Philip Kremer

§1. Introduction. When truth-theoretic paradoxes are generated, two factors seem to be at play: the behaviour that truth intuitively has; and the facts about which singular terms refer to which sentences, and so on. For example, paradoxicality might be partially attributed to the contingent fact that the singular term, "the italicized sentence on page one", refers to the sentence, The italicized sentence on page one is not true. Factors of this second kind might be represented by a ground model: an interpretation of all the names, function symbols, and predicates in the potentially self-referential language under study, with the exception of the predicate "x is true". Formally, suppose that L is an uninterpreted first order language. M = 〈D, I〉 is a classical model for L if D is a nonempty set; and I is a function assigning to each name of L a member of D, to each n-place function symbol of L a function from Dn to D, and to each nonlogical n-place predicate of L a function from Dn to {t, f}. Suppose furthermore that L+ is obtained by adding a new one-place predicate T to L, and that L has a quote name ‘A’ for each sentence A of L+. We follow Gupta and Belnap [5] in defining S =df {A: A is a sentences of L+}. A classical model M = 〈D, I〉 for L is a ground model for L iff both S ⊆ D, and I(‘A’) = A for each A ∈ S. A classical ground model for L is a representation of the supposedly unproblematic fragment of L+: a representation of which terms refer to which objects, and of which objects have which nonsemantic properties.

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics.

In The Revision Theory of Truth (MIT Press), Gupta and Belnap (1993) claim as an advantage of their approach to truth "its consequence that truth behaves like an ordinary classical concept under certain conditions—conditions that can roughly be characterized as those in which there is no vicious reference in the language." To clarify this remark, they define Thomason models, nonpathological models in which truth behaves like a classical concept, and investigate conditions under which a model is Thomason: they argue that a model is Thomason when there is no vicious reference in it. We extend their investigation, considering notions of nonpathologicality and senses of "no vicious reference" generated both by revision theories of truth and by fixedpoint theories of truth. We show that some of the fixed-point theories have an advantage analogous to that which Gupta and Belnap claim for their approach, and that at least one revision theory does not. This calls into question the claim that the revision theories have a distinctive advantage in this regard.

In response to the liar’s paradox, Kripke developed the fixed-point semantics for languages expressing their own truth concepts. Kripke’s work suggests a number of related fixed-point theories of truth for such languages. Gupta and Belnap develop their revision theory of truth in contrast to the fixed-point theories. The current paper considers three natural ways to compare the various resulting theories of truth, and establishes the resulting relationships among these theories. The point is to get a sense of the lay of the land amid a variety of options. Our results will also provide technical fodder for the methodological remarks of the companion paper to this one.

In "The Logical Structure of Linguistic Commitment I" (The Journal of Philosophical Logic 23 (1994), 369-400), we sketch a linguistic theory (inspired by Brandom's Making it Explicit) which includes an "expressivist" account of the implication connective, →: the role of → is to "make explicit" the inferential proprieties among possible commitments which proprieties determine, in part, the significances of sentences. This motivates reading (A → B) as "commitment to A is, in part, commitment to B". Our project is to study the logic of →. LSLC I approximates (A → B) as "anyone committed to A is committed to B", ignoring issues of whether A is relevant to B. The present paper includes considerations of relevance, motivating systems of relevant commitment entailment related to the systems of commitment entailment of LSLC I. We also consider the relevance logics that result from a commitment reading of Fine's semantics for relevance logics, a reading that Fine suggests

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 \ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. Indeed, van Benthem et al. show that S4 \ S4 is the bimodal logic of the particular product space , leaving open the question of whether S4 \ S4 is also complete for the product space . We answer this question in the negative

Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and □ can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system be a topological space X together with a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □ , and two temporal modalities, ○ and * , both interpreted using the continuous function f. In particular, ○ expresses f’s action on X from one moment to the next, and * expresses the asymptotic behaviour of f

Philip Kremer, Department of Philosophy, McMaster University Note: The following version of this paper does not contain the proofs of the stated theorems. A longer version, complete with proofs, is forthcoming. §1. Introduction. In "The truth is never simple" and its addendum, Burgess conducts a breathtakingly comprehensive survey of the complexity of the set of truths which arise when you add a truth predicate to arithmetic, and interpret that predicate according to the fixed point semantics or the revision-theoretic semantics for languages expressing their own truth concepts. Burgess considers various sets that can be said to represent truth in this context, and shows that their complexity ranges from Π11 or Σ11 to Π12 or Σ12. Thus, enriching arithmetic with a truth predicate increases its complexity, which is otherwise only ∆11

We begin to fill a lacuna in the relevance logic enterprise by providing a foundational analysis of identity in relevance logic. We consider rival interpretations of identity in this context, settling on the relevant indiscernibility interpretation, an interpretation related to Dunn's relevant predication project. We propose a general test for the stability of an axiomatisation of identity, relative to this interpretation, and we put various axiomatisations to this test. We fill our discussion out with both formal and philosophical remarks on identity in relevance logic