INDICATIVE CONDITIONALS, RESTRICTED QUANTIFICATION, AND NAIVE TRUTH

This paper extends Kripke’s theory of truth to a language with a variably strict conditional operator, of the kind that Stalnaker and others have used to represent ordinary indicative conditionals of English. It then shows how to combine this with a different and independently motivated conditional operator, to get a substantial logic of restricted quantification within naive truth theory.

Two-valued logics for naive truth theory

It is part of the current wisdom that the Liar and similar semanticparadoxes can be taken care of by the use of certain non-classicalmultivalued logics. In this paper I want to suggest that bivalent logic can do just as well. This is accomplished by using a non-deterministic matrix to define the negation connective. I show that the systems obtained in this way support a transparent truth predicate. The paper also contains some remarks on the conceptual interest of such systems.

NAIVE TRUTH AND RESTRICTED QUANTIFICATION: SAVING TRUTH A WHOLE LOT BETTER

Restricted quantification poses a serious and under-appreciated challenge for nonclassical approaches to both vagueness and the semantic paradoxes. It is tempting to explain “All A are B” as “For all x, if x is A then x is B”; but in the nonclassical logics typically used in dealing with vagueness and the semantic paradoxes (even those where ‘if … then’ is a special conditional not definable in terms of negation and disjunction or conjunction), this definition of restricted quantification fails to deliver important principles of restricted quantification that we’d expect. If we’re going to use a nonclassical logic, we need one that handles restricted quantification better.The challenge is especially acute for naive theories of truth—roughly, theories that take True(〈A〉) to be intersubstitutable with A, even when A is a “paradoxical” sentence such as a Liar-sentence. A naive truth theory inevitably involves a somewhat nonclassical logic; the challenge is to get a logic that’s compatible with naive truth and also validates intuitively obvious claims involving restricted quantification (for instance, “If S is a truth stated by Jones, and every truth stated by Jones was also stated by Smith, then S is a truth stated by Smith”). No extant naive truth theory even comes close to meeting this challenge, including the theory I put forth in Saving Truth from Paradox. After reviewing the motivations for naive truth, and elaborating on some of the problems posed by restricted quantification, I will show how to do better. (I take the resulting logic to be appropriate for vagueness too, though that goes beyond the present paper.)In showing that the resulting logic is adequate to naive truth, I will employ a somewhat novel fixed point construction that may prove useful in other contexts.