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On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

Luck Darnière & Markus Junker

Archive for Mathematical Logic 49 (7-8):743-771 (2010)

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Leo Esakia on Duality in Modal and Intuitionistic Logics.Guram Bezhanishvili (ed.) - 2014 - Dordrecht, Netherland: Springer.details This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) Download Export citation Bookmark 3 citations
Decidability of admissibility: On a problem by Friedman and its solution by Rybakov.Jeroen P. Goudsmit - 2021 - Bulletin of Symbolic Logic 27 (1):1-38.details Rybakov proved that the admissible rules of $\mathsf {IPC}$ are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility. Download Export citation Bookmark
Decidability of Admissibility: On a Problem by Friedman and its Solution by Rybakov.Jeroen P. Goudsmit - 2021 - Bulletin of Symbolic Logic 27 (1):1-38.details Rybakov (1984a) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility. Download Export citation Bookmark
The Admissible Rules of ${{mathsf{BD}_{2}}}$ and ${mathsf{GSc}}$.Jeroen P. Goudsmit - 2018 - Notre Dame Journal of Formal Logic 59 (3):325-353.details The Visser rules form a basis of admissibility for the intuitionistic propositional calculus. We show how one can characterize the existence of covers in certain models by means of formulae. Through this characterization, we provide a new proof of the admissibility of a weak form of the Visser rules. Finally, we use this observation, coupled with a description of a generalization of the disjunction property, to provide a basis of admissibility for the intermediate logics BD2 and GSc. Download Export citation Bookmark 2 citations
Finitely generated free Heyting algebras: the well-founded initial segment.R. Elageili & J. K. Truss - 2012 - Journal of Symbolic Logic 77 (4):1291-1307.details In this paper we describe the well-founded initial segment of the free Heyting algebra ��α on finitely many, α, generators. We give a complete classification of initial sublattices of ��₂ isomorphic to ��₁ (called 'low ladders'), and prove that for 2 < α < ω, the height of the well-founded initial segment of ��α. Download Export citation Bookmark 3 citations
Hereditarily Structurally Complete Intermediate Logics: Citkin’s Theorem Via Duality.Nick Bezhanishvili & Tommaso Moraschini - 2023 - Studia Logica 111 (2):147-186.details A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer (...) Download Export citation Bookmark 1 citation

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