Philip Kremer

Consider the following sentence: (1) is not true. It has long been known that the sentence, (1), produces a paradox, the socalled liar’s paradox: it seems impossible consistently to maintain that (1) is true, and impossible consistently to maintain that (1) is not true: if (1) is true, then (1) says, truly, that (1) is not true so that (1) is not true; on the other hand, if (1) is not true, then what (1) says is the case, i.e., (1) is true. Given such a paradox, one might be sceptical of the notion of truth, or at least of the prospects of giving a scientifically respectable account of truth. Alfred Tarski’s great accomplishment was to show how to give — contra this scepticism — a formal definition of truth for a wide class of formalized languages. Tarski did not, however, show how to give a definition of truth for languages (such as English) that contain their own truth predicates. He thought that this could not be done, precisely because of the liar’s paradox. More generally, Tarski reckoned that any language with its own truth predicate would be inconsistent, as long as it obeyed the rules of standard classical logic, and had the ability to refer to its own sentences. As we will see in our remarks on Theorem 2.1 in Section 2.3, Tarski was not quite right: there are consistent classical interpreted languages that refer to their own sentences and have their own truth predicates. (This point originates in Gupta 1982 and is strengthened in Gupta and Belnap 1993.) Given the close connection between meaning and truth, it is widely held that any semantics for a language L, i.e., any theory of meaning for L, will be closely related to a theory of truth for : indeed, it is commonly held that something like a Tarskian theory of truth for will be a central part of a semantics for L. Thus, the impossibility of giving a Tarskian theory of truth for languages with their own truth predicates threatens the project of giving a semantics for languages with their own truth predicates. We had to wait until the work of Kripke 1975 and of Martin & Woodruff 1975 for a systematic formal proposal of a semantics for languages with their own truth predicates. The basic thought is simple: take the offending sentences, such as (1), to be neither true nor false. Kripke, in particular, shows how to implement this thought for a wide variety of languages, in effect employing a semantics with three values, true, false and neither. It is safe to say that Kripkean approaches have replaced Tarskian pessimism as the new orthodoxy concerning languages with their own truth predicates. One of the main rivals to the three-valued semantics is the Revision Theory of Truth, or RTT, independently conceived by Hans Herzberger and Anil Gupta, and first presented in publication in 1982. The first monographs on the topic are by Yaqūb and the locus classicus, Gupta & Belnap, both in 1993. The Revision Theory of Truth (RTT) is designed to model the kind of reasoning that the liar sentence leads to, within a two-valued context. The central idea is the idea of a revision process: a process by which we revise hypotheses about the truth-value of one or more sentences. The present article’s purpose is to outline the Revision Theory of Truth.

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived.

In the topological semantics for modal logic, S4 is well known to be complete for the rational line and for the real line: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete but strongly complete, for the rational line. But no similarly easy amendment is available for the real line. In an earlier paper, we proved a general theorem: S4 is strongly complete for any dense-in-itself metric space. Strong completeness for the real line is a special case. In the current paper, we give a proof of strong completeness tailored to the special case of the real line: the current proof is simpler and more accessible than the proof of the more general result and involves slightly different techniques. We proceed in two steps: first, we show that S4 is strongly complete for the space of finite and infinite binary sequences, equipped with a natural topology; and then we show that there is an interior map from the real line onto this space.

In the topological semantics, quantified intuitionistic logic, QH, is known to be strongly complete not only for the class of all topological spaces but also for some particular topological spaces — for example, for the irrational line, ${\Bbb P}$, and for the rational line, ${\Bbb Q}$, in each case with a constant countable domain for the quantifiers. Each of ${\Bbb P}$ and ${\Bbb Q}$ is a separable zero-dimensional dense-in-itself metrizable space. The main result of the current article generalizes these known results: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality ≤ the space’s weight; consequently, QH is strongly complete for any separable zero-dimensional dense-in-itself metrizable space with a constant countable domain. We also prove a result that follows from earlier work of Moerdijk: if we allow varying domains for the quantifiers, then QH is strongly complete for any dense-in-itself metrizable space with countable domains.

The simplest bimodal combination of unimodal logics \ and \ is their fusion, \, axiomatized by the theorems of \ for \ and of \ for \, and the rules of modus ponens, necessitation for \ and for \, and substitution. Shehtman introduced the frame product \, as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced the topological product \, as the logic of the products of certain topological spaces. For almost all well-studies logics, we have \, for example, \. Van Benthem et al. show, by contrast, that \. It is straightforward to define the product of a topological space and a frame: the result is a topologized frame, i.e., a set together with a topology and a binary relation. In this paper, we introduce topological-frame products \ of modal logics, providing a complete axiomatization of \, whenever \ is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include \ and \, but not \ or \. We leave open the problem of axiomatizing \, \, and other related logics. When \, our result confirms a conjecture of van Benthem et al. concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.

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.

There is an intuition, notoriously difficult to formalise, that only some predicates express real properties. J. M. Dunn formalises this intuition with relevance logic, proposing a notion of relevant predication. For each first order formula Ax, Dunn specifies another formula that is intuitively interpreted as "Ax expresses a real property". Chapter I calls such an approach an object language approach, since the claim that Ax expresses a real property is rendered as a formula in the object language. On a metalanguage approach, on the other hand, the claim that Ax expresses a real property would be metalinguistic, mentioning but not using the formula Ax. ;Our Introduction begins by investigating Dunn's motivation for relevant predication, and argues that it implicitly presupposes some interpretation of identity. Indeed, part of the dissertation's work is to use relevant predication as a key to a coherent account of relevant identity. ;Chapter I re-motivates relevant predication. We consider P. Geach's distinction between real and "Cambridge" change, and suggest that there is an underlying distinction between real and Cambridge predicates. We further argue that no classical object language approach can formalise this distinction. So we turn to relevance logic, remotivating relevant predication, with Geach's distinction in mind. ;In light of this new motivation, we argue that relevant prediction relies on a logically weak notion of identity, according to which "x = y" means "x and y share all relevant properties". Identity is taken up in a more technical setting in Chapters IV and V. ;Subsequent chapters investigate technical issues that flow from Chapter I. Chapter II investigates metalinguistic grammatical characterisations of relevant predication. Chapter III concerns the semantics of first order relevance logic. Chapter IV investigates the identity in R and its relationship to relevant predication. In particular, Chapter IV argues against introducing the axiom, $\to$ Ay), in relevance logic, since the resulting systems are unstable, in a sense motivated by the notion of relevant identity underlying Dunn's notion of relevant predication. Finally Chapter V investigates the possibility of a coherent theory of identity in R's modal relatives ${\bf R}\sp\square$ and E.

The topological semantics for modal logic interprets a standard modal propositional language in topological spaces rather than Kripke frames: the most general logic of topological spaces becomes S4. But other modal logics can be given a topological semantics by restricting attention to subclasses of topological spaces: in particular, S5 is logic of the class of almost discrete topological spaces, and also of trivial topological spaces. Dynamic Topological Logic interprets a modal language enriched with two unary temporal connectives, next and henceforth. DTL interprets the extended language in dynamic topological systems: a DTS is a topological space together with a continuous function used to interpret the temporal connectives. In this paper, we axiomatize four conservative extensions of S5, and show them to be the logic of continuous functions on almost discrete spaces, of homeomorphisms on almost discrete spaces, of continuous functions on trivial spaces and of homeomorphisms on trivial spaces

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. In this paper, we axiomatize the topological product of S4 and S5, which is strictly between S4 ⊗ S5 and S4 × S5. We also apply our techniques to (1) proving a conjecture of van Benthem et al concerning the logic of products of Alexandrov spaces with arbitrary topological spaces; and (2) solving a problem in quantified modal logic: in particular, it is known that standard quantified S4 without identity, QS4, is complete in Kripke semantics with expanding domains; we show that QS4 is complete not only in topological semantics with constant domains (which was already shown by Rasiowa and Sikorski), but wrt the topological space Q with a constant countable domain.

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space

We critically investigate and refine Dunn's relevant predication, his formalisation of the notion of a real property. We argue that Dunn's original dialectical moves presuppose some interpretation of relevant identity, though none is given. We then re-motivate the proposal in a broader context, considering the prospects for a classical formalisation of real properties, particularly of Geach's implicit distinction between real and ''Cambridge'' properties. After arguing against these prospects, we turn to relevance logic, re-motivating relevant predication with Geach's distinction in mind. Finally we draw out some consequences of Dunn's proposal for the theory of identity in relevance logic.

Let $\mathcal{L}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality $\square$ and a temporal modality $\bigcirc$ , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language $\mathcal{L}$ by interpreting $\mathcal{L}$ in dynamic topological systems, i.e. ordered pairs $\langle X, f\rangle$ , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result

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 Q × Q, leaving open the question of whether S4 ⊕ S4 is also complete for the product space R × R. We answer this question in the negative.

The simplest combination of unimodal logics \ into a bimodal logic is their fusion, \, axiomatized by the theorems of \. Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product \. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product \, using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of \

§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.