Results for 'modal relations'

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  1.  7
    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 (...)
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  2.  21
    Modal Logics for Reasoning About Infinite Unions and Intersections of Binary Relations.Natasha Alechina, Philippe Balbiani & Dmitry Shkatov - 2012 - Journal of Applied Non-Classical Logics 22 (4):275 - 294.
    (2012). Modal logics for reasoning about infinite unions and intersections of binary relations. Journal of Applied Non-Classical Logics: Vol. 22, No. 4, pp. 275-294. doi: 10.1080/11663081.2012.705960.
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  3.  1
    Modal Definability Based on Łukasiewicz Validity Relations.Bruno Teheux - 2016 - Studia Logica 104 (2):343-363.
    We study two notions of definability for classes of relational structures based on modal extensions of Łukasiewicz finitely-valued logics. The main results of the paper are the equivalent of the Goldblatt-Thomason theorem for these notions of definability.
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  4.  5
    A Hierarchy of Modal Logics with Relative Accessibility Relations.Philippe Balbiani & Ewa Orlowska - 1999 - Journal of Applied Non-Classical Logics 9 (2-3):303-328.
    ABSTRACT In this paper we introduce and investigate various classes of multimodal logics based on frames with relative accessibility relations. We discuss their applicability to representation and analysis of incomplete information. We provide axiom systems for these logics and we prove their completeness.
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  5.  25
    On Modal Logics Characterized by Models with Relative Accessibility Relations: Part I.Stéphane Demri & Dov Gabbay - 2000 - Studia Logica 65 (3):323-353.
    This work is divided in two papers (Part I and Part II). In Part I, we study a class of polymodal logics (herein called the class of "Rare-logics") for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability are established by faithfully translating (...)
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  6.  12
    On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II.Stéphane Demri & Dov Gabbay - 2000 - Studia Logica 66 (3):349-384.
    This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal (...)
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  7.  28
    Modal Logics with Linear Alternative Relations.Krister Segerberg - 1970 - Theoria 36 (3):301-322.
  8.  36
    Mosaics and Step-by-Step| Remarks onA Modal Logic of Relations' by Venema & Marx.Robin Hirsch & Ian Hodkinson - 1999 - In E. Orłowska (ed.), Logic at Work. Heidelberg.
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  9.  8
    Modal Logics with Functional Alternative Relations.Krister Segerberg - 1986 - Notre Dame Journal of Formal Logic 27 (4):504-522.
  10.  21
    A Correlation Between Modal Reduction Principles and Properties of Relations.Frederic B. Fitch - 1973 - Journal of Philosophical Logic 2 (1):97 - 101.
  11.  7
    Witold A. Pogorzelski, Piotr Wojtylak/Cn-Defini-Tions of Propositional Connectives 1 Su Gao, Peter Gerdes/Computably Enumerable Equiva-Lence Relations 27 Yoshihito Tanaka/Model Existence in Non-Compact Modal[REVIEW]Mary-Anne Williams, Thomas Meyer, Basic Infobase Change, David Billington & Andrew Rock - 2001 - Studia Logica 67:439-440.
  12. Davis Chandler. Modal Operators, Equivalence Relations, and Projective Algebras, American Journal of Mathematics, Vol. 76 , Pp. 747–762. [REVIEW]Gebhard Fuhrken - 1959 - Journal of Symbolic Logic 24 (3):253.
  13. Review: Chandler Davis, Modal Operators, Equivalence Relations, and Projective Algebras. [REVIEW]Gebhard Fuhrken - 1959 - Journal of Symbolic Logic 24 (3):253-253.
  14. Mosaics and Step-by-Step. Remarks on “A Modal Logic of Relations”.Robin Hirsch, Ian Hodkinson, Maarten Marx, Szabolsc Mikulás & Mark Reynolds - 1999 - In E. Orłowska (ed.), Logic at Work. Heidelberg.
     
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  15. Mentalism and Modal Logic: A Study in the Relations Between Logical and Metaphysical Systems.Moshe Kroy - 1976 - Athenaion.
     
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  16. A Modal Approach to Dynamic Ontology: Modal Mereotopology.Dimiter Vakarelov - 2008 - Logic and Logical Philosophy 17 (1-2):163-183.
    In this paper we show how modal logic can be applied in the axiomatizations of some dynamic ontologies. As an example we consider the case of mereotopology, which is an extension of mereology with some relations of topological nature like contact relation. We show that in the modal extension of mereotopology we may define some new mereological and mereotopological relations with dynamic nature like stable part-of and stable contact. In some sense such “stable” relations can (...)
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  17.  45
    On the 3d Visualisation of Logical Relations.Hans Smessaert - 2009 - Logica Universalis 3 (2):303-332.
    The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations (...)
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  18.  25
    Means-End Relations and a Measure of Efficacy.Jesse Hughes, Albert Esterline & Bahram Kimiaghalam - 2006 - Journal of Logic, Language and Information 15 (1-2):83-108.
    Propositional dynamic logic (PDL) provides a natural setting for semantics of means-end relations involving non-determinism, but such models do not include probabilistic features common to much practical reasoning involving means and ends. We alter the semantics for PDL by adding probabilities to the transition systems and interpreting dynamic formulas 〈α〉 ϕ as fuzzy predicates about the reliability of α as a means to ϕ. This gives our semantics a measure of efficacy for means-end relations.
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  19.  23
    Identical Twins, Deduction Theorems, and Pattern Functions: Exploring the Implicative BCsK Fragment of S. [REVIEW]Lloyd Humberstone - 2007 - Journal of Philosophical Logic 36 (5):435 - 487.
    We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and (...)
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  20.  11
    Гіпотеза мови мислення дж. фодора й епістемічна логіка.Konstantin Rayhert - 2016 - Схід 3 (143):88-93.
    The study is to show the similarities between J. Fodor's Language of Thought hypothesis and epistemic modal logic. According to the J. Fodor's hypothesis there is the language of thought that is the meta-language in which mental representations of attitudes of organism to propositions expressed in object-language are formulated. These attitudes are called "propositional attitudes". In the hypothesis propositional attitudes are thoughts and relations between organism and proposition. Propositional attitudes are of interest for epistemic modal logics. In (...)
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  21.  20
    Identical Twins, Deduction Theorems, and Pattern Functions: Exploring the Implicative BCsK Fragment of S. [REVIEW]Lloyd Humberstone - 2006 - Journal of Philosophical Logic 35 (5):435 - 487.
    We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and (...)
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  22. First-Order Modal Logic in the Necessary Framework of Objects.Peter Fritz - 2016 - Canadian Journal of Philosophy 46 (4-5):584-609.
    I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only (...)
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  23. Chrysippus' Modal Logic and Its Relation to Philo and Diodorus.Susanne Bobzien - 1993 - In K. Doering & Th Ebert (eds.), Dialektiker und Stoiker. Franz Steiner. pp. 63--84.
    ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and (...) theorems, and to make clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)
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  24. Modal-Epistemic Arithmetic and the Problem of Quantifying In.Jan Heylen - 2013 - Synthese 190 (1):89-111.
    The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the problems of (...)
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  25. Haecceitism for Modal Realists.Sam Cowling - 2012 - Erkenntnis 77 (3):399-417.
    In this paper, I examine the putative incompatibility of three theses: (1) Haecceitism, according to which some maximal possibilities differ solely in terms of the non-qualitative or de re possibilities they include; (2) Modal correspondence, according to which each maximal possibility is identical with a unique possible world; (3) Counterpart theory, according to which de re modality is analyzed in terms of counterpart relations between individuals. After showing how the modal realism defended by David Lewis resolves this (...)
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  26.  8
    Actuality, Tableaux, and Two-Dimensional Modal Logic.Fabio Lampert - forthcoming - Erkenntnis:1-41.
    In this paper we present tableau methods for two-dimensional modal logics. Althoughmodels for such logics are well known, proof systems remain rather unexplored as mostof their developments have been purely axiomatic. The logics herein considered containfirst-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actualor reference worlds, and the lower indices representing worlds of evaluation — first and second dimensions, respectively. The tableaux modulate over (...)
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  27. Foreknowledge and Free Will.Norman M. Swartz - 2004 - Internet Encyclopedia of Philosophy.
    Suppose it were known, by someone else, what you are going to choose to do tomorrow. Wouldn't that entail that tomorrow you must do what it was known in advance that you would do? In spite of your deliberating and planning, in the end, all is futile: you must choose exactly as it was earlier known that you would. The supposed exercise of your free will is ultimately an illusion. Historically, the tension between foreknowledge and the exercise of free will (...)
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  28.  44
    Die stoische Modallogik (Stoic Modal Logic).Susanne Bobzien - 1986 - Königshausen & Neumann.
    ABSTRACT: Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the modalities; modalities as properties of propositions; contingent propositions; the relation (...)
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  29.  3
    Actuality, Tableaux, and Two-Dimensional Modal Logic.Fabio Lampert - forthcoming - Erkenntnis:1-41.
    In this paper we present tableau methods for two-dimensional modal logics. Althoughmodels for such logics are well known, proof systems remain rather unexplored as mostof their developments have been purely axiomatic. The logics herein considered containfirst-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actualor reference worlds, and the lower indices representing worlds of evaluation — first and second dimensions, respectively. The tableaux modulate over (...)
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  30.  31
    The Theory of Relations, Complex Terms, and a Connection Between Λ and Ε Calculi.Edward N. Zalta - manuscript
    This paper introduces a new method of interpreting complex relation terms in a second-order quantified modal language. We develop a completely general second-order modal language with two kinds of complex terms: one kind for denoting individuals and one kind for denoting n-place relations. Several issues arise in connection with previous, algebraic methods for interpreting the relation terms. The new method of interpreting these terms described here addresses those issues while establishing an interesting connection between λ and ε (...)
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  31.  35
    Encoding Modal Logics in Logical Frameworks.Arnon Avron, Furio Honsell, Marino Miculan & Cristian Paravano - 1998 - Studia Logica 60 (1):161-208.
    We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deduction-style proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq (...)
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  32.  20
    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, (...)
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  33.  15
    A Modal Logic for Non-Deterministic Discourse Processing.Tim Fernando - 1999 - Journal of Logic, Language and Information 8 (4):445-468.
    A modal logic for translating a sequence of English sentences to a sequence of logical forms is presented, characterized by Kripke models with points formed from input/output sequences, and valuations determined by entailment relations. Previous approaches based (to one degree or another) on Quantified Dynamic Logic are embeddable within it. Applications to presupposition and ambiguity are described, and decision procedures and axiomatizations supplied.
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  34.  22
    Dynamic Tableaux for Dynamic Modal Logics.Jonas De Vuyst - 2013 - Dissertation, Vrije Universiteit Brussel
    In this dissertation we present proof systems for several modal logics. These proof systems are based on analytic (or semantic) tableaux. -/- Modal logics are logics for reasoning about possibility, knowledge, beliefs, preferences, and other modalities. Their semantics are almost always based on Saul Kripke’s possible world semantics. In Kripke semantics, models are represented by relational structures or, equivalently, labeled graphs. Syntactic formulas that express statements about knowledge and other modalities are evaluated in terms of such models. -/- (...)
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  35.  16
    A Naturalist's Approach to Modal Intuitions.Gunnar Björnsson - 2004 - In Erik Weber Tim De Mey (ed.), Modal Epistemology.
    Modal inquiry is plagued by methodological problems. The best-developed views on modal semantics and modal ontology take modalstatements to be true in virtue of relations between possible worlds. Unfortunately, such views turn modal epistemology into a mystery, and this paper is about ways to avoid that problem. It looks at different remedies suggested by Quine, Blackburn and Peacocke and finds them all wanting. But although Peacocke’s version of the popular conceptualist approach fails to give a (...)
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  36.  50
    Standard Gödel Modal Logics.Xavier Caicedo & Ricardo O. Rodriguez - 2010 - Studia Logica 94 (2):189-214.
    We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations (...)
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  37.  13
    Lower Bounds for Modal Logics.Pavel Hrubeš - 2007 - Journal of Symbolic Logic 72 (3):941 - 958.
    We give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ₁, ψ₂,... s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel—Löb's logic, S and S4. We also determine some speed-up relations between different systems of modal logic on formulas (...)
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  38. Expressive Completeness in Modal Language.Allen Hazen - 1976 - Journal of Philosophical Logic 5 (1):25--46.
    The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special class of entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as ‘disguised’ quantifiers, (...)
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  39. Theories of Properties, Relations, and Propositions.George Bealer - 1979 - Journal of Philosophy 76 (11):634-648.
    This is the only complete logic for properties, relations, and propositions (PRPS) that has been formulated to date. First, an intensional abstraction operation is adjoined to first-order quantifier logic, Then, a new algebraic semantic method is developed. The heuristic used is not that of possible worlds but rather that of PRPS taken at face value. Unlike the possible worlds approach to intensional logic, this approach yields a logic for intentional (psychological) matters, as well as modal matters. At the (...)
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  40.  8
    Modal Structuralism and Theism.Silvia Jonas - forthcoming - In Fiona Ellis (ed.), New Models of Religious Understanding. Oxford: Oxford University Press.
    Drawing an analogy between modal structuralism about mathematics and theism, I o er a structuralist account that implicitly de nes theism in terms of three basic relations: logical and metaphysical priority, and epis- temic superiority. On this view, statements like `God is omniscient' have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical (...)
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  41.  68
    Many-Valued Modal Logics II.Melvin Fitting - unknown
    Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible — in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common (...) language, and on what sentences will they agree? This problem can be reformulated as one about many-valued Kripke models, allowing many-valued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the many-valued version and the multiple expert one will be formally established. Finally we will axiomatize many-valued modal logics, and sketch a proof of completeness. (shrink)
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  42.  43
    Summation Relations and Portions of Stuff.Maureen Donnelly & Thomas Bittner - 2009 - Philosophical Studies 143 (2):167 - 185.
    According to the prevalent 'sum view' of stuffs, each portion of stuff is a mereological sum of its subportions. The purpose of this paper is to re-examine the sum view in the light of a modal temporal mereology which distinguishes between different varieties of summation relations. While admitting David Barnett's recent counter-example to the sum view, we show that there is nonetheless an important sense in which all portions of stuff are sums of their subportions. We use our (...)
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  43.  33
    A Reconstruction of Aristotle's Modal Syllogistic.Marko Malink - 2006 - History and Philosophy of Logic 27 (2):95-141.
    Ever since ?ukasiewicz, it has been opinio communis that Aristotle's modal syllogistic is incomprehensible due to its many faults and inconsistencies, and that there is no hope of finding a single consistent formal model for it. The aim of this paper is to disprove these claims by giving such a model. My main points shall be, first, that Aristotle's syllogistic is a pure term logic that does not recognize an extra syntactic category of individual symbols besides syllogistic terms and, (...)
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  44.  4
    The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon.Hans Smessaert - 2012 - Logica Universalis 6 (1-2):171-199.
    Peters and Westerståhl (Quantifiers in Language and Logic, 2006), and Westerståhl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the “classical” Aristotelian squares of opposition and the “modern” Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality relation of internal negation (SNEG). Furthermore, (...)
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  45.  51
    Models for Normal Intuitionistic Modal Logics.Milan Božić & Kosta Došen - 1984 - Studia Logica 43 (3):217 - 245.
    Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. (...)
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  46.  29
    Application of Modal Logic to Programming.Vaughan R. Pratt - 1980 - Studia Logica 39 (2-3):257 - 274.
    The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.
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  47.  41
    Tableaus for Many-Valued Modal Logic.Melvin Fitting - 1995 - Studia Logica 55 (1):63 - 87.
    We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.
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  48.  32
    Models for Stronger Normal Intuitionistic Modal Logics.Kosta Došen - 1985 - Studia Logica 44 (1):39 - 70.
    This paper, a sequel to Models for normal intuitionistic modal logics by M. Boi and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown (...)
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  49.  37
    Fundamental Axioms for Preference Relations.Bengt Hansson - 1968 - Synthese 18 (4):423 - 442.
    The basic theory of preference relations contains a trivial part reflected by axioms A1 and A2, which say that preference relations are preorders. The next step is to find other axims which carry the theory beyond the level of the trivial. This paper is to a great part a critical survey of such suggested axioms. The results are much in the negative — many proposed axioms imply too strange theorems to be acceptable as axioms in a general theory (...)
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  50.  61
    An Update on “Might”.Jaap van der Does, Willem Groeneveld & Frank Veltman - 1997 - Journal of Logic, Language and Information 6 (4):361-380.
    This paper is on the update semantics for might of Veltman. Threeconsequence relations are introduced and studied in an abstract setting.Next we present sequent-style systems for each of the consequence relations.We show the logics to be complete and decidable. The paper ends with asyntactic cut elimination result.
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