ThAi says something not only about whatever A talks about, but it also says something about language. Still, the T-scheme in this form is the cornerstone of many sophisticated theories of truth, such …