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Philip Kremer [43]Philip Alcuin Kremer [1]
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Philip Kremer
University of Toronto, St. George
  1.  11
    Dynamic Topological Logic.Philip Kremer & Grigori Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
    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 (...)
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  2.  10
    Strong Completeness of S4 for Any Dense-in-Itself Metric Space.Philip Kremer - 2013 - 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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  3.  72
    Comparing Fixed-Point and Revision Theories of Truth.Philip Kremer - 2009 - 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. 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 (...)
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  4.  29
    Indeterminacy of Fair Infinite Lotteries.Philip Kremer - 2014 - 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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  5.  20
    Quantified Modal Logic on the Rational Line.Philip Kremer - 2014 - Review of Symbolic Logic 7 (3):439-454.
  6.  1
    Dynamic Topological Logic.Philip Kremer & Giorgi Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
    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 (...)
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  7.  50
    How Truth Behaves When There’s No Vicious Reference.Philip Kremer - 2010 - Journal of Philosophical Logic 39 (4):345-367.
    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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  8.  46
    On the Complexity of Propositional Quantification in Intuitionistic Logic.Philip Kremer - 1997 - 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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  9.  44
    Some Supervaluation-Based Consequence Relations.Philip Kremer & Michael Kremer - 2003 - 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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  10.  35
    Quantifying Over Propositions in Relevance Logic: Nonaxiomatisability of Primary Interpretations of ∀P and ∃P.Philip Kremer - 1993 - Journal of Symbolic Logic 58 (1):334-349.
  11.  85
    Supervaluation Fixed-Point Logics of Truth.Philip Kremer & Alasdair Urquhart - 2008 - 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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  12.  38
    Matching Topological and Frame Products of Modal Logics.Philip Kremer - 2016 - Studia Logica 104 (3):487-502.
    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 (...)
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  13.  15
    The Gupta-Belnap Systems ${Rm S}^#$ and ${Rm S}^*$ Are Not Axiomatisable.Philip Kremer - 1993 - Notre Dame Journal of Formal Logic 34 (4):583-596.
  14.  78
    Relevant Identity.Philip Kremer - 1999 - 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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  15.  48
    The Modal Logic of Continuous Functions on the Rational Numbers.Philip Kremer - 2010 - 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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  16.  4
    The Incompleteness of S4 {Bigoplus} S4 for the Product Space.Philip Kremer - 2015 - Studia Logica 103 (1):219-226.
    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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  17.  35
    Dynamic Topological S5.Philip Kremer - 2009 - Annals of Pure and Applied Logic 160 (1):96-116.
    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. (...)
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  18.  34
    The Logical Structure of Linguistic Commitment I: Four Systems of Non-Relevant Commitment Entailment. [REVIEW]Mark Norris Lance & Philip Kremer - 1994 - Journal of Philosophical Logic 23 (4):369 - 400.
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  19. Quantifying Over Propositions in Relevance Logic: Nonaxiomatisability of Primary Interpretations of $\Forall P$ and $\Exists P$.Philip Kremer - 1993 - Journal of Symbolic Logic 58 (1):334-349.
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  20.  52
    The Logical Structure of Linguistic Commitment II: Systems of Relevant Commitment Entailment. [REVIEW]Mark Lance & Philip Kremer - 1996 - 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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  21. Axiomatizing the Next-Interior Fragment of Dynamic Topological Logic.Philip Kremer, Grigori Mints & V. Rybakov - 1997 - Bulletin of Symbolic Logic 3:376-377.
  22.  3
    Completeness of Second-Order Propositional S4 and H in Topological Semantics.Philip Kremer - 2018 - Review of Symbolic Logic 11 (3):507-518.
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  23.  6
    The Guptα-Belnαp Systems S and S* Are Not Axiomatisable.Philip Kremer - 1993 - Notre Dame Journal of Formal Logic 34 (4):583-596.
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  24.  25
    The Topological Product of S4 and S.Philip Kremer - unknown
    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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  25.  19
    Relevant Predication: Grammatical Characterisations. [REVIEW]Philip Kremer - 1989 - 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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  26.  21
    Defining Relevant Implication in a Propositionally Quantified S.Philip Kremer - 1997 - 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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  27.  16
    Propositional Quantification in the Topological Semantics for S.Philip Kremer - 1997 - 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.  17
    The Revision Theory of Truth.Philip Kremer - 2008 - Stanford Encyclopedia of Philosophy.
  29.  47
    Dunn's Relevant Predication, Real Properties and Identity.Philip Kremer - 1997 - 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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  30.  10
    The Modal Logic of Continuous Functions on Cantor Space.Philip Kremer - 2006 - 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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  31.  6
    Topological-Frame Products of Modal Logics.Philip Kremer - 2018 - Studia Logica 106 (6):1097-1122.
    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 (...)
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  32.  70
    Mathematical Logic.Philip Kremer - unknown
    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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  33.  32
    The Incompleteness of S4 ⊕ S4 for the Product Space R × R.Philip Kremer - unknown
    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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  34.  14
    John Woods. Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences, Cambridge University Press, Cambridge, New York, 2003, Xviii+ 362 Pp. [REVIEW]Philip Kremer - 2004 - Bulletin of Symbolic Logic 10 (1):116-118.
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  35.  41
    The Truth is Sometimes Simple.Philip Kremer - manuscript
    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 (...)
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  36.  38
    Does Truth Behave Like a Classical Concept When There is No Vicious Reference?Philip Kremer - unknown
    §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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  37.  10
    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]Richard Bradley, Roya Sorensen, Mirror Notation & Philip Kremer - 1999 - Journal of Philosophical Logic 28:661-662.
  38.  8
    Montréal, Québec, Canada May 17–21, 2006.Jeremy Avigad, Sy Friedman, Akihiro Kanamori, Elisabeth Bouscaren, Philip Kremer, Claude Laflamme, Antonio Montalbán, Justin Moore & Helmut Schwichtenberg - 2007 - Bulletin of Symbolic Logic 13 (1).
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  39.  3
    Defining Relevant Implication in a Propositionally Quantified S4.Philip Kremer - 1997 - 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 $\mathbf{S4}$, 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 $\mathbf{S4\pi}+$.
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  40.  3
    On the Complexity of Propositional Quantification in Intuitionistic Logic.Philip Kremer - 1997 - Journal of Symbolic Logic 62 (2):529-544.
    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.
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  41.  4
    Editorial Introduction.Philip Kremer & Heinrich Wansing - 2010 - Journal of Philosophical Logic 39 (4):341-344.
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  42. Anil Gupta and Nuel Belnap, The Revision Theory of Truth. [REVIEW]Philip Kremer - 1995 - Philosophy in Review 15 (1):39-42.
  43. Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences. [REVIEW]Philip Kremer - 2004 - Bulletin of Symbolic Logic 10 (1):115-117.