•  416
    Abductive reasoning in neural-symbolic systems
    with Artur S. D’Avila Garcez, Oliver Ray, and John Woods
    Topoi 26 (1): 37-49. 2007.
    Abduction is or subsumes a process of inference. It entertains possible hypotheses and it chooses hypotheses for further scrutiny. There is a large literature on various aspects of non-symbolic, subconscious abduction. There is also a very active research community working on the symbolic (logical) characterisation of abduction, which typically treats it as a form of hypothetico-deductive reasoning. In this paper we start to bridge the gap between the symbolic and sub-symbolic approaches to abdu…Read more
  •  376
    We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix with entries in {0,1}, we allow for one or more blank squares in the matrix, say $a_{i,j} =?.$ The method allows us to decide whether to declare $a_{i,j} = 0$ or $a_{i,j} = 1$ or $a_{i,j} =?$ undecided. This algorithmic method is then applied to modelling several legal and practical…Read more
  •  140
    Advice on Abductive Logic
    with John Woods
    Logic Journal of the IGPL 14 (2): 189-219. 2006.
    One of our purposes here is to expose something of the elementary logical structure of abductive reasoning, and to do so in a way that helps orient theorists to the various tasks that a logic of abduction should concern itself with. We are mindful of criticisms that have been levelled against the very idea of a logic of abduction; so we think it prudent to proceed with a certain diffidence. That our own account of abduction is itself abductive is methodological expression of this diffidence. A s…Read more
  •  123
    Handbook of the History of Logic (edited book)
    with John Woods and Akihiro Kanamori
    Elsevier. 2004.
    Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, includ…Read more
  •  116
    A theory of hypermodal logics: Mode shifting in modal logic (review)
    Journal of Philosophical Logic 31 (3): 211-243. 2002.
    A hypermodality is a connective □ whose meaning depends on where in the formula it occurs. The paper motivates the notion and shows that hypermodal logics are much more expressive than traditional modal logics. In fact we show that logics with very simple K hypermodalities are not complete for any neighbourhood frames
  •  100
    Roadmap for preferential logics
    with Karl Schlechta
    Journal of Applied Non-Classical Logics 19 (1): 43-95. 2009.
    We give a systematic overview of semantical and logical rules in non monotonic and related logics. We show connections and sometimes subtle differences, and also compare such rules to uses of the notion of size.
  •  90
    Independence — Revision and Defaults
    with Karl Schlechta
    Studia Logica 92 (3): 381-394. 2009.
    We investigate different aspects of independence here, in the context of theory revision, generalizing slightly work by Chopra, Parikh, and Rodrigues, and in the context of preferential reasoning.
  •  88
    A logical account of formal argumentation
    with Martin W. A. Caminada
    Studia Logica 93 (2-3): 109-145. 2009.
    In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment
  •  76
    Combining Temporal Logic Systems
    with Marcelo Finger
    Notre Dame Journal of Formal Logic 37 (2): 204-232. 1996.
    This paper investigates modular combinations of temporal logic systems. Four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analyzed, namely soundness, completeness, and decidability. Each combination method comprises three submethods that combine the languages, the inference systems, and the semantics of…Read more
  •  75
    Adding a temporal dimension to a logic system
    with Marcelo Finger
    Journal of Logic, Language and Information 1 (3): 203-233. 1992.
    We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with Since and Until) in a special way. We refer to this method as adding a temporal dimension to L or just temporalising L. We show that the logic system T(L) preserves several properties of the original temporal logic like soundness, completeness, …Read more
  •  75
    Fuzzy logics based on [0,1)-continuous uninorms
    with George Metcalfe
    Archive for Mathematical Logic 46 (5-6): 425-449. 2007.
    Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special …Read more
  •  68
    Sequential Dynamic Logic
    with Alexander Bochman
    Journal of Logic, Language and Information 21 (3): 279-298. 2012.
    We introduce a substructural propositional calculus of Sequential Dynamic Logic that subsumes a propositional part of dynamic predicate logic, and is shown to be expressively equivalent to propositional dynamic logic. Completeness of the calculus with respect to the intended relational semantics is established.
  •  65
    Context-dependent Abduction and Relevance
    with Rolf Nossum and John Woods
    Journal of Philosophical Logic 35 (1): 65-81. 2006.
    Based on the premise that what is relevant, consistent, or true may change from context to context, a formal framework of relevance and context is proposed in which • contexts are mathematical entities • each context has its own language with relevant implication • the languages of distinct contexts are connected by embeddings • inter-context deduction is supported by bridge rules • databases are sets of formulae tagged with deductive histories and the contexts they belong to • abduction and rev…Read more
  •  63
    Cut and pay
    with Marcelo Finger
    Journal of Logic, Language and Information 15 (3): 195-218. 2006.
    In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides an approx…Read more
  •  59
    There are several areas in logic where the monotonicity of the consequence relation fails to hold. Roughly these are the traditional non-monotonic systems arising in Artificial Intelligence (such as defeasible logics, circumscription, defaults, ete), numerical non-monotonic systems (probabilistic systems, fuzzy logics, belief functions), resource logics (also called substructural logics such as relevance logic, linear logic, Lambek calculus), and the logic of theory change (also called belief re…Read more
  •  59
    A Comment on Work by Booth and Co-authors
    with Karl Schlechta
    Studia Logica 94 (3): 403-432. 2010.
    Booth and his co-authors have shown in [2], that many new approaches to theory revision (with fixed K ) can be represented by two relations, , where is a sub-relation of < . They have, however, left open a characterization of the infinite case, which we treat here.
  •  57
    Reactive preferential structures and nonmonotonic consequence
    with Karl Schlechta
    Review of Symbolic Logic 2 (2): 414-450. 2009.
    We introduce Information Bearing Relation Systems (IBRS) as an abstraction of many logical systems. These are networks with arrows recursively leading to other arrows etc. We then define a general semantics for IBRS, and show that a special case of IBRS generalizes in a very natural way preferential semantics and solves open representation problems for weak logical systems. This is possible, as we can the strong coherence properties of preferential structures by higher arrows, that is, arrows, w…Read more
  •  57
    Many-Dimensional Modal Logics: Theory and Applications (edited book)
    Elsevier North Holland. 2003.
    Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a…Read more
  •  55
    Obligations and prohibitions in Talmudic deontic logic
    with M. Abraham and U. Schild
    Artificial Intelligence and Law 19 (2-3): 117-148. 2011.
    This paper examines the deontic logic of the Talmud. We shall find, by looking at examples, that at first approximation we need deontic logic with several connectives: O T A Talmudic obligation F T A Talmudic prohibition F D A Standard deontic prohibition O D A Standard deontic obligation. In classical logic one would have expected that deontic obligation O D is definable by $O_DA \equiv F_D\neg A$ and that O T and F T are connected by $O_TA \equiv F_T\neg A$ This is not the case in the Talmud f…Read more
  •  55
    Two dimensional Standard Deontic Logic [including a detailed analysis of the 1985 Jones–Pörn deontic logic system]
    with Mathijs Boer, Xavier Parent, and Marija Slavkovic
    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
  •  53
    Handbook of Philosophical Logic (edited book)
    with Franz Guenthner
    Kluwer Academic Publishers. 1989.
    The first edition of the Handbook of Philosophical Logic (four volumes) was published in the period 1983-1989 and has proven to be an invaluable reference work ...
  •  53
    Analytic Calculi for Product Logics
    with George Metcalfe and Nicola Olivetti
    Archive 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.
  •  52
    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 (of extensions) to such networks. We offer three different approaches to obtaining semantics. 1. The translation approach This uses the m…Read more
  •  51
    What is a Logical System? (edited book)
    Oxford University Press. 1994.
    This superb collection of papers focuses on a fundamental question in logic and computation: What is a logical system? With contributions from leading researchers--including Ian Hacking, Robert Kowalski, Jim Lambek, Neil Tennent, Arnon Avron, L. Farinas del Cerro, Kosta Dosen, and Solomon Feferman--the book presents a wide range of views on how to answer such a question, reflecting current, mainstream approaches to logic and its applications. Written to appeal to a diverse audience of readers, W…Read more
  •  51
    Resource-origins of Nonmonotonicity
    with John Woods
    Studia Logica 88 (1): 85-112. 2008.
    Formal nonmonotonic systems try to model the phenomenon that common sense reasoners are able to “jump” in their reasoning from assumptions Δ to conclusions C without their being any deductive chain from Δ to C. Such jumps are done by various mechanisms which are strongly dependent on context and knowledge of how the actual world functions. Our aim is to motivate these jump rules as inference rules designed to optimise survival in an environment with scant resources of effort and time. We begin w…Read more
  •  50
    Uncertainty Rules in Talmudic Reasoning
    with Moshe Koppel
    History and Philosophy of Logic 32 (1): 63-69. 2011.
    The Babylonian Talmud, compiled from the 2nd to 7th centuries C.E., is the primary source for all subsequent Jewish laws. It is not written in apodeictic style, but rather as a discursive record of (real or imagined) legal (and other) arguments crossing a wide range of technical topics. Thus, it is not a simple matter to infer general methodological principles underlying the Talmudic approach to legal reasoning. Nevertheless, in this article, we propose a general principle that we believe helps …Read more
  •  50
    Sameness and individuation
    with J. M. Moravcsik
    Journal of Philosophy 70 (16): 513-526. 1973.