This important book provides a new unifying methodology for logic. It replaces the traditional view of logic as manipulating sets of formulas with the notion of structured families of labelled formulas with algebraic structures. This approach has far reaching consequences for the methodology of logics and their semantics, and the book studies the main features of such systems along with their applications. It will interest logicians, computer scientists, philosophers and linguists.

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 many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. To study (...) the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

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.

This long awaited book gives a thorough account of the mathematical foundations of Temporal Logic, one of the most important areas of logic in computer science.The book, which consists of fifteen chapters, moves on from giving a solid introduction in semantical and axiomatic approaches to temporal logic to covering the central topics of predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositionalquantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results.Much of the research presented here (...) is frontline in the new results and in the unifying methodology. This is an indispensable reference work for both the pure logician and the theoretical computer scientist. (shrink)

From the point of view of non-classical logics, Heyting's implication is the smallest implication for which the deduction theorem holds. This book studies properties of logical systems having some of the classical connectives and implication in the neighbourhood of Heyt ing's implication. I have not included anything on entailment, al though it belongs to this neighbourhood, mainly because of the appearance of the Anderson-Belnap book on entailment. In the later chapters of this book, I have included material that might be (...) of interest to the intuitionist mathematician. Originally, I intended to include more material in that spirit but I decided against it. There is no coherent body of material to include that builds naturally on the present book. There are some serious results on topological models, second order Beth and Kripke models, theories of types, etc., but it would require further research to be able to present a general theory, possibly using sheaves. That would have postponed pUblication for too long. I would like to dedicate this book to my colleagues, Professors G. Kreisel, M.O. Rabin and D. Scott. I have benefited greatly from Professor Kreisel's criticism and suggestions. Professor Rabin's fun damental results on decidability and undecidability provided the powerful tools used in obtaining the majority of the results reported in this book. Professor Scott's approach to non-classical logics and especially his analysis of the Scott consequence relation makes it possible to present Heyting's logic as a beautiful, integral part of non-classical logics. (shrink)

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, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (shrink)

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 ...

This paper is part of a research program centered around argumentation networks and offering several research directions for argumentation networks, with a view of using such networks for integrating logics and network reasoning. In Section 1 we introduce our program manifesto. In Section 2 we motivate and show how to substitute one argumentation network as a node in another argumentation network. Substitution is a purely logical operation and doing it for networks, besides developing their theory further, also helps us see (...) how to bring logic and networks closer together. Section 3 develops the formal properties of the new kind of network and Section 4 offers general discussion and comparison with the literature. (shrink)

In this paper, we prove the correspondence between complete extensions in abstract argumentation and 3-valued stable models in logic programming. This result is in line with earlier work of [6] that identified the correspondence between the grounded extension in abstract argumentation and the well-founded model in logic programming, as well as between the stable extensions in abstract argumentation and the stable models in logic programming.

This paper studies methodologically robust options for giving logical contents to nodes in abstract argumentation networks. It defines a variety of notions of attack in terms of the logical contents of the nodes in a network. General properties of logics are refined both in the object level and in the metalevel to suit the needs of the application. The network-based system improves upon some of the attempts in the literature to define attacks in terms of defeasible proofs, the so-called rule-based (...) systems. We also provide a number of examples and consider a rigorous case study, which indicate that our system does not suffer from anomalies. We define consequence relations based on a notion of defeat, consider rationality postulates, and prove that one such consequence relation is consistent. (shrink)

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 reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences. (shrink)

Modern applications of logic in mathematics, computer science, and linguistics use combined systems of different types of logic working together. This book develops a method for combining--or fibring--systems by breaking them into simple components which can be manipulated easily and recombined.

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, What is a Logical System? will (...) excite discussion among students, teachers, and researchers in mathematics, logic, computer science, philosophy, and linguistics. (shrink)

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 methodology of ' Logic by translation'. We translate faithfully the new networks into ordinary Dung networks with more nodes and extract the semantics from the translation. 2. The labelling approach This method regards the arrows as additional entities to be attacked and to mount attacks and applies a variation of the usual machinery of Camindada like labelling to the network. The new concept we need to employ here is that of 'joint attacks'. 3. The logic programming approach We translate the higher level network into a logic program and obtain semantics for it through known semantics for logic programs. We then compare our methods with those of S. Modgil and P. M. Dung et al. (shrink)

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.

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers (...) in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations. (shrink)

In this paper, we introduce the methodology and techniques of metaargumentation to model argumentation. The methodology of meta-argumentation instantiates Dung's abstract argumentation theory with an extended argumentation theory, and is thus based on a combination of the methodology of instantiating abstract arguments, and the methodology of extending Dung's basic argumentation frameworks with other relations among abstract arguments. The technique of meta-argumentation applies Dung's theory of abstract argumentation to itself, by instantiating Dung's abstract arguments with meta-arguments using a technique called flattening. (...) We characterize the domain of instantiation using a representation technique based on soundness and completeness. Finally, we distinguish among various instantiations using the technique of specification languages. (shrink)

Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. The (...) tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)

The Handbook of the Logic of Argument and Inference is an authoritative reference work in a single volume, designed for the attention of senior undergraduates, graduate students and researchers in all the leading research areas concerned with the logic of practical argument and inference. After an introductory chapter, the role of standard logics is surveyed in two chapters. These chapters can serve as a mini-course for interested readers, in deductive and inductive logic, or as a refresher. Then follow two chapters (...) of criticism; one the internal critique and the other the empirical critique. The first deals with objections to standard logics (as theories of argument and inference) arising from the research programme in philosophical logic. The second canvasses criticisms arising from work in cognitive and experimental psychology. The next five chapters deal with developments in dialogue logic, interrogative logic, informal logic, probability logic and artificial intelligence. The last chapter surveys formal approaches to practical reasoning and anticipates possible future developments. Taken as a whole the Handbook is a single-volume indication of the present state of the logic of argument and inference at its conceptual and theoretical best. Future editions will periodically incorporate significant new developments. (shrink)

Given an argumentation network we associate with it a modal formula representing the 'logical content' of the network. We show a one-to-one correspondence between all possible complete Caminada labellings of the network and all possible models of the formula.

We describe the state of the Talmudic Logic project as of end of 2019. The Talmud is the most comprehensive and fundamental work of Jewish religious law, employing a large number of logical components centuries ahead of their time. In many cases the basic principles are not explicitly formulated, which makes it difficult to formalize and make available to the modern student of Logic. This project on Talmudic Logic, aims to present logical analysis of Talmudic reasoning using modern logical tools. (...) We investigate principles of Talmudic Logic and publish a series of books, one book or more for each principle. The series begins with the systematic analysis of Talmudic inference rules. The first book shows that we can present Talmudic reasoning intuitions as a systematic logical system basic to modern non-deductive reasoning, such as Argumentum A Fortiori, Abduction and Analogy. The second book offers a systematic common sense method for intuitively defining sets and claims that this method adequately models the Talmudic use of the rules Klal uPrat. These books also criticize modern Talmudic research methodology. Later books deal with additional topics like Deontic logic, and Temporal logic, Agency and processes in the Talmud and more. The aims of the project are two fold: 1. To import into the Talmudic study modern logical methods with a view to help understand complicated Talmudic passages, which otherwise cannot be addressed. 2. To export from the Talmud new logical principles which are innovative and useful to modern contemporary logic. (shrink)

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 ' Logic by translation'. We translate faithfully the new networks into ordinary Dung networks with more nodes and extract the semantics from the translation. 2. The labelling approach This method regards the arrows as additional entities to be attacked and to mount attacks and applies a variation of the usual machinery of Camindada like labelling to the network. The new concept we need to employ here is that of 'joint attacks'. 3. The logic programming approach We translate the higher level network into a logic program and obtain semantics for it through known semantics for logic programs. We then compare our methods with those of S. Modgil and P. M. Dung et al. (shrink)

This volume serves as a detailed introduction for those new to the field as well as a rich source of new insights and potential research agendas for those already engaged with the philosophy of economics.

The properties of negation, in combination with those of other logical operations and structural features of the deductibility relation, serve as gateways among logical systems. Negation therefore plays an important role in selecting logical systems for particular applications. This volume provides a thorough treatment of this concept, based on contributions written by authors from various branches of logic. The resulting 14 research papers address a variety of topics including negation in relevant logics; a defense of dialetheic theory of negation; stable (...) negation in logic programming; antirealism and falsity; and negation, denial, and language change in philosophical logic. Suited to scholars and graduate students in the fields of philosophy, logic mathematics, computer science, and linguistics. Annotation copyrighted by Book News, Inc., Portland, OR. (shrink)

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, which do not go (...) to points, but to arrows themselves. (shrink)

Agents act on the basis of their beliefs and these beliefs change as they interact with other agents. In this book the authors propose and explain general logical tools for handling change. These tools include preferential reasoning, theory revision, and reasoning in inheritance systems, and the authors use these tools to examine nonmonotonic logic, deontic logic, counterfactuals, modal logic, intuitionistic logic, and temporal logic. This book will be of benefit to researchers engaged with artificial intelligence, and in particular agents, multiagent (...) systems and nonmonotonic logic. (shrink)

In this paper we show that some versions of Dung’s abstract argumentation frames are equivalent to classical propositional logic. In fact, Dung’s attack relation is none other than the generalised Peirce–Quine dagger connective of classical logic which can generate the other connectives ${\neg, \wedge, \vee, \to}$ of classical logic. After establishing the above correspondence we offer variations of the Dung argumentation frames in parallel to variations of classical logic, such as resource logics, predicate logic, etc., etc., and create resource argumentation (...) frames, predicate argumentation frames, etc., etc. We also offer the notion of logic proof as a geometrical walk along the nodes of a Dung network and thus we are able to offer a geometrical abstraction of the notion of inference based argumentation. Thus our paper is also a contribution to the question: “What is a logical system” in as much as it integrates logic with abstract argumentation networks. (shrink)

Agenda Relevance is the first volume in the authors' omnibus investigation of the logic of practical reasoning, under the collective title, A Practical Logic of Cognitive Systems. In this highly original approach, practical reasoning is identified as reasoning performed with comparatively few cognitive assets, including resources such as information, time and computational capacity. Unlike what is proposed in optimization models of human cognition, a practical reasoner lacks perfect information, boundless time and unconstrained access to computational complexity. The practical reasoner is (...) therefore obliged to be a cognitive economizer and to achieve his cognitive ends with considerable efficiency. Accordingly, the practical reasoner avails himself of various scarce-resource compensation strategies. He also possesses neurocognitive traits that abet him in his reasoning tasks. Prominent among these is the practical agent's striking (though not perfect) adeptness at evading irrelevant information and staying on task. On the approach taken here, irrelevancies are impediments to the attainment of cognitive ends. Thus, in its most basic sense, relevant information is cognitively helpful information. Information can then be said to be relevant for a practical reasoner to the extent that it advances or closes some cognitive agenda of his. The book explores this idea with a conceptual detail and nuance not seen the standard semantic, probabilistic and pragmatic approaches to relevance; but wherever possible, the authors seek to integrate alternative conceptions rather than reject them outright. A further attraction of the agenda-relevance approach is the extent to which its principal conceptual findings lend themselves to technically sophisticated re-expression in formal models that marshal the resources of time and action logics and label led deductive systems. Agenda Relevance is necessary reading for researchers in logic, belief dynamics, computer science, AI, psychology and neuroscience, linguistics, argumentation theory, and legal reasoning and forensic science, and will repay study by graduate students and senior undergraduates in these same fields. Key features: relevance action and agendas practical reasoning belief dynamics non-classical logics labelled deductive systems. (shrink)

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural logics. The book (...) can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory. (shrink)