Results for 'Modal Heyting algebras'

969 found
Order:
  1. Bi-Heyting algebras, toposes and modalities.Gonzalo E. Reyes & Houman Zolfaghari - 1996 - Journal of Philosophical Logic 25 (1):25 - 43.
    The aim of this paper is to introduce a new approach to the modal operators of necessity and possibility. This approach is based on the existence of two negations in certain lattices that we call bi-Heyting algebras. Modal operators are obtained by iterating certain combinations of these negations and going to the limit. Examples of these operators are given by means of graphs.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   20 citations  
  2.  30
    Heyting Algebras: Duality Theory.Leo Esakia - 2019 - Cham, Switzerland: Springer Verlag.
    This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   31 citations  
  3.  26
    Heyting Algebras with Operators.Yasusi Hasimoto - 2001 - Mathematical Logic Quarterly 47 (2):187-196.
    In this paper, we will give a general description of subdirectly irreducible Heyting algebras with operators under some weak conditions, which includes the finite case, the normal case and the case for Boolean algebras with diamond operator. This can be done by normalizing these operators. This answers the question posed in Wolter [4].
    Direct download  
     
    Export citation  
     
    Bookmark   2 citations  
  4.  90
    Fatal Heyting Algebras and Forcing Persistent Sentences.Leo Esakia & Benedikt Löwe - 2012 - Studia Logica 100 (1-2):163-173.
    Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  5. Varieties of monadic Heyting algebras. Part I.Guram Bezhanishvili - 1998 - Studia Logica 61 (3):367-402.
    This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   21 citations  
  6.  52
    Symmetrical Heyting algebras with a finite order type of operators.Luisa Iturrioz - 1995 - Studia Logica 55 (1):89 - 98.
    The main purpose of this paper is to introduce a class of algebraic structures related to many-valued ukasiewicz algebras. They are symmetrical Heyting algebras with a set of modal operators indexed by a finite completely symmetric poset. A representation theorem is given for these (not functionally complete) algebras.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  7.  39
    n‐linear weakly Heyting algebras.Sergio A. Celani - 2006 - Mathematical Logic Quarterly 52 (4):404-416.
    The present paper introduces and studies the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋn of n-linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety [MATHEMATICAL SCRIPT CAPITAL W]ℋ2 that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic (...) introduced in [2]. (shrink)
    Direct download  
     
    Export citation  
     
    Bookmark   3 citations  
  8.  71
    Varieties of monadic Heyting algebras part II: Duality theory.Guram Bezhanishvili - 1999 - Studia Logica 62 (1):21-48.
    In this paper we continue the investigation of monadic Heyting algebras which we started in [2]. Here we present the representation theorem for monadic Heyting algebras and develop the duality theory for them. As a result we obtain an adequate topological semantics for intuitionistic modal logics over MIPC along with a Kripke-type semantics for them. It is also shown the importance and the effectiveness of the duality theory for further investigation of monadic Heyting (...) and logics over MIPC. (shrink)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  9.  58
    Varieties of monadic Heyting algebras. Part III.Guram Bezhanishvili - 2000 - Studia Logica 64 (2):215-256.
    This paper is the concluding part of [1] and [2], and it investigates the inner structure of the lattice (MHA) of all varieties of monadic Heyting algebras. For every n , we introduce and investigate varieties of depth n and cluster n, and present two partitions of (MHA), into varieties of depth n, and into varieties of cluster n. We pay a special attention to the lower part of (MHA) and investigate finite and critical varieties of monadic (...) algebras in detail. In particular, we prove that there exist exactly thirteen critical varieties in (MHA) and that it is decidable whether a given variety of monadic Heyting algebras is finite or not. The representation of (MHA) is also given. All these provide us with a satisfactory insight into (MHA). Since (MHA) is dual to the lattice NExtMIPC of all normal extensions of the intuitionistic modal logic MIPC, we also obtain a clearer picture of the lattice structure of intuitionistic modal logics over MIPC. (shrink)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  10.  70
    Frontal Operators in Weak Heyting Algebras.Sergio A. Celani & Hernán J. San Martín - 2012 - Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation τ(a)b(ba){\tau(a) \leq b \vee (b \rightarrow a)}, for all a,bA{a, b \in A}. These operators were studied from an algebraic, logical and topological point (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  11.  25
    Frontal Operators in Weak Heyting Algebras.Sergio A. Celani & Hern?N. J. San Mart?N. - 2012 - Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator r preserving finite meets which also satisfies the equation?? b V, for all a,b? A. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  12.  41
    The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic ★.Leo Esakia - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):349-366.
    In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   21 citations  
  13.  51
    The universal modality, the center of a Heyting algebra, and the Blok–Esakia theorem.Guram Bezhanishvili - 2010 - Annals of Pure and Applied Logic 161 (3):253-267.
    We introduce the bimodal logic , which is the extension of Bennett’s bimodal logic by Grzegorczyk’s axiom □→p)→p and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of , thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logic WS5.C, which is the extension of WS5 by the axiom →, and the bimodal logic , which is the extension of Shehtman’s bimodal logic by (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  14.  50
    Not Every Splitting Heyting or Interior Algebra is Finitely Presentable.Alex Citkin - 2012 - Studia Logica 100 (1-2):115-135.
    We give an example of a variety of Heyting algebras and of a splitting algebra in this variety that is not finitely presentable. Moreover, we show that the corresponding splitting pair cannot be defined by any finitely presentable algebra. Also, using the Gödel-McKinsey-Tarski translation and the Blok-Esakia theorem, we construct a variety of Grzegorczyk algebras with similar properties.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  15. Intuitionistic Modal Algebras.Sergio A. Celani & Umberto Rivieccio - 2024 - Studia Logica 112 (3):611-660.
    Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  16.  62
    Model completions and r-Heyting categories.Silvio Ghilardi & Marek Zawadowski - 1997 - 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. (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  17.  35
    Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic.Minghui Ma, Alessandra Palmigiano & Mehrnoosh Sadrzadeh - 2014 - Annals of Pure and Applied Logic 165 (4):963-995.
    In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   9 citations  
  18.  55
    Modalities as interactions between the classical and the intuitionistic logics.Michał Walicki - 2006 - Logic and Logical Philosophy 15 (3):193-215.
    We give an equivalent formulation of topological algebras, interpreting S4, as boolean algebras equipped with intuitionistic negation. The intuitionistic substructure—Heyting algebra—of such an algebra can be then seen as an “epistemic subuniverse”, and modalities arise from the interaction between the intuitionistic and classical negations or, we might perhaps say, between the epistemic and the ontological aspects: they are not relations between arbitrary alternatives but between intuitionistic substructures and one common world governed by the classical (propositional) logic. As (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark  
  19.  38
    Krull dimension in modal logic.Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan & Jan van Mill - 2017 - Journal of Symbolic Logic 82 (4):1356-1386.
    We develop the theory of Krull dimension forS4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for aT1-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  20.  62
    Compatibility and accessibility: lattice representations for semantics of non-classical and modal logics.Wesley Holliday - 2022 - In David Fernández Duque & Alessandra Palmigiano, Advances in Modal Logic, Vol. 14. College Publications. pp. 507-529.
    In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  21.  65
    Bounded distributive lattices with strict implication.Sergio Celani & Ramon Jansana - 2005 - Mathematical Logic Quarterly 51 (3):219-246.
    The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as (...)
    Direct download  
     
    Export citation  
     
    Bookmark   29 citations  
  22.  41
    Connected modal logics.Guram Bezhanishvili & David Gabelaia - 2011 - Archive for Mathematical Logic 50 (3-4):287-317.
    We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  23.  49
    Modal Consequence Relations Extending $mathbf{S4.3}$: An Application of Projective Unification.Wojciech Dzik & Piotr Wojtylak - 2016 - Notre Dame Journal of Formal Logic 57 (4):523-549.
    We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  24. Belief Modalities Defined by Nuclei.Thomas Mormann - manuscript
    Abstract. The aim of this paper is to show that the topological interpretation of knowledge as an interior kernel operator K of a topological space (X, OX) comes along with a partially ordered family of belief modalities B that fit K in the sense that the pairs (K, B) satisfy all axioms of Stalnaker’s KB logic of knowledge and belief with the exception of the contentious axiom of negative introspection (NI). The new belief modalities B introduced in this paper are (...)
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark  
  25.  23
    On modal and intuitionistic logics: Guram Bezhanishvili : Leo Esakia on duality in modal and intuitionistic logics. Dordrecht: Springer, 2014, 334pp, €107.09 HB.Costas Dimitracopoulos - 2014 - Metascience 24 (2):337-340.
    The volume under review contains work dedicated to the memory of Leo Esakia, who died in 2010, after having worked for over 40 years towards developing duality theory for modal and intuitionistic logics. The collection comprises ten technical contributions that follow the first chapter, in which the reader can find information on Esakia’s studies and career, as well as a complete list of his research publications. In the sequel, we will refer briefly to each of these ten chapters, following (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  26.  57
    Physical Properties as Modal Operators in the Topos Approach to Quantum Mechanics.Hector Freytes, Graciela Domenech & Christian de Ronde - 2014 - Foundations of Physics 44 (12):1357-1368.
    In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark  
  27.  34
    Computable Heyting Algebras with Distinguished Atoms and Coatoms.Nikolay Bazhenov - 2023 - Journal of Logic, Language and Information 32 (1):3-18.
    The paper studies Heyting algebras within the framework of computable structure theory. We prove that the class _K_ containing all Heyting algebras with distinguished atoms and coatoms is complete in the sense of the work of Hirschfeldt et al. (Ann Pure Appl Logic 115(1-3):71-113, 2002). This shows that the class _K_ is rich from the computability-theoretic point of view: for example, every possible degree spectrum can be realized by a countable structure from _K_. In addition, there (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  28.  16
    Leo Esakia on Duality in Modal and Intuitionistic Logics.Guram Bezhanishvili (ed.) - 2014 - Dordrecht, Netherland: Springer.
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  29.  38
    Locales, Nuclei, and Dragalin Frames.Guram Bezhanishvili & Wesley Holliday - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté, Advances in Modal Logic, Volume 11. CSLI Publications. pp. 177-196.
    It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a poset, forming what (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  30.  38
    A Catalog ofWeak Many-Valued Modal Axioms and their Corresponding Frame Classes.Costas D. Koutras - 2003 - Journal of Applied Non-Classical Logics 13 (1):47-71.
    In this paper we provide frame definability results for weak versions of classical modal axioms that can be expressed in Fitting's many-valued modal languages. These languages were introduced by M. Fitting in the early '90s and are built on Heyting algebras which serve as the space of truth values. The possible-worlds frames interpreting these languages are directed graphs whose edges are labelled with an element of the underlying Heyting algebra, providing us a form of many-valued (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  31.  40
    Frame constructions, truth invariance and validity preservation in many-valued modal logic.Pantelis E. Eleftheriou & Costas D. Koutras - 2005 - Journal of Applied Non-Classical Logics 15 (4):367-388.
    In this paper we define and examine frame constructions for the family of manyvalued modal logics introduced by M. Fitting in the '90s. Every language of this family is built on an underlying space of truth values, a Heyting algebra H. We generalize Fitting's original work by considering complete Heyting algebras as truth spaces and proceed to define a suitable notion of H-indexed families of generated subframes, disjoint unions and bounded morphisms. Then, we provide an algebraic (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  32. Autoreferential semantics for many-valued modal logics.Zoran Majkic - 2008 - Journal of Applied Non-Classical Logics 18 (1):79-125.
    In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices (Heyting algebras) is based on relative (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  33.  51
    Topos Semantics for Higher-Order Modal Logic.Steve Awodey, Kohei Kishida & Hans-Cristoph Kotzsch - 2014 - Logique Et Analyse 228:591-636.
    We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known interpretation of higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE, but rather by a suitable complete Heyting algebra H. The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  34.  55
    Canonicity and Completeness Results for Many-Valued Modal Logics.Costas D. Koutras, Christos Nomikos & Pavlos Peppas - 2002 - Journal of Applied Non-Classical Logics 12 (1):7-42.
    We prove frame determination results for the family of many-valued modal logics introduced by M. Fitting in the early '90s. Each modal language of this family is based on a Heyting algebra, which serves as the space of truth values, and is interpreted on an interesting version of possible-worlds semantics: the modal frames are directed graphs whose edges are labelled with an element of the underlying Heyting algebra. We introduce interesting generalized forms of the classical (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  35.  19
    On Heyting Algebras with Negative Tense Operators.Federico G. Almiñana, Gustavo Pelaitay & William Zuluaga - 2023 - Studia Logica 111 (6):1015-1036.
    In this paper, we will study Heyting algebras endowed with tense negative operators, which we call tense H-algebras and we proof that these algebras are the algebraic semantics of the Intuitionistic Propositional Logic with Galois Negations. Finally, we will develop a Priestley-style duality for tense H-algebras.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  36.  60
    Inquisitive Heyting Algebras.Vít Punčochář - 2021 - Studia Logica 109 (5):995-1017.
    In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures and provide several alternative characterizations of these algebras. For instance, it is shown that a (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  37.  85
    Expansions of Semi-Heyting Algebras I: Discriminator Varieties.H. P. Sankappanavar - 2011 - Studia Logica 98 (1-2):27-81.
    This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   9 citations  
  38.  13
    Degree of Satisfiability in Heyting Algebras.Benjamin Merlin Bumpus & Zoltan A. Kocsis - forthcoming - Journal of Symbolic Logic:1-19.
    We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top $ is no larger than $\frac {2}{3}$. Finally, (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  39.  55
    Locally Finite Reducts of Heyting Algebras and Canonical Formulas.Guram Bezhanishvili & Nick Bezhanishvili - 2017 - Notre Dame Journal of Formal Logic 58 (1):21-45.
    The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  40.  35
    Formulas of one propositional variable in intuitionistic logic with the Solovay modality.Leo Esakia & Revaz Grigolia - 2008 - Logic and Logical Philosophy 17 (1-2):111-127.
    A description of the free cyclic algebra over the variety of Solovay algebras, as well as over its pyramid locally finite subvarieties is given.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  41.  28
    Semi-Heyting Algebras and Identities of Associative Type.Juan M. Cornejo & Hanamantagouda P. Sankappanavar - 2019 - Bulletin of the Section of Logic 48 (2).
    An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ ≈ x ∧ y, x ∧ ≈ x ∧ [ → ], and x → x ≈ 1.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  42.  62
    Finitely generated free Heyting algebras: the well-founded initial segment.R. Elageili & J. K. Truss - 2012 - Journal of Symbolic Logic 77 (4):1291-1307.
    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.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  43.  26
    On self‐distributive weak Heyting algebras.Mohsen Nourany, Shokoofeh Ghorbani & Arsham Borumand Saeid - 2023 - Mathematical Logic Quarterly 69 (2):192-206.
    We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  44. Decidability problem for finite Heyting algebras.Katarzyna Idziak & Pawel M. Idziak - 1988 - Journal of Symbolic Logic 53 (3):729-735.
    The aim of this paper is to characterize varieties of Heyting algebras with decidable theory of their finite members. Actually we prove that such varieties are exactly the varieties generated by linearly ordered algebras. It contrasts to the result of Burris [2] saying that in the case of whole varieties, only trivial variety and the variety of Boolean algebras have decidable first order theories.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  45.  2
    On Geometric Implications.Amirhossein Akbar Tabatabai - 2025 - Studia Logica 113 (1):79-108.
    It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  46.  20
    On Geometric Implications.Amirhossein Akbar Tabatabai - forthcoming - Studia Logica:1-30.
    It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  47.  21
    One-Variable Fragments of First-Order Logics.Petr Cintula, George Metcalfe & Naomi Tokuda - 2024 - Bulletin of Symbolic Logic 30 (2):253-278.
    The one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts $\mathrm {S5}$ and $\mathrm {MIPC}$ of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  48.  48
    On some Classes of Heyting Algebras with Successor that have the Amalgamation Property.José L. Castiglioni & Hernán J. San Martín - 2012 - Studia Logica 100 (6):1255-1269.
    In this paper we shall prove that certain subvarieties of the variety of Salgebras (Heyting algebras with successor) has amalgamation. This result together with an appropriate version of Theorem 1 of [L. L. Maksimova, Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras, Algebra i Logika, 16(6):643-681, 1977] allows us to show interpolation in the calculus IPC S (n), associated with these varieties.We use that every algebra in any of the varieties of S-algebras studied (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  49.  43
    Intuitionistic Propositional Logic with Galois Negations.Minghui Ma & Guiying Li - 2023 - Studia Logica 111 (1):21-56.
    Intuitionistic propositional logic with Galois negations ( IGN\mathsf {IGN} ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair (¬,)(\lnot,{\sim }) and dual Galois pair (¬˙,˙)(\dot{\lnot },\dot{\sim }) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for IGN\mathsf {IGN} are developed. A Hilbert-style axiomatic system HN\mathsf {HN} is given for IGN\mathsf {IGN}, and Galois negation logics are defined as extensions of \(\mathsf (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  50.  32
    Contrapositionally complemented Heyting algebras and intuitionistic logic with minimal negation.Anuj Kumar More & Mohua Banerjee - 2023 - Logic Journal of the IGPL 31 (3):441-474.
    Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
1 — 50 / 969