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  1. Tomasz Połacik (2008). Back and Forth Between First-Order Kripke Models. Logic Journal of the Igpl 16 (4):335-355.
    We introduce the notion of bisimulation for first-order Kripke models. It is defined as a relation that satisfies certain zig-zag conditions involving back-and-forth moves between nodes of Kripke models and, simultaneously, between the domains of their underlying structures. As one of our main results, we prove that if two Kripke models bisimulate to a certain degree, then they are logically equivalent with respect to the class of formulae of the appropriate complexity. Two applications of the notion introduced in the paper (...)
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  2. Tomasz Połacik (1998). Propositional Quantification in the Monadic Fragment of Intuitionistic Logic. Journal of Symbolic Logic 63 (1):269-300.
    We study the monadic fragment of second order intuitionistic propositional logic in the language containing the standard propositional connectives and propositional quantifiers. It is proved that under the topological interpretation over any dense-in-itself metric space, the considered fragment collapses to Heyting calculus. Moreover, we prove that the topological interpretation over any dense-in-itself metric space of fragment in question coincides with the so-called Pitts' interpretation. We also prove that all the nonstandard propositional operators of the form q $\mapsto \exists$ p (q (...)
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  3. Tomasz Połacik (1998). Pitts' Quantifiers Are Not Topological Quantification. Notre Dame Journal of Formal Logic 39 (4):531-544.
    We show that Pitts' modeling of propositional quantification in intuitionistic logic (as the appropriate interpolants) does not coincide with the topological interpretation. This contrasts with the case of the monadic language and the interpretation over sufficiently regular topological spaces. We also point to the difference between the topological interpretation over sufficiently regular spaces and the interpretation of propositional quantifiers in Kripke models.
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  4. Tomasz Połacik (1994). Second Order Propositional Operators Over Cantor Space. Studia Logica 53 (1):93 - 105.
    We consider propositional operators defined by propositional quantification in intuitionistic logic. More specifically, we investigate the propositional operators of the form $A^{\ast}\colon p\mapsto \exists q)$ where A is one of the following formulae: $\vee \neg \neg q,\rightarrow ,\rightarrow )\rightarrow \vee \neg \neg q)$ . The equivalence of $A^{\ast}$ to $\neg \neg p$ is proved over the standard topological interpretation of intuitionistic second order propositional logic over Cantor space. We relate topological interpretations of second order intuitionistic propositional logic over Cantor space (...)
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