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Positive modal logic

Studia Logica 55 (2):301 - 317 (1995)

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  1. (1 other version)Modes of Adjointness.M. Menni & C. Smith - 2013 - Journal of Philosophical Logic (2-3):1-27.
    The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues related to the calculi and not on the significance of adjunctions in modal logic. It then seems a worthy enterprise (both (...)
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  • One Variable Relevant Logics are S5ish.Nicholas Ferenz - 2024 - Journal of Philosophical Logic 53 (4):909-931.
    Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula $$\mathcal {A}$$ A of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt [15] semantics for (...)
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  • Positive modal logic beyond distributivity.Nick Bezhanishvili, Anna Dmitrieva, Jim de Groot & Tommaso Moraschini - 2024 - Annals of Pure and Applied Logic 175 (2):103374.
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  • On Equational Completeness Theorems.Tommaso Moraschini - 2022 - Journal of Symbolic Logic 87 (4):1522-1575.
    A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence relative to some class of algebras. We characterize logics admitting an equational completeness theorem that are either locally tabular or have some tautology. In particular, it is shown that a protoalgebraic logic admits an equational completeness theorem precisely when it has two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem is shown to (...)
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  • Monotone Subintuitionistic Logic: Duality and Transfer Results.Jim de Groot & Dirk Pattinson - 2022 - Notre Dame Journal of Formal Logic 63 (2):213-242.
    We consider subintuitionistic logics as an extension of positive propositional logic with a binary modality, interpreted over ordered and unordered monotone neighborhood frames, with a range of frame conditions. This change in perspective allows us to apply tools and techniques from the modal setting to subintuitionistic logics. We provide a Priestley-style duality, and transfer results from the (classical) logic of monotone neighborhood frames to obtain completeness, conservativity, and a finite model property for the basic logic, extended with a number of (...)
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  • Varieties of Relevant S5.Shawn Standefer - 2023 - Logic and Logical Philosophy 32 (1):53–80.
    In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal (...)
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  • Quantified Modal Relevant Logics.Nicholas Ferenz - 2023 - Review of Symbolic Logic 16 (1):210-240.
    Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...)
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  • Positive logic with adjoint modalities: Proof theory, semantics, and reasoning about information: Positive logic with adjoint modalities.Mehrnoosh Sadrzadeh - 2010 - Review of Symbolic Logic 3 (3):351-373.
    We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of _T_, _S4_, and _S5_, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of such (...)
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  • Positive Monotone Modal Logic.Jim de Groot - 2021 - Studia Logica 109 (4):829-857.
    Positive monotone modal logic is the negation- and implication-free fragment of monotone modal logic, i.e., the fragment with connectives and. We axiomatise positive monotone modal logic, give monotone neighbourhood semantics based on posets, and prove soundness and completeness. The latter follows from the main result of this paper: a duality between so-called \-spaces and the algebraic semantics of positive monotone modal logic. The main technical tool is the use of coalgebra.
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  • Kripke completeness of strictly positive modal logics over meet-semilattices with operators.Stanislav Kikot, Agi Kurucz, Yoshihito Tanaka, Frank Wolter & Michael Zakharyaschev - 2019 - Journal of Symbolic Logic 84 (2):533-588.
    Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same (...)
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  • First-Degree Entailment and its Relatives.Yaroslav Shramko, Dmitry Zaitsev & Alexander Belikov - 2017 - Studia Logica 105 (6):1291-1317.
    We consider a family of logical systems for representing entailment relations of various kinds. This family has its root in the logic of first-degree entailment formulated as a binary consequence system, i.e. a proof system dealing with the expressions of the form \, where both \ and \ are single formulas. We generalize this approach by constructing consequence systems that allow manipulating with sets of formulas, either to the right or left of the turnstile. In this way, it is possible (...)
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  • A modal translation for dual-intuitionistic logic.Yaroslav Shramko - 2016 - Review of Symbolic Logic 9 (2):251-265.
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  • On the Deductive System of the Order of an Equationally Orderable Quasivariety.Ramon Jansana - 2016 - Studia Logica 104 (3):547-566.
    We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.
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  • An Intriguing Logic with Two Implicational Connectives.Lloyd Humberstone - 2000 - Notre Dame Journal of Formal Logic 41 (1):1-40.
    Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two (...)
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  • 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 $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension |${\textrm {ILM}}$|-|${\vee }$| for c|$\vee (...)
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  • Between Hilbert and Gentzen: four-valued consequence systems and structural reasoning.Yaroslav Shramko - 2022 - Archive for Mathematical Logic 61 (5):627-651.
    Structural reasoning is simply reasoning that is governed exclusively by structural rules. In this context a proof system can be said to be structural if all of its inference rules are structural. A logic is considered to be structuralizable if it can be equipped with a sound and complete structural proof system. This paper provides a general formulation of the problem of structuralizability of a given logic, giving specific consideration to a family of logics that are based on the Dunn–Belnap (...)
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  • Hilbert-style axiomatization of first-degree entailment and a family of its extensions.Yaroslav Shramko - 2021 - Annals of Pure and Applied Logic 172 (9):103011.
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  • The Strong Version of a Sentential Logic.Ramon Jansana, Josep Maria Font & Hugo Albuquerque - 2017 - Studia Logica 105 (4):703-760.
    This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic examples are (...)
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  • Sequent Systems for Negative Modalities.Ori Lahav, João Marcos & Yoni Zohar - 2017 - Logica Universalis 11 (3):345-382.
    Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used (...)
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  • Restricted Arrow.C. M. Asmus - 2009 - Journal of Philosophical Logic 38 (4):405-431.
    In this paper I present a range of substructural logics for a conditional connective ↦. This connective was original introduced semantically via restriction on the ternary accessibility relation R for a relevant conditional. I give sound and complete proof systems for a number of variations of this semantic definition. The completeness result in this paper proceeds by step-by-step improvements of models, rather than by the one-step canonical model method. This gradual technique allows for the additional control, lacking in the canonical (...)
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  • 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 pseudo-complements as in (...)
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  • Kripke-Completeness and Sequent Calculus for Quasi-Boolean Modal Logic.Minghui Ma & Juntong Guo - forthcoming - Studia Logica:1-30.
    Quasi-Boolean modal algebras are quasi-Boolean algebras with a modal operator satisfying the interaction axiom. Sequential quasi-Boolean modal logics and the relational semantics are introduced. Kripke-completeness for some quasi-Boolean modal logics is shown by the canonical model method. We show that every descriptive persistent quasi-Boolean modal logic is canonical. The finite model property of some quasi-Boolean modal logics is proved. A cut-free Gentzen sequent calculus for the minimal quasi-Boolean logic is developed and we show that it has the Craig interpolation property.
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  • Knowledge and ignorance in Belnap–Dunn logic.Daniil Kozhemiachenko & Liubov Vashentseva - forthcoming - Logic Journal of the IGPL.
    In this paper, we argue that the usual approach to modelling knowledge and belief with the necessity modality |$\Box $| does not produce intuitive outcomes in the framework of the Belnap–Dunn logic (⁠|$\textsf{BD}$|⁠, alias |$\textbf{FDE}$|—first-degree entailment). We then motivate and introduce a nonstandard modality |$\blacksquare $| that formalizes knowledge and belief in |$\textsf{BD}$| and use |$\blacksquare $| to define |$\bullet $| and |$\blacktriangledown $| that formalize the unknown truth and ignorance as not knowing whether, respectively. Moreover, we introduce another modality (...)
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  • Lambek Calculus with Conjugates.Igor Sedlár & Andrew Tedder - 2020 - Studia Logica 109 (3):447-470.
    We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.
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  • Proof systems for various fde-based modal logics.Sergey Drobyshevich & Heinrich Wansing - 2020 - Review of Symbolic Logic 13 (4):720-747.
    We present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing and Odintsov & Wansing, as well as the modal logic KN4 with strong implication introduced in Goble. In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system (...)
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  • Proof Theory for Functional Modal Logic.Shawn Standefer - 2018 - Studia Logica 106 (1):49-84.
    We present some proof-theoretic results for the normal modal logic whose characteristic axiom is \. We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively. We close by proving equivalences with a Fitch-style proof system for revision theory.
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  • Constant Domain Quantified Modal Logics Without Boolean Negation.Greg Restall - 2005 - Australasian Journal of Logic 3:45-62.
    his paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a two-place modal operator, the relevant conditional. The completeness proof is adapted from a proof for classical modal predicate logic (I follow James Garson’s 1984 presentation of the completeness proof quite closely), but (...)
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  • Four-valued Logic.Katalin Bimbó & J. Michael Dunn - 2001 - Notre Dame Journal of Formal Logic 42 (3):171-192.
    Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system (...)
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  • From positive PDL to its non-classical extensions.Igor Sedlár & Vít Punčochář - 2019 - Logic Journal of the IGPL 27 (4):522-542.
    We provide a complete binary implicational axiomatization of the positive fragment of propositional dynamic logic. The intended application of this result are completeness proofs for non-classical extensions of positive PDL. Two examples are discussed in this article, namely, a paraconsistent extension with modal De Morgan negation and a substructural extension with the residuated operators of the non-associative Lambek calculus. Informal interpretations of these two extensions are outlined.
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  • Algorithmic correspondence and canonicity for non-distributive logics.Willem Conradie & Alessandra Palmigiano - 2019 - Annals of Pure and Applied Logic 170 (9):923-974.
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  • The Fmla-Fmla Axiomatizations of the Exactly True and Non-falsity Logics and Some of Their Cousins.Yaroslav Shramko, Dmitry Zaitsev & Alexander Belikov - 2019 - Journal of Philosophical Logic 48 (5):787-808.
    In this paper we present a solution of the axiomatization problem for the Fmla-Fmla versions of the Pietz and Rivieccio exactly true logic and the non-falsity logic dual to it. To prove the completeness of the corresponding binary consequence systems we introduce a specific proof-theoretic formalism, which allows us to deal simultaneously with two consequence relations within one logical system. These relations are hierarchically organized, so that one of them is treated as the basic for the resulting logic, and the (...)
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  • Compatibility operators in abstract algebraic logic.Hugo Albuquerque, Josep Maria Font & Ramon Jansana - 2016 - Journal of Symbolic Logic 81 (2):417-462.
    This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz (...)
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  • Elementary definability and completeness in general and positive modal logic.Ernst Zimmermann - 2003 - Journal of Logic, Language and Information 12 (1):99-117.
    The paper generalises Goldblatt's completeness proof for Lemmon–Scott formulas to various modal propositional logics without classical negation and without ex falso, up to positive modal logic, where conjunction and disjunction, andwhere necessity and possibility are respectively independent.Further the paper proves definability theorems for Lemmon–Scottformulas, which hold even in modal propositional languages without negation and without falsum. Both, the completeness theorem and the definability theoremmake use only of special constructions of relations,like relation products. No second order logic, no general frames are (...)
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  • An Escape From Vardanyan’s Theorem.Ana de Almeida Borges & Joost J. Joosten - 2023 - Journal of Symbolic Logic 88 (4):1613-1638.
    Vardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$ —the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf (...)
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  • Axiomatization of Crisp Gödel Modal Logic.Ricardo Oscar Rodriguez & Amanda Vidal - 2021 - Studia Logica 109 (2):367-395.
    In this paper we consider the modal logic with both $$\Box $$ and $$\Diamond $$ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra $$[0,1]_G$$. We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most (...)
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  • Order- dual realational semantics for non-distributive propositional logics.Chrysafis Hartonas - 2016 - Logic Journal of the IGPL 25 (2):145-182.
    This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n -ary additive (...)
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  • Full Models for Positive Modal Logic.Ramon Jansana - 2002 - Mathematical Logic Quarterly 48 (3):427-445.
    The positive fragment of the local modal consequence relation defined by the class of all Kripke frames is studied in the context ofAlgebraic Logic. It is shown that this fragment is non-protoalgebraic and that its class of canonically associated algebras according to the criteria set up in [7] is the class of positive modal algebras. Moreover its full models are characterized as the models of the Gentzen calculus introduced in [3].
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  • A New Semantics for Positive Modal Logic.S. Celani & R. Jansana - 1997 - Notre Dame Journal of Formal Logic 38 (1):1-18.
    The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering on the set of possible worlds and an accessibility relation connected to the quasi ordering by the conditions (1) that the composition of with is included in the composition of with and (2) the analogous for the inverse of and . This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions (...)
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  • Neighbourhood Semantics for Modal Relevant Logics.Nicholas Ferenz & Andrew Tedder - 2023 - Journal of Philosophical Logic 52 (1):145-181.
    In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.
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  • Hennessy-Milner and Van Benthem for Instantial Neighbourhood Logic.Jim de Groot - 2022 - Studia Logica 110 (3):717-743.
    We investigate bisimulations for instantial neighbourhood logic and an \-indexed collection of its fragments. For each of these logics we give a Hennessy-Milner theorem and a Van Benthem-style characterisation theorem.
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  • Weakening of Intuitionistic Negation for Many-valued Paraconsistent da Costa System.Zoran Majkić - 2008 - Notre Dame Journal of Formal Logic 49 (4):401-424.
    In this paper we propose substructural propositional logic obtained by da Costa weakening of the intuitionistic negation. We show that the positive fragment of the da Costa system is distributive lattice logic, and we apply a kind of da Costa weakening of negation, by preserving, differently from da Costa, its fundamental properties: antitonicity, inversion, and additivity for distributive lattices. The other stronger paraconsistent logic with constructive negation is obtained by adding an axiom for multiplicative property of weak negation. After that, (...)
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  • A General Framework for $$ {FDE}$$ FDE -Based Modal Logics.Sergey Drobyshevich - 2020 - Studia Logica 108 (6):1281-1306.
    We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the (...)
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  • Duality for Coalgebras for Vietoris and Monadicity.Marco Abbadini & Ivan di Liberti - forthcoming - Journal of Symbolic Logic:1-34.
    We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.
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  • Modal operators for meet-complemented lattices.José Luis Castiglioni & Rodolfo C. Ertola-Biraben - 2017 - Logic Journal of the IGPL 25 (4):465-495.
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  • A Sahlqvist theorem for distributive modal logic.Mai Gehrke, Hideo Nagahashi & Yde Venema - 2004 - Annals of Pure and Applied Logic 131 (1-3):65-102.
    In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist (...)
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  • Negation in the Context of Gaggle Theory.J. Michael Dunn & Chunlai Zhou - 2005 - Studia Logica 80 (2):235-264.
    We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn 's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn 's original one. Ku is the minimal logic that has a (...)
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  • Varieties of positive modal algebras and structural completeness.Tommaso Moraschini - 2019 - Review of Symbolic Logic 12 (3):557-588.
    Positive modal algebras are the$$\left\langle { \wedge, \vee,\diamondsuit,\square,0,1} \right\rangle $$-subreducts of modal algebras. We prove that the variety of positive S4-algebras is not locally finite. On the other hand, the free one-generated positive S4-algebra is shown to be finite. Moreover, we describe the bottom part of the lattice of varieties of positive S4-algebras. Building on this, we characterize structurally complete varieties of positive K4-algebras.
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  • Intuitionistic propositional logic with Galois connections.Wojciech Dzik, Jouni Järvinen & Michiro Kondo - 2010 - Logic Journal of the IGPL 18 (6):837-858.
    In this work, an intuitionistic propositional logic with a Galois connection is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but Glivenko's Theorem does not hold. (...)
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  • Completeness and incompleteness for anodic modal logics.Juliana Bueno-Soler - 2009 - Journal of Applied Non-Classical Logics 19 (3):291-310.
    We propose a new approach to positive modal logics, hereby called anodic modal logics. Our treatment is completely positive since the language has neither negation nor any falsum or minimal particle. The elimination of the minimal particle of the language requires introducing the new concept of factual sets and factual deductions which permit us to talk about deductions in the actual world. We start from a positive fragment of the standard system K, denoted by K⊃, ∧, ◊, which is a (...)
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  • Modal and temporal extensions of non-distributive propositional logics.Chrysafis Hartonas - 2016 - Logic Journal of the IGPL 24 (2):156-185.
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