## Labelled Deductive Systems

D. Gabbay. *Labelled Deductive Systems*, Oxford: Clarendon Press, 1996.

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#### Abstract:

Labelled deductive systems (LDSs) form a general framework, proposed by the author of this book, to treat a great variety of logical systems. Labels are used in order to describe structures richer than sets of formulas, making it possible to manipulate information, databases, etc. The notion of deduction is considered from the viewpoint of a pruned (monotonicity is rejected) variant of Tarski's consequence relation. The author says this notion was put forward in A. Tarski's paper "On the concept of logical consequence" [Actualites Sci. Indust. 394 (1936), 1--11; JFM 62.1050.05], but that paper is a presentation of the semantical notion of consequence and does not present an axiomatization of this notion, which Tarski had given earlier [Ann. Polon. Math. 7 (1928), 270--272; JFM 55.0038.15]. The bibliography also lacks several other basic references [e.g., R. Wojcicki, Theory of logical calculi, Kluwer Acad. Publ., Dordrecht], and serious misprints appear in the references to A. Rose's papers. The book presents the formal machinery of LDSs: proof theory, semantics, and algebraic LDSs, with many examples: modal and tense logics, fuzzy and many-valued logics, linear and relevant logics, nonmonotonic logic, etc. (some important nonclassical logics such as paraconsistent logic are not taken into account). No genuinely new results about these logics are presented, nor are their philosophical foundations discussed. These criticisms notwithstanding, this book will be of great interest for anyone who wants to grasp, together with a number of formal techniques, a multiplicity of logical systems, mostly related to computer science and artificial intelligence. (The first chapter, entitled "What is a logical system?", has also appeared elsewhere [in What is a logical system?, 179--216, Oxford Univ. Press, New York, 1994].)

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@BOOK{gabbay96:_label_deduc_system, title = {Labelled Deductive Systems}, publisher = {Oxford: Clarendon Press}, year = {1996}, author = {D. Gabbay}, abstract = { Labelled deductive systems (LDSs) form a general framework, proposed by the author of this book, to treat a great variety of logical systems. Labels are used in order to describe structures richer than sets of formulas, making it possible to manipulate information, databases, etc. The notion of deduction is considered from the viewpoint of a pruned (monotonicity is rejected) variant of Tarski's consequence relation. The author says this notion was put forward in A. Tarski's paper "On the concept of logical consequence" [Actualites Sci. Indust. 394 (1936), 1--11; JFM 62.1050.05], but that paper is a presentation of the semantical notion of consequence and does not present an axiomatization of this notion, which Tarski had given earlier [Ann. Polon. Math. 7 (1928), 270--272; JFM 55.0038.15]. The bibliography also lacks several other basic references [e.g., R. Wojcicki, Theory of logical calculi, Kluwer Acad. Publ., Dordrecht], and serious misprints appear in the references to A. Rose's papers. The book presents the formal machinery of LDSs: proof theory, semantics, and algebraic LDSs, with many examples: modal and tense logics, fuzzy and many-valued logics, linear and relevant logics, nonmonotonic logic, etc. (some important nonclassical logics such as paraconsistent logic are not taken into account). No genuinely new results about these logics are presented, nor are their philosophical foundations discussed. These criticisms notwithstanding, this book will be of great interest for anyone who wants to grasp, together with a number of formal techniques, a multiplicity of logical systems, mostly related to computer science and artificial intelligence. (The first chapter, entitled "What is a logical system?", has also appeared elsewhere [in What is a logical system?, 179--216, Oxford Univ. Press, New York, 1994].) } }

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