Gentzen Calculi for Modal Propositional Logic
The book is about Gentzen calculi for (the main systems of) modal logic. It is divided into three parts. In the first part we introduce and discuss the main philosophical ideas related to proof theory, and we try to identify criteria for distinguishi
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TRENDS IN LOGIC Studia Logica Library VOLUME 32 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Wieslaw Dziobiak, University of Puerto Rico at Mayagüez, USA Melvin Fitting, City University of New York, USA Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany
SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time. Volume Editor Heinrich Wansing
For further volumes: http://www.springer.com/series/6645
Francesca Poggiolesi
Gentzen Calculi for Modal Propositional Logic
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Francesca Poggiolesi VRIJE UNIVERSITEIT BRUSSEL CLWF/LW Pleinlaan 2 B-1050 Brussels Belgium [email protected]
ISBN 978-90-481-9669-2 e-ISBN 978-90-481-9670-8 DOI 10.1007/978-90-481-9670-8 Springer Dordrecht Heidelberg London New York c Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Modal logic, the logic created to formalise the concepts of possibility and necessity, has a long and important history that begins with Aristotle, who claims that Since there is a difference according as something belongs, necessarily belongs, or may belong to something else (for many things belong indeed, but not necessarily, others neither necessarily nor indeed at all, but it is possible for them to belong), it is clear that there will be different syllogisms to prove each of these relations, and syllogisms with differently related terms, one syllogism concluding from what is necessary, another from what is, a third from what is possible. [2, Book I, part 8]
For many years modal logic was exclusively considered a (Hilbertian) syntactic tool. It was not until Kripke’s discoveries of the early 1960s that modal logic was opened up to semantic research, and it is in this context that its reputation and develo
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