This book is intended to be a survey of the most important results in mathematical logic for philosophers. It is a survey of results which have philosophical significance and it is intended to be accessible to philosophers. I have assumed the mathematical sophistication acquired· in an introductory logic course or in reading a basic logic text. In addition to proving the most philosophically significant results in mathematical logic, I have attempted to illustrate various methods of proof. For example, the completeness (...) of quantification theory is proved both constructively and non-constructively and relative ad vantages of each type of proof are discussed. Similarly, constructive and non-constructive versions of Godel's first incompleteness theorem are given. I hope that the reader· will develop facility with the methods of proof and also be caused by reflect on their differences. I assume familiarity with quantification theory both in under standing the notations and in finding object language proofs. Strictly speaking the presentation is self-contained, but it would be very difficult for someone without background in the subject to follow the material from the beginning. This is necessary if the notes are to be accessible to readers who have had diverse backgrounds at a more elementary level. However, to make them accessible to readers with no background would require writing yet another introductory logic text. Numerous exercises have been included and many of these are integral parts of the proofs. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

Modal logic is one of the most widely applied logical formalisms. Systems of modal logic are being used in many disciplines, ranging from artificial intelligence, computer science, mathematics, formal grammar and semantics to philosophy. This volume presents substantial recent advances in the relational and the algorithmic treatment of modal logics. It contains papers from the fifth conference on "Advances in Modal logic," held in Manchester (UK) in September 2004. Written by leading experts in the field, the present book is (...) indispensable for any advanced student and researcher in pure and applied modal logic. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science.

Modal Logic, originally conceived as the logic of necessity and possibility, has developed into a powerful mathematical and computational discipline. It is the main source of formal languages aimed at analyzing complex notions such as common knowledge and formal provability. Modal and modal-like languages also provide us with families of restricted description languages for relational and topological structures; they are being used in many disciplines, ranging from artificial intelligence, computer science and mathematics via natural language syntax and semantics to philosophy. (...) This volume presents a broad and up-to-date view of the field, with contributions covering both the foundations of modal logic itself and each of the aforementioned application areas. Complemented with an editorial introduction covering the roots of modal logic, this book is indispensable for any advanced student and researcher in non-classical logic and its applications. (shrink)

This title introduces students to non-classical logic, syllogistic, to quantificational and modal logic. The book includes exercises throughout and a glossary of terms and symbols. Taking students beyond classical mathematical logic, "Philosophical Logic" is a wide-ranging introduction to more advanced topics in the study of philosophical logic. Starting by contrasting familiar classical logic with constructivist or intuitionist logic, the book goes on to offer concise but easy-to-read introductions to such subjects as quantificational and syllogistic logic, modal logic and set (...) theory. Chapters of this title include: Sentential Logic; Quantificational Logic; Sentential Modal Logic; Quantification and Modality; Set Theory; Incompleteness; An Introduction to Term Logic; and, Modal Term Logic. In addition, the book includes a list of symbols and a glossary of terms for ease of reference and exercises throughout help students master the topics covered in the book. (shrink)

The ŁΠ and logics were introduced by Godo, Esteva and Montagna. These logics extend many other known propositional and predicate logics, including the three mainly investigated ones (Gödel, product and Łukasiewicz logic). The aim of this paper is to show some advances in this field. We will see further reduction of the axiomatic systems for both logics. Then we will see many other logics contained in the ŁΠ family of logics (namely logics induced (...) by the continuous finitely constructed t-norms and Takeuti and Titani's fuzzy predicate logic). (shrink)

Modal logic originated in philosophy as the logic of necessity and possibility. Now it has reached a high level of mathematical sophistication and has many applications in a variety of disciplines, including theoretical and applied computer science, artificial intelligence, the foundations of mathematics, and natural language syntax and semantics. This volume represents the proceedings of the first international workshop on Advances in Modal Logic, held in Berlin, Germany, October 8-10, 1996. It offers an up-to-date perspective on the field, with contributions (...) covering its proof theory, its applications in knowledge representation, computing and mathematics, as well as its theoretical underpinnings. (shrink)

Proc. of the 8th International Conference on Advances in Modal Logic, (AiML'2010).

Linear logic, introduced in 1986 by J.-Y. Girard, is based upon a fine grain analysis of the main proof-theoretical notions of logic. The subject develops along the lines of denotational semantics, proof nets and the geometry of interaction. Its basic dynamical nature has attracted computer scientists, and various promising connections have been made in the areas of optimal program execution, interaction nets and knowledge representation. This book is the refereed proceedings of the first international meeting on linear logic held at (...) Cornell University, in June 1993. Survey papers devoted to specific areas of linear logic, as well as an extensive general introduction to the subject by J.-Y. Girard, have been added, so as to make this book a valuable tool both for the beginner and for the advanced researcher. (shrink)

A model of normative provisions and related axioms represented by using RDF/owl are presented as a contribution to implement the semantic web in the legal domain. In particular, a pattern able to implement the Hohfeldian legal fundamental relations between provisions using OWL-DL expressivity is proposed. Moreover, a query-based approach able to deal with relations between provision instances is described. An example of advanced access and reasoning over provisions using the proposed approach, as well as a prototype architecture of a (...) provision retrieval system, are shown. The main benefit is represented by the ability of the approach to keep the complexity of the problem within a description logic computational tractability. (shrink)

1. Lemmon’s stratification. By a “tense logic” I mean a system with the following features: it contains sentential variables which stand for sentences which in some cases are true at some times and false at others; it contains the usual truth-functions, whose truth-conditions are given the obvious modifications, e.g. Np is true when and only when p is false, Kpq is true when and only when both its conjuncts are; and it contains two additional functions which may be interpreted as (...) ‘It will be the case that p’ and ‘It has been the case that p’, the former being true when and only when the plain p will be true later on and the latter when and only when the plain p has been true at some earlier time. (shrink)

This book is a collection of articles on research that attempts to connect logic and mathematics with empirical and cognition. There have been various such a...

Intensional logic has emerged, since the 1960' s, as a powerful theoretical and practical tool in such diverse disciplines as computer science, artificial intelligence, linguistics, philosophy and even the foundations of mathematics. The present volume is a collection of carefully chosen papers, giving the reader a taste of the frontline state of research in intensional logics today. Most papers are representative of new ideas and/or new research themes. The collection would benefit the researcher as well as the student. This (...) book is a most welcome addition to our series. The Editors CONTENTS PREFACE IX JOHAN VAN BENTHEM AND NATASHA ALECHINA Modal Quantification over Structured Domains PATRICK BLACKBURN AND WILFRIED MEYER-VIOL Modal Logic and Model-Theoretic Syntax 29 RUY J. G. B. DE QUEIROZ AND DOV M. GABBAY The Functional Interpretation of Modal Necessity 61 VLADIMIR V. RYBAKOV Logics of Schemes for First-Order Theories and Poly-Modal Propositional Logic 93 JERRY SELIGMAN The Logic of Correct Description 107 DIMITER VAKARELOV Modal Logics of Arrows 137 HEINRICH WANSING A Full-Circle Theorem for Simple Tense Logic 173 MICHAEL ZAKHARYASCHEV Canonical Formulas for Modal and Superintuitionistic Logics: A Short Outline 195 EDWARD N. ZALTA 249 The Modal Object Calculus and its Interpretation NAME INDEX 281 SUBJECT INDEX 285 PREFACE Intensional logic has many faces. In this preface we identify some prominent ones without aiming at completeness. (shrink)

Since ancient times, philosophers have recognised that truth comes in many 'modes', so that a proposition can be not only true or false, but also, for example, 'necessary' or 'possible'. These ideas led to the modern field of modal logic, a lively area of research at the interface of philosophy, mathematics and computer science. -/- Nowadays, the term 'modal logic' is understood in a broad sense, allowing it to encompass logics for reasoning about seemingly unrelated phenomena such as knowledge, (...) obligations, time, space, and proofs, among many others. Contemporary research in modal logic draws on techniques from many disciplines, including complexity theory, combinatorics, universal algebra, category theory, topology, and proof theory. -/- These proceedings record the papers presented at Advances in Modal Logic 2024, the 15th in a series of biennial conferences that aim to report on important new developments in pure and applied modal logic. Topics in this issue include epistemic modal logic, constructive and many-valued modal logic, unification, alge aic and neighbourhood semantics, proof theory and complexity of modal logics, conditional and quantified modal logic. (shrink)

This paper briefly outlines some advancements in paraconsistent logics for modelling knowledge representation and reasoning. Emphasis is given on the so-called Logics of Formal Inconsistency (LFIs), a class of paraconsistent logics that formally internalize the very concept(s) of consistency and inconsistency. A couple of specialized systems based on the LFIs will be reviewed, including belief revision and probabilistic reasoning. Potential applications of those systems in the AI area of KRR are tackled by illustrating some examples that emphasizes (...) the importance of a fine-tuned treatment of consistency in modelling reputation systems, preferences, argumentation, and evidence. (shrink)

We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme a ϕ → a ϕ. In the intended semantics, the plain..

This is a short introduction to a special issue of the Journal on Logic, Language and Information, dealing with "recent advances in logical and algebraic approaches to computational linguistics".

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge (...) that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)

Constraint programming is the fruit of several decades of research carried out in mathematical logic, automated deduction, operations research and arti?cial intelligence. The tools and programming languages arising from this research?eldhaveenjoyedrealsuccessintheindustrialworldastheycontributetosolving hard combinatorial problems in diverse domains such as production planning, communication networks, robotics and bioinformatics. This volume contains the extended and reviewed versions of a selection of papers presented at the Joint ERCIM/CoLogNET International Workshop on Constraint Solving and Constraint Logic Programming, which was held from June 30 to July (...) 2, 2003. The venue chosen for the seventh edition of this annual workshop was the Computer and Automation Research Institute of the Hungarian Academy of Sciences in Budapest, Hungary. This institute is one of the 20 members of the Working Group on Constraints of the European Research Consortium for Informatics and Mathematics. For many participants this workshop provided the?rst opportunity to visit their ERCIM partner in Budapest. CoLogNET is the European-funded network of excellence dedicated to s- porting and enhancing cooperation and research on all areas of computational logic, and continues the work done previously by the Compulog Net. In part- ular, the aim of the logic and constraint logic programming area of CoLogNET is to foster and support all research activities related to logic programming and constraint logic programming. The editors would like to take the opportunity and thank all the authors who submitted papers to this volume, as well as the reviewers for their helpful work. (shrink)

We study a propositional bimodal logic consisting of two S4 modalities £ and [a], together with the interaction axiom scheme a £ϕ → £ aϕ. In the intended semantics, the plain £ is given the McKinsey-Tarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation a. The interaction axiom expresses the property that the Ra relation is lower semi-continuous with respect to the topology. The (...) class of topological Kripke frames characterised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a differential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although tableau completeness and decidability are still open questions. (shrink)

This book introduces the reader to relevant logic and provides the subject with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used in deriving its conclusion. The logic is placed in the context of possible world semantics and situation semantics, which are then applied to provide an understanding of the various logical particles and natural language conditionals. The book ends by examining various applications of relevant logic and (...) presenting some interesting open problems. It will be of interest to a range of readers including advanced students of logic, philosophical and mathematical logicians, and computer scientists. (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The inductive logic developed in the second and third essays is limited in important ways. For example: (a) the logic makes no provision for missing or misleading data; (b) it gives the scientist no control over the evidence reaching him; (c) revision-based scientist must work with theories written in the cramped idiom of firstorder logic; (d) the idea of efficient induction is only weakly expressed (in terms of “dominance”).