Results for 'Non-deterministic matrices'

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  1. Non-deterministic Matrices and Modular Semantics of Rules.Arnon Avron - unknown
    We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set (...)
     
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  2.  5
    Non-Deterministic Matrices: Theory and Applications to Algebraic Semantics.Ana Claudia de Jesus Golzio - 2022 - Bulletin of Symbolic Logic 28 (2):260-261.
  3.  33
    Finite non-deterministic semantics for some modal systems.Marcelo E. Coniglio, Luis Fariñas del Cerro & Newton M. Peron - 2015 - Journal of Applied Non-Classical Logics 25 (1):20-45.
    Trying to overcome Dugundji’s result on uncharacterisability of modal logics by finite logical matrices, Kearns and Ivlev proposed, independently, a characterisation of some modal systems by means of four-valued multivalued truth-functions , as an alternative to Kripke semantics. This constitutes an antecedent of the non-deterministic matrices introduced by Avron and Lev . In this paper we propose a reconstruction of Kearns’s and Ivlev’s results in a uniform way, obtaining an extension to another modal systems. The first part (...)
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  4. Non-deterministic algebraization of logics by swap structures1.Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana Claudia Golzio - 2020 - Logic Journal of the IGPL 28 (5):1021-1059.
    Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. (...)
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  5.  23
    Non-deterministic Conditionals and Transparent Truth.Federico Pailos & Lucas Rosenblatt - 2015 - Studia Logica 103 (3):579-598.
    Theories where truth is a naive concept fall under the following dilemma: either the theory is subject to Curry’s Paradox, which engenders triviality, or the theory is not trivial but the resulting conditional is too weak. In this paper we explore a number of theories which arguably do not fall under this dilemma. In these theories the conditional is characterized in terms of non-deterministic matrices. These non-deterministic theories are similar to infinitely-valued Łukasiewicz logic in that they are (...)
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  6.  10
    Axiomatizing non-deterministic many-valued generalized consequence relations.Sérgio Marcelino & Carlos Caleiro - 2019 - Synthese 198 (S22):5373-5390.
    We discuss the axiomatization of generalized consequence relations determined by non-deterministic matrices. We show that, under reasonable expressiveness requirements, simple axiomatizations can always be obtained, using inference rules which can have more than one conclusion. Further, when the non-deterministic matrices are finite we obtain finite axiomatizations with a suitable generalized subformula property.
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  7.  82
    Modal logic with non-deterministic semantics: Part I—Propositional case.Marcelo E. Coniglio, Luis Fariñas del Cerro & Newton Peron - 2020 - Logic Journal of the IGPL 28 (3):281-315.
    Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. (...)
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    Modal logic with non-deterministic semantics: Part I—Propositional case.Marcelo E. Coniglio, Fariñas Del Cerro Luis & Marques Peron Newton - 2020 - Logic Journal of the IGPL 28 (3):281-315.
    Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices, in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the axiom was replaced by the deontic axiom. In this paper, we propose even weaker (...)
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  9. Logical Non-determinism as a Tool for Logical Modularity: An Introduction.Arnon Avron - unknown
    It is well known that every propositional logic which satisfies certain very natural conditions can be characterized semantically using a multi-valued matrix ([Los and Suszko, 1958; W´ ojcicki, 1988; Urquhart, 2001]). However, there are many important decidable logics whose characteristic matrices necessarily consist of an infinite number of truth values. In such a case it might be quite difficult to find any of these matrices, or to use one when it is found. Even in case a logic does (...)
     
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  10.  10
    Modal Logic With Non-Deterministic Semantics: Part II—Quantified Case.Marcelo E. Coniglio, Luis Fariñasdelcerro & Newton Marques Peron - 2022 - Logic Journal of the IGPL 30 (5):695-727.
    In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositional non-normal modal logics as an alternative to the standard Kripke possible world semantics. This kind of modal system characterized by finite non-deterministic matrices was originally proposed by Ju. Ivlev in the 70s. The aim of this second paper is to introduce a formal non-deterministic semantical framework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown that several (...)
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  11.  17
    Multi-valued Calculi for Logics Based on Non-determinism.Arnon Avron & Beata Konikowska - 2005 - Logic Journal of the IGPL 13 (4):365-387.
    Non-deterministic matrices are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the Rasiowa-Sikorski decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of (...)
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  12. Strong Cut-Elimination, Coherence, and Non-deterministic Semantics.Arnon Avron - unknown
    An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion (...)
     
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  13. A Triple Correspondence in Canonical Calculi: Strong Cut-Elimination, Coherence, and Non-deterministic Semantics.Arnon Avron & Anna Zamansky - unknown
    An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion (...)
     
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  14.  31
    Errata and Addenda to ‘Finite non-deterministic semantics for some modal systems’.Marcelo E. Coniglio, Luis Fariñas del Cerro & Newton M. Peron - 2016 - Journal of Applied Non-Classical Logics 26 (4):336-345.
    In this note, an error in the axiomatization of Ivlev’s modal system Sa+ which we inadvertedly reproduced in our paper “Finite non-deterministic semantics for some modal systems”, is fixed. Additionally, some axioms proposed in were slightly modified. All the technical results in which depend on the previous axiomatization were also fixed. Finally, the discussion about decidability of the level valuation semantics initiated in is taken up. The error in Ivlev’s axiomatization was originally pointed out by H. Omori and D. (...)
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  15. Towards an hyperalgebraic theory of non-algebraizable logics.Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana C. Golzio - 2016 - CLE E-Prints 16 (4):1-27.
    Multialgebras (or hyperalgebras) have been very much studied in the literature. In the realm of Logic, they were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) as a useful semantics tool for characterizing some logics (in particular, several logics of formal inconsistency or LFIs) which cannot be characterized by a single finite matrix. In particular, these LFIs are not algebraizable by any method, including Blok and Pigozzi general theory. Carnielli and Coniglio introduced (...)
     
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  16.  3
    Non-transitive Correspondence Analysis.Yaroslav Petrukhin & Vasily Shangin - forthcoming - Journal of Logic, Language and Information:1-27.
    The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent logic (...)
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  17.  7
    From Inconsistency to Incompatibility.Marcelo E. Coniglio & Guilherme V. Toledo - forthcoming - Logic and Logical Philosophy:1-36.
    The aim of this article is to generalize logics of formal inconsistency (LFIs) to systems dealing with the concept of incompatibility, expressed by means of a binary connective. The basic idea is that having two incompatible formulas to hold trivializes a deduction, and as a special case, a formula becomes consistent (in the sense of LFIs) when it is incompatible with its own negation. We show how this notion extends that of consistency in a non-trivial way, presenting conservative translations for (...)
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  18. Swap structures semantics for Ivlev-like modal logics.Marcelo E. Coniglio & Ana Claudia Golzio - 2019 - Soft Computing 23 (7):2243-2254.
    In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure (...)
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  19.  6
    Two Decision Procedures for da Costa’s $$C_n$$ C n Logics Based on Restricted Nmatrix Semantics.Marcelo E. Coniglio & Guilherme V. Toledo - 2022 - Studia Logica 110 (3):601-642.
    Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and Cila. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa's logic C_1) but the whole (...)
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  20.  26
    Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics.Ofer Arieli, Arnon Avron & Anna Zamansky - 2011 - Studia Logica 97 (1):31 - 60.
    Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This (...)
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  21.  27
    On Non-Deterministic Quantification.Thomas Macaulay Ferguson - 2014 - Logica Universalis 8 (2):165-191.
    This paper offers a framework for extending Arnon Avron and Iddo Lev’s non-deterministic semantics to quantified predicate logic with the intent of resolving several problems and limitations of Avron and Anna Zamansky’s approach. By employing a broadly Fregean picture of logic, the framework described in this paper has the benefits of permitting quantifiers more general than Walter Carnielli’s distribution quantifiers and yielding a well-behaved model theory. This approach is purely objectual and yields the semantical equivalence of both α-equivalent formulae (...)
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  22. Recovery operators, paraconsistency and duality.Walter A. Carnielli, Marcelo E. Coniglio & Abilio Rodrigues Filho - 2020 - Logic Journal of the IGPL 28 (5):624-656.
    There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the (...)
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  23. Non-deterministic Semantics for Logics with a Consistency Operator.Arnon Avron - unknown
    In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics (...)
     
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  24.  65
    A Non-deterministic View on Non-classical Negations.Arnon Avron - 2005 - Studia Logica 80 (2-3):159-194.
    We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...)
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  25. Non-deterministic algebras and algebraization of logics.Ana Claudia Golzio & Marcelo E. Coniglio - 2015 - Filosofia da Linguagem E da Lógica (Philosophy of Language and Philosophy of Logic, in Portuguese).
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  26.  44
    Non-deterministic inductive definitions.Benno van den Berg - 2013 - Archive for Mathematical Logic 52 (1-2):113-135.
    We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call “non-deterministic inductive definitions”. We give applications to formal topology as well as a predicative justification of this principle.
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  27.  5
    Non-deterministic semantics for dynamic topological logic.David Fernández - 2009 - Annals of Pure and Applied Logic 157 (2-3):110-121.
    Dynamic Topological Logic () is a combination of , under its topological interpretation, and the temporal logic interpreted over the natural numbers. is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples , where is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of over spaces (...)
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  28.  4
    Non-deterministic Logic of Informal Provability has no Finite Characterization.Pawel Pawlowski - 2021 - Journal of Logic, Language and Information 30 (4):805-817.
    Recently, in an ongoing debate about informal provability, non-deterministic logics of informal provability BAT and CABAT were developed to model the notion. CABAT logic is defined as an extension of BAT logics and itself does not have independent and decent semantics. The aim of the paper is to show that, semantically speaking, both logics are rather complex and they can be characterized by neither finitely many valued deterministic semantics nor possible word semantics including neighbourhood semantics.
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  29.  11
    Non-Deterministic Inductive Definitions and Fullness.Takako Nemoto & Hajime Ishihara - 2016 - In Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter. pp. 163-170.
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  30.  8
    Gödel, Non-Deterministic Systems, and Hermetic Automata.William H. Desmonde - 1971 - International Philosophical Quarterly 11 (1):49-74.
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  31.  15
    Non deterministic classical logic: the $lambdamu^{++}$-calculus.Karim Nour - 2002 - Mathematical Logic Quarterly 48 (3):357-366.
    In this paper, we present an extension of $lambdamu$-calculus called $lambdamu^{++}$-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel-or.
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  32.  14
    Non Deterministic Classical Logic: The λμ++ ‐calculus.Karim Nour - 2002 - Mathematical Logic Quarterly 48 (3):357-366.
    In this paper, we present an extension of λμ-calculus called λμ++-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel-or.
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  33.  6
    Non Deterministic Classical Logic: The -calculus.Karim Nour - 2002 - Mathematical Logic Quarterly 48 (3):357-366.
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  34. First-order swap structures semantics for some Logics of Formal Inconsistency.Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana Claudia Golzio - 2020 - Journal of Logic and Computation 30 (6):1257-1290.
    The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case of QmbC, (...)
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  35. Many-valued non-deterministic semantics for first-order Logics of Formal (In)consistency.Arnon Avron - unknown
    A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large (...)
     
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  36. Many-valued non-deterministic semantics for first-order logics of formal (in)consistency.Arnon Avron - manuscript
    A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large (...)
     
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  37. 5-valued Non-deterministic Semantics for The Basic Paraconsistent Logic mCi.Arnon Avron - 2008 - Studies in Logic, Grammar and Rhetoric 14 (27).
    One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5-valued characteristic nondeterministic matrix for mCi. This provides a quite non-trivial example for the utility and effectiveness of the use of non-deterministic many-valued semantics.
     
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  38. Propensities in a non-deterministic physics.N. Gisin - 1991 - Synthese 89 (2):287 - 297.
    Propensities are presented as a generalization of classical determinism. They describe a physical reality intermediary between Laplacian determinism and pure randomness, such as in quantum mechanics. They are characterized by the fact that their values are determined by the collection of all actual properties. It is argued that they do not satisfy Kolmogorov axioms; other axioms are proposed.
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  39.  14
    Equivalents of the finitary non-deterministic inductive definitions.Ayana Hirata, Hajime Ishihara, Tatsuji Kawai & Takako Nemoto - 2019 - Annals of Pure and Applied Logic 170 (10):1256-1272.
    We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the (...)
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  40. Cut-Elimination and Quantification in Canonical Systems.Anna Zamansky & Arnon Avron - 2006 - Studia Logica 82 (1):157-176.
    Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with (...)
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  41.  41
    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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  42. Darwin's doubt, non-deterministic Darwinism and the cognitive science of religion.Robin Attfield - 2010 - Philosophy 85 (4):465-483.
    Alvin Plantinga, echoing a worry of Charles Darwin which he calls 'Darwin's doubt', argues that given Darwinian evolutionary theory our beliefs are unreliable, since they are determined to be what they are by evolutionary pressures and could have had no other content. This papers surveys in turn deterministic and non-deterministic interpretations of Darwinism, and concludes that Plantinga's argument poses a problem for the former alone and not for the latter. Some parallel problems arise for the Cognitive Science of (...)
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  43.  77
    Multi-valued Semantics: Why and How.Arnon Avron - 2009 - Studia Logica 92 (2):163-182.
    According to Suszko's Thesis,any multi-valued semantics for a logical system can be replaced by an equivalent bivalent one. Moreover: bivalent semantics for families of logics can frequently be developed in a modular way. On the other hand bivalent semantics usually lacks the crucial property of analycity, a property which is guaranteed for the semantics of multi-valued matrices. We show that one can get both modularity and analycity by using the semantic framework of multi-valued non-deterministic matrices. We further (...)
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  44. Minimality and Non-determinism in Multi-context Systems.Floris Roelofsen & Luciano Serafini - 2005 - In B. Kokinov A. Dey (ed.), Modeling and Using Context. Springer. pp. 424--435.
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  45. Canonical calculi with (n,k)-ary quantifiers.Arnon Avron - unknown
    Propositional canonical Gentzen-type systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [2] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [23] extends (...)
     
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  46.  85
    Cut-free ordinary sequent calculi for logics having generalized finite-valued semantics.Arnon Avron, Jonathan Ben-Naim & Beata Konikowska - 2007 - Logica Universalis 1 (1):41-70.
    . The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general (...)
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  47. Algebra of Non-deterministic Programs: Demonic Operations, Orders and Axioms.Robin Hirsch, Szabolcs Mikulás & Tim Stokes - 2022 - Logic Journal of the IGPL 30 (5):886-906.
    Demonic composition, demonic refinement and demonic union are alternatives to the usual ‘angelic’ composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and the demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckiĭ has shown that the isomorphism class of algebras of (...)
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  48.  58
    Computation, chaos and non-deterministic symbolic computation: The chinese room problem solved?Robert W. Kentridge - 2001 - Psycoloquy 12 (50).
  49.  12
    Cognition, Chaos and Non-Deterministic Symbolic Computation: The Chinese Room Problem Solved.R. W. Kentridge - 1993 - Think (misc) 2:44-47.
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    Tree-Like Proof Systems for Finitely-Many Valued Non-deterministic Consequence Relations.Pawel Pawlowski - 2020 - Logica Universalis 14 (4):407-420.
    The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics. Using the framework it is possible to generate strongly complete proof systems with respect to any finitely valued deterministic and non-deterministic logic. I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.
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