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61Dynamic topological logicAnnals of Pure and Applied Logic 131 (1-3): 133-158. 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
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50Propositional Quantification in the Topological Semantics for SNotre Dame Journal of Formal Logic 38 (2): 295-313. 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 second-order 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
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Anil Gupta and Nuel Belnap, The Revision Theory of Truth (review)Philosophy in Review 15 (1): 39-42. 1995.
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89The modal logic of continuous functions on the rational numbersArchive for Mathematical Logic 49 (4): 519-527. 2010.Let ${{\mathcal L}^{\square\circ}}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish 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
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92Indeterminacy of fair infinite lotteriesSynthese 191 (8): 1757-1760. 2014.In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued 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
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7Defining Relevant Implication in a Propositionally Quantified S4Journal of Symbolic Logic 62 (4): 1057-1069. 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
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46Shehtman 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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29The Guptα-Belnαp Systems S and S* are not AxiomatisableNotre Dame Journal of Formal Logic 34 (4): 583-596. 1993.
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93On the complexity of propositional quantification in intuitionistic logicJournal of Symbolic Logic 62 (2): 529-544. 1997.We define a propositionally quantified intuitionistic logic Hπ + by a natural extension of Kripke's semantics for propositional intutionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π +, S4π +, S4.2π +, K4π +, Tπ +, Kπ + and Bπ +, studied by Fine
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66Dynamic topological S5Annals of Pure and Applied Logic 160 (1): 96-116. 2009.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 enrich…Read more
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39The Gupta-Belnap systems ${\rm S}^\#$ and ${\rm S}^*$ are not axiomatisableNotre Dame Journal of Formal Logic 34 (4): 583-596. 1993.
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Axiomatizing the next-interior fragment of dynamic topological logicBulletin of Symbolic Logic 3 376-377. 1997.
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32The modal logic of continuous functions on cantor spaceArchive for Mathematical Logic 45 (8): 1021-1032. 2006.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 axiom…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 |