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  1. Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.) (2012). Advances in Modal Logic 9. College Publications.
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  2. Guram Bezhanishvili, Silvio Ghilardi & Mamuka Jibladze (2010). An Algebraic Approach to Subframe Logics. Modal Case. Notre Dame Journal of Formal Logic 52 (2):187-202.
    We prove that if a modal formula is refuted on a wK4-algebra ( B ,□), then it is refuted on a finite wK4-algebra which is isomorphic to a subalgebra of a relativization of ( B ,□). As an immediate consequence, we obtain that each subframe and cofinal subframe logic over wK4 has the finite model property. On the one hand, this provides a purely algebraic proof of the results of Fine and Zakharyaschev for K4 . On the other hand, it (...)
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  3. Silvio Ghilardi (2010). Continuity, Freeness, and Filtrations. Journal of Applied Non-Classical Logics 20 (3):193-217.
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  4. Franz Baader & Silvio Ghilardi (2007). Connecting Many-Sorted Theories. Journal of Symbolic Logic 72 (2):535 - 583.
    Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called E-connection approach for combining modal logics to an (...)
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  5. Guram Bezhanishvili & Silvio Ghilardi (2007). An Algebraic Approach to Subframe Logics. Intuitionistic Case. Annals of Pure and Applied Logic 147 (1):84-100.
    We develop duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces. We show that these binary relations are in 1–1 correspondence with subframes of Heyting spaces. We introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear iff it is a subframe variety, and that it is dense nuclear iff it is a cofinal subframe variety. We give an alternative proof that every subframe variety (...)
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  6. Silvio Ghilardi (2004). Unification, Finite Duality and Projectivity in Varieties of Heyting Algebras. Annals of Pure and Applied Logic 127 (1-3):99-115.
    We investigate finitarity of unification types in locally finite varieties of Heyting algebras, giving both positive and negative results. We make essential use of finite dualities within a conceptualization for E-unification theory 733–752) relying on the algebraic notion of a projective object.
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  7. Silvio Ghilardi & Lorenzo Sacchetti (2004). Filtering Unification and Most General Unifiers in Modal Logic. Journal of Symbolic Logic 69 (3):879-906.
    We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).
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  8. Silvio Ghilardi & Daniele Mundici (2003). Foreword. Studia Logica 73 (1):1-1.
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  9. Silvio Ghilardi (2000). Best Solving Modal Equations. Annals of Pure and Applied Logic 102 (3):183-198.
    We show that some common varieties of modal K4-algebras have finitary unification type, thus providing effective best solutions for equations in free algebras. Applications to admissible inference rules are immediate.
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  10. Silvio Ghilardi (1999). Unification in Intuitionistic Logic. Journal of Symbolic Logic 64 (2):859-880.
    We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.
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  11. Silvio Ghilardi & Pierangelo Miglioli (1999). On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics. Studia Logica 63 (3):353-385.
    By using algebraic-categorical tools, we establish four criteria in order to disprove canonicity, strong completeness, w-canonicity and strong w-completeness, respectively, of an intermediate propositional logic. We then apply the second criterion in order to get the following result: all the logics defined by extra-intuitionistic one-variable schemata, except four of them, are not strongly complete. We also apply the fourth criterion in order to prove that the Gabbay-de Jongh logic D1 is not strongly w-complete.
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  12. Silvio Ghilardi & Giancarlo Meloni (1997). Constructive Canonicity in Non-Classical Logics. Annals of Pure and Applied Logic 86 (1):1-32.
    Sufficient syntactic conditions for canonicity in intermediate and intuitionistic modal logics are given. We present a new technique which does not require semantic first-order reduction and which is constructive in the sense that it works in an intuitionistic metatheory through a model without points which is classically isomorphic to the usual canonical model.
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  13. Silvio Ghilardi & Marek Zawadowski (1997). Model Completions and R-Heyting Categories. Annals of Pure and Applied Logic 88 (1):27-46.
    Under some assumptions on an equational theory S , we give a necessary and sufficient condition so that S admits a model completion. These assumptions are often met by the equational theories arising from logic. They say that the dual of the category of finitely presented S-algebras has some categorical stucture. The results of this paper combined with those of [7] show that all the 8 theories of amalgamable varieties of Heyting algebras [12] admit a model completion. Further applications to (...)
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  14. Silvio Ghilardi & Giancarlo Meloni (1996). Relational and Partial Variable Sets and Basic Predicate Logic. Journal of Symbolic Logic 61 (3):843-872.
    In this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are (...)
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  15. Silvio Ghilardi (1995). An Algebraic Theory of Normal Forms. Annals of Pure and Applied Logic 71 (3):189-245.
    In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...)
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  16. Silvio Ghilardi & Marek Zawadowski (1995). A Sheaf Representation and Duality for Finitely Presented Heyting Algebras. Journal of Symbolic Logic 60 (3):911-939.
    A. M. Pitts in [Pi] proved that HA op fp is a bi-Heyting category satisfying the Lawrence condition. We show that the embedding $\Phi: HA^\mathrm{op}_\mathrm{fp} \longrightarrow Sh(\mathbf{P_0,J_0})$ into the topos of sheaves, (P 0 is the category of finite rooted posets and open maps, J 0 the canonical topology on P 0 ) given by $H \longmapsto HA(H,\mathscr{D}(-)): \mathbf{P_0} \longrightarrow \text{Set}$ preserves the structure mentioned above, finite coproducts, and subobject classifier, it is also conservative. This whole structure on HA op (...)
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  17. Silvio Ghilardi & Marek Zawadowski (1995). Undefinability of Propositional Quantifiers in the Modal System S. Studia Logica 55 (2):259 - 271.
    We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and.
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  18. Silvio Ghilardi (1992). Quantified Extensions of Canonical Propositional Intermediate Logics. Studia Logica 51 (2):195 - 214.
    The quantified extension of a canonical prepositional intermediate logic is complete with respect to the generalization of Kripke semantics taking into consideration set-valued functors defined on a category.
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  19. Silvio Ghilardi (1991). Incompleteness Results in Kripke Semantics. Journal of Symbolic Logic 56 (2):517-538.
    By means of models in toposes of C-sets (where C is a small category), necessary conditions are found for the minimum quantified extension of a propositional (intermediate, modal) logic to be complete with respect to Kripke semantics; in particular, many well-known systems turn out to be incomplete.
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  20. Giovanna Corsi & Silvio Ghilardi (1989). Directed Frames. Archive for Mathematical Logic 29 (1):53-67.
    Predicate extensions of the intermediate logic of the weak excluded middle and of the modal logic S4.2 are introduced and investigated. In particular it is shown that some of them are characterized by subclasses of the class of directed frames with either constant or nested domains.
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  21. Silvio Ghilardi (1989). Presheaf Semantics and Independence Results for Some Non-Classical First-Order Logics. Archive for Mathematical Logic 29 (2):125-136.
    The logicD-J of the weak exluded middle with constant domains is proved to be incomplete with respect to Kripke semantics, by introducing models in presheaves on an arbitrary category. Additional incompleteness results are obtained for the modal systems with nested domains extendingQ-S4.1.
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