# Philip Kremer

Areas of Specialization 1 more
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##### Topological-Frame Products of Modal Logics Studia Logica 106 (6): 1097-1122. 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 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 …Read more
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##### On the Complexity of Propositional Quantification in Intuitionistic Logic Journal of Symbolic Logic 62 (2): 529-544. 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.
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##### The Incompleteness of S4 {bigoplus} S4 for the Product Space Studia Logica 103 (1): 219-226. 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
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##### Defining Relevant Implication in a Propositionally Quantified S4 Journal 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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##### Dynamic topological logic with Giorgi Mints Annals 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
• ##### Real Properties, Relevance Logic, and Identity Dissertation, 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