-
58Analytic Calculi for Product LogicsArchive for Mathematical Logic 43 (7): 859-889. 2004.Product logic Π is an important t-norm based fuzzy logic with conjunction interpreted as multiplication on the real unit interval [0,1], while Cancellative hoop logic CHL is a related logic with connectives interpreted as for Π but on the real unit interval with 0 removed (0,1]. Here we present several analytic proof systems for Π and CHL, including hypersequent calculi, co-NP labelled calculi and sequent calculi.
-
54Two dimensional Standard Deontic Logic [including a detailed analysis of the 1985 Jones–Pörn deontic logic system]Synthese 187 (2): 623-660. 2012.This paper offers a two dimensional variation of Standard Deontic Logic SDL, which we call 2SDL. Using 2SDL we can show that we can overcome many of the difficulties that SDL has in representing linguistic sets of Contrary-to-Duties (known as paradoxes) including the Chisholm, Ross, Good Samaritan and Forrester paradoxes. We note that many dimensional logics have been around since 1947, and so 2SDL could have been presented already in the 1970s. Better late than never! As a detailed case study i…Read more
-
38Semantics for Higher Level Attacks in Extended Argumentation Frames Part 1: OverviewStudia Logica 93 (2-3): 357-381. 2009.In 2005 the author introduced networks which allow attacks on attacks of any level. So if a → b reads a attacks 6, then this attack can itself be attacked by another node c. This attack itself can attack another node d. This situation can be iterated to any level with attacks and nodes attacking other attacks and other nodes. In this paper we provide semantics to such networks. We offer three different approaches to obtaining semantics. 1. The translation approach This uses the methodology of ' …Read more
-
39Completeness properties of heyting's predicate calculus with respect to re modelsJournal of Symbolic Logic 41 (1): 81-94. 1976.
-
Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1: Logical FoundationsStudia Logica 55 (3): 449-451. 1995.
-
10Representation of the Montague Semantics as a Form of the Suppes Semantics with Applications to the Problem of the Introduction of the Passive Voice, the Tenses, and Negation as TransformationsIn Jaakko Hintikka (ed.), Approaches to Natural Language, D. Reidel Publishing. pp. 395--409. 1973.
-
45Algorithmic proof methods and cut elimination for implicational logics part I: Modal implicationStudia Logica 61 (2): 237-280. 1998.In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility.…Read more
-
63The undecidability of intuitionistic theories of algebraically closed fields and real closed fieldsJournal of Symbolic Logic 38 (1): 86-92. 1973.
-
12Languages, Meta-languages and METATEM, A Discussion PaperLogic Journal of the IGPL 4 (2): 255-272. 1996.Meta-languages are vital to the development and usage of formal systems, and yet the nature of meta-languages and associated notions require clarification. Here we attempt to provide a clear definition of the requirements for a language to be a meta-language, together with consideration of issues of proof theory, model theory and interpreters for such a language
-
14Products of modal logics. Part 2: relativised quantifiers in classical logicLogic Journal of the IGPL 8 (2): 165-210. 2000.In the first part of this paper we introduced products of modal logics and proved basic results on their axiomatisability and the f.m.p. In this continuation paper we prove a stronger result - the product f.m.p. holds for products of modal logics in which some of the modalities are reflexive or serial. This theorem is applied in classical first-order logic, we identify a new Square Fragment of the classical logic, where the basic predicates are binary and all quantifiers are relativised, and for…Read more
-
13A Language For Handling Hypothetical Updates And InconsistencyLogic Journal of the IGPL 4 (3): 385-416. 1996.In this paper we propoee a logic programming language which supports hypothetical updates together with integrity constraints. The language makes use of a revision mechanism, which is needed to restore consistency when an update violates some integrity constraint. The revision policy adopted is based on the simple idea that more recent information is preferred to earlier one. We show how this language can be used to represent and perform several types of defeasible reasoning. We develop a logica…Read more
-
35Expressive functional completeness in tense logic (preliminary report)In U. Mönnich (ed.), Aspects of Philosophical Logic, Dordrecht. pp. 91--117. 1981.
-
12Handbook of Philosophical Logic, Volume II. Extensions of Classical LogicPhilosophical Quarterly 36 (142): 101. 1986.
-
40The decision problem for some finite extensions of the intuitionistic theory of abelian groupsStudia Logica 34 (1): 59-67. 1975.
-
13Defeasible inheritance systems and reactive diagramsLogic Journal of the IGPL 17 (1): 1-54. 2008.Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x → y and x ↛ y . Given a diagram D, one can ask the formal question of “is there a valid path between node x and node y?” Depending on the existence of a valid path we can answer the question “x is a y” or “x is not a y”. The answer to the above question is determined through a complex inductive algorithm on paths between arbitrary pairs of points in the graph. This paper aims to simplify and interpret…Read more
-
18Kripke Saul A.. Semantical considerations for modal logics. Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23-26 August, 1962, Acta Philosophica Fennica 1963, pp. 83–94 (review)Journal of Symbolic Logic 34 (3): 501-501. 1969.
-
11The Functional Interpretation of the Existential QuantifierLogic Journal of the IGPL 3 (2-3): 243-290. 1995.
-
Sampling logic and argumentationJournal of the Indian Council of Philosophical Research 28 (2): 233-255. 2010.
-
57A neural cognitive model of argumentation with application to legal inference and decision makingJournal of Applied Logic 12 (2): 109-127. 2014.Formal models of argumentation have been investigated in several areas, from multi-agent systems and artificial intelligence (AI) to decision making, philosophy and law. In artificial intelligence, logic-based models have been the standard for the representation of argumentative reasoning. More recently, the standard logic-based models have been shown equivalent to standard connectionist models. This has created a new line of research where (i) neural networks can be used as a parallel computati…Read more
-
44Size and logicReview of Symbolic Logic 2 (2): 396-413. 2009.We show how to develop a multitude of rules of nonmonotonic logic from very simple and natural notions of size, using them as building blocks