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Author: Dov M. Gabbay
Publisher: Springer Science & Business Media
ISBN: 9401729778
Category : Philosophy
Languages : en
Pages : 304
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Book Description
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.
Author: Dov M. Gabbay
Publisher: Springer Science & Business Media
ISBN: 9401729778
Category : Philosophy
Languages : en
Pages : 304
Get Book
Book Description
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.
Author: M. Fitting
Publisher: Springer Science & Business Media
ISBN: 9789027715739
Category : Mathematics
Languages : en
Pages : 574
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Book Description
"Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems. We present proof procedures of each of these types for the most common normal modal logics: S5, S4, B, T, D, K, K4, D4, KB, DB, and also G, the logic that has become important in applications of modal logic to the proof theory of Peano arithmetic. Further, we present a similar variety of proof procedures for an even larger number of regular, non-normal modal logics (many introduced by Lemmon). We also consider some quasi-regular logics, including S2 and S3. Virtually all of these proof procedures are studied in both propositional and first-order versions (generally with and without the Barcan formula). Finally, we present the full variety of proof methods for Intuitionistic logic (and of course Classical logic too). We actually give two quite different kinds of tableau systems for the logics we consider, two kinds of Gentzen sequent calculi, and two kinds of natural deduction systems. Each of the two tableau systems has its own uses; each provides us with different information about the logics involved. They complement each other more than they overlap. Of the two Gentzen systems, one is of the conventional sort, common in the literature.
Author: Anil Nerode
Publisher: Springer Science & Business Media
ISBN: 1468402110
Category : Computers
Languages : en
Pages : 383
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Book Description
In writing this book, our goal was to produce a text suitable for a first course in mathematical logic more attuned than the traditional textbooks to the recent dramatic growth in the applications of logic to computer science. Thus our choice of topics has been heavily influenced by such applications. Of course, we cover the basic traditional topics - syntax, semantics, soundness, completeness and compactness - as well as a few more advanced results such as the theorems of Skolem-Lowenheim and Herbrand. Much of our book, however, deals with other less traditional topics. Resolution theorem proving plays a major role in our treatment of logic, especially in its application to Logic Programming and PROLOG. We deal extensively with the mathematical foundations of all three of these subjects. In addition, we include two chapters on nonclassical logic- modal and intuitionistic - that are becoming increasingly important in computer science. We develop the basic material on the syntax and se mantics (via Kripke frames) for each of these logics. In both cases, our approach to formal proofs, soundness and completeness uses modifications of the same tableau method introduced for classical logic. We indicate how it can easily be adapted to various other special types of modal log ics. A number of more advanced topics (including nonmonotonic logic) are also briefly introduced both in the nonclassical logic chapters and in the material on Logic Programming and PROLOG.
Author: Thomas Piecha
Publisher: Springer
ISBN: 331922686X
Category : Philosophy
Languages : en
Pages : 283
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Book Description
This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.
Author: Yiannis N. Moschovakis
Publisher: Springer Science & Business Media
ISBN: 1461228220
Category : Mathematics
Languages : en
Pages : 607
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Book Description
The volume is the outgrowth of a workshop with the same title held at MSRI in the week of November 13-17, 1989, and for those who did not get it, Logic from Computer Science is the converse of Logic in Computer Science, the full name of the highly successful annual LICS conferences. We meant to have a conference which would bring together the LICS commu nity with some of the more traditional "mathematical logicians" and where the emphasis would be on the flow of ideas from computer science to logic rather than the other way around. In a LICS talk, sometimes, the speaker presents a perfectly good theorem about (say) the A-calculus or finite model theory in terms of its potential applications rather than its (often more ob vious) intrinsic, foundational interest and intricate proof. This is not meant to be a criticism; the LICS meetings are, after all, organized by the IEEE Computer Society. We thought, for once, it would be fun to see what we would get if we asked the speakers to emphasize the relevance of their work for logic rather than computer science and to point out what is involved in the proofs. I think, mostly, it worked. In any case, the group of people represented as broad a selection of logicians as I have seen in recent years, and the quality of the talks was (in my view) exceptionally, unusually high. I learned a lot and (I think) others did too.
Author: Hans Jürgen Ohlbach
Publisher: Springer Science & Business Media
ISBN: 9401145741
Category : Philosophy
Languages : en
Pages : 431
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Book Description
th This volume is dedicated to Dov Gabbay who celebrated his 50 birthday in October 1995. Dov is one of the most outstanding and most productive researchers we have ever met. He has exerted a profound influence in major fields of logic, linguistics and computer science. His contributions in the areas of logic, language and reasoning are so numerous that a comprehensive survey would already fill half of this book. Instead of summarizing his work we decided to let him speak for himself. Sitting in a car on the way to Amsterdam airport he gave an interview to Jelle Gerbrandy and Anne-Marie Mineur. This recorded conversation with him, which is included gives a deep insight into his motivations and into his view of the world, the Almighty and, of course, the role of logic. In addition, this volume contains a partially annotated bibliography of his main papers and books. The length of the bibliography and the broadness of the topics covered there speaks for itself.
Author: Steven Homer
Publisher: Springer
ISBN: 3540471359
Category : Mathematics
Languages : en
Pages : 158
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Book Description
The courses given at the 1st C.I.M.E. Summer School of 1988 dealt with the main areas on the borderline between applied logic and theoretical computer science. These courses are recorded here in five expository papers: S. Homer: The Isomorphism Conjecture and its Generalization.- A. Nerode: Some Lectures on Intuitionistic Logic.- R.A. Platek: Making Computers Safe for the World. An Introduction to Proofs of Programs. Part I. - G.E. Sacks: Prolog Programming.- A. Scedrov: A Guide to Polymorphic Types.
Author: Aybüke Özgün
Publisher: Springer Nature
ISBN: 3030984796
Category : Mathematics
Languages : en
Pages : 367
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Book Description
This book constitutes the refereed proceedings of the 13th International Tbilisi Symposium on Logic, Language and Computation, TbiLLC 2019, held in Batumi, Georgia, in September 2019. The volume contains 17 full revised papers presented at the conference from 17 submissions. The scientific program consisted of tutorials, invited lectures, contributed talks, and two workshops. The symposium offered two tutorials in language and logic and aimed at students as well as researchers working in the other areas: · Language: Sign language linguistics. State of the art, by Fabian Bross (University of Stuttgart, Germany) · Logic: Axiomatic Semantics, by Graham E. Leigh (University of Gothenburg, Sweden)
Author: Gerard Meurant
Publisher: Elsevier
ISBN: 0080933920
Category : Computers
Languages : en
Pages : 1280
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Book Description
The second part of this Handbook presents a choice of material on the theory of automata and rewriting systems, the foundations of modern programming languages, logics for program specification and verification, and some chapters on the theoretic modelling of advanced information processing.
Author: Fabrizio Cariani
Publisher: Springer
ISBN: 3319086154
Category : Mathematics
Languages : en
Pages : 285
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Book Description
This volume presents the refereed proceedings of the 12th International Conference on Deontic Logic and Normative Systems, DEON 2014, held in Ghent, Belgium, in July 2014. The 17 revised papers and the 2 invited papers included in this volume were carefully reviewed and selected from 31 submissions. Topics covered include challenges from natural language for deontic logic; the relationship between deontic and other types of modality: epistemic modality, imperatives, supererogatory, etc.; the deontic paradoxes; the modeling of normative concepts other than obligation and permission, e.g., values; the game-theoretical aspects of deontic reasoning; the emergence of norms; norms from a conversational and pragmatic point of view; and norms and argumentation.
