Solovay-Type Theorems for Circular Definitions

Abstract

We present an extension of the basic revision theory of circular definitions with a unary operator, 2. We present a Fitch-style proof system that is sound and complete with respect to the extended semantics. The logic of the box gives rise to a simple modal logic, and we relate provability in the extended proof system to this modal logic via a completeness theorem, using interpretations over circular definitions, analogous to Solovay's completeness theorem for G L using arithmetical interpretations. We adapt our proof to a special class of circular definitions as well as to the first-order case. §1. Introduction. One of the important discoveries in provability logic is the connection between Peano arithmetic (PA) and the modal logic G L, first demonstrated by Robert Solovay. 1 Solovay showed that G L is complete with respect to provability in PA under all so-called arithmetical interpretations. These interpretations connect necessity in the modal language to provability predicates in the arithmetical language.Revision theory is a general theory of circular definitions. It was originally developed as a theory of truth by Anil Gupta and, independently, Hans Herzberger, with important early contributions by Nuel Belnap. 2 The theory was generalized to a theory of circular definitions in Gupta (1988-89), which was further elaborated in Gupta & Belnap (1993). 3 In this paper, we will show that there is a connection, similar to the one between PA and G L, between the circular definitions of revision theory and a particular modal logic, which we call "RT ," for revision theory. 4 We will prove that RT is complete with respect to validity under all D-interpretations for all sets of circular definitions D.The modal logic RT arises naturally from an extension of revision theory that we will present below. Gupta & Standefer (2014) present an extension of revision theory that uses different primitives, which have independent philosophical interest. The extension presented here adds a unary connective, 2, to basic revision theory. This connective can be glossed as saying, roughly, "according to the current hypothesis," or, in the context of a revision sequence, "at the previous stage." The modal logic RT is the logic one obtains from viewing the box simply as a modal operator.The addition of 2 to revision theory increases the expressive power of the theory. The most striking demonstration of the increase in expressive power is that the box permits

Conditionals in Theories of Truth

“…Hence, no definition meets the requirement for finiteness, as this requirement is set out in section 3.1. We can, by liberalizing the requirement, obtain a notion of finite definition suitable for L + (see Standefer [68]). Here, however, we have chosen to restrict the theorem to those definitions formulated in L which count as finite definitions of L .…”

“…In the interest of brevity, we omit proofs in this essay. The reader will find proofs as well as a more detailed presentation of the theory in Standefer[66,68].…”

“…See Standefer[68] 28. Here and in the rest of the present subsection, we assume that the ground language is classical.…”

Guest Editors’ Introduction

“…In this direction, it is worth mentioning the recent paper [27] by Gupta and Standefer, who have studied the logic of the rule of revision via conditionals devised to reflect it. The paper also explores an interpretation of the proposed conditionals by means of a modal operator, a connection that had been considered already in [55], and which has been further investigated proof-theoretically in [57]. The behavior of truth in languages with no vicious self-reference, which was investigated in [26, ch.…”

Non-Classical Circular Definitions

“…SeeGupta and Belnap [1993, 253-255] for discussion. It has, however, recently been shown how to extend the revision theory with new biconditionals that validate versions of all the defining clauses; for details, seeStandefer [2015] andGupta and Standefer [2016].29 The following is based on the ccpo frameworks described in andVisser [2004].30 I will generally suppress mention of assignments to variables and use the notation A(d) to indicate evaluation under an assignment whose values on the free variables of A, x, are the objectsd.Australasian Journal of Logic (14:1) 2017, Article no. 6…”