Disquotational Truth and Analyticity

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences. yet not in the scope of negation, the system with the reflection schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to 80. §I. Disquotational truth. According to disquotationalism, the meaning of the truth predicate is governed by the (local) disquotation sentences: Because of the liar paradox, only sentences 4 not containing the truth predicate T are allowed in the disquotation scheme. One can strengthen the disquotation sentences by requiring their uniformity. I call these stronger variants the uniform disquotation sentences: The dot above x is used, as usually, for indicating that the numeral for x is formally substituted for the free variable x. While the local disquotation sentences yield a disquotational theory of truth (considered as a unary predicate), their uniform counterparts yield a disquotational theory of satisfaction, where satisfaction is a relation between formulas 4(v) and objects x. As I am dealing exclusively with arithmetic, truth may be conceived as a unary predicate because closed terms (numerals) are available for all numbers. Even if the disquotation sentences, or their uniform strengthenings, are combined with the axioms of PA or a similar theory, they are disappointingly weak; they do not establish any new mathematical insights. Tarski [30] observed the deductive weakness of the disquotation sentences as axioms. Philosophers express this weakness by saying that the disquotation sentences do not prove infinite 'generalizations.' Since many disquotationalists claim that the expression of infinite conjunctions is Received March 15.2000; revised October 7,2000; accepted March 15.2001.