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Profile: Philip Kremer (University of Toronto)
  1. Philip Kremer, Does Truth Behave Like a Classical Concept When There is No Vicious Reference?
    §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 (...)
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  2. Philip Kremer, Mathematical Logic.
    modality , 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..
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  3. Philip Kremer, Matching Topological Products and Frame Products of Modal Logics.
    The simplest combination of unimodal logics L1 and L2 into a bimodal logic is their fusion, L1 ⊗ L2, axiomatized by the theorems of L1 for 1 and of L2 for 2. 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 L1 × L2 of L1 and L2. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological (...)
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  4. Philip Kremer, The Incompleteness of S4 ⊕ S4 for the Product Space R × R.
    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 (...)
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  5. Philip Kremer, The Truth is Sometimes Simple.
    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" (1986) and its addendum (1988), 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 (...)
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  6. Philip Kremer, The Topological Product of S4 and S.
    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 (...)
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  7. Philip Kremer (forthcoming). Quantified Modal Logic on the Rational Line. Review of Symbolic Logic:1-16.
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  8. Philip Kremer (2014). Indeterminacy of Fair Infinite Lotteries. Synthese 191 (8):1757-1760.
    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 ... probability function with the same standard part but infinitesimal differences.” They illustrate (...)
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  9. Philip Kremer (2013). Strong Completeness of S4 for Any Dense-in-Itself Metric Space. Review of Symbolic Logic 6 (3):545-570.
    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 (...)
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  10. Philip Kremer (2010). How Truth Behaves When There's No Vicious Reference. Journal of Philosophical Logic 39 (4):344 - 367.
    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 (...)
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  11. Philip Kremer (2010). The Modal Logic of Continuous Functions on the Rational Numbers. Archive for Mathematical Logic 49 (4):519-527.
    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 and Nerode have axiomatized a logic S4C, and have shown (...)
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  12. Philip Kremer & Heinrich Wansing (2010). Editorial Introduction. Journal of Philosophical Logic 39 (4):341-344.
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  13. Philip Kremer (2009). Comparing Fixed-Point and Revision Theories of Truth. Journal of Philosophical Logic 38 (4):363 - 403.
    In response to the liar’s paradox, Kripke developed the fixed-point semantics for languages expressing their own truth concepts. (Martin and Woodruff independently developed this semantics, but not to the same extent as Kripke.) 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 (...)
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  14. Philip Kremer (2009). Dynamic Topological S5. Annals of Pure and Applied Logic 160 (1):96-116.
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  15. Philip Kremer (2008). Supervaluation Fixed-Point Logics of Truth. Journal of Philosophical Logic 37 (5):407 - 440.
    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 (...)
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  16. Philip Kremer, The Revision Theory of Truth. Stanford Encyclopedia of Philosophy.
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  17. Jeremy Avigad, Sy Friedman, Akihiro Kanamori, Elisabeth Bouscaren, Philip Kremer, Claude Laflamme, Antonio Montalbán, Justin Moore & Helmut Schwichtenberg (2007). Montréal, Québec, Canada May 17–21, 2006. Bulletin of Symbolic Logic 13 (1).
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  18. Philip Kremer (2006). The Modal Logic of Continuous Functions on Cantor Space. Archive for Mathematical Logic 45 (8):1021-1032.
    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 (...)
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  19. Philip Kremer & Grigori Mints (2005). Dynamic Topological Logic. Annals of Pure and Applied Logic 131 (1-3):133-158.
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  20. Philip Kremer (2004). John Woods. Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences, Cambridge University Press, Cambridge, New York, 2003, Xviii+ 362 Pp. [REVIEW] Bulletin of Symbolic Logic 10 (1):116-118.
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  21. Philip Kremer & Michael Kremer (2003). Some Supervaluation-Based Consequence Relations. Journal of Philosophical Logic 32 (3):225-244.
    In this paper, we define some consequence relations based on supervaluation semantics for partial models, and we investigate their properties. For our main consequence relation, we show that natural versions of the following fail: upwards and downwards Lowenheim-Skolem, axiomatizability, and compactness. We also consider an alternate version for supervaluation semantics, and show both axiomatizability and compactness for the resulting consequence relation.
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  22. Richard Bradley, Roya Sorensen, Mirror Notation & Philip Kremer (1999). Colin Oakes/Interpretations of Intuitionist Logic in Non-Normal 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 28:661-662.
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  23. Philip Kremer (1999). Relevant Identity. Journal of Philosophical Logic 28 (2):199-222.
    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 (...)
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  24. Philip Kremer (1997). Defining Relevant Implication in a Propositionally Quantified S. Journal of Symbolic Logic 62 (4):1057-1069.
    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+.
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  25. Philip Kremer (1997). Dunn's Relevant Predication, Real Properties and Identity. Erkenntnis 47 (1):37-65.
    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. (...)
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  26. Philip Kremer (1997). On the Complexity of Propositional Quantification in Intuitionistic Logic. Journal of Symbolic Logic 62 (2):529-544.
    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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  27. Philip Kremer (1997). Propositional Quantification in the Topological Semantics for S. Notre Dame Journal of Formal Logic 38 (2):295-313.
    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 , (...)
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  28. Philip Kremer, Grigori Mints & V. Rybakov (1997). Axiomatizing the Next-Interior Fragment of Dynamic Topological Logic. Bulletin of Symbolic Logic 3:376-377.
     
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  29. Mark Lance & Philip Kremer (1996). The Logical Structure of Linguistic Commitment II: Systems of Relevant Commitment Entailment. [REVIEW] Journal of Philosophical Logic 25 (4):425 - 449.
    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 (...)
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  30. Philip Kremer (1995). Anil Gupta and Nuel Belnap, The Revision Theory of Truth Reviewed By. Philosophy in Review 15 (1):39-42.
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  31. Mark Norris Lance & Philip Kremer (1994). The Logical Structure of Linguistic Commitment I: Four Systems of Non-Relevant Commitment Entailment. [REVIEW] Journal of Philosophical Logic 23 (4):369 - 400.
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  32. Philip Kremer (1993). Quantifying Over Propositions in Relevance Logic: Nonaxiomatisability of Primary Interpretations of ∀P and ∃P. Journal of Symbolic Logic 58 (1):334-349.
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  33. Philip Kremer (1993). The Guptα-Belnαp Systems S and S* Are Not Axiomatisable. Notre Dame Journal of Formal Logic 34 (4).
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  34. Philip Kremer (1993). The Gupta-Belnap Systems ${Rm S}^#$ and ${Rm S}^*$ Are Not Axiomatisable. Notre Dame Journal of Formal Logic 34 (4):583-596.
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  35. Philip Kremer (1989). Relevant Predication: Grammatical Characterisations. [REVIEW] Journal of Philosophical Logic 18 (4):349 - 382.
    This paper reformulates and decides a certain conjecture in Dunn's 'Relevant Predication 1: The Formal Theory' (Journal of Philosophical Logic 16, 347-381, 1987). This conjecture of Dunn's relates his object-language characterisation of a property's being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in 'Relevant Implication and Relevant Functions' (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et (...)
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