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Theory of Semi-Instantiation in Abstract Argumentation Gabbay, Dov M. in To appear Logica Universalis (2015) We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas ... [more ▼] We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature. [less ▲] Detailed reference viewed: 58 (1 UL)Topological Aspects of Matrix Abduction 2 Gabbay, Dov M. ; in The Road to Universal Logic - Festschrift for the 50th Birthday of Jean-Yves Béziau Volume II, Chapter 17 (2015) In the last few years, several publications discussed a new method of matrix completion, called matrix abduction, which is particularly useful in the context of Talmudic logic and legal ruling systems in ... [more ▼] In the last few years, several publications discussed a new method of matrix completion, called matrix abduction, which is particularly useful in the context of Talmudic logic and legal ruling systems in general. Given a matrix AA with entries that are either in {0,1}{0,1} or blank, the method allows us to decide whether each blank entry should be 0 or 1, or remain undecided. Unlike existing matrix completion methods, which are based on notions of analogy using distance (analogy to nearest neighbors), the new method takes a different approach and completes the matrix using only topological criteria; as a result, the outcome of the process is often significantly different. This chapter will focus specifically on one of these criteria, involving the representation of the finite partially ordered set as minimally generated multisets (ordered by inclusion) for which there is no known exact polynomial-time evaluation algorithm. We examine the existing efficient greedy algorithm for this criteria, its connection the similar known concepts dimension and two dimension of a poset (partial ordered set) then derive new useful properties and an algorithm. We start by proving that the exact evaluation of the criteria is NP-complete both in the general case and in the limited context of matrix abduction. We then discuss one-point removal properties of the problematic criteria, which may aid in solving specific cases and be utilized by new algorithms. Finally, we present a new greedy algorithm that offers a significant improvement over the existing one, and discuss the possibility of reusing existing algorithms that approximate 2-dimension of a poset. [less ▲] Detailed reference viewed: 106 (0 UL)Probabilistic Argumentation: An Equational Approach Gabbay, Dov M. ; in Logica Universalis (2015), 9(3), 345--382 There is a generic way to add any new feature to a system. It involves (1) identifying the basic units which build up the system and (2) introducing the new feature to each of these basic units. In the ... [more ▼] There is a generic way to add any new feature to a system. It involves (1) identifying the basic units which build up the system and (2) introducing the new feature to each of these basic units. In the case where the system is argumentation and the feature is probabilistic we have the following. The basic units are: (a) the nature of the arguments involved; (b) the membership relation in the set S of arguments; (c) the attack relation; and (d) the choice of extensions. Generically to add a new aspect (probabilistic, or fuzzy, or temporal, etc) to an argumentation network ⟨S,R⟩⟨S,R⟩ can be done by adding this feature to each component (a–d). This is a brute-force method and may yield a non-intuitive or meaningful result. A better way is to meaningfully translate the object system into another target system which does have the aspect required and then let the target system endow the aspect on the initial system. In our case we translate argumentation into classical propositional logic and get probabilistic argumentation from the translation. Of course what we get depends on how we translate. In fact, in this paper we introduce probabilistic semantics to abstract argumentation theory based on the equational approach to argumentation networks. We then compare our semantics with existing proposals in the literature including the approaches by M. Thimm and by A. Hunter. Our methodology in general is discussed in the conclusion. [less ▲] Detailed reference viewed: 158 (10 UL)Reactive standard deontic logic Gabbay, Dov M. ; in Journal of Logic and Computation (2015), 25(1), 117--157 Detailed reference viewed: 110 (4 UL)The attack as strong negation, part I Gabbay, Dov M. in Logic Journal of the IGPL (2015) We add strong negation N to classical logic and interpret the attack relation of `x attacks y' in argumentation as (x?Ny). We write a corresponding object level (using N only) classical theory for each ... [more ▼] We add strong negation N to classical logic and interpret the attack relation of `x attacks y' in argumentation as (x?Ny). We write a corresponding object level (using N only) classical theory for each argumentation network and show that the classical models of this theory correspond exactly to the complete extensions of the argumentation network. We show by example how this approach simplifies the study of abstract argumentation networks. We compare with other translations of abstract argumentation networks into logic, such as classical predicate logic or modal logics, or logic programming, and we also compare with Abstract Dialectical Frameworks. [less ▲] Detailed reference viewed: 107 (2 UL)Probabilistic Argumentation. An Equational Approach Gabbay, Dov M. ; in Logica Universalis (2015), abs/1503.05501 There is a generic way to add any new feature to a system. It involves 1) identifying the basic units which build up the system and 2) introducing the new feature to each of these basic units. In the case ... [more ▼] There is a generic way to add any new feature to a system. It involves 1) identifying the basic units which build up the system and 2) introducing the new feature to each of these basic units. In the case where the system is argumentation and the feature is probabilistic we have the following. The basic units are: a. the nature of the arguments involved; b. the membership relation in the set S of arguments; c. the attack relation; and d. the choice of extensions. Generically to add a new aspect (probabilistic, or fuzzy, or temporal, etc) to an argumentation network hS, R i can be done by adding this feature to each component a – d. This is a brute-force method and may yield a non-intuitive or meaningful result. A better way is to meaningfully translate the object system into another target system which does have the aspect required and then let the target system endow the aspect on the initial system. In our case we translate argumentation into classical propositional logic and get probabilistic argumentation from the translation. Of course what we get depends on how we translate. In fact, in this paper we introduce probabilistic semantics to abstract argumentation theory based on the equational approach to argumentation networks. We then compare our semantics with existing proposals in the literature including the approaches by M. Thimm and by A. Hunter. Our methodology in general is discussed in the conclusion. [less ▲] Detailed reference viewed: 86 (5 UL)Equilibrium States in Numerical Argumentation Networks Gabbay, Dov M. ; in Logica Universalis (2015), 9(4), 411--473 Detailed reference viewed: 113 (1 UL)The Attack as Intuitionistic Negation Gabbay, Dov M. ; in Logic Journal of the IGPL (2015), abs/1510.00077 Detailed reference viewed: 93 (6 UL)Topological Aspects of Matrix Abduction 1 ; Gabbay, Dov M. in Koslow, Arnold; Buchsbaum, Arthur (Eds.) The Road to Universal Logic: Festschrift for the 50th Birthday of Jean-Yves Béziau Volume II (2015) A new method of abduction, matrix abduction, has been introduced in Abraham, M., Gabbay, D., Schild, U.: Talmudic argumentum a fortiori inference rule (Kal Vachomer) using matrix abduction. Studia Logica ... [more ▼] A new method of abduction, matrix abduction, has been introduced in Abraham, M., Gabbay, D., Schild, U.: Talmudic argumentum a fortiori inference rule (Kal Vachomer) using matrix abduction. Studia Logica 92(3), 281–364 (2009). This method describes the Kal Vachomer and the Binyan Abh rules by using microscopic parameters which exist in the inputs of these rules. In order to find these parameters the method needs to calculate the minimal number of parameters that will describe the logical rule. In the current chapter, the matrix abduction method is formulated by Partially Orderd Sets (Posets). Consequently it is shown that the minimal number of parameters similarly defined to the dimension and k-dimension of Posets and a new poset dimension is defined which is the Kal Vachomer Dimension. In addition, several theorems and bounds of this dimension are shown. [less ▲] Detailed reference viewed: 133 (0 UL)A neural cognitive model of argumentation with application to legal inference and decision making ; Gabbay, Dov M. ; in Journal of Applied Logic (2014), 12(2), 109--127 Detailed reference viewed: 120 (1 UL)Systems of Interacting Argumentation Networks Gabbay, Dov M. in Journal of Logics and their Applications (2014) Detailed reference viewed: 40 (0 UL)Products of modal logics and tensor products of modal algebras Gabbay, Dov M. ; ; in Journal of Applied Logic (2014), 12(4), 570--583 Detailed reference viewed: 108 (1 UL)The Handling of Loops in Talmudic Logic, with Application to Odd and Even Loops in Argumentation ; Gabbay, Dov M. ; in HOWARD-60: A Festschrift on the Occasion of Howard Barringer's 60th Birthday (2014) Detailed reference viewed: 55 (0 UL)Platonic Realism and Talmudic Reasoning ; ; Gabbay, Dov M. et al in Abraham, Michael; Gabbay, Dov M.; Schild, Uri (Eds.) Platonic Realism and Talmudic Reasoning (2014) Detailed reference viewed: 48 (0 UL)The paradoxes of permission an action based solution Gabbay, Dov M. ; Gammaitoni, Loïc ; Sun, Xin in Journal of Applied Logic (2014), 12(2), 179191 The aim of this article is to construct a deontic logic in which the free choice postulate allow (Ross, 1941) [11] would be consistent and all the implausible result mentioned in (Hanson, in press) [5 ... [more ▼] The aim of this article is to construct a deontic logic in which the free choice postulate allow (Ross, 1941) [11] would be consistent and all the implausible result mentioned in (Hanson, in press) [5] will be blocked. To achieve this we first developed a new theory of action. Then we build a new deontic logic in which the deontic action operator and the deontic proposition operator are explicitly distinguished. [less ▲] Detailed reference viewed: 116 (16 UL)Reactivity and Grammars: An Exploration ; ; Gabbay, Dov M. in Language, Culture, Computation. Computing - Theory and Technology - Essays Dedicated to Yaacov Choueka on the Occasion of His 75th Birthday Part I (2014) Detailed reference viewed: 104 (0 UL)Intuitionistic basis for IOL Parent, Xavier ; van der Torre, Leon ; Gabbay, Dov M. in Hansson, Sven Ove (Ed.) David Makinson's outsanding contribution to non-classical logic (2014) Detailed reference viewed: 129 (20 UL)What is a Logical System? An Evolutionary View: 1964-2014 Gabbay, Dov M. in Gabbay, Dov M.; Siekmann; Woods (Eds.) Computational Logic (2014) Detailed reference viewed: 123 (1 UL)Toward a Linguistic Interpretation of Deontic Paradoxes - Beth-Reichenbach Semantics Approach for a New Analysis of the Miners Scenario Gabbay, Dov M. ; ; Sun, Xin et al in Deontic Logic and Normative Systems - 12th International Conference DEON 2014, Ghent, Belgium, July 12-15, 2014. Proceedings (2014) . A linguistic analysis of deontic paradoxes can be used to further develop deontic logic. In this paper we provide a Beth-Reichenbach semantics to analyze deontic paradoxes, and we illustrate it on the ... [more ▼] . A linguistic analysis of deontic paradoxes can be used to further develop deontic logic. In this paper we provide a Beth-Reichenbach semantics to analyze deontic paradoxes, and we illustrate it on the single agent decision problem of the miners scenario. We also introduce extensions with reactive arrows and actions, which can be used to give a linguistic interpretation of multi-agent dialogues. [less ▲] Detailed reference viewed: 115 (4 UL)An equational approach to the merging of argumentation networks Gabbay, Dov M. ; in Journal of Logic and Computation (2014), 24(6), 1253--1277 Detailed reference viewed: 114 (1 UL) |
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