Giles’s Game and the Proof Theory of Łukasiewicz Logic

Fermüller, Christian G.; Metcalfe, George

doi:10.1007/s11225-009-9185-2

Cited by 26 publications

(17 citation statements)

References 32 publications

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“…Two main omissions are proof theory [86][87][88] and game-theoretic semantics [89][90][91]. Quite a few areas of current research have only been mentioned in the list of applications above and in the bibliographical remarks collected at the end of each chapter.…”

Section: Applications and Further Readingmentioning

confidence: 99%

Advanced Łukasiewicz calculus and MV-algebras

Mundici¹

2011

176

240

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SCOPE OF THE SERIESTrends 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.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. Cover design: eStudio Calamar, Berlin/FigueresPrinted on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To CeciliaJust as the Z-module structure of ðG; 1Þ is missing in the MV-algebra A ¼ CðG; 1Þ; several fundamental notions and constructs available in the framework of MV-algebras and Łukasiewicz logic hardly make any sense for unital '-groups, despite the latter are categorically equivalent to MV-algebras. Thus, the equational definability of the class of MV-algebras gives us a way of introducing free and finitely presented objects-while the class of unital '-groups is not even definable vii in first-order logic. Induction on the complexity of Łukasiewicz formulas, combined with their geometric representation as McNaughton functions, is a main tool to explore syntactic and semantic consequence in Ł ? , and the fundamental logic property of interpolation. Formulas in Ł ? denote continuously valued events, just as boolean formulas denote yes-no events; coherent probability assessments on these events yield Rényi conditionals, which would make no sense for unital '-groups. r-complete MV-algebras provide a natural framework for generalizations of many classical results originally proved for r-complete boolean algebras, such as the theorem of Loomis-Sikorski and Poincaré's recurrence theorem. Several main techniques and results of probability theory, that Carathéodory reformulated in the language of r-complete boolean algebras, have nontrivial MV-algebraic generalizations. Bases originate as algebraically invariant counterparts of disjunctive Schauder normal forms in Łukasiewicz logic; an MV-algebra has a basis iff it is finitely presented. Classical first-order logic with identity has a generalization to a Łukasiewicz first-order logic Ł xx with [0,1]-valued identity. Models of Ł xx are suitable sets X of unit vectors in a Hilbert space H, and the identity degree of any two vectors u; v 2 X is their scalar product; functions and relations on X satisfy suitable continuity properties.Since this book is devoted to these genuine MV-algebraic and logical topics, its overlap with books on '-groups, with or witho...

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Section: Applications and Further Readingmentioning

confidence: 99%

Advanced Łukasiewicz calculus and MV-algebras

Mundici¹

2011

176

240

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“…This is already the case for all logical rules considered so far. However, as shown in [6,7], by making this principle explicit we arrive at a rule for strong conjunction, that is missing in Giles:…”

Section: Definitionmentioning

confidence: 99%

“…Theorem 4 (essentially Giles, but see also [7]) A L-sentence F is evaluated to v M (F ) = x in interpretation M iff the G-game for F under risk value assignment · M is (1 − x)-valued for me.…”

Section: This Allows Us To Characterize Strong Lukasiewicz Logic L Bymentioning

confidence: 99%

“…Relative to the proof of Theorem 4 (see [10,9,7] and it remains to show that my optimal way to reduce the exhibited quantified formula to instances as required by rule (R Π k m ) results in a risk that corresponds to the specified truth function. In order to do so remember that the principle of limited liability is in place.…”

Section: Proportionality Quantifiersmentioning

confidence: 99%

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Randomized game semantics for semi-fuzzy quantifiers

Fermüller

Roschger

2013

Logic Journal of IGPL

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“…Remark 2 Also modal logics can be characterized by appropriate semantic games (see, e.g., [8]). A less well-known, but also very fertile game theoretic approach to semantics is Giles's game for Lukasiewicz logic [11,12,9]. Giles developed his game semantics for reasoning under uncertainty independent from Hintkka.…”

Section: Theorem 1 (Hintikka 2 )mentioning

confidence: 99%

On matrices, Nmatrices and games

Fermüller¹

2013

J Logic Computation

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Hintikka's semantic game for classical logic is generalized to the family of all finite valued matrices. This in turn serves as a springboard for developing game semantics for all propositional formulas with respect to arbitrary finite non-deterministic matrices. In this approach a new concept of non-deterministic valuation, called 'liberal valuation', emerges that augments the usually employed static and dynamic valuations in a natural manner. Liberal valuation is shown to correspond to unrestricted semantic games, while the characterization of static and dynamic valuations involves certain restrictions of the game that are handled by an interactive pruning procedure.

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Giles’s Game and the Proof Theory of Łukasiewicz Logic

Cited by 26 publications

References 32 publications

Advanced Łukasiewicz calculus and MV-algebras

Advanced Łukasiewicz calculus and MV-algebras

Randomized game semantics for semi-fuzzy quantifiers

On matrices, Nmatrices and games

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