
15Quantified intuitionistic logic over metrizable spacesReview of Symbolic Logic 12 (3): 405425. 2019.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 zerodimensional denseinitself metrizable space. The main result of the current article genera…Read more

11Completeness of secondorder propositional s4 and H in topological semanticsReview of Symbolic Logic 11 (3): 507518. 2018.

14TopologicalFrame Products of Modal LogicsStudia Logica 106 (6): 10971122. 2018.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 twodimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced the topological …Read more

6Dynamic topological logicAnnals of Pure and Applied Logic 131 (13): 133158. 2005.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 topolo…Read more

Real Properties, Relevance Logic, and IdentityDissertation, University of Pittsburgh. 1994.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 langu…Read more

44Shehtman 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 …Read more

13On the Complexity of Propositional Quantification in Intuitionistic LogicJournal of Symbolic Logic 62 (2): 529544. 1997.We define a propositionally quantified intuitionistic logic $\mathbf{H}\pi +$ by a natural extension of Kripke's semantics for propositional intutionistic logic. We then show that $\mathbf{H}\pi+$ is recursively isomorphic to full second order classical logic. $\mathbf{H}\pi+$ is the intuitionistic analogue of the modal systems $\mathbf{S}5\pi +, \mathbf{S}4\pi +, \mathbf{S}4.2\pi +, \mathbf{K}4\pi +, \mathbf{T}\pi +, \mathbf{K}\pi +$ and $\mathbf{B}\pi +$, studied by Fine.

92Comparing fixedpoint and revision theories of truthJournal of Philosophical Logic 38 (4): 363403. 2009.In response to the liar’s paradox, Kripke developed the fixedpoint semantics for languages expressing their own truth concepts. Kripke’s work suggests a number of related fixedpoint theories of truth for such languages. Gupta and Belnap develop their revision theory of truth in contrast to the fixedpoint 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 …Read more

153Relevant identityJournal of Philosophical Logic 28 (2): 199222. 1999.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 o…Read more

72modality , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language L by interpreting L in dynamic topological systems, i.e. ordered pairs X, f , where X is a topological space and f is a..

47§1. Introduction. When truththeoretic 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:…Read more

17Colin oakes/interpretations of intuitionist logic in nonnormal modal logics 47–60 Aviad heifetz/iterative and fixed point common belief 61–79 dw mertz/the logic of instance ontology 81–111 (review)Journal of Philosophical Logic 28 661662. 1999.

61The logical structure of linguistic commitment II: Systems of relevant commitment entailment (review)Journal of Philosophical Logic 25 (4). 1996.In "The Logical Structure of Linguistic Commitment I" (The Journal of Philosophical Logic 23 (1994), 369400), 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"…Read more

13The Incompleteness of S4 {bigoplus} S4 for the Product SpaceStudia Logica 103 (1): 219226. 2015.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 …Read more

Paradox and paraconsistency: Conflict resolution in the abstract sciences (review)Bulletin of Symbolic Logic 10 (1): 115117. 2004.

69How Truth Behaves When There’s No Vicious ReferenceJournal of Philosophical Logic 39 (4): 345367. 2010.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 …Read more

23Defining relevant implication in a propositionally quantified SJournal of Symbolic Logic 62 (4): 10571069. 1997.R. K. Meyer once gave precise form to the question of whether relevant implication can be defined in any modal system, and his answer was `no'. In the present paper, we extend S4, first with propositional quantifiers, to the system S4π+; and then with definite propositional descriptions, to the system S4π+ lp . We show that relevant implication can in some sense be defined in the modal system S4π+ lp , although it cannot be defined in S4π+

46Philip 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…Read more

30Strong completeness of s4 for any denseinitself metric spaceReview of Symbolic Logic 6 (3): 545570. 2013.In the topological semantics for modal logic, S4 is wellknown 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 denseinitself 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 complete…Read more

47Matching Topological and Frame Products of Modal LogicsStudia Logica 104 (3): 487502. 2016.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 twodimensional: he defined 2d Cartesian products of 1d 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…Read more

28Dynamic topological logicAnnals of Pure and Applied Logic 131 (13): 133158. 2005.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 topolo…Read more

Anil Gupta and Nuel Belnap, The Revision Theory of Truth (review)Philosophy in Review 15 (1): 3942. 1995.

25Propositional Quantification in the Topological Semantics for SNotre Dame Journal of Formal Logic 38 (2): 295313. 1997.Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems , , : given a Kripke frame, the quantifiers range over all the sets of possible worlds. is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub , …Read more

77The modal logic of continuous functions on the rational numbersArchive for Mathematical Logic 49 (4): 519527. 2010.Let ${{\mathcal L}^{\square\circ}}$ be a propositional language with standard Boolean connectives plus two modalities: an S4ish topological modality □ and a temporal modality ◦, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language ${{\mathcal L}^{\square\circ}}$ by interpreting ${{\mathcal L}^{\square\circ}}$ in dynamic topological systems, i.e., ordered pairs 〈X, f〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren …Read more

64Indeterminacy of fair infinite lotteriesSynthese 191 (8): 17571760. 2014.In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use nonstandard analysis to construct a family of nicelybehaved hyperrationalvalued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... pr…Read more

7Defining Relevant Implication in a Propositionally Quantified S4Journal of Symbolic Logic 62 (4): 10571069. 1997.R. K. Meyer once gave precise form to the question of whether relevant implication can be defined in any modal system, and his answer was `no'. In the present paper, we extend $\mathbf{S4}$, first with propositional quantifiers, to the system $\mathbf{S4\pi}+$; and then with definite propositional descriptions, to the system $\mathbf{S4\pi}+^{lp}$. We show that relevant implication can in some sense be defined in the modal system $\mathbf{S4\pi}+^{lp}$, although it cannot be defined in $\mathbf{…Read more

31Shehtman 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 pape…Read more
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Areas of Specialization
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Modal Logic 
Quantified Modal Logic 
Semantics for Modal Logic 
Intuitionistic Logic 
Relevance Logic 
Liar Paradox 