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  1. 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 nodec space. This, in turn, can (...)
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  • Stable Modal Logics.Guram Bezhanishvili, Nick Bezhanishvili & Julia Ilin - 2018 - Review of Symbolic Logic 11 (3):436-469.
  • Deciding Regular Grammar Logics with Converse Through First-Order Logic.Stéphane Demri & Hans De Nivelle - 2005 - Journal of Logic, Language and Information 14 (3):289-329.
    We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with (...)
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  • Modal Logics with Belnapian Truth Values.Serge P. Odintsov & Heinrich Wansing - 2010 - Journal of Applied Non-Classical Logics 20 (3):279-304.
    Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to (...)
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  • Temporal Interpretation of Monadic Intuitionistic Quantifiers.Guram Bezhanishvili & Luca Carai - forthcoming - Review of Symbolic Logic:1-24.
    We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” and “sometime in the past”. It is well known that Prior’s intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf (...)
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  • On Intermediate Inquisitive and Dependence Logics: An Algebraic Study.Davide Emilio Quadrellaro - 2022 - Annals of Pure and Applied Logic 173 (10):103143.
  • Axiomatic and Dual Systems for Constructive Necessity, a Formally Verified Equivalence.Lourdes del Carmen González-Huesca, Favio E. Miranda-Perea & P. Selene Linares-Arévalo - 2019 - Journal of Applied Non-Classical Logics 29 (3):255-287.
    We present a proof of the equivalence between two deductive systems for constructive necessity, namely an axiomatic characterisation inspired by Hakli and Negri's system of derivations from assumptions for modal logic , a Hilbert-style formalism designed to ensure the validity of the deduction theorem, and the judgmental reconstruction given by Pfenning and Davies by means of a natural deduction approach that makes a distinction between valid and true formulae, constructively. Both systems and the proof of their equivalence are formally verified (...)
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  • Truth and Falsehood: An Inquiry Into Generalized Logical Values.Yaroslav Shramko & Heinrich Wansing - 2011 - Dordrecht, Netherland: Springer.
    The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...)
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  • Neighborhood Semantics for Modal Logic.Eric Pacuit - 2017 - Cham, Switzerland: Springer.
    This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for neighborhood models and other model-theoretic (...)
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  • Natural Deduction, Hybrid Systems and Modal Logics.Andrzej Indrzejczak - 2010 - Dordrecht, Netherland: Springer.
    This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.
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  • 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 algebras and its generalizations (...)
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  • Proof Theory and Algebra in Logic.Hiroakira Ono - 2019 - Singapore: Springer Singapore.
    This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate (...)
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  • Modal Definability: Two Commuting Equivalence Relations.Yana Rumenova & Tinko Tinchev - 2022 - Logica Universalis 16 (1):177-194.
    We prove that modal definability with respect to the class of all structures with two commuting equivalence relations is an undecidable problem. The construction used in the proof shows that the same is true for the subclass of all finite structures. For that reason we prove that the first-order theories of these classes are undecidable and reduce the latter problem to the former.
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  • Admissibility of Π2-Inference Rules: Interpolation, Model Completion, and Contact Algebras.Nick Bezhanishvili, Luca Carai, Silvio Ghilardi & Lucia Landi - 2023 - Annals of Pure and Applied Logic 174 (1):103169.
  • On the Proof Complexity of Logics of Bounded Branching.Emil Jeřábek - 2023 - Annals of Pure and Applied Logic 174 (1):103181.
  • Johan van Benthem on Logic and Information Dynamics.Alexandru Baltag & Sonja Smets (eds.) - 2014 - Cham, Switzerland: Springer International Publishing.
    This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...)
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  • Post Completeness in Congruential Modal Logics.Peter Fritz - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. College Publications. pp. 288-301.
    Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As (...)
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  • The Intensional Side of Algebraic-Topological Representation Theorems.Sara Negri - 2017 - Synthese 198 (Suppl 5):1121-1143.
    Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...)
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  • Modal Logic.James W. Garson - 2009 - Stanford Encyclopedia of Philosophy.
  • Tarskian Consequence Relations Bilaterally: Some Familiar Notions.Sergey Drobyshevich - 2019 - Synthese 198 (S22):5213-5240.
    This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a fragment, (...)
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  • What Can You Say? Measuring the Expressive Power of Languages.Alexander Kocurek - 2018 - Dissertation, University of California, Berkeley
    There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how these investigations (...)
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  • On the Proof Theory of Infinitary Modal Logic.Matteo Tesi - forthcoming - Studia Logica:1-32.
    The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by introducing a labelled sequent calculus which is sound and complete with respect to Kripke semantics. We establish the structural properties of the system, namely admissibility of the structural rules and of the cut rule. Finally, we show how to embed common knowledge in the infinitary calculus and we discuss first-order (...)
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  • Unification with Parameters in the Implication Fragment of Classical Propositional Logic.Philippe Balbiani & Mojtaba Mojtahedi - 2022 - Logic Journal of the IGPL 30 (3):454-464.
    In this paper, we show that the implication fragment of classical propositional logic is finitary for unification with parameters.
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  • On Intermediate Justification Logics.Nicholas Pischke - forthcoming - Logic Journal of the IGPL.
    We study arbitrary intermediate propositional logics extended with a collection of axioms from justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev’s, Fitting’s or Lehmann and Studer’s models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying intermediate logic. Further, by a modification of a method of Fitting, (...)
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  • Algebraic Semantics for Modal and Superintuitionistic Non-Monotonic Logics.David Pearce & Levan Uridia - 2013 - Journal of Applied Non-Classical Logics 23 (1-2):147-158.
    The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?
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  • The Omega-Rule Interpretation of Transfinite Provability Logic.David Fernández-Duque & Joost J. Joosten - 2018 - Annals of Pure and Applied Logic 169 (4):333-371.
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  • Modal Logics of Metric Spaces.Guram Bezhanishvili, David Gabelaia & Joel Lucero-Bryan - 2015 - Review of Symbolic Logic 8 (1):178-191.
  • Tabularity and Post-Completeness in Tense Logic.Qian Chen & Minghui Ma - forthcoming - Review of Symbolic Logic:1-18.
    A new characterization of tabularity in tense logic is established, namely, a tense logic L is tabular if and only if $\mathsf {tab}_n^T\in L$ for some $n\geq 1$. Two characterization theorems for the Post-completeness in tabular tense logics are given. Furthermore, a characterization of the Post-completeness in the lattice of all tense logics is established. Post numbers of some tense logics are shown.
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  • Topological Completeness of Logics Above S4.Guram Bezhanishvili, David Gabelaia & Joel Lucero-Bryan - 2015 - Journal of Symbolic Logic 80 (2):520-566.
  • An Extension of Jónsson‐Tarski Representation and Model Existence in Predicate Non‐Normal Modal Logics.Yoshihito Tanaka - 2022 - Mathematical Logic Quarterly 68 (2):189-201.
  • Prior’s OIC Nonconservativity Example Revisited.Lloyd Humberstone - 2014 - Journal of Applied Non-Classical Logics 24 (3):209-235.
    In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...)
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  • Contrapositionally Complemented Heyting Algebras and Intuitionistic Logic with Minimal Negation.Anuj Kumar More & Mohua Banerjee - forthcoming - Logic Journal of the IGPL.
    Two algebraic structures, the contrapositionally complemented Heyting algebra and the contrapositionally $\vee $ complemented Heyting algebra, 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 corresponding to ccHas and its extension ${\textrm {ILM}}$-${\vee }$ for c$\vee $cHas are then investigated. (...)
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  • Unification Types in Euclidean Modal Logics.Majid Alizadeh, Mohammad Ardeshir, Philippe Balbiani & Mojtaba Mojtahedi - forthcoming - Logic Journal of the IGPL.
    We prove that $\textbf {K}5$ and some of its extensions that do not contain $\textbf {K}4$ are of unification type $1$.
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  • Remarks About the Unification Types of Some Locally Tabular Normal Modal Logics.Philippe Balbiani, ÇiĞdem Gencer, Maryam Rostamigiv & Tinko Tinchev - forthcoming - Logic Journal of the IGPL.
    It is already known that unifiable formulas in normal modal logic $\textbf {K}+\square ^{2}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{2}\bot $ are unitary. In this paper, we prove that for all $d{\geq }3$, unifiable formulas in normal modal logic $\textbf {K}+\square ^{d}\bot $ are either finitary or unitary and unifiable formulas in normal modal logic $\textbf {Alt}_{1}+\square ^{d}\bot $ are unitary.
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  • Improving Strong Negation.Satoru Niki - forthcoming - Review of Symbolic Logic:1-27.
    Strong negation is a well-known alternative to the standard negation in intuitionistic logic. It is defined virtually by giving falsity conditions to each of the connectives. Among these, the falsity condition for implication appears to unnecessarily deviate from the standard negation. In this paper, we introduce a slight modification to strong negation, and observe its comparative advantages over the original notion. In addition, we consider the paraconsistent variants of our modification, and study their relationship with non-constructive principles and connexivity.
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  • An Algebraic Approach to Inquisitive and -Logics.Nick Bezhanishvili, Gianluca Grilletti & Davide Emilio Quadrellaro - forthcoming - Review of Symbolic Logic:1-41.
    This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety (...)
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  • A Graded Semantics for Counterfactuals.Libor Běhounek & Ondrej Majer - 2021 - Synthese 199 (5-6):11963-11994.
    This article presents an extension of Lewis’ analysis of counterfactuals to a graded framework. Unlike standard graded approaches, which use the probabilistic framework, we employ that of many-valued logics. Our principal goal is to provide an adequate analysis of the main background notion of Lewis’ approach—the one of the similarity of possible worlds. We discuss the requirements imposed on the analysis of counterfactuals by the imprecise character of similarity and concentrate in particular on robustness, i.e., the requirement that small changes (...)
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  • The Range of Modal Logic: An Essay in Memory of George Gargov.Johan van Benthem - 1999 - Journal of Applied Non-Classical Logics 9 (2-3):407-442.
    ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that ‘modal logic’ (...)
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  • One Modal Logic to Rule Them All?Wesley H. Holliday & Tadeusz Litak - 2018 - In Guram Bezhanishvili, Giovanna D'Agostino, George Metcalfe & Thomas Studer (eds.), Advances in Modal Logic, Vol. 12. London: College Publications. pp. 367-386.
    In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to (...)
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  • On the Finite Model Property of Intuitionistic Modal Logics Over MIPC.Takahito Aoto & Hiroyuki Shirasu - 1999 - Mathematical Logic Quarterly 45 (4):435-448.
    MIPC is a well-known intuitionistic modal logic of Prior and Bull . It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property whenever L is Kripke-complete and universal.
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  • A Topological Completeness Theorem for a Weak Version of Stalnaker's Logic of Knowledge and Belief.Thomas Mormann - manuscript
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  • A Note on the Complexity of S4.2.Aggeliki Chalki, Costas D. Koutras & Yorgos Zikos - 2021 - Journal of Applied Non-Classical Logics 31 (2):108-129.
    S4.2 is the modal logic of directed partial pre-orders and/or the modal logic of reflexive and transitive relational frames with a final cluster. It holds a distinguished position in philosophical...
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  • On the Logic of Belief and Propositional Quantification.Yifeng Ding - 2021 - Journal of Philosophical Logic 50 (5):1143-1198.
    We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. (...)
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  • On the Modal Logic of the Non-Orthogonality Relation Between Quantum States.Shengyang Zhong - 2018 - Journal of Logic, Language and Information 27 (2):157-173.
    It is well known that the non-orthogonality relation between the states of a quantum system is reflexive and symmetric, and the modal logic \ is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies each (...)
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  • On Sahlqvist Formulas in Relevant Logic.Guillermo Badia - 2018 - Journal of Philosophical Logic 47 (4):673-691.
    This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.
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  • A Computational Learning Semantics for Inductive Empirical Knowledge.Kevin T. Kelly - 2014 - In Alexandru Baltag & Sonja Smets (eds.), Johan van Benthem on Logic and Information Dynamics. Springer International Publishing. pp. 289-337.
    This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...)
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  • The Power of Belnap: Sequent Systems for SIXTEEN ₃. [REVIEW]Heinrich Wansing - 2010 - Journal of Philosophical Logic 39 (4):369 - 393.
    The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented.
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  • Zolin and Pizzi: Defining Necessity From Noncontingency.Lloyd Humberstone - 2013 - Erkenntnis 78 (6):1275-1302.
    The point of the present paper is to draw attention to some interesting similarities, as well as differences, between the approaches to the logic of noncontingency of Evgeni Zolin and of Claudio Pizzi. Though neither of them refers to the work of the other, each is concerned with the definability of a (normally behaving, though not in general truth-implying) notion of necessity in terms of noncontingency, standard boolean connectives and additional but non-modal expressive resources. The notion of definability involved is (...)
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  • Inquisitive Logic.Ivano Ciardelli & Floris Roelofsen - 2011 - Journal of Philosophical Logic 40 (1):55-94.
    This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).
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  • Three Dual Ontologies.Chris Brink & Ingrid Rewitzky - 2002 - Journal of Philosophical Logic 31 (6):543-568.
    In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aiin of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world (...)
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