Labelled Non Classical Logics

I am very happy to have this opportunity to introduce Luca Vigano's book on Labelled Non-Classical Logics. I put forward the methodology of labelled deductive systems to the participants of Logic Colloquium'90 (Labelled Deductive systems, a Position Paper, In J. Oikkonen and J. Vaananen, editors, Logic Colloquium '90, Volume 2 of Lecture Notes in Logic, pages 66-68, Springer, Berlin, 1993), in an attempt to bring labelling as a recognised and significant component of our logic culture. It was a response to earlier isolated uses of labels by various distinguished authors, as a means to achieve local proof theoretic goals. Labelling was used in many different areas such as resource labelling in relevance logics, prefix tableaux in modal logics, annotated logic programs in logic programming, proof tracing in truth maintenance systems, and various side annotations in higher-order proof theory, arithmetic and analysis. This widespread local use of labels was an indication of an underlying logical pattern, namely the simultaneous side-by-side manipulation of several kinds of logical information. It was clear that there was a need to establish the labelled deductive systems methodology. Modal logic is one major area where labelling can be developed quickly and sys tematically with a view of demonstrating its power and significant advantage. In modal logic the labels can play a double role.

This book is an introduction to non-classical propositional logics. It brings together for the first time in a textbook a range of topics in logic, many of them of relatively recent origin, including modal, conditional, intuitionist, many-valued, paraconsistent, relevant and fuzzy logics. The material is unified by the underlying theme of world-semantics. All of the topics are explained clearly and accessibly, using devices such as tableaux proofs, and their relation to current philosophical issues and debates is discussed. Students with a basic understanding of classical logic will find this an invaluable introduction to an area that has become of central importance in both logic and philosophy, but which, until now, could be studied only through the research literature. It will interest those studying logic, those who need to know about non-classical logics because of their philosophical importance, and, more widely, readers working in mathematics and computer science.

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.

Classical logic is traditionally introduced by itself, but that makes it seem arbitrary and unnatural. This text introduces classical alongside several nonclassical logics (relevant, constructive, quantative, paraconsistent).

This book introduces a general framework, called Compiled Labelled Deductive System (CLDS), for a unified presentation of logics. It shows how this technique is applied to various families of non-classical logics widely used in Computer Science and Artificial Intelligence. It includes complete natural deduction proof systems, together with proofs and examples, for the families of propositional and predicate modal logics, propositional conditional logics of normality, multiplicative linear logic and for Lukasiewicz fuzzy logic. The CLDS framework generalises the notion of a logical theory to a structured theory consisting of labelled formulas and relations between the labels. This formalisation makes the logics more amenable to applications. Moreover, the framework provides a semantic approach based on a first-order translation, which can be uniformly applied to any logic whose semantics is first-order axiomatisable. Such a semantic approach also facilitates the development of automated theorem provers for non-classical logics.

This volume brings together a group of logic-minded philosophers and philosophically oriented logicians to address a diversity of topics on the structural analysis of non-classical logics. It mainly focuses on the construction of different types of models for various non-classical logics of current interest, including modal logics, epistemic logics, dynamic logics, and observational predicate logic. The book presents a wide range of applications of two well-known approaches in current research: (i) structural modeling of certain philosophical issues in the framework of non-classic logics, such as admissible models for modal logic, structural models for modal epistemology and for counterfactuals, and epistemological models for common knowledge and for public announcements; (ii) conceptual analysis of logical properties of, and formal semantics for, non-classical logics, such as sub-formula property, truthmaking, epistemic modality, behavioral strategies, speech acts and assertions. The structural analysis provided in this volume will appeal not only to graduate students and experts in non-classic logics, but also to readers from a wide range of disciplines, including computer science, cognitive science, linguistics, game theory and theory of action, to mention a few.

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to (abstract) algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics.

Quantification and modalities have always been topics of great interest for logicians. These two themes emerged from philosophy and language in ancient times; they were studied by traditional informal methods until the 20th century. In the last century the tools became highly mathematical, and both modal logic and quantification found numerous applications in Computer Science. At the same time many other kinds of nonclassical logics were investigated and applied to Computer Science. Although there exist several good books in propositional modal logics, this book is the first detailed monograph in nonclassical first-order quantification. It includes results obtained during the past thirty years. The field is very large, so we confine ourselves with only two kinds of logics: modal and superintuitionistic. The main emphasis of Volume 1 is model-theoretic, and it concentrates on descriptions of different sound semantics and completeness problem --- even for these seemingly simple questions we have our hands full. The major part of the presented material has never been published before. Some results are very recent, and for other results we either give new proofs or first proofs in full detail.