R-Calculus, IV: Propositional Logic 9811986320, 9789811986321

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Perspectives in Formal Induction, Revision and Evolution

Wei Li Yuefei Sui

R-Calculus, IV: Propositional Logic

Perspectives in Formal Induction, Revision and Evolution Editor-in-Chief Wei Li, Beijing, China Series Editors Jie Luo, Beijing, China Yuefei Sui, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

Perspectives in Formal Induction, Revision and Evolution is a book series focusing on the logics used in computer science and artificial intelligence, including but not limited to formal induction, revision and evolution. It covers the fields of formal representation, deduction, and theories or meta-theories of induction, revision and evolution, where the induction is of the first level, the revision is of the second level, and the evolution is of the third level, since the induction is at the formula stratum, the revision is at the theory stratum, and the evolution is at the logic stratum. In his book “The Logic of Scientific Discovery”, Karl Popper argues that a scientific discovery consists of conjecture, theory, refutation, and revision. Some scientific philosophers do not believe that a reasonable conjecture can come from induction. Hence, induction, revision and evolution have become a new territory for formal exploration. Focusing on this challenge, the perspective of this book series differs from that of traditional logics, which concerns concepts and deduction. The series welcomes proposals for textbooks, research monographs, and edited volumes, and will be useful for all researchers, graduate students, and professionals interested in the field.

Wei Li · Yuefei Sui

R-Calculus, IV: Propositional Logic

Wei Li Beihang University Beijing, China

Yuefei Sui Institute of Computing Technology Chinese Academy of Sciences Beijing, China

ISSN 2731-3689 ISSN 2731-3697 (electronic) Perspectives in Formal Induction, Revision and Evolution ISBN 978-981-19-8632-1 ISBN 978-981-19-8633-8 (eBook) https://doi.org/10.1007/978-981-19-8633-8 Jointly published with Science Press The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press. © Science Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface to the Series

Classical mathematical logics (propositional logic, first-order logic and modal logic) and applied logics (temporal logic, dynamic logic, situation calculus, etc.) concern deduction, a logical process from universal statements to particular statements. Description logics concern concepts which and deduction compose of general logics. Induction and belief revision are the topics of general logics and philosophical logics, and evolution is a new research area in computer science. To formalize induction, revision and evolution is a goal of this series. Revision is omnipresent in sciences. A new theory usually is a revision of an old theory or several old theories. Copernicus’ heliocentric theory is a revision of the Tychonic system; the theory of relativity is a revision of the classical theory of movement; the quantum theory is a revision of classical mechanics; etc. The AGM postulates is a set of conditions a reasonable revision operator should satisfy. Professor Li proposed a calculus for first-order logic, called R-calculus, which is sound and complete with respect to maximal consistent subsets. R-calculus has several variants which can be used in other logics, such as nonmonotonic logics, can propose new problems in the classical logics, and will be used in bigdata. Popper proposed in his book The Logic of Scientific Discovery that a scientific discovery consists of four aspects: conjecture, theory, refutation and revision. Some scientific philosophers refuted that a reasonable conjecture should come from induction. Hence, induction, revision and evolution become a new territory to be discovered in a formal way. An induction process is from particular statements to universal statements. A typical example is the mathematical induction, which is a set of nontrivial axioms in Peano arithmetic which makes Peano arithmetic incomplete with respect to the standard model of Peano arithmetic. A logic for induction is needed to guide data mining in artificial intelligence. Data mining is a canonical induction process, which mines rules from data. In biology, Evolution is change in the heritable traits of biological populations over successive generations. In sciences, Darwin’s evolution theory is an evolution of intuitive theories of plants and animals. In logic, an evolution is a generating process of combining two logics into a new logic, where the new logic should have the traits v

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of two logics. Hence, we should define the corresponding heritable traits of logics, sets of logics and sequences of logics. Simply speaking, predicate modal logic is an evolution of propositional modal logic and predicate logic, and there are many new problems in predicate modal logic to be solved, such as the constant domain semantics, the variant domain semantics. The series should focus on formal representation, deduction, theories or metatheories of induction, revision and evolution, where the induction is of the first level, the revision is of the second level and the evolution is of the third level, because the induction is at the formula stratum, the revision is at the theory stratum and the evolution is at the logic stratum. The books in the series differ in level: some are overviews and some highly specialized. Here, the books of overviews are for undergraduated students; and the highly specialized ones are for graduated students and researchers in computer science and mathematical logic. Beijing, China October 2016

Wei Li Jie Luo Fangming Song Yuefei Sui Ju Wang Wujia Zhu

Preface

Propositional logic is basic. R-calculus is a belief revision operator satisfying AGM postulates. Combining propositional logic and R-calculus produces a new point of view to consider belief revision. For any i ∈ {0, 1}, a theory  is Ti -valid if for any assignment v, v(A) = i for some formula A ∈ ; and is Ti -valid if there is an assignment v such that v(A) = i for each formula A ∈ . For any Q 1 , Q 2 ∈ {E, A} a sequent  ⇒  is G Q 1 Q 2 -valid if for any assignment v, either v(A) = 0 for Q 1 -formula A ∈ , or v(B) = 1 for Q 2 -formula B ∈ ; and is G Q 1 Q 2 -valid if there is an assignment v such that v(A) = 1 for Q 1 -formula A ∈ , and v(B) = 1 for Q 2 -formula B ∈ . There are variant definitions of the validity of a sequent. For example, for any Q 1 , Q 2 ∈ {A, E} and i, j ∈ {0, 1}, a sequent  is G Q 1 i Q 2 j -valid if for any assignment v, either Q 1 A ∈ (v(A) = i) or Q 2 B ∈ (v(B) = j); and is G Q 1 i Q 2 j -valid if there is an assignment v such that Q 1 A ∈ (v(A) = i), and Q 2 B ∈ (v(B) = j). The negation of G Q 1 i Q 2 j -validity is G Q¯1 (1−i) Q¯2 (1− j) -validity, where E¯ = A and ¯ = E. A For any Q 1 , Q 2 ∈ {A, E} and i, j ∈ {0, 1}, there are sound and complete Gentzen deduction systems G Q 1 Q 2 / Q 1 Q 2 and G Q 1 i Q 2 j / Q 1 i Q 2 j for G Q 1 Q 1 / Q 1 Q 2 - and G Q 1 i Q 2 j / Q 1 i Q 2 j -validity,respectively, which are monotonic in  iff Q 1 = E, and nonmonotonic in  iff Q 2 = A. R-calculus is a Gentzen-typed deduction system which is nonmonotonic, and is a concrete belief revision operator which is proved to satisfy AGM postulates and DP postulates. Corresponding to Gentzen deduction system G Q 1 i Q 2 j /G Q 1 i Q 2 j , there is a sound and complete R-calculus R Q 1 i Q 2 j /R Q 1 i Q 2 j , respectively. R Q 1 i Q 2 j preserves the ⊆-minimal change. Let R, Q, P denote the ⊆-, - and  -minimal change, respectively. For any Y1 , Y2 ∈ {R, Q}, there are sound and complete R-calculi RY1 Q 1 iY2 Q 2 j and RY1 Q 1 iY2 Q 2 j , and R-calculus RPQ 1 iPQ 2 j /PQ 1 iPQ 2 j is sound and complete for disjunctive/conjunctive normal sequents, respectively, where sequent  ⇒  is in disjunctive (conjunctive) normal form if each formula in

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 is in conjunctive (disjunctive) normal form and each formula in  is in conjunctive (disjunctive) normal form. Applications of R-calculus in -propositional logic and logic of supersequents are also given. Beijing, China October 2021

Wei Li Yuefei Sui

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Variant Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 R-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 R-Calculi in This Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 R-Calculi for Weak Propositional Logics . . . . . . . . . . . . . . . . 1.4.2 R-Calculi for Tableau Proof/Gentzen Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 R-Calculi for Deduction Systems G Q 1 Q 2 /G Q 1 Q 2 . . . . . . . . . . 1.4.4 R-Calculi for Deduction Systems G Q 1 i Q 2 j /G Q 1 i Q 2 j . . . . . . . 1.4.5 R-Calculi for Deduction Systems GY1 Q 1 iY2 Q 2 j /GY1 Q 1 iY2 Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Applications of R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 R-Calculus for -Propositional Logic . . . . . . . . . . . . . . . . . . 1.5.2 R-Calculi for Supersequents . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 5 5

9 9 10 11 12 14

2 R-Calculus for Simplified Propositional Logics . . . . . . . . . . . . . . . . . . . . 2.1 Weak Propositional Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Gentzen Deduction System G¬ . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Gentzen Deduction System G¬ . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Gentzen Deduction System G∧ . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Gentzen Deduction System G∧ . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Gentzen Deduction System G∨ . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Gentzen Deduction System G∨ . . . . . . . . . . . . . . . . . . . . . . . . 2.2 R-Calculus for Weak Propositional Logics . . . . . . . . . . . . . . . . . . . . . 2.2.1 R-Calculus R¬ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 R-Calculus R¬ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 R-Calculus R∧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 R-Calculus R∧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 16 18 19 21 22 23 24 25 26 27 28

5 7 8

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2.2.5 R-Calculus R∨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 R-Calculus R∨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Variant Propositional Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Gentzen Deduction System G¬→ . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Gentzen Deduction System G¬→ . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Gentzen Deduction System G⊗⊕ . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Gentzen Deduction System G⊕⊗ . . . . . . . . . . . . . . . . . . . . . . . 2.4 R-Calculi for Variant Propositional Logics . . . . . . . . . . . . . . . . . . . . . 2.4.1 R-Calculus R¬→ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 R-Calculus R¬→ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 R-Calculus R⊕⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 R-Calculus R⊕⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 31 31 35 35 41 42 43 44 46 49 52

3 R-Calculi for Tableau/Gentzen Deduction Systems . . . . . . . . . . . . . . . . 3.1 Tableau Proof Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Tableau Proof System Tf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Tableau Proof System Tt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Tableau Proof System Tt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Tableau Proof System Tf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 R-Calculi for Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 R-Calculus Sf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 R-Calculus St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 R-Calculus St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 R-Calculus Sf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gentzen Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Gentzen Deduction System Gt . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Gentzen Deduction System Gf . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Gentzen Deduction System Gt . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Gentzen Deduction System Gf . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 R-Calculi for Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 R-Calculus Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 R-Calculus Rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 R-Calculus Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 R-Calculus Rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 56 58 60 63 65 66 67 69 71 73 74 75 78 79 83 84 85 87 90 95 98 99

4 R-Calculi R Q 1 Q 2 /R Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Gentzen Deduction Systems G Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 R-Calculi R Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 103 103 105 106 106 107 110

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4.2.3 Deduction Systems R Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Gentzen Deduction Systems G Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 R-Calculi R Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Deduction Systems R Q 1 Q 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112 115 115 116 116 117 117 119 123 123 124

5 R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Gentzen Deduction Systems GE0∗ /GE1∗ . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Monotonicity of GE0∗ and GA1∗ . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Gentzen Deduction Systems G Q 1 i Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 R-Calculi R Q 1 i Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Gentzen Deduction Systems G Q 1 i Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 R-Calculi R Q 1 i Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 127 127 129 131 132 134 134 141 142 142 143 146 147 147 148 151 152 152 153 156 156 157 160

6 R-Calculi: RY1 Q 1 i Y2 Q 2 j /RY1 Q 1 i Y2 Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Variant R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 R-Calculus Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 R-Calculus Rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 R-Calculus Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 R-Calculus Qf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 R-Calculus Pt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 R-Calculus Pf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 R-Calculi RY1 Q 1 iY2 Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 162 162 163 165 166 167 169 169

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6.2.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 R-Calculi RY1 Q 1 iY2 Q 2 j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Deduction Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170 170 177 178 179 179 179 180 180

7 R-Calculi for Supersequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Supersequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Supersequent Gentzen Deduction System G+ . . . . . . . . . . . . 7.2 Reduction of Supersequents to Sequents . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Gentzen Deduction System Gft . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Gentzen Deduction System Gff . . . . . . . . . . . . . . . . . . . . . . . . 7.3 R-Calculus R+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Rft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Rff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Gentzen Deduction System G− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Reduction of Supersequents to Sequents . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Gentzen Deduction System Gff . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Gentzen Deduction System Gft . . . . . . . . . . . . . . . . . . . . . . . . 7.6 R-Calculus R− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Rff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Rft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 185 185 189 190 190 191 195 198 200 202 202 203 204 208 210 213 214

8 R-Calculi for -Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 -Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Tableau Proof Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Tt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Tf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8.2.3 Tableau Proof Systems T t /Tf . . . . . . . . . . . . . . . . . . . . . . . . ∗ 8.3 R-Calculi R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 R-Calculus Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 R-Calculus Rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Other Minimal Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 R-Calculus R t ...................................... 8.4.2 R-Calculus Q t ...................................... 8.4.3 R-Calculus P t ...................................... 8.5 Gentzen Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Gentzen Deduction System Gt . . . . . . . . . . . . . . . . . . . . . . . .

215 216 216 218 218 219 221 221 221 225 228 229 232 236 238 238 242

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8.6 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 R-Calculus Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 R-Calculus Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 R-Calculus Pt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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245 245 248 250 252 252

Chapter 1

Introduction

1.1 Propositional Logic Propositional logic is basic, based on which we developed first-order logic and modal logic, which compose of classical logics. Based on these classical logics, other logics are developed, such as temporal logic, Hoare logic, dynamic logic, computational tree logic, situation calculus, etc. There are three kinds of the deduction systems in propositional logic: • Axiomatic deduction system, with three axioms and one deduction rule with which we can deduce all the valid formulas, and with which it is not easy to find a proof in the axiomatic system because so a few axioms are available. • Gentzen deduction system, with two deduction rules for each logical connectives and one axiom, with which we also can deduce all the valid formulas and a formal proof of a valid formula is easy to be found, just by decomposing formulas into atoms by the deduction rules. • Between axiomatic deduction system and Gentzen deduction system is natural deduction system, with less deduction rules than the Gentzen one, which is much easier to be mastered than the axiomatic one and harder than the Gentzen one. To vary propositional logic, a simple way is to use less logical connectives. Traditionally, there are logical connectives in propositional logic: ¬, ∧, ∨, →, ↔, where {¬, ∧}, {¬, ∨}, {¬, →} are minimal such that another connective can be represented. Even though logic L ∧ containing one connective {∧} is not complete in the expression, it would be interesting to give a sound and complete deduction system for L ∧ . In the second chapter, we will give sound, complete Gentzen deduction systems G¬ , G∧ and G∨ for L ¬ , L ∧ and L ∨ , respectively and sound, complete Rcalculi R¬ , R∧ and R∨ for sequents. To show how to get a sound and complete Gentzen deduction system from truth-value tables for logical connectives, we will © Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_1

1

2

1 Introduction

give sound and complete Gentzen deduction systems G¬→ /G⊕⊗ to show how we build G¬→ /G⊕⊗ by truth-value tables for ¬, → /⊕, ⊗; and sound and complete R-calculi R¬→ /R⊕⊗ for sequents.

1.2 Variant Deduction Systems Let ,  be sets of formulas. A theory  is • Tt -valid, denoted by |=t , if for any assignment v, v(A) = 1 for some A ∈ ; • Tf -valid, denoted by |=f , if for any assignment v, v(B) = 0 for some B ∈ ; • Tt -valid, denoted by |=t , if there is an assignment v such that v(A) = 1 for every A ∈ ; • Tf -valid, denoted by |=f , if there is an assignment v such that v(B) = 0 for every B ∈ . |=t |=f |=t |=f

 if AvE A ∈ (v(A) = 1)  if BvEB ∈ (v(B) = 0)  if EvAA ∈ (v(A) = 1)  if EvBB ∈ (v(B) = 0).

There are sound and complete tableau proof systems Tt , Tf , Tt , Tf for Tt , Tf , Tt , Tf -valid theories , respectively. That is, for any theory , t f t f

 iff  iff  iff  iff

|=t  |=f  |=t  |=f .

A sequent  ⇒  is • Gt -valid, denoted by |=t  ⇒ , if for any assignment v, either v(A) = 0 for some A ∈  or v(B) = 1 for some B ∈ ; • Gf -valid, denoted by |=f  ⇒ , if for any assignment v, either v(A) = 1 for some A ∈  or v(B) = 0 for some B ∈ . Dually, a co-sequent  →  is • Gt -valid, denoted by |=t  → , if there is an assignment v such that v(A) = 1 for every A ∈  and v(B) = 0 for every B ∈ ; • Gf -valid, denoted by |=f  → , if there is an assignment v such that v(A) = 0 for every A ∈  and v(B) = 1 for every B ∈ . |=t |=f |=t |=f

   

⇒  if Av(E A ∈ (v(A) = 0) or EB ∈ (v(B) = 1)) ⇒  if Av(E A ∈ (v(A) = 1) or EB ∈ (v(B) = 0)) →  if Ev(AA ∈ (v(A) = 1)&AB ∈ (v(B) = 0)) →  if Ev(AA ∈ (v(A) = 0)&AB ∈ (v(B) = 1)).

1.2 Variant Deduction Systems

3

Gt /Gf -validity is complementary to Gt /Gf -one. There are sound and complete Gentzen deduction systems Gt , Gf , Gt , Gf for Gt , Gf -valid sequents  ⇒ /Gt , Gf -valid co-sequents  → , respectively. That is, for any sequent  ⇒  and co-sequent  → , t f t f

   

⇒  iff ⇒  iff →  iff →  iff

|=t  ⇒  |=f  ⇒  |=t  →  |=f  → .

Let Q 1 , Q 2 ∈ {A, E}. A sequent  ⇒  is G Q 1 Q 2 -valid, denoted by |= Q 1 Q 2  ⇒ , if for any assignment v, either Q 1 A ∈ (v(A) = 0) or Q 2 B ∈ (v(B) = 1). A co-sequent  →  is G Q 1 Q 2 -valid, denoted by |= Q 1 Q 2  → , if there is an assignment v such that Q 1 A ∈ (v(A) = 1) and Q 2 B ∈ (v(B) = 0). |= Q 1 Q 2  ⇒  if Av(Q 1 A ∈ (v(A) = 0) or Q 2 B ∈ (v(B) = 1)) |= Q 1 Q 2  →  if Ev(Q 1 A ∈ (v(A) = 1)&Q 2 B ∈ (v(B) = 0)). G Q 1 Q 2 -validity is complementary to G Q 1 Q 2 , where A = E and E = A. There are sound and complete Gentzen deduction systems G Q 1 Q 2 /G Q 1 Q 2 for G Q 1 Q 2 -valid sequents/G Q 1 Q 2 -valid co-sequents, respectively, which are monotonic in  if and only if Q 1 = E, and nonmonotonic in  if and only if Q 2 = A. Let i, j ∈ {0, 1}. A sequent  ⇒  is G Q 1 i Q 2 j -valid, denoted by |= Q 1 i Q 2 j  ⇒ , if for any assignment v, either Q 1 A ∈ (v(A) = i) or Q 2 B ∈ (v(B) = j). A co-sequent  →  is G Q 1 i Q 2 j -valid, denoted by |= Q 1 i Q 2 j  → , if there is an assignment v such that Q 1 A ∈ (v(A) = i) and Q 2 B ∈ (v(B) = j). |= Q 1 i Q 2 j  ⇒  if Av(Q 1 A ∈ (v(A) = i) or Q 2 B ∈ (v(B) = j)) |= Q 1 i Q 2 j  →  if Ev(Q 1 A ∈ (v(A) = i)&Q 2 B ∈ (v(B) = j)). G Q 1 i Q 2 j -validity is complementary to G Q 1 (1−i)Q 2 (1− j) . There are sound and complete Gentzen deduction systems G Q 1 i Q 2 j /G Q 1 i Q 2 j for G Q 1 i Q 2 j -valid sequents/G Q 1 i Q 2 j -valid co-sequents, respectively, which are nonmonotonic in  if and only if Q 1 = A, and monotonic in  if and only if Q 2 = E. That is, for sequent  ⇒  and co-sequent  → , Q 1 i Q 2 j  ⇒  iff |= Q 1 i Q 2 j  ⇒  Q 1 i Q 2 j  →  iff |= Q 1 i Q 2 j  → . Let R, Q, P denote the ⊆-, - and -minimal change, respectively. For Y1 , Y2 ∈ {R, Q, P}, there are sound and complete Gentzen deduction systems GY1 Q 1 iY2 Q 2 j / GY1 Q 1 iY2 Q 2 j for GY1 Q 1 iY2 Q 2 j -valid sequents/GY1 Q 1 iY2 Q 2 j -valid co-sequents, respec-

4

1 Introduction

tively, which are nonmonotonic in  if and only if Q 1 = A, and monotonic in  if and only if Q 2 = E. That is, for sequent  ⇒  and co-sequent  → , Y1 Q 1 iY2 Q 2 j  ⇒  iff |= Q 1 iY2 Q 2 j  ⇒  Y1 Q 1 iY2 Q 2 j  →  iff |=Y1 Q 1 iY2 Q 2 j  → .

1.3 R-Calculus The first author (Li 2007) developed a Gentzen-typed deduction system in first-order logic (Alchourrón et al. 1985), called R-calculus, to reduce a configuration | into a consistent theory  ∪ , where  is a minimal change of  by , i.e., a maximal set of  which is consistent with , where  is a set of atoms, and ,  are consistent theories of first-order logic. Here, | corresponds to iterating revision  ◦ . Hence, the deduction system gives a concrete revision operator which is shown to satisfy AGM postulates (Alchourrón et al. 1985; Fermé and Hansson 2011). R-calculus consists of the following rules and axioms: • Structural rules: (contraction L ) , A, A| ⇒ , A| (contraction R ) |B, B,  ⇒ |A,  (interchange L ) , A1 , A2 | ⇒ , A2 , A1 | (interchange R ) |B1 , B2 ,  ⇒ |B2 , B1 , ; • R-axiom:

(¬) , ¬A|A,  ⇒ , ¬A|;

• R-elimination rule: 1 , A B A →T B 2 , B C |C, 2 ⇒ |2 ; |1 , A, 2 ⇒ |1 , 2 • R-logical deduction rules: |B1 ,  ⇒ | |B2 ,  ⇒ | (R2∧ ) |B1 ∧ B2 ,  ⇒ | |B1 ∧ B2 ,  ⇒ | |B1 ,  ⇒ | |¬B1 ,  ⇒ | (R∨ ) |B2 ,  ⇒ | (R→ ) |B2 ,  ⇒ | |B1 ∨ B2 ,  ⇒ | |B1 → B2 ,  ⇒ | |B(t),  ⇒ | |B(x),  ⇒ | A E (R ) (R ) |Ax B(x),  ⇒ | |Ex B(x),  ⇒ | (R1∧ )

where t is a term, and x is a variable not freely occur in  and .

1.4 R-Calculi in This Volume

5

The deduction rules in R-calculus are corresponding to those in Gentzen deduction system of first-order logic. In Gentzen deduction system, a sequent  ⇒  is reduced to atomic sequents   ⇒  by using the rules of the right-hand side and of the lefthand side, where   ⇒  is atomic if   ,  are sets of atoms and   ⇒  is an axiom if and only if   ∩  = ∅. In R-calculus, a configuration |A,  is reduced to literal configurations |l,  by using the deduction rules for logical symbols (logical connectives and quantifiers in first-order logic) (Li and Sui 2017a, 2018), where |l,  ⇒ , l| is an axiom if and only if   ¬l; otherwise, |l,  ⇒ |, i.e., l is deleted from theory {l} ∪ .

1.4 R-Calculi in This Volume We consider three kinds of R-calculi: one for weak propositional logics, one for propositional logic and one for supersequents and -propositional logic.

1.4.1 R-Calculi for Weak Propositional Logics We consider three weak propositional logics L ¬ , L ∧ , L ∨ , and give sound and complete • Gentzen deduction systems G¬ , G∧ , G∨ for G¬ , G∧ , G∨ -valid sequents, respectively, • R-calculi R¬ , R∧ , R∨ for R¬ , R∧ , R∨ -valid reductions, respectively; • Gentzen deduction systems G¬ , G∧ , G∨ for G¬ , G∧ , G∨ -valid co-sequents, respectively, • R-calculi R¬ , R∧ , R∨ for R¬ , R∧ , R∨ -valid co-reductions, respectively. Moreover, we consider two propositional logics L ¬→ , L ⊕⊗ , and give sound and complete • Gentzen deduction systems G¬→ , G⊕⊗ for G¬→ , G⊕⊗ -valid sequents, respectively, • R-calculi R¬→ , R⊕⊗ for R¬→ , R⊕⊗ -valid sequents, respectively, • Gentzen deduction systems G¬→ , G⊕⊗ for G¬→ , G⊕⊗ -valid co-sequents, respectively, • R-calculi R¬→ , R⊕⊗ for R¬→ , R⊕⊗ -valid co-reductions, respectively,

1.4.2 R-Calculi for Tableau Proof/Gentzen Deduction Systems Corresponding to tableau proof systems and Gentzen deduction systems there are sound and complete R-calculi.

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1 Introduction

Given a theory  and formula A, • Corresponding to Tt -validity, using  to revise A obtains theory , A , denoted by |=t |A ⇒ , A , where A = A iff  ∪ {A} is Tt -valid, where ⇒ means “reduced to". • Corresponding to Tf -validity, using  to revise B obtains theory , B  , denoted by |=f |B ⇒ , B  , where B  = B iff  ∪ {B} is Tf -valid. • Corresponding to Tt -validity, using  to revise A ∈  obtains theory  − {A }, denoted by |=t |A ⇒  − {A }, where A = A iff  − {A} is Tt -valid. • Corresponding to Tf -validity, using  to revise B ∈  obtains theory  − {B  }, denoted by |=f |B ⇒  − {B  }, where B  = B iff  − {B} is Tf -valid. There are sound and complete R-calculi St , Sf , St , Sf for preserving Tt , Tf , Tt , Tf -validities, respectively. That is, for any reduction |A ⇒   /|B ⇒  , t f t f

|A ⇒   iff |=t |A ⇒   |B ⇒  iff |=f |B ⇒  |A ⇒   iff |=t |A ⇒   |B ⇒  iff |=f |B ⇒  .

Given a sequent  ⇒ /co-sequent  →  and pair (A, B) of formulas, • Corresponding to Gt -validity, using  →  to revise (A, B) obtains a sequent , A → , B  , denoted by |=t  → |(A, B) ⇒ , A → , B  , if A = A iff , A →  is Gt -valid and B  = B iff , A → , B is Gt -valid. • Corresponding to Gf -validity, using  →  to revise (A, B) obtains a sequent , A → , B  , denoted by |=f  → |(A, B) ⇒ , A → , B  , if A = A iff , A →  is Gf -valid and B  = B iff , A → , B is Gf -valid. • Corresponding to Gt -validity, using  ⇒  to revise (A, B), where A ∈  and B ∈ , obtains a sequent  − {A } ⇒  − {B  }, denoted by |=t  ⇒ |(A, B) ⇒  − {A } ⇒  − {B  }, if A = A iff  − {A} ⇒  is Gt -valid and B  = B iff  − {A } ⇒  − {B} is Gt -valid. • Corresponding to Gf -validity, using  ⇒  to revise (A, B), where A ∈  and B ∈ , obtains a sequent  − {A } ⇒  − {B  }, denoted by |=f  ⇒ |(A, B) ⇒  − {A } ⇒  − {B  },

1.4 R-Calculi in This Volume

7

if A = A iff  − {A} ⇒  is Gf -valid and B  = B iff  − {A } ⇒  − {B} is Gf -valid. There are sound and complete R-calculi Rt , Rf /Rt , Rf for preserving Gt , Gf / Gt , Gf -validities, respectively. That is, for any reduction  ⇒ |(A, B) ⇒   ⇒  /co-reduction  → |(A, B) ⇒   →  , t f t f

   

⇒ |(A, B) ⇒   ⇒  iff ⇒ |(A, B) ⇒   ⇒  iff → |(A, B) ⇒   →  iff → |(A, B) ⇒   →  iff

|=t  ⇒ |(A, B) ⇒   ⇒  |=f  ⇒ |(A, B) ⇒   ⇒  |=t  → |(A, B) ⇒   →  |=f  → |(A, B) ⇒   →  .

1.4.3 R-Calculi for Deduction Systems G Q 1 Q 2 /G Q 1 Q 2 Let Q 1 , Q 2 ∈ {A, E}. Given a sequent  ⇒  and pair (A, B) of formulas, the result of  ⇒  G Q 1 Q 2 revising (A, B) is   ⇒  , denoted by |= Q 1 Q 2  ⇒ |(A, B) ⇒   ⇒  , if   ⇒  =

where

⎧  ±1 A ⇒  ±2 B if |= Q 1 Q 2  ±1 A ⇒  ±2 B ⎪ ⎪ ⎨ if |= Q 1 Q 2 ± A ⇒   ±1 A ⇒  B if |= Q 1 Q 2  ⇒  ±2 B  ⇒  ± ⎪ 2 ⎪ ⎩ ⇒ otherwise, 

−{A} if Q 1 = E and A ∈  ∪{A} if Q 1 = A,  −{B} if Q 2 = E and B ∈  ±2 B = ∪{B} if Q 2 = A.

±1 A =

Given a co-sequent  →  and pair (A, B) of formulas, the result of  →  G Q 1 Q 2 -revising (A, B) is   →  , denoted by |= Q 1 Q 2  → |(A, B) ⇒   →  , if   →  =

⎧ ⎪ ⎪  ±1 A →  ±2 B if |= Q 1 Q 2  ±1 A →  ±2 B ⎨ if |= Q 1 Q 2 ± A →   ±1 A →  B if |= Q 1 Q 2  →  ±2 B   →  ± ⎪ 2 ⎪ ⎩  →  otherwise.

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1 Introduction

There is a sound and complete R-calculus R Q 1 Q 2 /R Q 1 Q 2 for preserving G Q 1 Q 2 / G Q 1 Q 2 -validity, respectively. That is, for any reduction  ⇒ |(A, B) ⇒   ⇒  /co-reduction  → |(A, B) ⇒   →  , Q 1 Q 2  ⇒ |(A, B) ⇒   ⇒  iff |= Q 1 Q 2  ⇒ |(A, B) ⇒   ⇒  Q 1 Q 2  → |(A, B) ⇒   →  iff |= Q 1 Q 2  → |(A, B) ⇒   →  .

1.4.4 R-Calculi for Deduction Systems G Q 1 i Q 2 j /G Q 1 i Q 2 j Let Q 1 , Q 2 ∈ {A, E} and i, j ∈ {0, 1}. Given a sequent  ⇒ Q 1 i Q 2 j  and pair (A, B) of formulas, the result of  ⇒  G Q 1 i Q 2 j -revising (A, B) is   ⇒  , denoted by |= Q 1 i Q 2 j  ⇒ |(A, B) ⇒   ⇒  , if   ⇒  = ⎧  ±1 A ⇒  ±2 B if  ±1 A ⇒  ±2 B is G Q 1 i Q 2 j -valid ⎪ ⎪ ⎨ otherwise, if  ±1 A ⇒  is G Q 1 i Q 2 j -valid  ±1 A ⇒  B otherwise, if  ⇒  ±2 B is G Q 1 i Q 2 j -valid  ⇒  ± ⎪ 2 ⎪ ⎩ ⇒ otherwise, where



∪{A} if Q 1 = A  −{A} if Q 1 = E and A ∈ , ∪{B} if Q 2 = A ±2 B = −{B} if Q 2 = E and B ∈ .

±1 A =

Given a co-sequent  → Q 1 i Q 2 j  and pair (A, B) of formulas, the result of  →  G Q 1 i Q 2 j -revising (A, B) is   →  , denoted by |= Q 1 i Q 2 j  → |(A, B) ⇒   →  , if   →  = ⎧  ±1 A →  ±2 B if  ±1 A →  ±2 B is G Q 1 i Q 2 j -valid ⎪ ⎪ ⎨  ±1 A →  otherwise, if  ±1 A →  is G Q 1 i Q 2 j -valid otherwise, if  →  ±2 B is G Q 1 i Q 2 j -valid ⎪  →  ±2 B ⎪ ⎩  →  otherwise. There is a sound and complete R-calculus R Q 1 i Q 2 j /R Q 1 i Q 2 j for preserving /G Q 1 i Q 2 j -validity. That is, for any reduction  ⇒ |(A, B) ⇒   ⇒  /coG reduction  → |(A, B) ⇒   →  , Q1i Q2 j

1.5 Applications of R-Calculi

9

Q 1 i Q 2 j  ⇒ |(A, B) ⇒   ⇒  iff |= Q 1 i Q 2 j  ⇒ |(A, B) ⇒   ⇒  Q 1 i Q 2 j  → |(A, B) ⇒   →  iff |= Q 1 i Q 2 j  → |(A, B) ⇒   →  .

1.4.5 R-Calculi for Deduction Systems GY1 Q 1 i Y2 Q 2 j /GY1 Q 1 i Y2 Q 2 j R-calculus R Q 1 i Q 2 j is with respect to the ⊆-minimal change R. There are -minimal change Q and -minimal change P, (Li and Sui 2013, 2014) and correspondingly there are R-calculus RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j for Y1 , Y2 ∈ {R, Q, P}. All these R-calculi RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j are sound and complete with respect to the ⊆-minimal change R, -minimal change Q and -minimal change P, respectively. That is, for any reduction  ⇒ |(A, B) ⇒   ⇒  /co-reduction  → |(A, B) ⇒   →  , Y1 Q 1 iY2 Q 2 j  ⇒ |(A, B) ⇒   ⇒  iff |=Y1 Q 1 iY2 Q 2 j  ⇒ |(A, B) ⇒   ⇒  Y1 Q 1 iY2 Q 2 j  → |(A, B) ⇒   →  iff |=Y1 Q 1 iY2 Q 2 j  → |(A, B) ⇒   →  . About these Gentzen deduction systems and R-calculi, we have the following equivalences: Gentzen deduction system Gt = GEE Gt = GAA Gf = GE1E0 Gf = GA0A1 G Q 1 Q 2 = G Q 1 0Q 2 1 G Q 1 Q 2 = G Q 1 1Q 2 0 G Q 1 i Q 2 j = GRQ 1 iRQ 2 j G Q 1 i Q 2 j = GRQ 1 iRQ 2 j

R-calculus Rt = REE Rt = RAA Rf = RE1E0 Rf = RA0A1 R Q 1 Q 2 = R Q 1 0Q 2 1 R Q 1 Q 2 = R Q 1 1Q 2 0 R Q 1 i Q 2 j = RRQ 1 iRQ 2 j R Q 1 i Q 2 j = RRQ 1 iRQ 2 j

1.5 Applications of R-Calculi We consider two applications of R-calculi to -propositional logic and supersequents (Li and Sui 2017b; Li et al. 2017a, b), where the former is a logic taking logical connective → as a nonlogical symbol, and the latter is a pseudo-extension of sequents, that is, a supersequent is equivalent to a sequent.

10

1 Introduction

1.5.1 R-Calculus for -Propositional Logic By taking → as a nonlogical symbol (denoted by ), we can develop -propositional logic, and corresponding sound and complete Gentzen deduction systems. The logic can be used in default logic and description logics. In propositional logic, we have A1 ∧ A2 → B ⇐ A1 → B∨A2 → B A1 ∨ A2 → B ≡ A1 → B∧A2 → B A → B1 ∧ B2 ≡ A → B1 ∧A → B2 A → B1 ∨ B2 ⇐ A → B1 ∨A → B2 ;

A1 ∧ A2 → B ≡ A1 → B∧A2 → B A1 ∨ A2 → B ≡ A1 → B∨A2 → B A → B1 ∧ B2 ≡ A → B1 ∨A → B2 A → B1 ∨ B2 ≡ A → B1 ∧A → B2 ;

where ∧/∨ is corresponding to ∧/∨ in semantics. A:B in default logic (Li et al. 2017a; Reiter Interpreting A  B as default B 1980), we have A1 ∧ A2 : B B A1 ∨ A2 : B B A : B1 ∧ B2 B1 ∧ B2 A : B1 ∨ B2 B1 ∧ B2

A1 : B A2 : B ∨ B B A1 : B A2 : B ≡ ∧ B B A : B1 A : B2 ≡ ∧ B1 B2 A : B1 A : B2 ⇐ ∨ ; B1 B2 ⇐

A1 ∧ A2 : B B A1 ∨ A2 : B ¬ B A : B1 ∧ B2 ¬ B1 ∧ B2 A : B1 ∨ B2 ¬ B1 ∧ B2

¬

A1 : B A2 : B ∧¬ B B A1 : B A2 : B ≡¬ ∨¬ B B A : B1 A : B2 ≡¬ ∨¬ B1 B2 A : B1 A : B2 ≡¬ ∧¬ . B1 B2 ≡¬

Interpreting A  B as the subsumption statement C  D in description logics, we have C1  C2  D ⇐ C1  D∨C2  D C1  C2  D ≡ C1  D∧C2  D C  D1  D2 ≡ C  D1 ∧C  D2 C  D1  D2 ⇐ C  D1 ∨C  D2 ;

C1  C2  D ≡ C1  D∧C2  D C1  C2  D ≡ C1  D∨C2  D C  D1  D2 ≡ C  D1 ∨C  D2 C  D1  D2 ≡ C  D1 ∧C  D2 .

For R-calculus of -propositional logic, we consider three kinds of minimal changes: (i) Subset-minimal (⊆-minimal) change. (ii) Pseudo-subformula-minimal ( -minimal) change, where is the pseudosubformula relation, just as the subformula relation ≤, where a formula A is a pseudo-subformula of B if eliminating some subformulas in B results in A. (iii) Deduction-based minimal ( -minimal) change, where a theory  is a minimal change of  by γ (denoted by |=U |γ ⇒ ), if   ∪ {γ } is consistent, and for any theory  with  ≺  , γ either   and  , or  is inconsistent.

1.5 Applications of R-Calculi

11

For ∗ ∈ {t, f}, there are tableau proof systems T∗ /T ∗ , Gentzen deduction sys∗  /G∗ and R-calculi Rt , Q , P for tableau proof systems and Rt , Qt , Pt tems G t t for Gentzen deduction systems sound and complete with respect to ⊆-, -, - and -minimal changes, respectively. That is, for any theory , ∗  iff |=∗    ∗  iff |=∗ , and for any sequent  ⇒ /co-sequent  → , ∗  ⇒  iff |=∗  ⇒    ∗   →  iff |=∗   → , and for any Y ∈ {R, Q, P} and any reduction |A ⇒   / |A ⇒   , tY |A ⇒   iff |=tY |A ⇒   tY |A ⇒   iff |=tY |A ⇒   , and any reduction  ⇒ |(A, B) ⇒   ⇒  /  → |(A, B) ⇒   →  , tY  ⇒ |(A, B) ⇒   ⇒  iff |=tY  ⇒ |(A, B) ⇒   ⇒  tY  → |(A, B) ⇒   ⇒  iff |=tY  → |(A, B) ⇒   ⇒  .

1.5.2 R-Calculi for Supersequents A supersequent δ is of form | ⇒ | , where , , , are sets of formulas. δ is valid, denoted by |=+ δ, if for any assignment v, either v(A) = 0 for some A ∈ , or v(B) = 1 for some B ∈ , or v(C) = 1 for some C ∈ , or v(D) = 0 for some D ∈ . That is, both each formula in  having truth-value 1 and each formula in  having truth-value 0 imply either some formula in  has truth-value 1 or some formula in has truth-value 0. A co-supersequent δ is of form | → | . δ is valid, denoted by |=− δ, if there is an assignment v such that v(A) = 1 for each A ∈ , v(B) = 0 for each B ∈ , v(C) = 0 for each C ∈  and v(D) = 1 for each D ∈ . | ⇒ | f|t ⇒ t|f | → | t|f → f|t The validity of | ⇒ | / | → | is equivalent to that of , ⇒ , / , → , . There is a sound and complete Gentzen deduction system G+ /G− for supersequents/co-supersequents, respectively. That is, for any supersequent/cosupersequent δ,

12

1 Introduction

+ | ⇒ | iff |=+ | ⇒ | − | → | iff |=− | → | . A supersequent δ is reduced to sequents:  ⇒ , Gft  ⇒ , Gff  ⇒ , Gtt  ⇒ , Gtf , and A co-supersequent δ is reduced to co-sequents:  → ,  → ,  → ,  → ,

Gtf Gtt Gff Gft .

Correspondingly there are sound and complete R-calculi R+ /R− for supersequents/co-supersequents, which are reduced to the following sound and complete R-calculi for sequents:  ⇒ |(A, C) ⇒   ⇒   ,  ⇒ |(A, D) ⇒   ⇒  ,  ⇒ |(B, C) ⇒  ⇒   ,  ⇒ |(B, D) ⇒  ⇒  , and

Rft Rff Rtt Rtf ,

 → |(A, C) ⇒   →   , Rtf  → |(A, D) ⇒   →  , Rtt  → |(B, C) ⇒  →   , Rff  → |(B, D) ⇒  →  , Rft .

1.6 Notations All the deduction systems and R-calculi in this book are listed in the following table: • For theories we have the following systems: Deduction systems R-calculi Tt , Tf /Tt , Tf St , Sf /St , Sf t f   t f T , T /Tt , Tf R , R   Rt , Q t , Pt • For sequents we have the following systems:

1.6 Notations

13

Deduction systems G¬ , G∧ , G∨ G¬ , G∧ , G∨ G¬→ , G⊕⊗ G¬→ , G⊕⊗ Gt , Gf /Gt , Gf G Q 1 Q 2 /G Q 1 Q 2 G Q 1 i Q 2 j /G Q 1 i Q 2 j GY1 Q 1 iY2 Q 2 j /GY1 Q 1 iY2 Q 2 j G+ /G− Gff , Gff Gff , Gff t G

R-calculi R¬ , R∧ , R∨ R¬ , R∧ , R∨ R¬→ , R⊕⊗ R¬→ , R⊕⊗ Rt , Rf /Rt , Rf R Q 1 Q 2 /R Q 1 Q 2 R Q 1 i Q 2 j /R Q 1 i Q 2 j RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j R+ /R− Rff , Rff Rff , Rff Rt , Qt , Pt

Our notation is standard (W. Li 2010; Mendelson 1964; Takeuti 1987) and given as needed. We use , ,  to denote theories; A, B, C formulas in propositional logic, and | a configuration which is equivalent to  ◦  in AGM’s belief revision. In the meta-language, we use ∼, &, or, A, E to express ¬, ∧, ∨, ∀, ∃, respectively.   δ1 δ1 We use δ2 to denote that δ1 and δ2 imply δ; and δ2 to denote that δ1 implies δ δ δ  δ and δ2 implies δ. In proof tree of completeness theorem, we use 1 to denote that δ2  δ1 δ1 and δ2 are at a same subnode of the tree; and to denote that δ1 and δ2 are at δ2 different subnodes of the tree. Hence, rules  ⇒ B1 ,  , A1 ⇒  (∧ R )  ⇒ B2 ,  , A1 ∧ A2 ⇒   ⇒ B1 ∧ B2 ,  , A ⇒  2 (∧2L ) , A1 ∧ A2 ⇒  (∧1L )

in traditional Gentzen deduction system are represented in this book as following two rules   , A1 ⇒   ⇒ B1 ,  (∨ L ) , A2 ⇒  (∨ R )  ⇒ B2 ,  , A1 ∧ A2 ⇒   ⇒ B1 ∧ B2 ,  In a proof tree ξ of completeness theorem, A1 ∧ A2 and B1 ∧ B2 have the following subnodes, respectively:

14

1 Introduction

 

 1 , A 1 ⇒ 1 if 1 , A1 ∧ A2 ⇒ 1 ∈ ξ  1 , A 2 ⇒ 1 1 ⇒ B1 , 1 if 1 ⇒ B1 ∧ B2 , 1 ∈ ξ 1 ⇒ B2 , 1

Let ,  be sets of literals. We define incon() iff El(l, ¬l ∈ ) con() iff ∼ El(l, ¬l ∈ ) val() iff El(l, ¬l ∈ ) inval() iff ∼ El(l, ¬l ∈ ). Notice that incon() iff val() and con() iff inval().

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symbolic Logic 50, 510–530 (1985) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Li, W.: Mathematical logic, foundations for information science. In: Progress in Computer Science and Applied Logic, vol. 25, Birkhäuser (2010) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Li, W., Sui, Y.: An R-calculus for the propositional logic programming. In: Proceedings of International Conference on Computer Science and Information Technology, pp. 863–870 (2014) Li, W., Sui, Y.: The R-calculus and the finite injury priority method. J. Comput. 12, 127–134 (2017) Li, W., Sui, Y.: The B4 -valued propositional logic with unary logical connectives ∼1 / ∼2 /¬. Front. Comput. Sci. 11, 887–894 (2017) Li, W., Sui, Y.: Multisequent Gentzen deduction systems for B22 -valued first-order logic. Artif. Intell. Res. 7, 53 (2018) Li, W., Sui, Y.: A computational framework for Karl Popper’s logic of scientific discovery. Sci. China Inf. Sci. 61, 042101:1–042101:10 (2018) Li, W., Sui, Y., Wang, Y.: The propositional normal default logic and the finite/infinite injury priority method. Sci. China Inf. Sci. 60, 092107 (2017) Li, W., Sui, Y., Wang, Y.: The lattice-modalized propositional logic: distributivity and modularity of the Gentzen deduction system GL. J. Intell. Fuzzy Syst. 33, 733–740 (2017) Mendelson, E.: Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California (1964) Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980) Takeuti, G.: Proof theory. In: Barwise, J. (ed.) Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987)

Chapter 2

R-Calculus for Simplified Propositional Logics

Assume that the logical language of propositional logic contains three logical symbols: ¬, ∧, ∨. There is a classical Gentzen deduction system G for propositional logic which is sound and complete with respect to classical semantics (Li 2010; Mendelson 1964; Takeuti 1987). This chapter firstly will give three logical sublanguages of propositional logic, where each language contains only one connective; and correspondingly there are • three weak propositional logics; • three Gentzen deduction systems which are sound and complete with respect to the classical interpretations of the corresponding logical connectives, respectively; and • three R-calculi which are sound and complete with respect to the ⊆-minimal change (Li 2007; Li and Sui 2013). Therefore, soundness and completeness are related only to the semantics of the concerned logics, no matter whether the logical language is complete with the semantics, such as the expressive completeness of L = {¬, ∧, ∨} in classical propositional logic, where for any n-ary truth-value function f : {0, 1}n → {0, 1}, there is a formula A in L such that for any assignment v and any a1 , . . . , an ∈ {0, 1}, if v( p1 ) = a1 , . . . , v( pn ) = an then f (a1 , . . . , an ) = v(A). There are three ways to deduce valid formulas in a logic: axiomatic systems, natural deduction systems and Gentzen-type deduction systems, where the latter are based on the semantics of logical connectives in the logic to be formalized. For each logical connective x ∈ {¬, ∧, ∨}, a Gentzen-type deduction system Gx will be given so that • soundness theorem is true, that is, for any sequent  ⇒ , if Gx  ⇒  then |=Gx  ⇒ ; and • completeness theorem is true, that is, for any sequent  ⇒ , if |=Gx  ⇒  then Gx  ⇒ ; © Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_2

15

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2 R-Calculus for Simplified Propositional Logics

and two R-calculi R x and Rx are given which are sound and complete with respect to ⊆-minimal change. Moreover, we will give two logical languages of propositional logic, which contain • logical connectives {¬, →} and {¬, ⊕, ⊗}; and • two Gentzen deduction systems G¬→ , G⊕⊗ are built which are sound and complete with respect to semantics of logical connectives, and • four R-calculi R¬→ , R⊕⊗ /R¬→ , R⊕⊗ are given which are sound and complete with respect to ⊆-minimal change (Li and Sui 2014).

2.1 Weak Propositional Logics We consider three weak propositional logics containing logical connective {¬}, {∧} and {∨}, respectively, and give sound and complete Gentzen deduction systems G¬ , G∧ , G∨ /G¬ , G∧ , G∨ and R-calculi R¬ , R∧ , R∨ /R¬ , R∧ , R∨ . The logical language of weak propositional logic Gx contains the following symbols: • variables: p0 , p1 , . . . ; • unary connective: x. Formulas are defined as follows: A ::= p|x A/A1 x A2 . Let v be an assignment, a function from variables to {0, 1}. The interpretation v(A) of formulas A in v is: ⎧ v( p) if ⎪ ⎪ ⎨ if 1 − v(A1 ) v(A) = ), v(A )} if min{v(A ⎪ 1 2 ⎪ ⎩ max{v(A1 ), v(A2 )} if

A= p A = ¬A1 and x = ¬ A = A1 ∧ A2 and x = ∧ A = A1 ∨ A2 and x = ∨

and A is satisfied in v, denoted by v |= A, if v(A) = 1.

2.1.1 Gentzen Deduction System G¬ Given a sequent δ =  ⇒ , we say that v satisfies δ, denoted by v |=¬ δ, if v |=  implies v |= , where v |=  if for each A ∈ , v |= A; and v |=  if for some B ∈ , v |= B. A sequent δ is valid, denoted by |=¬ δ, if for any assignment v, v |=¬ δ. Gentzen deduction system G¬ contains the following axiom and deduction rules.

2.1 Weak Propositional Logics

17

• Axiom: (A¬ )

 ∩  = ∅ , ⇒

where ,  are sets of variables. • Deduction rules: (¬ L )

 ⇒ A,  , B ⇒  (¬ R ) . , ¬A ⇒   ⇒ ¬B, 

Definition 2.1.1 A sequent  ⇒  is provable in G¬ , denoted by ¬  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of deduction rules in G¬ . Theorem 2.1.2 (Soundness theorem) Given a sequent  ⇒ , if ¬  ⇒  then |=¬  ⇒ . Proof We prove that each axiom is valid and each deduction rule preserves the validity. (A¬ ) Assume that p ∈  ∩ . Then, for any assignment v, either v( p) = 0 or v( p) = 1. Hence, v |=  or v |= . (¬ L ) Assume that for any assignment v, v |=  implies v |= A, . For any assignment v, assume that v |= ¬A, . Then, v |= , and by induction assumption, v |= A, . Since v |= A, v |= . (¬ R ) Assume that for any assignment v, v |= , B implies v |= . For any assignment v, assume that v |= . If v |= B then v |= ¬B, and v |= ¬B, ; otherwise, v |= , B and by induction assumption, v |= , and hence v |= ¬B, .  Theorem 2.1.3 (Completeness theorem) Given a sequent  ⇒ , if |=¬  ⇒  then ¬  ⇒ . Proof Let δ =  ⇒ . Define a linear tree, called the reduction tree for δ, denoted by T (δ), from which we can obtain either a proof of δ or a show of the nonvalidity of δ. This reduction tree T (δ) for δ contains a sequent at each node and is constructed in stages as follows. Stage 0: T0 (δ) = {δ}. Stage k(k > 0) : Tk (δ) is defined by cases. Case 0. If ,  have no logical connectives in them, write nothing above  ⇒ . Case 1. Every topmost sequent  ⇒  in Tk−1 (δ) has a common formula in  and . Then, stop. Case 2. Not case 1. Tk (δ) is defined as follows. Let  ⇒  be any topmost sequent of the tree which has been defined by stage k − 1. Subcase (¬ L ). Let ¬A be the formula in  whose outmost logical symbol is ¬, and to which no reduction has been applied in previous stages. Then, write down

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2 R-Calculus for Simplified Propositional Logics

 ⇒ A,  above  ⇒ . We say that a (¬ L ) reduction has been applied to ¬A. Subcase (¬ R ). Let ¬B be the formula in  whose outmost logical symbol is ¬, and to which no reduction has been applied in previous stages. Then, write down , B ⇒  above  ⇒ . We say that a (¬ R ) reduction has been applied to ¬B. Given a sequent δ, if T (δ) is ended with a sequent containing common formulas, then it is a routine to construct a proof of δ. Otherwise, T (δ) = δ1 , . . . , δn , there is no rule applicable for δn and δn = n ⇒ n has no common formulas in n and n . Let ∪ = {A ∈ i : i ⇒ i ∈ T (δ)}, ∪ = {B ∈ i : i ⇒ i ∈ T (δ)}. We define an assignment in which every statement A ∈ ∪ is true and every statement in B ∈ ∪ is false. Define v such that for any variable p, v( p) = 1 iff p ∈ ∪. By induction on the structure of formulas, we prove that if A ∈ ∪ then v |= A; and if B ∈ ∪ then v |= B. Case A = ¬A1 ∈ ∪. Assume that δ  ∈ T (δ) is the first one such that A ∈ δ  . Then, there are a δ  ∈ T (δ) and formula sets ,  such that δ  =   ⇒ ¬A1 ,  and δ  =   , A ⇒  . By induction assumption, v |=   , A ⇒ , and so v |=   ⇒ ¬A,  . Case B = ¬B1 ∈ ∪. Assume that δ  ∈ T (δ) is the first one such that B ∈ δ  . Then, there are a δ  ∈ T (δ) and formula sets   ,  such that δ  =   ⇒ ¬B1 ,  and δ  =   , B ⇒  . By induction assumption, v |=   , B ⇒  , and v |=   ⇒ ¬B,  . This completes the proof.  Notice that each formula A is of form (¬)i p, where p is a variable.

2.1.2 Gentzen Deduction System G¬ Given a co-sequent δ =  → , we say that δ is valid, denoted by v |=¬ δ, if there is an assignment v such that for each A ∈ , v(A) = 1, and for each B ∈ , v(B) = 0. Gentzen deduction system G¬ contains the following axiom and deduction rules.

2.1 Weak Propositional Logics

19

• Axiom: (A¬ )

∩=∅ ,  → 

where ,  are sets of variables. • Deduction rules: (¬ L )

 → A,  , B →  (¬ R ) , ¬A →   → ¬B, 

Definition 2.1.4 A co-sequent  →  is provable in G¬ , denoted by ¬  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in G¬ . Theorem 2.1.5 (Soundness and completeness theorem) For any co-sequent  → , ¬  →  iff |=¬  → . 

2.1.3 Gentzen Deduction System G∧ A sequent δ is valid, denoted by |=∧ δ, if for any assignment v, v |=  implies v |= , where v |=  if for each A ∈ , v(A) = 1; and v |=  if for some B ∈ , v(B) = 1. Gentzen deduction system G∧ contains the following • Axiom: (A∧ )

 ∩  = ∅ , ⇒

where ,  are sets of variables. • Deduction rules:   , A1 ⇒   ⇒ B1 ,  (∧ L ) , A2 ⇒  (∧ R )  ⇒ B2 ,  , A1 ∧ A2 ⇒   ⇒ B1 ∧ B2 ,   where δ imply δ.

δ1 δ2 means that either δ1 or δ2 implies δ; and

 δ

δ1 δ2 means that δ1 and δ2

Definition 2.1.6 A sequent  ⇒  is provable in G∧ , denoted by ∧  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in G∧ . Theorem 2.1.7 (Soundness theorem) For any sequent  ⇒ , if ∧  ⇒  then |=∧  ⇒ .

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2 R-Calculus for Simplified Propositional Logics

Proof We prove that each axiom is valid and each deduction rule preserves the validity. (A∧ ) Assume that p ∈  ∩ . Then, for any assignment v, either v( p) = 0 or v( p) = 1, i.e., either v |=  or v |= . (∧ L ) Assume that for any assignment v, v |= , A1 implies v |= . For any assignment v, assume that v |= A1 ∧ A2 , . Then, v |= A1 , , and by induction assumption, v |= . (∧ R ) Assume that for any assignment v, v |=  implies v |= B1 , ; and v |=  implies v |= B2 , . For any assignment v, assume that v |= . If v |= B1 or v |= B2 then v |= , which implies v |= B1 ∧ B2 , ; otherwise, v |= B1 ∧ B2 , and v |=  B1 ∧ B2 , . Theorem 2.1.8 (Completeness theorem) For any sequent  ⇒ , if |=∧  ⇒  then ∧  ⇒ . Proof Given a sequent  ⇒ , we construct a tree T such that either (i) For each branch ξ of T, a sequent  ⇒   at the leaf of ξ is an axiom, or (ii) There is an assignment v such that v |=  ⇒ . T is constructed as follows: • The root of T is  ⇒ ; • For a node ξ , if each sequent  ⇒   at ξ is atomic then the node is a leaf. • Otherwise, ξ has the direct children nodes containing the following sequents: ⎧ ⎨ 1 , A1 , A2 ⇒ 1 if 1 , A1 ∧ A2 ⇒ 1 ∈ ξ 1 ⇒ B1 , 1 if 1 ⇒ B1 ∧ B2 , 1 ∈ ξ ⎩ 1 ⇒ B2 , 1 

 δ1 δ where means that δ1 and δ2 are at a same subnode of the tree; and 1 means δ2 δ2   1 , A 1 ⇒ 1 that δ1 and δ2 are at different subnodes of the tree. Sometimes we use  1 , A 2 ⇒ 1 to denote 1 , A1 , A2 ⇒ 1 . Lemma 2.1.9 If for each branch ξ ⊆ T, there is a sequent  ⇒   ∈ ξ which is an axiom in G∧ then T is a proof tree of  ⇒ . Proof By the definition of T, T is a proof tree of  ⇒ .



Lemma 2.1.10 If there is a branch ξ ⊆ T such that each sequent  ⇒   ∈ ξ is not an axiom in G∧ then there is an assignment v such that v |=  ⇒ . Proof Let γ be the set of all the atomic sequents in ξ .

2.1 Weak Propositional Logics

Let

21

  L =  ⇒ ∈γ   ,   R =  ⇒ ∈γ  .

Define an assignment v as follows: for any variable p,  v( p) =

1 if p ∈  L 0 otherwise.

Then, v is well-defined and v |=   ⇒  , where   ⇒  is the sequent at the leaf node of ξ. We proved by induction on ξ that each sequent  ⇒   ∈ ξ is not satisfied by v. Case   ⇒  = 2 , A1 ∧ A2 ⇒  ∈ ξ. Then   ⇒  has a child node ∈ ξ containing sequent 2 , A1 , A2 ⇒  . By induction assumption, v |= 2 , A1 , A2 ⇒  . Hence, v |= 2 , A1 ∧ A2 ⇒  . Case   ⇒  =   ⇒ B1 ∧ B2 , 2 ∈ ξ. Then   ⇒  has a child node ∈ ξ containing sequent   ⇒ Bi , 2 . By induction assumption, v |=   ⇒ Bi , 2 .  Hence, v |=   ⇒ B1 ∧ B2 , 2 . Notice that each formula A is of form p1 ∧ · · · ∧ pn .

2.1.4 Gentzen Deduction System G∧ Given a co-sequent δ =  → , we say that δ is valid, denoted by |=∧ δ, if there is an assignment v such that for each A ∈ , v(A) = 1, and for each B ∈ , v(B) = 0. Gentzen deduction system G∧ contains the following • Axiom: (A∧ )

∩=∅ ,  → 

where ,  are sets of variables. • Deduction rules:   , A1 →   → B1 ,  (∧ L ) , A2 →  (∧ R )  → B2 ,  , A1 ∧ A2 →   → B1 ∧ B2 ,  Definition 2.1.11 A co-sequent  →  is provable in G∧ , denoted by ∧  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in G∧ . Theorem 2.1.12 (Soundness and completeness theorem) For any co-sequent  → , ∧  →  if and only if |=∧  → . 

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2.1.5 Gentzen Deduction System G∨ A sequent δ =  ⇒  is G∨ -valid, denoted by |=∨ δ, if for any assignment v, either v |=  or v |= , where v |=  if for some A ∈ , v(A) = 0; and v |=  if for some B ∈ , v(B) = 1. Gentzen deduction system G∨ contains the following • Axiom: (A∨ )

 ∩  = ∅ , ⇒

where ,  are sets of variables. • Deduction rules:   , A1 ⇒   ⇒ B1 ,  (∨ R )  ⇒ B2 ,  (∨ L ) , A2 ⇒  , A1 ∨ A2 ⇒   ⇒ B1 ∨ B2 , . Definition 2.1.13 A sequent  ⇒  is provable in G∨ , denoted by ∨  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in G∨ . Theorem 4.2 (Soundness theorem) For any sequent  ⇒ , if ∨  ⇒  then |=∨  ⇒ . Proof We prove that each axiom is valid, and each deduction rule preserves the validity. (A∨ ) Assume that p ∈  ∩  = ∅. Then, for any assignment v, either v( p) = 0 or v( p) = 1, i.e., either v |= , or v |= . (∨ L ) Assume that for any assignment v, v |= , A1 implies v |= ; and v |= A2 ,  implies v |= . Then, for any assignment v, assume that v |= , A1 ∨ A2 . Then, v |= A1 ∨ A2 . If v |= A1 then v |= A1 , , and by induction assumption, v |= ; and if v |= A2 then v |= A2 , , and by induction assumption, v |= . (∨ R ) Assume that for any assignment v, v |=  implies v |= Bi , . Then, for any  assignment v, assume that v |= . Then, v |= Bi , , and v |= B1 ∨ B2 , . Theorem 2.1.14 (Completeness theorem) For any sequent  ⇒ , if |=∨  ⇒   then ∨  ⇒ . Proof Given a sequent  ⇒ , we construct a tree T such that either (i) For each branch ξ of T, a sequent  ⇒   at the leaf of ξ is an axiom. (ii) There is an assignment v such that v |=  ⇒ .

2.1 Weak Propositional Logics

23

T is constructed as follows: • The root of T is  ⇒ . • For a node ξ , if each sequent  ⇒   at ξ is atomic then the node is a leaf. • Otherwise, ξ has the direct children nodes containing the following sequents: ⎧  1 , A 1 ⇒ 1 ⎪ ⎪ if 1 , A1 ∨ A2 ⇒ 1 ∈ ξ ⎨   1 , A 2 ⇒ 1 1 ⇒ B1 , 1 ⎪ ⎪ if 1 ⇒ B1 ∨ B2 , 1 ∈ ξ ⎩ 1 ⇒ B2 , 1 Lemma 2.1.15 If for each branch ξ ⊆ T, there is a sequent  ⇒   ∈ ξ which is an axiom in G∧ then T is a proof tree of  ⇒ . Proof By the definition of T, T is a proof tree of  ⇒ .



Lemma 2.1.16 If there is a branch ξ ⊆ T such that each sequent  ⇒   ∈ ξ is not an axiom in G∧ then there is an assignment v such that v |=  ⇒ . Proof Let γ be the set of all the atomic sequents in ξ . Let

  L =  ⇒ ∈γ   ,   R =  ⇒ ∈γ  .

Define an assignment v as follows: for any variable p,  v( p) =

1 if p ∈  L 0 otherwise.

Then, v is well-defined and v |=   ⇒  , where   ⇒  is the sequent at the leaf node of ξ. We proved by induction on ξ that each sequent  ⇒   ∈ ξ is not satisfied by v. Case   ⇒  = 2 , A1 ∨ A2 ⇒  ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequent 2 , Ai ⇒  . By induction assumption, v |= 2 , Ai ⇒  . Hence, v |= 2 , A1 ∨ A2 ⇒  . Case   ⇒  =   ⇒ B1 ∨ B2 , 2 ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequent   ⇒ B1 , B2 , 2 . By induction assumption, v |=   ⇒ B1 , B2 ,  2 . Hence, v |=   ⇒ B1 ∨ B2 , 2 . Notice that each formula A is of form p1 ∨ · · · ∨ pn .

2.1.6 Gentzen Deduction System G∨ Given a co-sequent δ =  → , we say that δ is G∨ -valid, denoted by |=∨ δ, if there is an assignment v such that for each A ∈ , v(A) = 1, and for each B ∈ , v(B) = 0.

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Gentzen deduction system G∨ contains the following • Axiom: (A∨ )

∩=∅ ,  → 

where ,  are sets of variables. • Deduction rules:   , A1 →   → B1 ,  (∨ L ) , A2 →  (∨ R )  → B2 ,  , A1 ∨ A2 →   → B1 ∨ B2 ,  Definition 2.1.17 A co-sequent  →  is provable in G∨ , denoted by ∨  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in G∨ . Theorem 2.1.18 (Soundness and completeness theorem) For any co-sequent  → , ∨  →  if and only if |=∨  → . 

2.2 R-Calculus for Weak Propositional Logics Correspondingly we have three R-calculi R¬ , R∧ , R∨ for sequents and R¬ , ikm, R∧ , R∨ for co-sequents. Let x ∈ {¬, ∧, ∨}. A sequent  ⇒  revises a formula pair (A, B), where A ∈  and B ∈ , and obtains a sequent   ⇒  , denoted by |=x  ⇒ |(A, B) ⇒   ⇒  , if 

 − {A} if |=x  − {A} ⇒  otherwise;   − {B} if |=x   ⇒  − {B}  =  otherwise.  =

A co-sequent  →  revises a formula pair (A, B) and obtains a co-sequent   →  , denoted by |=x δ = (A, B)| →  ⇒   →  , if 

, A if |=x , A →  otherwise;  , B if |=x   → , B   =  otherwise.

 =

2.2 R-Calculus for Weak Propositional Logics

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2.2.1 R-Calculus R¬ Let A ∈  and B ∈ . A reduction δ is valid, denoted by |=¬ δ, if 

 − {A} if |=¬  − {A} ⇒  otherwise   − {B} if |=¬   ⇒  − {B}  =  otherwise. 

 =

R-calculus R¬ consists of the following axioms and deduction rules: let A ∈  and B ∈ . • Axioms: ∼ E p  = p( p  ∈  ∩ ) E p  = p( p  ∈  ∩ ) L ) (A−  ⇒ |( p, q) ⇒  ⇒ |q  ⇒ |( p, q) ⇒  − { p} ⇒ |q   Eq  = q(q  ∈  ∩ ) R ∼ Eq = q(q ∈  ∩ ) R (A− )  (A0 )    ⇒ |q ⇒  ⇒   ⇒ |q ⇒   ⇒  − {q}

(A0L )

where ,  are sets of variables, p ∈  and q ∈ . • Deduction rules:  ⇒ |(λ, A; B) ⇒  ⇒ |B  ⇒ |(¬A, B) ⇒  ⇒ |B  ⇒ |(λ, A; B) ⇒  ⇒  − {A}|B L (¬− )  ⇒ |(¬A, B) ⇒  − {¬A} ⇒ |B   R  ⇒ |(B, λ) ⇒  ⇒  (¬0 )   ⇒ |¬B ⇒   ⇒    ⇒ |(B, λ) ⇒   − {B} ⇒  (¬−R )   ⇒ |¬B ⇒   ⇒  − {¬B}

(¬0L )

where  ⇒ |(λ, A; B) denotes ( ⇒ |(λ, A)|B. Definition 2.2.1 A reduction δ is provable in R¬ , denoted by ¬ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R¬ . Theorem 2.2.2 (Soundness and completeness theorem) For any reduction δ = (A, B)| ⇒  ⇒   ⇒  , ¬ δ iff |=¬ δ. 

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2.2.2 R-Calculus R¬ A reduction δ is valid, denoted by |=¬ δ, if 

, A if , A →  is G¬ -valid otherwise  , B if   → , B is G¬ -valid  =  otherwise. 

 =

R-calculus R¬ consists of the following axioms and deduction rules: • Axioms: p∈ p∈ / (A+ L)  → |( p, q) ⇒  → |q  → |( p, q) ⇒ , p → |q q ∈  q∈ /  + 0 (A (A R )  ) R  → |q ⇒   →    → |q ⇒   → , q (A0L )

where ,  are sets of variables, and p, q are atomic. • Deduction rules:  → |(λ, A; B) ⇒  → |B  → |(¬A, B) ⇒  → |B  → |(λ, A; B) ⇒  → , A|B (¬+ L)  → |(¬A, B) ⇒ , ¬A → |B |  → |(B, λ) ⇒   →  (¬0R )   → |¬B ⇒   →    → |(B, λ) ⇒   , B →  (¬+R )   → |¬B ⇒   → , ¬B (¬0L )

where (λ, A; B)| →  is B|((λ, A)| → ). Definition 2.2.3 A reduction δ is provable in R¬ , denoted by ¬ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R¬ . Theorem 2.2.4 (Soundness and completeness theorem) For any reduction δ = (A, B)| →  ⇒   →  , ¬ δ iff |=¬ δ. 

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27

2.2.3 R-Calculus R∧ Let A ∈  and B ∈ . A reduction δ is valid, denoted by |=∧ δ, if 

 − {A} if |=∧  − {A} ⇒  otherwise   − {B} if |=∧   ⇒  − {B}  =  otherwise. 

 =

R-calculus R∧ consists of the following axioms and deduction rules: • Axiom: ∼ E p  = p( p  ∈  ∩ ) E p  = p( p  ∈  ∩ ) L (A− )  ⇒ |( p, q) ⇒  ⇒ |q  ⇒ |( p, q) ⇒  − { p} ⇒ |q    Eq  = q(q  ∈   ∩ ) R ∼ Eq = q(q ∈  ∩ ) R (A (A0 ) ) −   ⇒ |q ⇒   ⇒    ⇒ |q ⇒   ⇒  − {q} (A0L )

where ,  are sets of variables, and p, q are variables. • Deduction rules:   ⇒ |(A1 , B) ⇒  ⇒ |B (∧0L )  ⇒ |(A2 , B) ⇒  ⇒ |B  ⇒ |(A1 ∧A2 , B) ⇒  ⇒ |B  ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B L (∧− )  ⇒ |(A2 , B) ⇒  − {A2 } ⇒ |B  ⇒ |(A1 ∧ A2 , B) ⇒ B| − {A1 ∧ A2 } ⇒     ⇒ |B1 ⇒   ⇒  (∧0R )   ⇒  − {B1 }|B2 ⇒   ⇒  − {B1 }   ⇒ |B1 ∧ B2 ⇒   ⇒    ⇒ |B1 ⇒   ⇒  − {B1 } R (∧− )   ⇒  − {B1 }|B2 ⇒   ⇒  − {B1 , B2 }   ⇒ |B1 ∧ B2 ⇒   ⇒  − {B1 ∧ B2 } Definition 2.2.5 A reduction δ is provable in R∧ , denoted by ∧ δ, if there is a sequence {δ1 , . . . , δn } such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R∧ . Theorem 2.2.6 (Soundness and completeness theorem) For any reduction δ = (A, B)| ⇒  ⇒   ⇒  , ∧ δ iff |=∧ δ. 

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2.2.4 R-Calculus R∧ A reduction δ is valid, denoted by |=∧ δ, if 

, A if , A →  is G∧ -valid otherwise  , B if   → , B is G∧ -valid  =  otherwise. 

 =

R-calculus R∧ consists of the following axioms and deduction rules: • Axioms: p∈ p∈ / L (A+ )  → |( p, q) ⇒  → |q  → |( p, q) ⇒ , p → |q q ∈  q∈ /  R R (A (A0 )  ) +  → |q ⇒   →    → |q ⇒   → , q (A0L )

where ,  are sets of variables and p, q are variables. • Deduction rules:   → |(A1 , B) ⇒  → |B (∧0L ) , A1 → |(A2 , B) ⇒ , A1 → |B  → |(A1 ∧A2 , B) ⇒  → |B  → |(A1 , B) ⇒ , A1 → |B L (∧+ ) , A1 → |(A2 , B) ⇒ , A1 , A2 → |B  → |(A1 ∧ A2 , B) ⇒ , A1 ∧ A2 → |B    → |B1 ⇒   →  (∧0R )   → |B2 ⇒   →    → |B1 ∧B2 ⇒   →    → |B1 ⇒   → , B1 R (∧+ )   → |B2 ⇒   → , B2   → |B1 ∧ B2 ⇒   → , B1 ∧ B2 Definition 2.2.7 A reduction δ is provable in R∧ , denoted by ∧ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R∧ . Theorem 2.2.8 (Soundness and completeness theorem) For any reduction  → |(A, B) ⇒   →  , ∧  → |(A, B) ⇒   →  iff |=∧  → |(A, B) ⇒   →  . 

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29

2.2.5 R-Calculus R∨ Let A ∈  and B ∈ . A reduction δ is valid, denoted by |=∨ δ, if 

 − {A} if |=∨  − {A} ⇒  otherwise   − {B} if |=∨   ⇒  − {B}  =  otherwise. 

 =

R-calculus R∨ consists of the following axioms and deduction rules: • Axioms: ∼ E p  = p( p  ∈  ∩ ) E p  = p( p  ∈  ∩ ) (A0L )  ⇒ |( p, q) ⇒  ⇒ |q  ⇒ |( p, q) ⇒  − { p} ⇒ |q   Eq  = q(q  ∈  ∩ ) R R ∼ Eq = q(q ∈  ∩ ) (A (A− )  ) 0  ⇒ |q ⇒   ⇒    ⇒ |q ⇒   ⇒  − {q} L (A− )

where ,  are sets of variables, and p, q are variables. • Deduction rules:   ⇒ |(A1 , B) ⇒  ⇒ |B L (∨− )  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 } ⇒ |B  ⇒ |(A1 ∨A2 , B) ⇒  ⇒ |B  ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B (∨0L )  − {A1 } ⇒ (A2 , B) ⇒  − {A1 , A2 } ⇒ |B  ⇒ |(A1 ∨ A2 , B) ⇒  − {A1 ∨ A2 } ⇒ |B    ⇒ |B1 ⇒   ⇒  (∧−R )   ⇒ |B2 ⇒   ⇒    ⇒ |B1 ∨ B2 ⇒   ⇒    ⇒ |B1 ⇒   ⇒  − {B1 } R (∨0 )   ⇒ |B2 ⇒   ⇒  − {B2 }   ⇒ |B1 ∨ B2 ⇒   ⇒  − {B1 ∨ B2 } Definition 2.2.9 A reduction δ is provable in R∨ , denoted by ∧ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R∨ . Theorem 2.2.10 (Soundness and completeness theorem) For any reduction  ⇒ |(A, B) ⇒   ⇒  , ∨  ⇒ |(A, B) ⇒   ⇒  iff |=∨  ⇒ |(A, B) ⇒   ⇒  . 

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2.2.6 R-Calculus R∨ A reduction δ =  → |(A, B) ⇒   →  is valid, denoted by |=∨ δ, if 

, A if , A →  is G∨ -valid otherwise  , B if   → , B is G∨ -valid  =  otherwise. 

 =

R-calculus R∨ consists of the following axioms and deduction rules: • Axioms: p∈ p∈ / L (A+ )  → |( p, q) ⇒  → |q  → |( p, q) ⇒ , p → |q q ∈  q∈ /  R R (A (A0 )  ) +  → |q ⇒   →    → |q ⇒   → , q (A0L )

where ,  are sets of variables, where p, q are variables. • Deduction rules:   → |(A1 , B) ⇒  → |B (∨0L )  → |(A2 , B) ⇒  → |B  → |(A1 ∨A2 , B) ⇒  → |B  → |(A1 , B) ⇒ , A1 → |B L (∨+ )  → |(A2 , B) ⇒ , A2 → |B  → |(A1 ∨ A2 , B) ⇒ , A1 ∨ A2 → |B    → |B1 ⇒   →  (∨0R )   → , B1 |B2 ⇒   → , B1   → |B1 ∨B2 ⇒   →    → |B1 ⇒   → , B1 R (∨+ )   → , B1 |B2 ⇒   → , B1 , B2   → |B1 ∨ B2 ⇒   → , B1 ∨ B2 Definition 2.2.11 A reduction δ is provable in R∨ , denoted by ∨ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R∨ . Theorem 2.2.12 (Soundness and completeness theorem) For any reduction δ =  → |(A, B) ⇒   →  ,  ∨ δ iff |=∨ δ.

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31

2.3 Variant Propositional Logics We consider two variant propositional logics • containing logical connectives ¬, → or ¬, ⊕, ⊗, and • give Gentzen deduction systems G¬→ , G⊕⊗ for sequents and G¬→ , G⊕⊗ for cosequents, and • R-calculi R¬→ , R⊕⊗ for sequents and R¬→ , R⊕⊗ for co-sequents.

2.3.1 Gentzen Deduction System G¬→ The logical language of propositional logic G¬→ contains the following symbols: • variables: p0 , p1 , . . . ; • unary connective: ¬; • binary connective: → . Formulas are defined as follows: A ::= p|¬A1 |A1 → A2 . Let v be an assignment, a function from variables to {0, 1}. The interpretation v(A) of formula A in assignment v is defined as follows: ⎧ if A = p ⎨ v( p) if A = ¬A1 v(A) = 1 − v(A1 ) ⎩ max{v(¬A1 ), v(A2 )} if A = A1 → A2 That is, the truth tables for logical connectives ¬ and → are given as follows: A ¬A 1 0 0 1

A1 → A2 1 0 1 10 11 0

Given two sets ,  of formulas, define v() = min{v(A) : A ∈ }, v() = max{v(B) : B ∈ }. Given a sequent δ =  ⇒ , we say that v satisfies δ, denoted by v |= δ, if v() = 1 implies v() = 1. A sequent δ is valid, denoted by |=¬→ δ, if for any assignment v, v |= δ.

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Gentzen deduction system G¬→ consists of the following axiom and deduction rules: • Axiom: (A¬→ )

 ∩  = ∅ , ⇒

where ,  are sets of variables. • Deduction rules:  ⇒ A,  , B ⇒  (¬ R ) , ¬A ⇒    ⇒ ¬B,  , ¬A1 ⇒   ⇒ ¬B1 ,  (→ L ) , A2 ⇒  (→ R )  ⇒ B2 ,  , A1 → A2 ⇒   ⇒ B1 → B2 , . (¬ L )

Definition 2.3.1 A sequent  ⇒  is provable, denoted by ¬→  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in G¬→ . Theorem 2.3.2 (Soundness theorem) For any sequent  ⇒ , if ¬→  ⇒  then |=¬→  ⇒ . Proof We prove that each axiom is valid and each deduction rule preserves the validity. (A¬→ ) Assume that p ∈  ∩ . Then, for any assignment v, either v( p) = 0 or v( p) = 1, i.e., either v |=  or v |= . (¬ L ) Assume that for any assignment v, v |=  implies v |= A, . Then, for any assignment v, assume that v |= , ¬A. v |= , and by induction assumption, v |= A, . Because v |= ¬A, v |= . (¬ R ) Assume that for any assignment v, v |= , B implies v |= . Then, for any assignment v, assume that v |= . If v |= ¬B then v |= ¬B, ; otherwise, v |= , B, and by induction assumption, v |= , which implies v |= ¬B, . (→ L ) Assume that for any assignment v, v |= , A2 implies v |=  v |= , ¬A1 implies v |= . Then, for any assignment v, assume that v |= , A1 → A2 . Then, either v |= A2 or v |= ¬A1 . For each case, we have that v |= . (→ R ) Assume that for any assignment v, either v |=  implies v |= ¬B1 , ; or v |=  implies v |= B2 , . Then, for any assignment v, assume that v |= . If v |=  then v |= B1 → B2 , ; otherwise, by induction assumption, either v |= ¬B1 ,  or v |= B2 , which implies v |= B1 → B2 , i.e., v |= B1 → B2 , . Theorem 2.3.3 (Completeness theorem) For any sequent  ⇒ , if |=¬→  ⇒  then ¬→  ⇒ .

2.3 Variant Propositional Logics

33

Proof Let δ =  ⇒ A. We will define a tree, called the reduction tree for δ, denoted by T (δ), from which we can obtain either a proof of δ or a show of the nonvalidity of δ. This reduction tree T (δ) for δ contains a sequent at each node and is constructed in stages as follows. Stage 0: T0 (δ) = {δ}. Stage k(k > 0) : Tk (δ) is defined by cases. Case 0. If  and  have any statement in common, write nothing above  ⇒ . Case 1. Every topmost sequent  ⇒  in Tk−1 (δ) has the same statement in  and . Then, stop. Case 2. Not case 1. Tk (δ) is defined as follows. Let  ⇒  be any topmost sequent of the tree which has been defined by stage k − 1. Subcase (¬ L ). Let ¬A1 , . . . , ¬An be all the formulas in  whose outmost logical symbol is ¬, and to which no reduction has been applied in previous stages. Then, write down  ⇒ A1 , . . . , An ,  above  ⇒ . We say that a (¬ L ) reduction has been applied to ¬A1 , . . . , ¬An . Subcase (¬ R ). Let ¬B1 , . . . , ¬Bn be all the formulas in  whose outmost logical symbol is ¬, and to which no reduction has been applied in previous stages. Then, write down , B1 , . . . , Bn ⇒  above  ⇒ . We say that a (¬ R ) reduction has been applied to ¬B1 , . . . , ¬Bn . Subcase (→ L ). Let A11 → A12 , . . . , An1 → An2 be all the statements in  whose outermost logical symbol is →, and to which no reduction has been applied in previous stages by any (→ L ). Then, write down all sequents of the form , E 1 , . . . , E n ⇒  above  ⇒ , where E i is either ¬Ai1 or Ai2 . We say that a (→ L ) reduction has been applied to A11 → A12 , . . . , An1 → An2 . Subcase (→ R ). Let B11 → B21 , . . . , B1n → B2n be all the statements in  whose outermost logical symbol is →, and to which no reduction has been applied in previous stages by any (→ R ). Then, write down  ⇒ ¬B11 , B11 , . . . , ¬B1n , B2n ,  above  ⇒ . We say that a (→ R ) reduction has been applied to B11 → B21 , . . . , B1n → B2n . So the collection of those sequents which are obtained by the above reduction process, together with the partial order obtained by this process, is the reduction tree for δ, denoted by T (δ). A sequence δ0 , . . . of sequents in T (δ) is a branch if δ0 = δ, δi+1 is immediately above δi .

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Given a sequent δ, if each branch of T (δ) is ended with a sequent containing common formulas, then it is a routine to construct a proof of δ. Otherwise, there is a branch σ = δ1 , . . . , δn of T (δ) such that there is no rule applicable for δn and δn = n ⇒ n has no common statements in n and n . Let ∪ = {A ∈ i : i ⇒ i ∈ σ }, ∪ = {B ∈ i : i ⇒ i ∈ σ }. We define an assignment in which every formula A ∈ ∪ is true and every formula in B ∈ ∪ is false. Define v such that for any variable p, v( p) = 1 iff p ∈ ∪. By the induction on the structure of formulas, we prove that v(A) = 1 if A ∈ ∪ and v(B) = 0 if B ∈ ∪. Case A = p ∈ ∪ is atomic. By the definition of v, v( p) = 1, i.e., v(A) = 1. Case B = q ∈ ∪ is atomic. By the definition of v, v(q) = 0, i.e., v(B) = 0. Case A = ¬A1 ∈ ∪. Let β be the least-length segment of σ such that β =   , ¬A ⇒  for some   and  . Then, there is a segment γ of σ such that β is a segment of γ and γ =   ⇒ A,  . By induction assumption, v |=   and v |= A,  , equivalently, v |=   , ¬A and v |=  . Case B = ¬B1 ∈ ∪. Let β be the least-length segment of σ such that β =   , ¬B ⇒  for some   and  . Then, there is a segment γ of σ such that β is a segment of γ and γ =   , B ⇒  . By induction assumption, v |=   , B and v |=  , equivalently, v |=   and v |= ¬B,  . Case A = A1 → A2 ∈ ∪. Let β be the least-length segment of σ such that β =   , A1 → A2 ⇒  for some   and  . Then, there is a segment γ of σ such that β is a segment of γ and γ is one of the following forms:   , ¬A1 ⇒  ,   , A 2 ⇒  . By induction assumption, either v |=   , ¬A1 or v |=   , A2 , and v |=  . Then, by definition of satisfaction, v |=   , A1 → A2 and v |=  . Case B = B1 → B2 ∈ ∪. Let β be the least-length segment of σ such that β =   ⇒ B1 → B2 ,  . Then, there is a segment γ of σ such that β is a segment of γ and γ =   ⇒ ¬B1 , B2 ,  . By the induction assumption, v |=   and v |= ¬B1 , B2 ,  , i.e., v |= B1 → B2 ,  . This completes the proof. 

2.3 Variant Propositional Logics

35

2.3.2 Gentzen Deduction System G¬→ A co-sequent  →  is valid, denoted by |=¬→  → , if there is an assignment v such that for each formula A ∈ , v(A) = 1, and for each formula B ∈ , v(B) = 0. Gentzen deduction system G¬→ consists of the following axiom and deduction rules: • Axiom: (A¬→ )

∩=∅ ,  → 

where ,  are sets of variables. • Deduction rules:  → A,  , B →  (¬ R ) , ¬A  →    → ¬B,  , ¬A1 →   → ¬B1 ,  (→ L ) , A2 →  (→ R )  → B2 ,  , A1 → A2 →   → B1 → B2 , . (¬ L )

Definition 2.3.4 A co-sequent  →  is provable in G¬→ , denoted by ¬→  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in G¬→ . Theorem 2.3.5 (Soundness and completeness theorem) For any co-sequent  → , ¬→  →  if and only if |=¬→  → . 

2.3.3 Gentzen Deduction System G⊗⊕ Traditionally there are two kinds of the semantics for the logical connectives ∨ and ∧ : the inclusive one and exclusive one, where in the exclusive one, ∨ and ∧ have the following truth tables: A1 ∨ A2 1 0 A1 ∧ A2 1 0 1 01 1 10 0 0 10 00 and in the inclusive one, ∨ and ∧ have the following ones: A1 ∨ A2 1 0 A1 ∧ A2 1 0 1 11 1 10 10 00 0 0

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We consider two connectives which truth tables are given as follows: A1 ⊕ A2 1 0 A1 ⊗ A2 1 0 1 01 1 10 10 01 0 0 The logical language of propositional logic G⊕⊗ contains the following symbols: • variables: p0 , p1 , . . . ; • unary connective: ¬; • binary connective: ⊕, ⊗. Formulas are defined as follows: A ::= p|¬A1 |A1 ⊕ A2 |A1 ⊗ A2 . Let v be an assignment, a function from variables to {0, 1}. The truth-value of formula A in an assignment v is defined as follows: ⎧ v( p) if ⎪ ⎪ ⎨ if 1 − v(A1 ) v(A) = f (v(A ), v(A )) if ⎪ ⊕ 1 2 ⎪ ⎩ f ⊗ (v(A1 ), v(A2 )) if

A= p A = ¬A1 A = A1 ⊕ A2 A = A1 ⊗ A2

where f ⊕ , f ⊗ : {0, 1}2 → {0, 1} are defined as follows: A ¬A 1 0 0 1

f⊕ 1 0 1 01 0 10

f⊗ 1 0 1 10 0 01

Given two sets ,  of formulas, define v() = min{v(A) : A ∈ }, v() = max{v(B) : B ∈ }. Given a sequent δ =  ⇒ , we say that v satisfies δ, denoted by v |= δ, if v() = 1 implies v() = 1. A sequent δ is valid, denoted by |=⊕⊗ δ, if for any assignment v, v |= δ. Gentzen deduction system G⊕⊗ consists of the following axiom and deduction rules. • Axiom: (A⊕⊗ )

incon() or val() or  ∩  = ∅ ⇒

where ,  are sets of literals.

2.3 Variant Propositional Logics

37

• Deduction rules: , A ⇒   ⇒ B,  (¬¬ R ) (¬¬ L ) ⎧, ⎧ ⇒ ¬¬B,   ¬¬A ⇒  , ¬A1 →   ⇒ B1 ,  ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ , ¬A  ⇒  2   ⇒ ¬B2 ,  R →  , A  ⇒ ¬B1 ,  ) (⊕ L ) ⎪ (⊕ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , A2 ⇒   ⇒ B2 ,  ,⎧A1 ⊕ A2 ⇒  ⎧ ⇒ B1 ⊕ B2 ,  →  , ¬A  ⇒ ¬B1 ,  ⎪ ⎪ 1 ⎪ ⎪ ⎨ ⎨ ⇒  , A  2   ⇒ ¬B2 ,  R →  , A  ⇒ B1 ,  ) (¬⊕ L ) ⎪ (¬⊕ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , ¬A2 ⇒   ⇒ B2 ,  , ¬(A1 ⊕ A2 ) ⇒   ⇒ ¬(B1 ⊕ B2 ),  and

⎧ ⎧ , ¬A1 →   ⇒ B1 ,  ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⇒  , A  2   ⇒ B2 ,  R →  , A  ⇒ ¬B1 ,  (⊗ L ) ⎪ ) (⊗ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , ¬A2 ⇒   ⇒ ¬B2 ,  ,⎧A1 ⊗ A2 ⇒  ⎧ ⇒ B1 ⊗ B2 ,  , ¬A  ⇒ ¬B1 ,  →  ⎪ ⎪ 1 ⎪ ⎪ ⎨ ⎨ ⇒  , ¬A  2   ⇒ B2 ,  R →  , A  ⇒ B1 ,  ) (¬⊗ L ) ⎪ (¬⊗ ⎪ ⎪ 1 ⎪ ⎩ ⎩ , A2 ⇒   ⇒ ¬B2 ,  , ¬(A1 ⊗ A2 ) ⇒   ⇒ ¬(B1 ⊗ B2 ), 

Definition 2.3.6 A sequent  ⇒  is provable in G⊕⊗ , denoted by ⊕⊗   , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in G⊕⊗ . Theorem 2.3.7 (Soundness theorem) For any sequent  ⇒ , if ⊕⊗  ⇒  then |=⊕⊗  ⇒ .  Proof We prove that each axiom is valid and each deduction rule preserves the validity. (A⊕⊗ ) Assume that incon(). Then, there is a literal l such that l, ¬l ∈ . For any assignment v, either v(l) = 0 or v(¬l) = 0, i.e., v |= . Assume that l ∈  ∩ . Then, for any assignment v, either v(l) = 0 or v(l) = 1, i.e., either v |=  or v |= .

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(¬ L ) Assume that for any assignment v, v |=  implies v |= A, . Then, for any assignment v, assume that v |= , ¬A. By induction assumption, v |= A, . Because v |= ¬A, v |= . (¬ R ) Assume that for any assignment v, v |= , B implies v |= . Then, for any assignment v, assume that v |= . If v |= ¬B then v |= ¬B, ; otherwise, v |= , B, and by induction assumption, v |= , which implies v |= ¬B, . (⊕ L ) Assume that for any assignment v, either v |= , ¬A1 implies v |=  v |= , ¬A2 implies v |=  or

v |= , A1 implies v |=  v |= , A2 implies v |= .

Then, for any assignment v, assume that v |= , A1 ⊕ A2 . Then, either v |= ¬A1 and v |= ¬A2 , or v |= A1 and v |= A2 . Hence, either v |= , ¬A1 and v |= , ¬A2 , or v |= , A1 and v |= , A2 . By induction assumption, v |= . (⊕ R ) Assume that for any assignment v, either v |=  implies v |= B1 ,  v |=  implies v |= ¬B2 , , or

v |=  implies v |= ¬B1 ,  v |=  implies v |= B2 , ,

Then, for any assignment v, assume that v |= . If either v |= B1 and v |= ¬B2 , or v |= ¬B1 and v |= B2 then v |= B1 ⊕ B2 , ; otherwise, by induction assumption, v |= , which implies v |= B1 ⊕ B2 , . (⊗ L ) Assume that for any assignment v, either v |= , ¬A1 implies v |=  v |= , A2 implies v |= , or

v |= , A1 implies v |=  v |= ¬A2 implies v |= .

Then, for any assignment v, assume that v |= , A1 ⊗ A2 . Then, either v |= ¬A1 and v |= A2 , or v |= A1 or v |= ¬A2 , i.e., either v |= , ¬A1 and v |= , A2 , or v |= , A1 and v |= , ¬A2 . By induction assumption, v |= . (⊗ R ) Assume that for any assignment v, either v |=  implies v |= B1 ,  v |=  implies v |= B2 , ,

2.3 Variant Propositional Logics

or

39

v |=  implies v |= ¬B1 ,  v |=  implies v |= ¬B2 , ,

Then, for any assignment v, assume that v |= . If either v |= B1 and v |= B2 , or v |= ¬B1 and v |= ¬B2 then v |= B1 ⊗ B2 , and v |= B1 ⊗ B2 , ; otherwise, by induction assumption, v |= , which implies v |= B1 ⊗ B2 , . Similar for other cases.  Theorem 2.3.8 (Completeness theorem) For any sequent  ⇒ , if |=⊕⊗  ⇒  then ⊕⊗  ⇒ . Proof Given a sequent  ⇒ , we construct a tree T such that either (i) For each branch ξ of T, a sequent  ⇒   at the leaf of ξ is an axiom. (ii) There is an assignment v such that v |=  ⇒ . T is constructed as follows: • The root of T is  ⇒ . • For a node ξ , if each sequent  ⇒   at ξ is literal then the node is a leaf. • Otherwise, ξ has the direct children nodes containing the following sequents: ⎧  1 , A ⇒ 1 ⎪ ⎪ ⎪ ⎪ 1⇒ B, 1 ⎪ ⎪ ⎡ ⎪ ⎪ 1 , ¬A1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎢ 1 , ¬A2 ⇒ 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣  1 , A 1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡   1 , A 2 ⇒ 1 ⎪ ⎪ 1 ⇒ B1 , 1 ⎪ ⎪ ⎪ ⎪ ⎢ 1 ⇒ ¬B2 , 1 ⎪ ⎪  ⎨⎢ ⎣ 1 ⇒ B1 , 1 ⎪ ⎪ ⎡  1 ⇒ ¬B2 , 1 ⎪ ⎪ 1 , ¬A1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎢  1 , A 2 ⇒ 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣  1 , A 1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡  1 , ¬A2 ⇒ 1 ⎪ ⎪ 1 ⇒ ¬B1 , 1 ⎪ ⎪ ⎪ ⎪ ⎢ 1 ⇒ ¬B2 , 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣ 1 ⇒ B1 , 1 ⎪ ⎪ ⎩ 1 ⇒ B2 , 1 and

if 1 , ¬¬A ⇒ 1 ∈ ξ if 1 ⇒ ¬¬B, 1 ∈ ξ if 1 , A1 ⊕ A2 ⇒ 1 ∈ ξ

if 1 ⇒ B1 ⊕ B2 , 1 ∈ ξ

if 1 , ¬(A1 ⊕ A2 ) ⇒ 1 ∈ ξ

if 1 ⇒ ¬(B1 ⊕ B2 ), 1 ∈ ξ

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2 R-Calculus for Simplified Propositional Logics

⎧⎡ 1 , ¬A1 ⇒ 1 ⎪ ⎪ ⎪ ⎢  1 , A 2 ⇒ 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎣  1 , A 1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡  1 , ¬A2 ⇒ 1 ⎪ ⎪ 1 ⇒ B1 , 1 ⎪ ⎪ ⎪ ⎪ ⎢ 1 ⇒ B2 , 1 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣ 1 ⇒ ¬B1 , 1 ⎪ ⎪ ⎨ ⎡  1 ⇒ ¬B2 , 1 1 , ¬A1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎢ 1 , ¬A2 ⇒ 1 ⎪ ⎪⎢ ⎪ ⎪ ⎣  1 , A 1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡   1 , A 2 ⇒ 1 ⎪ ⎪ 1 ⇒ ¬B1 , 1 ⎪ ⎪ ⎪ ⎪ ⎢ 1 ⇒ B2 , 1 ⎪ ⎪⎢ ⎪ ⎪ ⎣ 1 ⇒ B1 , 1 ⎪ ⎪ ⎩ 1 ⇒ ¬B2 , 1

if 1 , A1 ⊗ A2 ⇒ 1 ∈ ξ

if 1 ⇒ B1 ⊗ B2 , 1 ∈ ξ

if 1 , ¬(A1 ⊗ A2 ) ⇒ 1 ∈ ξ

if 1 ⇒ ¬(B1 ⊗ B2 ), 1 ∈ ξ

Lemma 2.3.9 If for each branch ξ ⊆ T, there is a sequent  ⇒   ∈ ξ which is an axiom in G⊕⊗ then T is a proof tree of  ⇒ . Proof By the definition of T, T is a proof tree of  ⇒  in G⊕⊗ .



Lemma 2.3.10 If there is a branch ξ ⊆ T such that each sequent  ⇒   ∈ ξ is not an axiom in G⊕⊗ then there is an assignment v such that v |=  ⇒ . Proof Let γ be the set of all the literal sequents in ξ .   L =  ⇒ ∈γ   ,   R =  ⇒ ∈γ  .

Let

Define an assignment v as follows: for any variable p,  v( p) =

1 if p ∈  L or ¬ p ∈  R 0 otherwise.

Then, v is well-defined and v |=   ⇒  , where   ⇒  is a sequent at the leaf node of ξ. We proved by induction on ξ that each sequent  ⇒   ∈ ξ is not satisfied by v. Case   ⇒  = 2 , ¬A ⇒  ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequent 2 ⇒ A,  . By induction assumption, v |= 2 ⇒ A,  . Hence, v |= 2 , ¬A ⇒  . Case   ⇒  =   ⇒ ¬B, 2 ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequent   , B ⇒ 2 . By induction assumption, v |=   , B ⇒ 2 . Hence, v |=   ⇒ ¬B, 2 . Case   ⇒  = 2 , A1 ⊕ A2 ⇒  ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequents either 2 , ¬A1 ⇒  or 2 , ¬A2 ⇒  ; and either 2 , A1 ⇒

2.3 Variant Propositional Logics

41

 or 2 , A2 ⇒  . By induction assumption, either v |= 2 , ¬A1 ⇒  or v |= 2 , ¬A2 ⇒  , and either v |= 2 , A1 ⇒  or v |= 2 , A2 ⇒  . Hence, v |=  2 , A 1 ⊕ A 2 ⇒  . Case   ⇒  =   ⇒ B1 ⊕ B2 , 2 ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequents either   ⇒ B1 , 2 or   ⇒ ¬B2 , 2 , and either   ⇒ ¬B1 , 2 or   ⇒ B2 , 2 . By induction assumption, either v |=   ⇒ B1 , 2 or v |=   ⇒ ¬B2 , 2 ; and either v |=   ⇒ ¬B1 , 2 or v |=   ⇒ B2 , 2 . Hence, v |=   ⇒ B1 ⊕ B2 , 2 . Case   ⇒  = 2 , A1 ⊗ A2 ⇒  ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequents either 2 , ¬A1 ⇒  or 2 , A2 ⇒  ; and either 2 , A1 ⇒  or 2 , ¬A2 ⇒  . By induction assumption, either v |= 2 , ¬A1 ⇒  or v |= 2 , A2 ⇒  , and either v |= 2 , A1 ⇒  or v |= 2 , ¬A2 ⇒  . Hence, v |= 2 , A1 ⊗ A2 ⇒  . Case   ⇒  =   ⇒ B1 ⊗ B2 , 2 ∈ ξ. Then   ⇒  has children nodes ∈ ξ containing sequents either   ⇒ B1 , 2 or   ⇒ B2 , 2 , and either   ⇒ ¬B1 , 2 or   ⇒ ¬B2 , 2 . By induction assumption, either v |=   ⇒ B1 , 2 or v |=   ⇒ B2 , 2 ; and either v |=   ⇒ ¬B1 , 2 or v |=   ⇒ ¬B2 , 2 . Hence, v |=   ⇒ B1 ⊗ B2 , 2 . Similar for other cases. 

2.3.4 Gentzen Deduction System G⊕⊗ A co-sequent  →  is valid, denoted by |=⊕⊗  → , if there is an assignment v such that for every A ∈ , v(A) = 1, and for every B ∈ , v(B) = 0. Gentzen deduction system G⊕⊗ consists of the following axiom and deduction rules. • Axiom: (A⊕⊗ )

con()&inval()& ∩  = ∅  → 

where ,  are sets of literals. • Deduction rules:  → A,  , A →  (¬ R ) , ¬A  →   ⎧ ⎧ →  ¬A,  →  , ¬A  → ¬B1 ,  ⎪ ⎪ 1 ⎪ ⎪ ⎨ ⎨  →  , A 2    → ¬B2 ,  , A1 →   → B1 ,  (⊕ L ) ⎪ (⊕ R ) ⎪ ⎪ ⎪ ⎩ ⎩ , ¬A2 →   → B2 ,  ,⎧A1 ⊕ A2 →  ⎧ → B1 ⊕ B2 ,  →  , ¬A  → ¬B1 ,  ⎪ ⎪ 1 ⎪ ⎪ ⎨ ⎨  →  , ¬A 2    → B2 ,  R , A  → B1 ,  →  ) (¬⊕ L ) ⎪ (¬⊕ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , A2 →   → ¬B2 ,  , ¬(A1 ⊕ A2 ) →   → ¬(B1 ⊕ B2 ),  (¬ L )

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2 R-Calculus for Simplified Propositional Logics

and

⎧ ⎧ , ¬A1 →   → ¬B1 ,  ⎪ ⎪ ⎪ ⎪ ⎨ ⎨  →  , ¬A 2    → B2 ,  , A1 →   → B1 ,  (⊗ L ) ⎪ (⊗ R ) ⎪ ⎪ ⎪ ⎩ ⎩ , A2 →   → ¬B2 ,  ,⎧A1 ⊗ A2 →  ⎧ → B1 ⊗ B2 ,   → ¬B1 ,  ⎪ , ¬A1 →  ⎪ ⎪ ⎪ ⎨ ⎨ , A  →  2    → ¬B2 ,   → B1 ,  (¬⊗ L ) ⎪ (¬⊗ R ) ⎪ ⎪ , A1 →  ⎪ ⎩ ⎩ , ¬A2 →   → B2 ,  , ¬(A1 ⊗ A2 ) →   → ¬(B1 ⊗ B2 ), 

Definition 2.3.11 A co-sequent  →  is provable in G⊕⊗ , denoted by ⊕⊗   , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in G⊕⊗ . Theorem 2.3.12 (Soundness and completeness theorem) For any co-sequent  → , ⊕⊗  →  if and only if |=⊕⊗  → . 

2.4 R-Calculi for Variant Propositional Logics Let x ∈ {{¬, →}, {¬, ⊕, ⊗}}. A sequent  ⇒  revises a formula pair (A, B) and obtains a sequent   ⇒  , denoted by |=x  ⇒ |(A, B) ⇒   ⇒  , if 

 − {A} if |=x  − {A} ⇒  otherwise;   − {B} if |=x   ⇒  − {B}  =  otherwise. 

 =

A co-sequent  →  revises a formula pair (A, B) and obtains a sequent   →  , denoted by |=x  → |(A, B) ⇒   →  , 



, A if |=x , A ⇒   otherwise;  , B if |=x   ⇒ , B  =  otherwise. 

 =

2.4 R-Calculi for Variant Propositional Logics

43

We will give R-calculi R¬→ , R⊕⊗ for sequents and R¬→ , R⊕⊗ for co-sequents, which are sound and complete, that is, x  ⇒ |(A, B) ⇒   ⇒  iff |=x  ⇒ |(A, B) ⇒   ⇒  x  → |(A, B) ⇒   →  iff |=x  → |(A, B) ⇒   →  .

2.4.1 R-Calculus R¬→ Let A ∈  and B ∈ . A reduction δ =  ⇒ |(A, B) ⇒   ⇒  is R¬→ -valid, denoted by |=¬→ δ, if 

 − {A} if |=¬→  − {A} ⇒  otherwise   − {B} if |=¬→   ⇒  − {B}   =  otherwise.

 =

R-calculus R¬→ consists of the following axioms and deduction rules: • Axioms: ∼ El  = l(l  , ¬l  ∈  or l  , ¬l  ∈  or l  ∈  ∩ )  ⇒ |(l, m) ⇒  ⇒ |m       L El = l(l , ¬l ∈  or l , ¬l ∈  or l ∈  ∩ ) ) (A−  ⇒ |(l, m) ⇒  − {l} ⇒ |m       R ∼ Em = m(m , ¬m ∈  or m , ¬m ∈  or m ∈  ∩ ) (A0 )   ⇒ |m ⇒   ⇒   = m(m  , ¬m  ∈  or m  , ¬m  ∈  or m  ∈  ∩ ) Em (A−R )   ⇒ |m ⇒   ⇒  − {m} (A0L )

where ,  are sets of literals and l, m are literals. • Deduction rules:  ⇒ |(A, B) ⇒  ⇒ |B  ⇒ |(¬¬A, B) ⇒  ⇒ |B L  ⇒ |(A, B) ⇒  − {A} ⇒ |B ) (¬¬−  ⇒ |(¬¬A, B) ⇒  − {¬¬A} ⇒ |B   ⇒ |B ⇒  ⇒  (¬¬0R )   ⇒ |¬¬B ⇒   ⇒    ⇒ |B ⇒   ⇒  − {B} (¬¬−R )  ⇒ |¬¬B ⇒  ⇒  − {¬¬B}

(¬¬0L )

44

2 R-Calculus for Simplified Propositional Logics

and 

 ⇒ |(¬A1 , B) ⇒  ⇒ |B  − {¬A1 } ⇒ |(A2 , B) ⇒  − {¬A1 } ⇒ |B  ⇒ |(A1 → A2 , B) ⇒  ⇒ |B  ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B L (→− )  − {¬A1 } ⇒ |(A2 , B) ⇒  − {¬A1 , A2 } ⇒ |B  ⇒ |(A1 → A2 , B) ⇒  − {A1 → A2 } ⇒ |B    ⇒ |¬B1 ⇒   ⇒  (→0R )   ⇒ |B2 ⇒   ⇒    ⇒ |B1 → B2 ⇒   ⇒    ⇒ |¬B1 ⇒   ⇒  − {¬B1 } R (→− )   ⇒ |B2 ⇒   ⇒  − {B2 }   ⇒ |B1 → B2 ⇒   ⇒  − {B1 → B2 }

(→0L )

and



 ⇒ |(A1 , B) ⇒  ⇒ |B  ⇒ |(¬A2 , B) ⇒  ⇒ |B  ⇒ |(¬(A1 →  A2 ), B) ⇒  ⇒ |B  ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B L)  ⇒ |(¬A2 , B) ⇒  − {¬A2 } ⇒ |B (¬ →−  ⇒ |(¬(A1 → A2 ), B) ⇒  − {¬(A1 → A2 )} ⇒ |B    ⇒ |B1 ⇒   ⇒  (¬ →0R )   ⇒  − {B1 }|¬B2 ⇒   ⇒  − {B1 }    ⇒ |(¬(A1 → 2 ), B) ⇒  ⇒ |B  A   ⇒ |B1 ⇒   ⇒  − {B1 } R)   ⇒  − {B1 }|¬B2 ⇒   ⇒  − {B1 , ¬B2 } (¬ →−   ⇒ |¬(B1 → B2 ) ⇒   ⇒  − {¬(B1 → B2 )}

(¬ →0L )

Definition 2.4.1 A reduction δ is provable in R¬→ , denoted by ¬→ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R¬→ . Theorem 2.4.2 (Soundness and completeness theorem) For any reduction δ = (A, B)| ⇒  ⇒   ⇒  , ¬→ δ iff |=¬→ δ.



2.4.2 R-Calculus R¬→ A reduction δ =  → |(A, B) ⇒   →  is R¬→ -valid, denoted by |=¬→ δ, if

2.4 R-Calculi for Variant Propositional Logics



, A if , A →  is G¬→ -valid  otherwise  , B if   → , B is G¬→ -valid   =  otherwise.

 =

R-calculus R¬→ consists of the following axioms and deduction rules: • Axioms:

¬l ∈  or l ∈   → |(l, m) ⇒  → |m ¬l ∈ / &l ∈ / L (A+ )  → |(l, m) ⇒ , l → |m m ∈   or ¬m ∈  (A0R )   → |m ⇒   →  / m∈ /   &¬m ∈ (A+R )    → |m ⇒  → , m

(A0L )

where ,  are sets of literals, and l, m are literals. • Deduction rules:  → |(A, B) ⇒  → |B  → |(¬¬A, B) ⇒  → |B L   → |(A, B) ⇒ , A  → |B ) (¬¬+  → |(¬¬A, B) ⇒ , ¬¬A → |B   R   → |B ⇒   →  (¬¬0 )   → |¬¬B ⇒   →    → |B ⇒   → , B (¬¬+R )  → |¬¬B ⇒  → , ¬¬B (¬¬0L )

and 

 → |(¬A1 , B) ⇒  → |B , ¬A1 → |(A2 , B) ⇒ , ¬A1 → |B  → |(A1 →  A2 , B) ⇒  → |B  → |(¬A1 , B) ⇒ , ¬A1 → |B L (→+ ) , ¬A1 → |(A2 , B) ⇒ , ¬A1 , A2 → |B  → |(A1 → A2 , B) ⇒ , A1 → A2 → |B    → |¬B1 ⇒   →  (→0R )   → |B2 ⇒   →    → |B1 → B2 ⇒   →    → |¬B1 ⇒   → , ¬B1 R (→+ )   → |B2 ⇒   → , B2   → |B1 → B2 ⇒   → , B1 → B2 (→0L )

45

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and 

 → |(A1 , B) ⇒  → |B  → |(¬A2 , B) ⇒  → |B  → |(¬(A1→ A2 ), B) ⇒  → |B  → |(A1 , B) ⇒ , A1 → |B L (¬ →+ )  → |(¬A2 , B) ⇒ , ¬A2 → |B  → |(¬(A1 → A2 ), B) ⇒ , ¬(A1 → A2 ) → |B    → |B1 ⇒   →  (¬ →0R )   → , B1 |¬B2 ⇒   → , B1   → |¬(B1→ B2 ) ⇒   →    → |B1 ⇒   → , B1 R (¬ →+ )   → , B1 |¬B2 ⇒   → , B1 , ¬B2   → |¬(B1 → B2 ) ⇒   → , ¬(B1 → B2 )

(¬ →0L )

Definition 2.4.3 A reduction δ is provable in R¬→ , denoted by ∧→ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R¬→ . Theorem 2.4.4 (Soundness and completeness theorem) For any reduction δ =  → |(A, B) ⇒   →  , ¬→ δ iff |=¬→ δ. 

2.4.3 R-Calculus R⊕⊗ Let A ∈  and B ∈ . A reduction δ =  ⇒ |(A, B) ⇒   ⇒  is R⊕⊗ -valid, denoted by |=⊕⊗ δ, if 

 − {A} if |=⊕⊗  − {A} ⇒   otherwise   − {B} if |=⊕⊗   ⇒  − {B}  =  otherwise. 

 =

R-calculus R⊕⊗ consists of the following axioms and deduction rules:

2.4 R-Calculi for Variant Propositional Logics

• Axioms: ∼ El  = l(l  , ¬l  ∈  or l  , ¬l  ∈  or l  ∈  ∩ )  ⇒ |(l, m) ⇒  ⇒ |m       L El = l(l , ¬l ∈  or l , ¬l ∈  or l ∈  ∩ ) (A− )  ⇒ |(l, m) ⇒  − {l} ⇒ |m       R ∼ Em = m(m , ¬m ∈  or m , ¬m ∈  or m ∈  ∩ ) (A0 )   ⇒ |m ⇒   ⇒   = m(m  , ¬m  ∈  or m  , ¬m  ∈  or m  ∈  ∩ ) Em (A−R )   ⇒ |m ⇒   ⇒  − {m} (A0L )

where ,  are sets of literals and l, m are literals. • Deduction rules:  ⇒ |(A, B) ⇒  ⇒ |B  ⇒ |(¬¬A, B) ⇒  ⇒ |B L  ⇒ |(A, B) ⇒  − {A} ⇒ |B ) (¬¬−  ⇒ |(¬¬A, B) ⇒  − {¬¬A} ⇒ |B   ⇒ |B ⇒  ⇒  (¬¬0R )   ⇒ |¬¬B ⇒   ⇒    ⇒ |B ⇒   ⇒  − {B} (¬¬−R )  ⇒ |¬¬B ⇒  ⇒  − {¬¬B}

(¬¬0L )

and ⎡

(⊕0L )

L ) (⊕−

(⊕0R )

(⊕−R )

 ⇒ |(¬A1 , B) ⇒  ⇒ |B ⎢  − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 } ⇒ |B ⎢ ⎣  ⇒ |(A1 , B) ⇒  ⇒ |B  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 } ⇒ |B  ⎧⇒  |(A1 ⊕ A2 , B) ⇒  ⇒ |B ⎪  ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B ⎪ ⎨   − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 , ¬A2 } ⇒ |B ⎪  ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B ⎪ ⎩  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 , A2 } ⇒ |B  ⇒ 1 ⊕ A2 , B) ⇒  − {A1 ⊕ A2 } ⇒ |B ⎡  |(A   ⇒ |¬B1 ⇒   ⇒  ⎢   ⇒  − {¬B1 }|B2 ⇒   ⇒  − {¬B1 } ⎢  ⎣  ⇒ |B1 ⇒   ⇒    ⇒  − {B1 }|¬B2 ⇒   ⇒  − {B1 }    ⎧ ⇒ |B1 ⊕ B2 ⇒   ⇒   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } ⎪ ⎪ ⎨   2 ⇒  ⇒  − {¬B1 , B2 }    ⇒  − {¬B1 }|B   ⇒ |B1 ⇒  ⇒  − {B1 } ⎪ ⎪ ⎩   ⇒  − {B1 }|¬B2 ⇒   ⇒  − {B1 , ¬B2 }   ⇒ |B1 ⊕ B2 ⇒   ⇒  − {B1 ⊕ B2 }

47

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and ⎡

(¬⊕0L )

L ) (¬⊕−

(¬⊕0R )

(¬⊕−R )

 ⇒ |(¬A1 , B) ⇒  ⇒ |B ⎢  − {¬A1 } ⇒ |(A2 , B) ⇒  − {¬A1 } ⇒ |B ⎢ ⎣  ⇒ |(A1 , B) ⇒  ⇒ |B  − {A1 } ⇒ |(¬A2 , B) ⇒  − {A1 } ⇒ |B  ⎧⇒  |(¬(A1 ⊕ A2 ), B) ⇒  ⇒ |B ⎪  ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B ⎪ ⎨   − {¬A1 } ⇒ |(A2 , B) ⇒  − {¬A1 , A2 } ⇒ |B ⎪  ⇒ |(A1 , B) ⇒ , A1 ⇒ |B ⎪ ⎩  − {A1 } ⇒ |(¬A2 , B) ⇒  − {A1 , ¬A2 } ⇒ |B  ⇒ 1 ⊕ A2 ), B) ⇒  − {¬(A1 ⊕ A2 )} ⇒ |B ⎡  |(¬(A   ⇒ |¬B1 ⇒   ⇒  ⎢   ⇒  − {¬B1 }|¬B2 ⇒   ⇒  − {¬B1 } ⎢  ⎣  ⇒ |B1 ⇒   ⇒    ⇒  − {B1 }|B2 ⇒   ⇒  − {B1 }    ⎧ ⇒ |¬(B1 ⊕ B2 ) ⇒  ⇒   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } ⎪ ⎪ ⎨   2 ⇒  ⇒  − {¬B1 , ¬B2 }    ⇒  − {¬B1 }|¬B   ⇒ |B1 ⇒  ⇒  − {B1 } ⎪ ⎪ ⎩   ⇒  − {B1 }|B2 ⇒   ⇒  − {B1 , B2 }   ⇒ |¬(B1 ⊕ B2 ) ⇒   ⇒  − {¬(B1 ⊕ B2 )}

and ⎡

(⊗0L )

L ) (⊗−

(⊗0R )

(⊗−R )

 ⇒ |(¬A1 , B) ⇒  ⇒ |B ⎢  − {¬A1 } ⇒ |(A2 , B) ⇒  − {¬A1 } ⇒ |B ⎢ ⎣  ⇒ |(A1 , B) ⇒  ⇒ |B  − {A1 } ⇒ |(¬A2 , B) ⇒  − {A1 } ⇒ |B  ⇒ ⎧  |(A1 ⊗ A2 , B) ⇒  ⇒ |B  ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B ⎪ ⎪ ⎨   − {¬A1 } ⇒ |(A2 , B) ⇒  − {¬A1 , A2 } ⇒ |B  ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B ⎪ ⎪ ⎩  − {A1 } ⇒ |(¬A2 , B) ⇒  − {A1 , ¬A2 } ⇒ |B  ⇒ 1 ⊗ A2 , B) ⇒  − {A1 ⊗ A2 } ⇒ |B ⎡  |(A   ⇒ |¬B1 ⇒   ⇒  ⎢   ⇒  − {¬B1 }|¬B2 ⇒   ⇒  − {¬B1 } ⎢  ⎣  ⇒ |B1 ⇒   ⇒    ⇒  − {B1 }|B2 ⇒   ⇒  − {B1 }    ⎧ ⇒ |B1 ⊗ B2 ⇒   ⇒   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } ⎪ ⎪ ⎨   2 ⇒  ⇒  − {¬B1 , ¬B2 }    ⇒  − {¬B1 }|¬B   ⇒ |B1 ⇒  ⇒  − {B1 } ⎪ ⎪ ⎩   ⇒  − {B1 }|B2 ⇒   ⇒  − {B1 , B2 }   ⇒ |B1 ⊗ B2 ⇒   ⇒  − {B1 ⊗ B2 }

2.4 R-Calculi for Variant Propositional Logics

49

and ⎡

(¬⊗0L )

L ) (¬⊗−

(¬⊗0R )

(¬⊗−R )

 ⇒ |(A1 , B) ⇒  ⇒ |B ⎢  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 } ⇒ |B ⎢ ⎣  ⇒ |(¬A1 , B) ⇒  ⇒ |B  − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 } ⇒ |B  ⎧⇒  |(¬(A1 ⊗ A2 ), B) ⇒  ⇒ |B ⎪  ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B ⎪ ⎨   − {A1 } ⇒ |(A2 , B) ⇒ B| − {A1 , A2 } ⇒ |B ⎪  ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B ⎪ ⎩  − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 , ¬A2 } ⇒ |B  ⇒ 1 ⊗ A2 , B) ⇒  − {¬(A1 ⊗ A2 )} ⇒ |B ⎡  |(¬(A   ⇒ |B1 ⇒   ⇒  ⎢   ⇒  − {B1 }|¬B2 ⇒   ⇒  − {B1 } ⎢  ⎣  ⇒ |¬B1 ⇒   ⇒    ⇒  − {¬B1 }|B2 ⇒   ⇒  − {¬B1 }    ⎧ ⇒ |¬(B1 ⊗ B2 ) ⇒  ⇒   ⇒ |B1 ⇒  ⇒  − {B1 } ⎪ ⎪ ⎨      ⇒  − {B1 }|¬B 2 ⇒  ⇒  − {B1 , ¬B2 }  ⇒ |¬B1 ⇒  ⇒  − {¬B1 } ⎪ ⎪ ⎩   ⇒  − {¬B1 }|B2 ⇒   ⇒  − {¬B1 , B2 }   ⇒ |¬(B1 ⊗ B2 ) ⇒   ⇒  − {¬(B1 ⊗ B2 )}

Definition 2.4.5 A reduction δ is provable in R⊕⊗ , denoted by ⊕⊗ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R⊕⊗ . Theorem 2.4.6 (Soundness and completeness theorem) For any reduction δ = (A, B)| ⇒  ⇒   ⇒  , ⊕⊗ δ iff |=⊕⊗ δ. 

2.4.4 R-Calculus R⊕⊗ A reduction δ = (A, B)| →  ⇒   →  is R⊕⊗ -valid, denoted by |=⊕⊗ δ, if 

, A if , A →  is G⊕⊗ -valid otherwise  , B if   → , B is G⊕⊗ -valid   =  otherwise.  =

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R-calculus R⊕⊗ consists of the following axioms and deduction rules: • Axioms:

¬l ∈  or l ∈   → |(l, m) ⇒  → |m ¬l ∈ / &l ∈ / L (A+ )  → |(l, m) ⇒ , l → |m m ∈   or ¬m ∈  R (A0 )   → |m ⇒   →  / m∈ /   &¬m ∈ (A+R )   → |m ⇒   → , m

(A0L )

where ,  are sets of literals, and l, m are literals. • Deduction rules:  → |(A, B) ⇒  → |B  → |(¬¬A, B) ⇒  → |B L   → |(A, B) ⇒ , A  → |B ) (¬¬+  → |(¬¬A, B) ⇒ , ¬¬A → |B    → |B ⇒   →  (¬¬0R )   → |¬¬B ⇒   →    → |B ⇒   → , B (¬¬+R )  → |¬¬B ⇒  → , ¬¬B (¬¬0L )

⎡

and (⊕0L )

L (⊕+ )

(⊕0R )

(⊕+R )

 → |(¬A1 , B) ⇒  → |B ⎢ , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 → |B ⎢ ⎣  → |(A1 , B) ⇒  → |B , A1 → |(A2 , B) ⇒ , A1 → |B   → ⎧  |(A1 ⊕ A2 , B) ⇒  → |B  → |(¬A1 , B) ⇒ , ¬A1 → |B ⎪ ⎪ ⎨  , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 , ¬A2 → |B  → |(A1 , B) ⇒ , A1 → |B ⎪ ⎪ ⎩ , ¬A1 → |(A2 , B) ⇒ , A1 , A2 → |B   → , A1 ⊕ A2 → |B ⎡  |(A1 ⊕ A2 , B) ⇒  → |¬B1 ⇒   →  ⎢   → , ¬B1 |B2 ⇒   → , ¬B1 ⎢  ⎣  → |B1 ⇒   →    → , B1 |¬B2 ⇒   → , B1    ⎧ → |B1 ⊕ B2 ⇒   →   → |¬B1 ⇒  → , ¬B1 ⎪ ⎪ ⎨      → , ¬B1 |B2 ⇒  → , ¬B1 , B2  → |B1 ⇒  → , B1 ⎪ ⎪ ⎩   → , B1 |¬B2 ⇒   → , B1 , ¬B2   → |B1 ⊕ B2 ⇒   → , B1 ⊕ B2

2.4 R-Calculi for Variant Propositional Logics

⎡

and (¬⊕0L )

L ) (¬⊕+

(¬⊕0R )

(¬⊕+R )

 → |(¬A1 , B) ⇒  → |B ⎢ , ¬A1 → |(A2 , B) ⇒ , ¬A1 → |B ⎢ ⎣  → |(A1 , B) ⇒  → |B , A1 → |(¬A2 , B) ⇒ , A1 → |B  ⎧ →  |(¬(A1 ⊕ A2 ), B) ⇒  → |B ⎪  → |(¬A1 , B) ⇒ , ¬A1 → |B ⎪ ⎨  , ¬A1 → |(A2 , B) ⇒ , ¬A1 , A2 → |B ⎪  → |(A1 , B) ⇒ , A1 → |B ⎪ ⎩ , A1 → |(¬A2 , B) ⇒ , A1 , ¬A2 → |B   → ⎡  |(¬(A1 ⊕ A2 ), B) ⇒ B|, ¬(A1 ⊕ A2 ) →   → |¬B1 ⇒  →  ⎢   → , ¬B1 |¬B2 ⇒   → , ¬B1 ⎢  ⎣  → |B1 ⇒   →    → , B1 |B2 ⇒   → , B1    ⎧ → |¬(B1 ⊕ B2 ) ⇒  →   → |¬B1 ⇒  → , ¬B1 ⎪ ⎪ ⎨      → , ¬B1 |¬B 2 ⇒  → , ¬B1 , ¬B2  → |B1 ⇒  → , B1 ⎪ ⎪ ⎩   → , B1 |B2 ⇒   → , B1 , B2   → |¬(B1 ⊕ B2 ) ⇒   → , ¬(B1 ⊕ B2 ) ⎡

and (⊗0L )

L (⊗+ )

(⊗0R )

(⊗+R )

 → |(¬A1 , B) ⇒  → |B ⎢ , ¬A1 → |(A2 , B) ⇒ , ¬A1 → |B ⎢ ⎣  → |(A1 , B) ⇒  → |B , A1 → |(¬A2 , B) ⇒ , A1 → |B   → ⎧  |(A1 ⊗ A2 , B) ⇒  → |B  → |(¬A1 , B) ⇒ , ¬A1 → |B ⎪ ⎪ ⎨  , ¬A1 → |(A2 , B) ⇒ , ¬A1 , A2 → |B  → |(A1 , B) ⇒ , A1 → |B ⎪ ⎪ ⎩ , A1 → |(¬A2 , B) ⇒ , A1 , ¬A2 → |B   → , A1 ⊗ A2 → |B ⎡  |(A1 ⊗ A2 , B) ⇒  → |¬B1 ⇒   →  ⎢   → , ¬B1 |¬B2 ⇒   → , ¬B1 ⎢  ⎣  → |B1 ⇒   →    → , B1 |B2 ⇒   → , B1    ⎧ → |B1 ⊗ B2 ⇒   →   → |¬B1 ⇒  → , ¬B1 ⎪ ⎪ ⎨      → , ¬B1 |¬B 2 ⇒  → , ¬B1 , ¬B2  → |B1 ⇒  → , B1 ⎪ ⎪ ⎩   → , B1 |B2 ⇒   → , B1 , B2   → |B1 ⊗ B2 ⇒   → , B1 ⊗ B2

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⎡

and (¬⊗0L )

L ) (¬⊗+

(¬⊗0R )

(¬⊗+R )

 → |(A1 , B) ⇒  → |B ⎢ , A1 → |(A2 , B) ⇒ , A1 → |B ⎢ ⎣  → |(¬A1 , B) ⇒  → |B , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 → |B  ⎧ →  |(¬(A1 ⊗ A2 ), B) ⇒  → |B ⎪  → |(A1 , B) ⇒ , A1 → |B ⎪ ⎨  , A1 → |(A2 , B) ⇒ , A1 , A2 → |B ⎪  → |(¬A1 , B) ⇒ , ¬A1 → |B ⎪ ⎩ , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 , ¬A2 → |B   → B) ⇒ , ¬(A1 ⊗ A2 ) → |B ⎡  |(¬(A1 ⊗ A2 ),  → |B1 ⇒   →  ⎢   → , B1 |¬B2 ⇒   → , B1 ⎢  ⎣  → |¬B1 ⇒   →    → , ¬B1 |B2 ⇒   → , ¬B1    ⎧ → |¬(B1 ⊗ B2 ) ⇒  →   → |B1 ⇒  → , B1 ⎪ ⎪ ⎨    → , B1 , ¬B2    → , B1 |¬B2 ⇒   → |¬B1 ⇒  → , ¬B1 ⎪ ⎪ ⎩   → , ¬B1 |B2 ⇒   → , ¬B1 , B2   → |¬(B1 ⊗ B2 ) ⇒   → , ¬(B1 ⊗ B2 )

Definition 2.4.7 A reduction δ is provable in R⊕⊗ , denoted by ⊕⊗ δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in R⊕⊗ . Theorem 2.4.8 (Soundness and completeness theorem) For any reduction δ = (A, B)| →  ⇒   →  , ⊕⊗ δ iff |=⊕⊗ δ.



References Li, W.: Mathematical logic, foundations for information science. In: Progress in Computer Science and Applied Logic, vol. 25, Birkhäuser (2010) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Li, W., Sui, Y.: An R-calculus for the propositional logic programming. In: Proceedings of International Conference on Computer Science and Information Technology, pp. 863–870 (2014) Mendelson, E.: Introduction to Mathematical Logic. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California (1964) Takeuti, G.: Proof theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987)

Chapter 3

R-Calculi for Tableau/Gentzen Deduction Systems

A logic consists of a logical language, syntax and semantics, where the logical language specifies what symbols can be used in the logic, and the symbols are decomposed into two classes: logical and nonlogical, where the logical symbols are the ones used in each language of the logic, and the nonlogical symbols are those which are different for different logical languages of the logic; the syntax specifies what strings of symbols are meaningful (formulas) in logic, and the semantics specifies the truth-values of formulas under an assignment (or a model). In propositional logic, there is no nonlogical symbol, because variables are logical. Hence, there is only one logical language of propositional logic. There are several choices of a set of logical connectives: {¬, ∧}, {¬, ∨}, {¬, →}, {¬, ∧, ∨}, {¬, ∧, ∨, →, ↔}. We choose {¬, ∧, ∨} as the set of logical connectives, and other connectives are defined as follows. A → B = ¬A ∨ B A ↔ B = (A → B) ∧ (B → A) = (¬A ∨ B) ∧ (¬B ∨ A). The semantics of propositional logic is defined by assignments, functions from variables to the {0, 1}-values. Li (2007) proposed a belief revision operator called R-calculus R in first-order logic, which satisfies the AGM postulates and is nonmonotonic (Clark 1987; Ginsberg 1987; Reiter 1980). These deduction rules in R-calculus are reduced to the following two deduction rules: (0)

  ¬A |A ⇒ 

(+)

  ¬A , |A ⇒ , A

© Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_3

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where (0) says that if A is inconsistent with  then the result of  revising A is ; and if A is consistent with  then the result of  revising A is  ∪ {A}. Therefore, R-calculus is monotonic by (−) and nonmonotonic by rule (+). A sequent  ⇒  is valid (Li 2010; Takeuti and Barwise 1987) if for any assignment v, v satisfying each formula in  implies v satisfying some formula in , equivalently, either v satisfies the negation of some formula in  or v satisfies some formula in . Dually, a co-sequent  →  is valid if there is an assignment v such that v satisfies each formula in  and the negation of each formula in . |=  ⇒  iff Av(EA ∈ (v(A) = 0) ∨ EB ∈ (v(B) = 1)) |=  →  iff Ev(AA ∈ (v(A) = 1)&AB ∈ (v(B) = 0)). When  is empty, sequent  ⇒ is valid iff  is f-valid; and when  is empty, sequent ⇒  is valid iff  is t-valid, where  is f-valid iff AvEA ∈ (v(A) = 0)  is t-valid iff AvEB ∈ (v(B) = 1). When  is empty, co-sequent  → is valid iff  is t-consistent; and when  is empty, co-sequent →  is valid iff  is f-consistent.  is t-consistent iff EvAA ∈ (v(A) = 1)  is f-consistent iff EvAB ∈ (v(B) = 0). tval(), fval() are monotonic in ; and tcon(), fcon() are nonmonotonic in . That is,  ⊆  & fval() ⇒ fval( )  ⊆  & tval() ⇒ tval( )  ⊆  & tcon( ) ⇒ tcon()  ⊆  & fcon( ) ⇒ fcon() There are sound and complete Gentzen deduction systems G and G for sequents and co-sequents, that is:   ⇒  iff |=  ⇒    →  iff |=  →  where G is monotonic and G nonmonotonic (Cao et al. 2016; Reiter 1980). A sequent  ⇒  is decomposed into two sequents  ⇒ and ⇒ ; and a cosequent  →  into two co-sequents  → and → . For these sequents and cosequents, we have four tableau proof systems in propositional logic (Hähnle 2001; Malinowski 2009; Urquhart and Gabbay 2001):

3 R-Calculi for Tableau/Gentzen Deduction Systems

• • • •

Tf Tt Tt Tf

55

:  ⇒ is Tf -provable iff  is f-valid; : ⇒  is Tt -provable iff  is t-valid; :  → is Tt -provable iff  is t-consistent; : →  is Tf -provable iff →  is f-consistent.

That is,

f  ⇒ t ⇒  t  → f → 

iff iff iff iff

|=f  ⇒ |=t ⇒  |=t  → |=f → ,

where f / t is complementary to t / f , respectively. G is complementary to G . Generally, we define Gt Gf Gt Gf

:|=t :|=f :|=t :|=f

   

⇒  iff Av(EA ∈ (v(A) = 0) or EB ∈ (v(B) = 1)) ⇒  iff Av(EA ∈ (v(A) = 1) or EB ∈ (v(B) = 0)) →  iff Ev(AA ∈ (v(A) = 1)&AB ∈ (v(B) = 0)) →  iff Ev(AA ∈ (v(A) = 0)&AB ∈ (v(B) = 1)).

Then, G = Gt and G = Gt . G is taken as a combination of Tf and Tt , and G a combination of Tt and Tf , denoted by G = Tf  Tt , Gf = Tt  Tf , G = Tt  Tf , Gf = Tf  Tt . For each tableau proof system, there is a corresponding R-calculus. The four R-calculi are classified into two classes: preserving t/f-validity or preserving t/fsatisfiability. Let ∗ ∈ {t, f}. • For T∗ -validity, there is an R-calculus S∗ to preserve the ∗-validity, that is, for any theory  and formulas A ∈ , (i) |A ⇒  is provable in S∗ if and only if  − {A} is not ∗-valid. (ii) |A ⇒  − {A} is provable in S∗ if and only if  − {A} is ∗-valid. • For T∗ -validity, there is an R-calculus S∗ to preserve the ∗-satisfiability, that is, for any theory  and formula B, (i) |B ⇒  is provable in S∗ if and only if  ∪ {B} is not ∗-consistent. (ii) |B ⇒ , B is provable in S∗ if and only if  ∪ {B} is ∗-consistent. These R-calculi are reduced to the following rules: • for Sf : for A ∈ , (0) • for St : for B ∈ ,

fval( − {A}) finval( − {A}) (−) |A ⇒  |A ⇒  − {A}

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(0)

tval( − {B}) tinval( − {B}) (−) |B ⇒  |B ⇒  − {B}

• for St : (+)

tincon(, A) tcon(, A) (0) |A ⇒ , A |A ⇒ 

(+)

fincon() fcon(, B) (0) |B ⇒ , B |B ⇒ 

• for Sf :

Let ∗ ∈ {t, f}. For sequents and co-sequents, there are R-calculi R∗ and R∗ , respectively, such that R∗ is a combination of Sf and St , and R∗ a combination of Sf and St , that is, Rt = Sf  St Rf = St  Sf Rt = St  Sf Rf = Sf  St .

3.1 Tableau Proof Systems Let a logical language of propositional logic contains the following symbols: • variables: p0 , p1 , . . . ; and • logical connectives: ¬, ∧, ∨. A formula A is a string of the following forms: A ::= p|¬A1 |A1 ∧ A2 |A1 ∨ A2 , where l ::= p|¬p is called a literal. An assignment v is a function from variables to B2 = {0, 1}. The truth-value v(A) of formula A under assignment v is ⎧ v(p) ⎪ ⎪ ⎨ 1 − v(A1 ) v(A) = min{v(A ⎪ 1 ), v(A2 )} ⎪ ⎩ max{v(A1 ), v(A2 )}

if A = p if A = ¬A1 if A = A1 ∧ A2 if A = A1 ∨ A2 .

A formula A is satisfied in v, denoted by v |= A, if v(A) = 1; and A is valid, denoted by |= A, if A is satisfied in any assignment v. Lemma 3.1.1 The following equivalences hold for v(A) = 0 :

3.1 Tableau Proof Systems

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A1 ∧ A2 ≡ A1 ∨A2 A1 ∨ A2 ≡ A1 ∧A2 ¬(A1 ∧ A2 ) ≡ ¬A1 ∧¬A2 ¬(A1 ∨ A2 ) ≡ ¬A1 ∨¬A2 Proof For any assignment v, (i) v(A1 ∧ A2 ) = 0 iff either v(A1 ) = 0 or v(A2 ) = 0; (ii) v(A1 ∨ A2 ) = 0 iff v(A1 ) = 0 and v(A2 ) = 0; (iii) v(¬(A1 ∧ A2 )) = 0 iff v(A1 ∧ A2 ) = 1, iff v(A1 ) = 1 and v(A2 ) = 1, iff v (¬A1 ) = 0 and v(¬A2 ) = 0; (iv) v(¬(A1 ∨ A2 )) = 0 iff v(A1 ∨ A2 ) = 1, iff either v(A1 ) = 1 or v(A2 ) = 1, iff   either v(¬A1 ) = 0 or v(¬A2 ) = 0. Lemma 3.1.2 The following equivalences hold for v(B) = 1: B1 ∧ B2 ≡ B1 ∧B2 B1 ∨ B2 ≡ B1 ∨B2 ¬(B1 ∧ B2 ) ≡ ¬B1 ∨¬B2 ¬(B1 ∨ B2 ) ≡ ¬B1 ∧¬B2 where ∧, ∨ are ∧, ∨ in semantics. Proof For any assignment v, (i) v(B1 ∧ B2 ) = 1 iff v(B1 ) = 1 and v(B2 ) = 1; (ii) v(B1 ∨ B2 ) = 1 iff either v(B1 ) = 1 or v(B2 ) = 1; (iii) v(¬(B1 ∧ B2 )) = 1 iff v(B1 ∧ B2 ) = 0, iff either v(B1 ) = 0 or v(B2 ) = 0, iff either v(¬B1 ) = 1 or v(¬B2 ) = 1; (iv) v(¬(B1 ∨ B2 )) = 1 iff v(B1 ∨ B2 ) = 0, iff v(B1 ) = 0 and v(B2 ) = 0, iff v   (¬B1 ) = 1 and v(¬B2 ) = 1. v(A) = 1 iff v(A) = 0, and v(A) = 0 iff v(A) = 1. Hence, we have the following equivalences (i) for v(A) = 1: A1 ∧ A2 ∼ = A1 ∨A2 A1 ∨ A2 ∼ = A1 ∧A2 ∼ ¬(A1 ∧ A2 ) = ¬A1 ∧¬A2 ¬(A1 ∨ A2 ) ∼ = ¬A1 ∨¬A2 (ii) for v(B) = 0:

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B1 ∧ B2 ∼ = B1 ∨ B2 ∼ = ∼ ¬(B1 ∧ B2 ) = ¬(B1 ∨ B2 ) ∼ =

B1 ∧B2 B1 ∨B2 ¬B1 ∨¬B2 ¬B1 ∧¬B2

A theory / is • Tf -valid, denoted by |=f  ⇒, if for any assignment v, there is a formula A ∈  such that v(A) = 0; • Tt -valid, denoted by |=t ⇒ , if for any assignment v, there is a formula B ∈  such that v(B) = 1. • Tt -valid, denoted by |=t  ⇒, if there is an assignment v such that for every formula A ∈ , v(A) = 1; • Tf -valid, denoted by |=f ⇒ , if there is an assignment v such that for every formula B ∈ , v(B) = 0. |=f  ⇒ iff AvEA ∈ (v(A) = 0) |=t ⇒  iff AvEB ∈ (v(B) = 1) |=t  ⇒ iff EvAC ∈ (v(C) = 1) |=f ⇒  iff EvAD ∈ (v(D) = 0).

3.1.1 Tableau Proof System Tf A sequent  ⇒ is Tf -valid, denoted by |=f  ⇒, if for any assignment v, there is a formula A ∈  such that v(A) = 0. That is,  ⇒ is valid if and only if  is unsatisfiable; that is, there is no assignment v such that v(A) = 1 for each formula A ∈ . Tableau proof system Tf consists of the following axiom and deductions: • Axiom: (Af )

Ep(p, ¬p ∈ ) , ⇒

where  is a set of literals. • Deduction rules: , A ⇒ (¬¬L ) ,  ¬¬A ⇒  , A1 ⇒ , ¬A1 ⇒ (∧L ) , A2 ⇒ (¬∧L ) , ¬A2 ⇒ , ,  A1 ∧ A2 ⇒  ¬(A1 ∧ A2 ) ⇒ , A1 ⇒ , ¬A1 ⇒ (∨L ) , A2 ⇒ (¬∨L ) , ¬A2 ⇒ , A1 ∨ A2 ⇒ , ¬(A1 ∨ A2 ) ⇒

3.1 Tableau Proof Systems

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Definition 3.1.3 A sequent  ⇒ is provable in Tf , denoted by f  ⇒, if there is a sequence 1 ⇒, . . . , n ⇒} of sequents such that n = , and for each 1 ≤ i ≤ n, i ⇒ is an axiom or is deduced from the previous sequents by one of the deduction rules in Tf . Theorem 3.1.4 (Soundness theorem) For any sequent  ⇒, if f  ⇒ then |=f ⇒. Proof We prove that each axiom is valid and each deduction rule preserves the validity. (Af ) Assume that there is a variable p such that p, ¬p ∈ . There is no assignment v such that v |= . Then, |=  ⇒ . (¬¬) Assume that there is no assignment v such that v |= , A1 ⇒ . Then, there is no assignment v such that v |= , ¬¬A1 ⇒ . (∧) Assume that there is no assignment v such that v |= , A1 ⇒ . Then, there is no assignment v such that v |= , A1 ∧ A2 ⇒ . (¬∧) Assume that there is no assignment v such that v |= , ¬A1 ⇒; and there is no assignment v such that v |= , ¬A2 ⇒ . Then, there is no assignment v such that v |= , ¬(A1 ∧ A2 ) ⇒ . (∨) Assume that there is no assignment v such that v |= , A1 ⇒; and there is no assignment v such that v |= , A2 ⇒ . Then, there is no assignment v such that v |= , A1 ∨ A2 ⇒ . (¬∨) Assume that there is no assignment v such that v |= , ¬A1 ⇒ . Then, there   is no assignment v such that v |= , ¬(A1 ∨ A2 ) ⇒ . Theorem 3.1.5 (Completeness theorem) For any sequent  ⇒, if |=f  ⇒ then f  ⇒ . Proof Given a sequent  ⇒, we construct a tree T such that either (i) for each branch ξ of T , there is a sequent  ⇒ at the leaf of ξ such that  ⇒ is an axiom, or (ii) there is an assignment v such that v |= . T is constructed as follows: • the root of T is  ⇒; • for a node ξ , if the sequent  ⇒ at ξ is literal then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following sequents: ⎧  ⎪ ⎪  1, A ⇒ ⎪ ⎪ 1 , A1 ⇒ ⎪ ⎪ ⎪ ⎪  ⎪ 1 , A2 ⇒ ⎪ ⎪ ⎪ ⎨ 1 , A1 ⇒  1 , A2 ⇒ ⎪ ⎪ 1 , ¬A1 ⇒ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , ¬A2 ⇒ ⎪ ⎪ 1 , ¬A1 ⇒ ⎪ ⎪ ⎩ 1 , ¬A2 ⇒

if 1 , ¬¬A ⇒∈ ξ if 1 , A1 ∧ A2 ⇒∈ ξ if 1 , A1 ∨ A2 ⇒∈ ξ if 1 , ¬(A1 ∧ A2 ) ⇒∈ ξ if 1 , ¬(A1 ∨ A2 ) ⇒∈ ξ

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 δ1 δ represents that δ1 , δ2 are at a same child node; and 1 represents that δ2 δ2 δ1 , δ2 are at different direct child nodes.  1 , A1 ⇒ is equivalent to 1 , A1 , A2 ⇒ . Hence, there is one and only Notice that 1 , A2 ⇒ one sequent at each node of the tree. where

Lemma 3.1.6 If for each branch ξ ⊆ T , there is a sequent  ⇒∈ ξ which is an axiom in Tf then T is a proof tree of  ⇒ . Proof By the definition of T , T is a proof tree of  ⇒ .

 

Lemma 3.1.7 If there is a branch ξ ⊆ T such that each sequent  ⇒∈ ξ is not an axiom in Tf then there is an assignment v such that v |= . Proof Let ξ be a branch of T such that each sequent  ⇒∈ ξ is not an axiom in Tf . Let  = {A : A ∈  ⇒∈ ξ }. Define an assignment v as follows: v(p) = 1 iff p ∈ . Then, v is well-defined, and for the sequent  ⇒ at the leaf node of ξ, v |=  . We prove by induction on nodes η of ξ that for each sequent  ⇒ at η, v |=  . Case  ⇒= 2 , ¬¬A ⇒∈ η. Then,  ⇒ has a child node ∈ ξ containing 2 , A ⇒ . By induction assumption, v |= 2 , A, which implies that v |= 2 , ¬¬A. Case  ⇒= 2 , A1 ∧ A2 ⇒∈ η. Then,  ⇒ has a child node ∈ ξ containing 2 , A1 ⇒ and 2 , A2 ⇒ . By induction assumption, v |= 2 , A1 and v |= 2 , A2 , i.e., v |= 2 , A1 ∧ A2 . Case  ⇒= 2 , A1 ∨ A2 ⇒∈ η. Then,  ⇒ has a child node ∈ ξ containing 2 , Ai ⇒ . By induction assumption, v |= 2 , Ai . Hence, v |= 2 , A1 ∨ A2 . Similar for other cases.  

3.1.2 Tableau Proof System Tt A sequent ⇒  is Tt -valid, denoted by |=t ⇒ , if for each assignment v, there is a B ∈  such that v(B) = 1. That is, ⇒  is valid if and only if ¬ is unsatisfiable. Tableau proof system Tt consists of the following axiom and deductions: • Axiom: (At ) where  is a set of literals.

Eq(q, ¬q ∈ ) , ⇒

3.1 Tableau Proof Systems

61

• Deduction rules: ⇒ B,  ⇒   ¬¬B,  ⇒ B1 ,  ⇒ ¬B1 ,  (∧R ) ⇒ B2 ,  (¬∧R ) ⇒ ¬B2 ,  ⇒ ⇒  B1 ∧ B2 ,   ¬(B1 ∧ B2 ),  ⇒ B1 ,  ⇒ ¬B1 ,  (∨R ) ⇒ B2 ,  (¬∨R ) ⇒ ¬B2 ,  ⇒ B1 ∨ B2 ,  ⇒ ¬(B1 ∨ B2 ),  (¬¬R )

Definition 3.1.8 A sequent ⇒  is provable in Tt , denoted by t ⇒  if there is a sequence {⇒ 1 , . . . , ⇒ n } of sequents such that n = , and for each 1 ≤ i ≤ n, ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in Tt . Theorem 3.1.9 (Soundness theorem) For any sequent ⇒ , if t ⇒  then |=t ⇒ . Proof We prove that each axiom is valid and each deduction rule preserves the validity. (At ) Assume that there is a variable q such that q, ¬q ∈ . Then, for any assignment v, v |= . Then, |=⇒ . (∧) Assume that for any assignment v, v |=⇒ B1 ,  and v |=⇒ B2 , . Then, for any assignment v, v |=⇒ B1 ∧ B2 , . (∨) Assume that for any assignment v, v |=⇒ Bi , . Then, for any assignment v, v |=⇒ B1 ∨ B2 , . (¬∧) Assume that for any assignment v, v |=⇒ ¬Bi , . Then, for any assignment v, v |=⇒ ¬(B1 ∧ B2 ), . (¬∨) Assume that for any assignment v, v |=⇒ ¬B1 ,  and v |=⇒ ¬B2 , .   Then, for any assignment v, v |=⇒ ¬(B1 ∨ B2 ), . Theorem 3.1.10 (Completeness theorem) For any sequent ⇒ , if |=t ⇒  then t ⇒ . Proof Given a sequent ⇒ , we construct a tree T such that either (i) for each branch ξ of T , the sequent ⇒  at the leaf of ξ is an axiom, or (ii) there is an assignment v such that v |= . T is constructed as follows: • the root of T is ⇒ ; • for a node ξ , if the sequent ⇒  at ξ is literal then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following sequents:

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⎧ ⇒ ⎪ ⎪  B, 1 ⎪ ⎪ ⇒ B1 , 1 ⎪ ⎪ ⎪ ⎪ ⇒ B2 , 1 ⎪ ⎪  ⎪ ⎪ ⎨ ⇒ B1 , 1  ⇒ B2 , 1 ⎪ ⎪ ⇒ ¬B1 , 1 ⎪ ⎪ ⎪ ⎪ ⇒ ⎪ ⎪  ¬B2 , 1 ⎪ ⎪ ⇒ ¬B1 , 1 ⎪ ⎪ ⎩ ⇒ ¬B2 , 1

if ⇒ ¬¬B, 1 ∈ ξ if ⇒ B1 ∧ B2 , 1 ∈ ξ if ⇒ B1 ∨ B2 , 1 ∈ ξ if ⇒ ¬(B1 ∧ B2 ), 1 ∈ ξ if ⇒ ¬(B1 ∨ B2 ), 1 ∈ ξ



⇒ B1 , 1 are taken as one sequent ⇒ B1 , B2 , 1 . Hence, ⇒ B2 , 1 there is one and only one sequent at each node of tree T .

Notice that two sequents

Lemma 3.1.11 If for each branch ξ ⊆ T , there is a sequent ⇒  ∈ ξ which is an axiom in Tt then T is a proof tree of ⇒ . Proof By the definition of T , T is a proof tree of ⇒ .

 

Lemma 3.1.12 If there is a branch ξ ⊆ T such that each sequent ⇒  ∈ ξ is not an axiom in Tt then there is an assignment v such that v |= . Proof Let ξ be a branch of T such that each sequent ⇒  ∈ ξ is not an axiom in Tt . Let  = {B : B ∈⇒  ∈ ξ }. Define an assignment v as follows: v(q) = 0 iff q ∈ . Then, v is well-defined, and for the sequent ⇒  at the leaf node of ξ, v |=  . We prove by induction on nodes η of ξ that for each sequent ⇒  at η, v |=  . Case ⇒  =⇒ ¬¬B, 2 ∈ η. Then, ⇒  has a child node ∈ ξ containing ⇒ B, 2 . By induction assumption, v |= B, 2 , which implies that v |= ¬¬B, 2 . Case ⇒  =⇒ B1 ∧ B2 , 2 ∈ η. Then, ⇒  has a child node ∈ ξ containing ⇒ Bi , 2 . By induction assumption, v |= Bi , 2 , i.e., v |= B1 ∧ B2 , 2 . Case ⇒  =⇒ B1 ∨ B2 , 2 ∈ η. Then, ⇒  has a child node ∈ ξ containing ⇒ B1 , 2 and ⇒ B2 , 2 . By induction assumption, v |= B1 , 2 and v |= B2 , 2 , i.e., v |= B1 ∨ B2 , 2 . Case ⇒  =⇒ ¬(B1 ∧ B2 ), 2 ∈ η. Then, ⇒  has a child node ∈ ξ containing ⇒ ¬B1 , 2 and ⇒ ¬B2 , 2 . By induction assumption, v |= ¬B1 , 2 and v |= ¬B2 , 2 , i.e., v |= ¬(B1 ∧ B2 ), 2 . Case ⇒  =⇒ ¬(B1 ∨ B2 ), 2 ∈ η. Then, ⇒  has a child node ∈ ξ containing ⇒ ¬Bi . By induction assumption, v |= ¬Bi , 2 , i.e., v |= ¬(B1 ∨ B2 ),   2 .

3.1 Tableau Proof Systems

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3.1.3 Tableau Proof System Tt A co-sequent  → is Tt -valid, denoted by |=  →, if there is an assignment v such that for each formula A ∈ , v |= A. Tableau proof system Tt consists of the following axiom and deductions: • Axiom: (At )

∼ Ep(p, ¬p ∈ ) ,  →

where  is a set of literals. • Deduction rules: , A → (¬¬L ) ,  ¬¬A →  , A1 → , ¬A1 → (∧L ) , A2 → (¬∧L ) , ¬A2 → , ,  A1 ∧ A2 →  ¬(A1 ∧ A2 ) → , A1 → , ¬A1 → (∨L ) , A2 → (¬∨L ) , ¬A2 → , A1 ∨ A2 → , ¬(A1 ∨ A2 ) → Definition 3.1.13 A co-sequent  → is Tt -provable, denoted by t  →, if there is a sequence {1 →, . . . , n →} of co-sequents such that n →=  →, and for each 1 ≤ i ≤ n, i → is an axiom or is deduced from the previous co-sequents by one of the deduction rules. Theorem 3.1.14 (Soundness theorem) For any co-sequent  →, if t  → then |=t  → . Proof We prove that each axiom is valid and each deduction rule preserves the validity. (At ) Assume that ∼ Ep(p, ¬p ∈ ). Then, there is an assignment v such that v |= , and defined as follows: for any variable p, v(p) = 1 iff p ∈ . Then, v is well-defined and v |=  ⇒ . (∧) Assume that there is an assignment v such that v |= , A1 ⇒ and v |= , A2 ⇒. Then, for this assignment v, v |= , A1 ∧ A2 ⇒ . (∨) Assume that there is an assignment v such that v |= , Ai ⇒ . Then, for this assignment v, v |= , A1 ∨ A2 ⇒ . (¬∧) Assume that there is an assignment v such that v |= , ¬Ai ⇒ . Then, for this assignment v, v |= , ¬(A1 ∧ A2 ) ⇒ . (¬∨) Assume that there is an assignment v such that v |= , ¬A1 ⇒ and v |=   , ¬A2 ⇒ . Then, for this assignment v, v |= , ¬(A1 ∨ A2 ) ⇒ .

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Theorem 3.1.15 (Completeness theorem) For any co-sequent  →, if |=t  → then t  → . Proof Given a co-sequent  →, we construct a tree T such that either (i) for each branch ξ of T , the co-sequent  → at the leaf of ξ is an axiom in Tf , or (ii) there is no assignment v such that v |= . T is constructed as follows: • the root of T is  →; • for a node ξ , if the co-sequent  → at ξ is literal then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following co-sequents: ⎧ 1 , A →1 ⎪ ⎪ ⎪ ⎪ 1 , A1 →1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , A2 →1 ⎪ ⎪ ⎨ 1 , A1 →1  1 , A2 →1 ⎪ ⎪ 1 , ¬A1 →1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , ¬A2 →1 ⎪ ⎪ 1 , ¬A1 →1 ⎪ ⎪ ⎩ 1 , ¬A2 →1

if 1 , ¬¬A →1 ∈ ξ if 1 , A1 ∧ A2 →1 ∈ ξ if 1 , A1 ∨ A2 →1 ∈ ξ if 1 , ¬(A1 ∧ A2 ) →1 ∈ ξ if 1 , ¬(A1 ∨ A2 ) →1 ∈ ξ

Lemma 3.1.16 If there is a branch ξ ⊆ T such that  →∈ ξ is an axiom in Tt then ξ is a proof of  → . Proof By the definition of T , ξ is a proof of  → .

 

Lemma 3.1.17 If for each branch ξ ⊆ T the co-sequent  →∈ ξ is not an axiom in Tt then T is a proof of sequent  ⇒ in Tf . Proof We prove by induction on nodes η of T that for any co-sequent  → at η, f  ⇒ . Case  → being literal. By assumption,  → is not an axiom in Tt , and then  ⇒ is an axiom in Tf . Case  →= 2 , ¬¬A →∈ η. Then,  → has a child node containing 2 , A → . By induction assumption, f 2 , A ⇒, and by (¬¬L ) ∈ Tf , f 2 , ¬¬A ⇒ . Case  →= 2 , A1 ∧ A2 →∈ η. Then,  → has a child node containing 2 , A1 , A2 → . By induction assumption, f 2 , A1 , A2 ⇒, and by (∧L ) ∈ Tf , i.e., f 2 , A1 ∧ A2 ⇒ . Case  →= 2 , ¬(A1 ∧ A2 ) →∈ η. Then,  → has child node containing 2 , ¬A1 → or 2 , ¬A2 → . By induction assumption, f 2 , ¬A1 ⇒, f 2 , ¬ A2 ⇒, and by (¬∧L ) ∈ Tf , i.e., f 2 , ¬(A1 ∧ A2 ) ⇒ . Similar for other cases.  

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3.1.4 Tableau Proof System Tf A co-sequent →  is Tf -valid, denoted by |=f → , if there is an assignment v such that for each formula B ∈ , v |= B. Tableau proof system Tf consists of the following axiom and deductions: • Axiom: (Af )

∼ Ep(p, ¬p ∈ ) , → 

where  is a set of literals. • Deduction rules: → B,  →   ¬¬B,  → B1 ,  → ¬B1 ,  (∧R ) → B2 ,  (¬∧R ) → ¬B2 ,   → B1 ∧ B2 ,   → ¬(B1 ∧ B2 ),  → B1 ,  → ¬B1 ,  (∨R ) → B2 ,  (¬∨R ) → ¬B2 ,  → B1 ∨ B2 ,  → ¬(B1 ∨ B2 ),  (¬¬R )

Definition 3.1.18 A co-sequent →  is provable in Tf , denoted by f →  if there is a sequence { → 1 , . . . , → n } of co-sequents such that n = , and for each 1 ≤ i ≤ n, → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules. Theorem 3.1.19 (Soundness theorem) For any co-sequent → , if f →  then   |=f → . Theorem 3.1.20 (Completeness theorem) For any co-sequent → , if |=f →  then f → . Proof Given a co-sequent → , we construct a tree T such that either (i) for each branch ξ of T , the co-sequent →  at the leaf of ξ is an axiom in Tt , or (ii) there is a branch ξ which is a proof of ⇒ . T is constructed as follows: • the root of T is → ; • for a node ξ , if the co-sequent →  at ξ is literal then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following co-sequents:

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⎧ → B, 1 ⎪ ⎪  ⎪ ⎪ → B1 , 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  → B2 , 1 ⎪ ⎪ ⎨ → B1 , 1  → B2 , 1 ⎪ ⎪ → ¬B1 , 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  → ¬B2 , 1 ⎪ ⎪ → ¬B1 , 1 ⎪ ⎪ ⎩ → ¬B2 , 1

if → ¬¬B, 1 ∈ ξ if → B1 ∧ B2 , 1 ∈ ξ if → B1 ∨ B2 , 1 ∈ ξ if → ¬(B1 ∧ B2 ), 1 ∈ ξ if → ¬(B1 ∨ B2 ), 1 ∈ ξ.

Lemma 3.1.21 If there is a branch ξ ⊆ T such that the co-sequent →  ∈ ξ is an axiom in Tf then ξ is a proof of → . Proof By the definition of T , ξ is a proof of → .

 

Lemma 3.1.22 If for each branch ξ ⊆ T , the co-sequent →  ∈ ξ is not an axiom in Tf then T is a proof of sequent ⇒  in Tt . Proof We prove by induction on nodes η of T that for each co-sequent →  at η, t ⇒  . Case →  being literal. By assumption, →  is not an axiom in Tf , and ⇒  is an axiom in Tt . Case →  = → ¬¬B, 2 ∈ η. Then, →  has a child node containing → A, 2 . By induction assumption, t ⇒ A, 2 , and by (¬¬R ) ∈ Tt , t ⇒ ¬¬B, 2 . Case →  = → B1 ∧ B2 , 2 ∈ η. Then, →  has children nodes containing → B1 , 2 and → B2 , 2 , respectively. By induction assumption, t ⇒ B1 , 2 and t ⇒ B2 , δ2 , and by (∧R ) ∈ Tt , i.e., t ⇒ B1 ∧ B2 , 2 . Case →  = → ¬(B1 ∧ B2 ), 2 ∈ η. Then, →  has a direct child node containing → ¬B1 , ¬B2 , 2 . By induction assumption, t ⇒ ¬B1 , ¬B2 , 2 , and by (¬∧R ) ∈ Tt , i.e., t ⇒ ¬(B1 ∧ B2 ), 2 . Similar for other cases.  

3.2 R-Calculi for Theories Let ∗ ∈ {t, f}. Given a theory X and formula X ∈ X, a reduction X|X ⇒ X − {X } is G∗ -valid, denoted by |=∗ X|X ⇒ X − {X }, if



X =

X if X − {X } is G∗ -valid λ otherwise.

Given a theory X and formula X , a reduction X|X ⇒ X, X is G∗ -valid, denoted by |=∗ X|X ⇒ X, X , if X =



X if X ∪ {X } is G∗ -valid λ otherwise.

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3.2.1 R-Calculus Sf Let A ∈ . A reduction |A ⇒  − {A } is Sf -valid, denoted by |=f |A ⇒  − {A }, if  A if |=f  − {A} A = λ otherwie. Intuitively, a formula A1 ∧ A2 is extractable from , if either A1 or A2 is extractable from ; and A1 ∨ A2 is extractable from  if A1 is extractable from  and A2 is extractable from  − {A1 }. A formula A1 ∧ A2 is not extractable from  if both A1 and A2 are not extractable from ; and A1 ∨ A2 is not extractable from  if either A1 is not extractable from  or A2 is not extractable from  − {A1 }. R-calculus Sf consists of the following axioms and deductions: • Axioms: (A− )

El = l(l , ¬l ∈ ) , |l ⇒  − {l}

(A0 )

∼ El = l(l , ¬l ∈ ) , |l ⇒ 

where  is a set of literals and l is a literal. • Deduction rules: |A ⇒  − {A} |A ⇒  (¬¬− ) (¬¬0 ) |¬¬A ⇒  − {¬¬A} |¬¬A ⇒   |A1 ⇒  − {A1 } |A1 ⇒  (∧− ) |A2 ⇒  − {A2 } (∧0 ) |A2 ⇒  |A |A  1 ∧ A2 ⇒  − {A1 ∧ A2 }  1 ∧ A2 ⇒  |A1 ⇒  − {A1 } |A1 ⇒  (∨− )  − {A1 }|A2 ⇒  − {A1 , A2 } (∨0 )  − {A1 }|A2 ⇒  − {A1 } |A1 ∨ A2 ⇒  − {A1 ∨ A2 } |A1 ∨ A2 ⇒  and



|¬A1 ⇒  − {¬A1 }  − {¬A1 }|¬A2 ⇒  − {¬A1 , ¬A2 } |¬(A1 ∧ A2 )⇒  − {¬(A1 ∧ A2 )} |¬A1 ⇒  (¬∧0 )  − {¬A1 }|¬A2 ⇒  − {¬A1 } |¬(A1 ∧ A2 ) ⇒   |¬A1 ⇒  − {¬A1 } (¬∨+ ) |¬A2 ⇒  − {¬A2 } |¬(A1 ∨ A2 ) ⇒  − {¬(A1 ∨ A2 )} |¬A1 ⇒  (¬∨− ) |¬A2 ⇒  |¬(A1 ∨ A2 ) ⇒ 

(¬∧− )

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Definition 3.2.1 A reduction |A ⇒ , A is provable in Sf , denoted by f |A ⇒  − {A }, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = |A ⇒  − {A }, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in Sf . Theorem 3.2.2 (Soundness theorems) For any reduction |A ⇒  − {A }, if f |A ⇒  then |= |A ⇒ ; and if f |A ⇒  − {A} then |= |A ⇒  − {A}. Proof We prove that each axiom is valid and each deduction rule preserves the validity. (A− ) Assume that  is a set of literals and there is a literal l = l such that l , ¬l ∈ . Then, f |l ⇒  − {l}, and |=f |l ⇒  − {l}, since  being Tf -valid implies  − {l} being Tf -valid. Assume that there is no such a literal l = l. Then, f |l ⇒ , and |=f |l ⇒ . (∧0 ) Assume that  − {A1 } being f-valid implies  being f-valid and  − {A2 } being f-valid implies  being f-valid. Then,  − {A1 ∧ A2 } being f-valid implies  being f-valid. (∧− ) Assume that  being f-valid implies  − {Ai } being f-valid. Then,  being f-valid implies  − {A1 ∧ A2 } being f-valid. (¬∧− ) Assume that |=f  implies |=f  − {¬A1 }; and |=f  − {¬A1 } implies |=f  − {¬A1 , ¬A2 }. Then, |=f  implies |=f  − {¬(A1 ∧ A2 )}. (¬∧0 ) Assume that |=f  implies |=f , and |=f  − {¬A1 } implies |=f  − {¬A1 }. Then, |=f  implies |=f . Similar for other cases.   Theorem 3.2.3 For any reduction |A ⇒ , A , if |= |A ⇒ , A then  |A ⇒ , A . Proof Given a reduction |A ⇒  − {A}, we construct a tree T such that either (i) T is a proof tree of |A ⇒  − {A}; i.e., for each branch ξ of T , the reduction γ at the leaf of ξ is an axiom of form (A− ), or (ii) there is a branch ξ of T such that ξ is a proof of |A ⇒ . T is constructed as follows: • the root of T is |A ⇒  − {A}; • for a node ξ , if each reduction at ξ is of form  |l ⇒  − {l } then the node is a leaf; • otherwise, ξ has the direct child containing the following reductions:

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69

⎧  ⎪  1 |A ⇒ 1 − {A} ⎪ ⎪ ⎪ 1 |A1 ⇒ 1 − {A1 } ⎪ ⎪ ⎪ ⎪ 1 |A2 ⇒ 1 − {A2 } ⎪ ⎪  ⎪ ⎪ ⎨ 1 |¬A1 ⇒ 1 − {¬A1 }  1 − {¬A1 }|¬A2 ⇒ 1 − {¬A1 , ¬A2 } ⎪ ⎪ 1 |A1 ⇒ 1 − {A1 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  1 − {A1 }|A2 ⇒ 1 − {A1 , A2 } ⎪ ⎪ ⎪ 1 |¬A1 ⇒ 1 − {¬A1 } ⎪ ⎩ 1 |¬A2 ⇒ 1 − {¬A2 }

if 1 (¬¬A) ∈ ξ if 1 (A1 ∧ A2 ) ∈ ξ if 1 (¬(A1 ∧ A2 )) ∈ ξ if 1 (A1 ∨ A2 ) ∈ ξ if 1 (¬(A1 ∨ A2 )) ∈ ξ.

where 1 (A) = 1 |A ⇒ 1 − {A}. Lemma 3.2.4 If for each branch ξ ⊆ T , there is a reduction  |A ⇒  |A ∈ ξ which is an axiom in Sf then T is a proof tree of |A ⇒ , A. Proof By the definition of T , T is a proof tree of |A ⇒ , A.

 

Lemma 3.2.5 If there is a branch ξ ⊆ T such that each reduction  |A ⇒  , A ∈ ξ is not an axiom in Sf , then ξ is a proof of |A ⇒  in Sf . Proof Let ξ be a branch of T such that each reduction  |A ⇒  , A ∈ ξ is not an axiom in Sf . We proved by induction on node η ∈ ξ that for each reduction  |A ⇒  , A ∈ η, f  |A ⇒  . Case  |l ⇒  , l ∈ η is a leaf node of ξ. Then, there is no literal l = l such that l , ¬l ∈  . By (Af0 ), f  |l ⇒  . Case δ = 2 |¬¬A ⇒ 2 , ¬¬A ∈ η. Then, η has a child node ∈ ξ containing reduction 2 |A ⇒ 2 , A. By induction assumption, f 2 |A ⇒ 2 , and by (¬¬0 ), f 2 |¬¬A ⇒ 2 . Case δ = 2 |A1 ∧ A2 ⇒ 2 , A1 ∧ A2 ∈ η. Then, η has a child node ∈ ξ containing reductions 2 |A1 ⇒ 2 , A1 and 2 |A2 ⇒ 2 , A2 . By induction assumption, f 2 |A1 ⇒ 2 , and f 2 |A2 ⇒ 2 . By (∧0 ), we have f 2 , A1 ∧ A2 ⇒ 2 . Case δ = 2 |A1 ∨ A2 ⇒ 2 , A1 ∨ A2 ∈ η. Then, η has a child node ∈ ξ containing reduction either 2 |A1 ⇒ 2 , A1 or 2 , A1 |A2 ⇒ 2 , A1 , A2 . By induction assumption, either f 2 |A1 ⇒ 2 or f 2 , A1 |A2 ⇒ 2 , A1 . By (∨0 ), we have f 2 , A1 ∨ A2 ⇒ . Similar for other cases.  

3.2.2 R-Calculus St Let B ∈ . A reduction |B ⇒  − {B } is St -valid, denoted by |=t |B ⇒  − {B }, if  B if |=t  − {B} B = λ otherwise.

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Intuitively, a formula B1 ∧ B2 is extractable from , if B1 is extractable from  and B2 is extractable from  − {B1 }; and B1 ∨ B2 is extractable from  if either B1 or B2 is extractable from . A formula B1 ∧ B2 is not extractable from  if either B1 is not extractable from  or B2 is not extractable from  − {B1 }; and B1 ∨ B2 is not extractable from  if both B1 and B2 are not extractable from . R-calculus St consists of the following axioms and deductions: • Axioms: (At− )

/ ) / ) Em = m(m , ¬m ∈ ∼ Em = m(m , ¬m ∈ , (At0 ) , |m ⇒  − {m} |m ⇒ 

where , m is literal. • Deduction rules: |B ⇒  − {B} |B ⇒  (¬¬− ) (¬¬0 )  |¬¬B ⇒  − {¬¬B}  |¬¬B ⇒  |B1 ⇒  − {B1 } |B1 ⇒  (∧− )  − {B1 }|B2 ⇒  − {B1 , B2 } (∧0 )  − {B1 }|B2 ⇒  − {B1 } |B |B  1 ∧ B2 ⇒  − {B1 ∧ B2 }  1 ∧ B2 ⇒  |B1 ⇒  − {B1 } |B1 ⇒  (∨− ) |B2 ⇒  − {B2 } (∨0 ) |B2 ⇒  |B1 ∨ B2 ⇒  − {B1 ∨ B2 } |B1 ∨ B2 ⇒  and



|¬B1 ⇒  − {¬B1 } |¬B2 ⇒  − {¬B2 } |¬(B1 ∧ B2 )⇒  − {¬(B1 ∧ B2 )} |¬B1 ⇒  (¬∧0 ) |¬B2 ⇒  |¬(B1 ∧ B2 ) ⇒   |¬B1 ⇒  − {¬B1 } (¬∨− )  − {¬B1 }|¬B2 ⇒  − {¬B1 , ¬B2 } |¬(B1 ∨ B2 )⇒  − {¬(B1 ∨ B2 )} |¬B1 ⇒  (¬∨− )  − {¬B1 }|¬B2 ⇒  − {¬B1 } |¬(B1 ∨ B2 ) ⇒  (¬∧− )

Definition 3.2.6 A reduction |B ⇒  − {B } is provable in St , denoted by t |B ⇒  − {B }, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = |B ⇒  − {B }, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in St .

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71

Theorem 3.2.7 (Soundness and completeness theorem) For any reduction |B ⇒  − {B }, |B ⇒  − {B } is provable in St if and only if |B ⇒  − {B } is valid, that is, t |B ⇒  − {B } iff |=t |B ⇒  − {B }.  

3.2.3 R-Calculus St A reduction |C ⇒ , C is St -valid, denoted by |=t |C ⇒ , C , if C =



C if |=t , C λ otherwise.

Intuitively, a formula C1 ∧ C2 is enumerable into , if C1 is enumerable into  and C2 is enumerable into  ∪ {C1 }; and C1 ∨ C2 is enumerable into  if either C1 or C2 is enumerable into . C1 ∧ C2 is not enumerable into , if either C1 is not enumerable into  or C2 is not enumerable into  ∪ {C1 }; and C1 ∨ C2 is not enumerable into  if both C1 and C2 are not enumerable into . R-calculus St consists of the following axiom and deductions: • Axioms: (A0t )

¬l ∈  ¬l ∈ / , (A+ , t) |l ⇒  |l ⇒ , l

where  is a set of literals and l is a literal. • Deduction rules: |C ⇒ , C (¬¬+ )  |¬¬C ⇒ , ¬¬C |C1 ⇒ , C1 (∧+ ) , C1 |C2 ⇒ , C1 , C2 |C  1 ∧ C2 ⇒ , C1 ∧ C2 |¬C1 ⇒ , ¬C1 (¬∧+ ) |¬C2 ⇒ , ¬C2  |¬(C1 ∧ C2 ) ⇒ , ¬(C1 ∧ C2 ) |C1 ⇒ , C1 (∨+ ) |C2 ⇒ , C2 |C  1 ∨ C2 ⇒ , C1 ∨ C2 |¬C1 ⇒ , ¬C1 (¬∨+ ) , ¬C1 |¬C2 ⇒ , ¬C1 , ¬C2 |¬(C1 ∨ C2 ) ⇒ , ¬(C1 ∨ C2 )

|C ⇒  (¬¬0 )  |¬¬C ⇒  |C1 ⇒  (∧0 ) , C1 |C2 ⇒ , C1 |C  1 ∧ C2 ⇒  |¬C1 ⇒  (¬∧0 ) |¬C2 ⇒   |¬(C1 ∧ C2 ) ⇒  |C1 ⇒  (∨0 ) |C2 ⇒  |C  1 ∨ C2 ⇒  |¬C1 ⇒  (¬∨0 ) |¬C2 ⇒  |¬(C1 ∨ C2 ) ⇒ 

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Definition 3.2.8 A reduction |C ⇒ , C is provable in St , denoted by t |C ⇒ , C , if there is a sequence {δ1 , . . . , δn } of reductions such that δn = |C ⇒ , C , and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in St . Theorem 3.2.9 (Soundness and completeness theorem) For any reduction |C ⇒ , C , |C ⇒ , C is provable in St iff |C ⇒ , C is valid. That is,

t |C ⇒ , C iff |=t |C ⇒ , C .

Proof Here, we give the proof of completeness part. Given a reduction |C ⇒ |C , we construct a trees T such that either (i) T is a proof tree of |C ⇒ , C; i.e., for each branch ξ of T , there is a reduction γ at the leaf of ξ which is an axiom of form (A+ t ), or (ii) there is a branch ξ ∈ T such that ξ is a proof of |C ⇒ . T is constructed as follows: • the root of T is |C ⇒ , C; • for a node ξ , if each reduction at ξ is of form  |l ⇒ , l then the node is a leaf; • otherwise, ξ has the direct child containing the following reductions: ⎧ ⎪ ⎪ 1 |C ⇒ 1 , C ⎪ ⎪ 1 |C1 ⇒ 1 , C1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , C1 |C2 ⇒ 1 , C1 , C2 ⎪ ⎪ ⎨ 1 |¬C1 ⇒ 1 , ¬C1  1 |¬C2 ⇒ 1 , ¬C2 ⎪ ⎪ 1 |C1 ⇒ 1 , C1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 |C2 ⇒ 1 , C2 ⎪ ⎪ 1 |¬C1 ⇒ 1 , ¬C1 ⎪ ⎪ ⎩ 1 , ¬C1 |¬C2 ⇒ 1 , ¬C1 , ¬C2

if 1 |¬¬C ⇒ 1 , ¬¬C ∈ ξ if 1 |C1 ∧ C2 ⇒ 1 , C1 ∧ C2 ∈ ξ if 1 |¬(C1 ∧ C2 ) ⇒ 1 , ¬(C1 ∧ C2 ) ∈ ξ if 1 |C1 ∨ C2 ⇒ 1 , C1 ∨ C2 ∈ ξ if 1 |¬(C1 ∨ C2 ) ⇒ 1 , ¬(C1 ∨ C2 ) ∈ ξ

Lemma 3.2.10 If for each branch ξ ⊆ T , there is a reduction  |C ⇒  |C ∈ ξ which is an axiom in St then T is a proof tree of |C ⇒ , C in St . Proof By the definition of T , T is a proof tree of |C ⇒ , C.

 

Lemma 3.2.11 If there is a branch ξ ⊆ T such that each reduction  |C ⇒  , C ∈ ξ is not an axiom in St , then ξ is a proof of |C ⇒ .

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73

Proof Let ξ be a branch of T such that any reduction  |C ⇒  , C ∈ ξ is not an axiom in St . We proved by induction on node η ∈ ξ that for each reduction  |C ⇒  , C ∈ η, f   |C ⇒  . Case  |l ⇒  , l ∈ η is a leaf node of ξ. Then, there is no literal l = l such that l , ¬l ∈  . Hence, t  |l ⇒  . Case δ = 2 |¬¬C ⇒ 2 , ¬¬C ∈ η. Then, η has a child node ∈ ξ containing reduction 2 |C ⇒ 2 , C. By induction assumption, t 2 |C ⇒ 2 , and by (¬¬0 ), t 2 |¬¬C ⇒ 2 . Case δ = 2 |C1 ∧ C2 ⇒ 2 , C1 ∧ C2 ∈ η. Then, η has a child node ∈ ξ containing reduction either 2 |C1 ⇒ 2 , C1 or 2 , C1 |C2 ⇒ 2 , C1 , C2 . By induction assumption, either t 2 |C1 ⇒ 2 , or t 2 , C1 |C2 ⇒ 2 , C1 . By (∧0 ), we have t 2 , C1 ∧ C2 ⇒ 2 . Case δ = 2 |C1 ∨ C2 ⇒ 2 , C1 ∨ C2 ∈ η. Then, η has a child node ∈ ξ containing reductions 2 |C1 ⇒ 2 , C1 and 2 |C2 ⇒ 2 , C2 . By induction assumption, t 2 |C1 ⇒ 2 and t 2 |C2 ⇒ 2 . By (∨0 ), we have t 2 , C1 ∨ C2 ⇒ . Similar for other cases.  

3.2.4 R-Calculus Sf A reduction |D ⇒ , D is Sf -valid, denoted by |=f |D ⇒ , D , if D =



D if |=f , D λ otherwise.

Intuitively, a formula D1 ∧ D2 is enumerable into , if either D1 or D2 is enumerable into ; and D1 ∨ D2 is enumerable into  if D1 is enumerable into  and D2 is enumerable into  ∪ {D1 }. A formula D1 ∧ D2 is not enumerable into , if both D1 and D2 are not enumerable into ; and D1 ∨ D2 is not enumerable into  if either D1 is not enumerable into  or D2 is not enumerable into  ∪ {D1 }. R-calculus Sf consists of the following axiom and deductions: • Axioms: (A0f )

¬l ∈  , |l ⇒ 

(A+ f)

where  is a set of literals and l is a literal. • Deduction rules:

¬l ∈ / . |l ⇒ , l

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|D ⇒ , D |¬¬D ⇒ , ¬¬D |D ⇒  (¬¬0 ) |¬¬D ⇒   |D1 ⇒ , D1 (∧+ ) |D2 ⇒ , D2 |D1 ∧ D2 ⇒, D1 ∧ D2 |D1 ⇒  (∧0 ) |D2 ⇒  |D1 ∧ D2 ⇒   |¬D1 ⇒ , ¬D1 (¬∧+ ) , ¬D1 |¬D2 ⇒ , ¬D1 , ¬D2 |¬(D1 ∧ D2 ) ⇒ , ¬(D1 ∧ D2 ) |¬D1 ⇒  (¬∧0 ) , ¬D1 |¬D2 ⇒ , ¬D1 |¬(D1 ∧ D2 ) ⇒   |D1 ⇒ , D1 (∨+ ) , D1 |D2 ⇒ , D1 , D2 |D1 ∨ D2 ⇒, D1 ∨ D2 |D1 ⇒  (∨0 ) , D1 |D2 ⇒ , D1 |D1 ∨ D2 ⇒   |¬D1 ⇒ , ¬D1 (¬∨+ ) |¬D2 ⇒ , ¬D2 |¬(D1 ∨ D2) ⇒ , ¬(D1 ∨ D2 ) |¬D1 ⇒  (¬∨0 ) |¬D2 ⇒  |¬(D1 ∨ D2 ) ⇒ 

(¬¬+ )

Definition 3.2.12 A reduction |D ⇒ , D is provable in Sf , denoted by  |D ⇒ , D , if there is a sequence {δ1 , . . . , δn } of reductions such that δn = |D ⇒ , D , and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in Sf . Theorem 3.2.13 (Soundness and completeness theorem) For any reduction |D ⇒ , D , |D ⇒ , D is provable in Sf if and only if |D ⇒ , D is Sf -valid. that is, f |D ⇒ , D iff |=f |D ⇒ , D .  

3.3 Gentzen Deduction Systems A sequent  ⇒  is

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• Gt -valid, denoted by |=t  ⇒ , if for any assignment v, either v(A) = 0 for some formula A ∈  or v(B) = 1 for some formula B ∈ ; • Gf -valid, denoted by |=f  ⇒ , if for any assignment v, either v(A) = 1 for some formula A ∈  or v(B) = 0 for some formula B ∈ . Gt : Av(EA ∈ (v(A) = 0) or EB ∈ (v(B) = 1)) Gf : Av(EA ∈ (v(A) = 1) or EB ∈ (v(B) = 0)). A co-sequent  →  is • Gt -valid, denoted by |=t  → , if there is an assignment v such that v(A) = 1 for every formula A ∈  and v(B) = 0 for every formula B ∈ ; • Gf -valid, denoted by |=f  → , if there is an assignment v such that v(A) = 0 for every formula A ∈  and v(B) = 1 for every formula B ∈ . Gt : Ev(AA ∈ (v(A) = 1)&AB ∈ (v(B) = 0)) Gf : Ev(AA ∈ (v(A) = 0)&AB ∈ (v(B) = 1)). A sequent  ⇒ /co-sequent  →  is atomic/literal if each formula in  and  is atomic/literal.

3.3.1 Gentzen Deduction System Gt Intuitively, if for any assignment v, either v |= , A1 ⇒  or v |= , A2 ⇒  then for any assignment v, v |= , A1 ∧ A2 ⇒ ; and if for any assignment v, v |=  ⇒ B1 ,  and v |=  ⇒ B2 ,  then for any assignment v, v |=  ⇒ B1 ∧ B2 , . If for any assignment v, v |= , A1 ⇒  and v |= , A2 ⇒  then for any assignment v, v |= , A1 ∨ A2 ⇒ ; and if for any assignment v, either v |=  ⇒ B1 ,  or v |=  ⇒ B2 ,  then for any assignment v, v |=  ⇒ B1 ∨ B2 , . Gentzen deduction system Gt consists of the following axiom and deductions: • Axiom: (At )

Ep(p, ¬p ∈ ) or Eq(q, ¬q ∈ ) or  ∩  = ∅ , ⇒

where ,  are sets of literals. • Deduction rules:

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, A ⇒   ⇒ B,  (¬¬L ) (¬¬R )  , ¬¬A ⇒    ⇒ ¬¬B,  , A1 ⇒   ⇒ B1 ,  (∧L ) , A2 ⇒  (∧R )  ⇒ B2 ,  ,   A1 ∧ A2 ⇒   ⇒ B1 ∧ B2 ,  , A1 ⇒   ⇒ B1 ,  L R ⇒  , A  ⇒ B2 ,  (∨ ) (∨ ) 2 , A1 ∨ A2 ⇒   ⇒ B1 ∨ B2 ,  and



 , ¬A1 ⇒   ⇒ ¬B1 ,  (¬∧L ) , ¬A2 ⇒  (¬∧R )  ⇒ ¬B2 ,  ,   ¬(A1 ∧ A2 ) ⇒   ⇒ ¬(B1 ∧ B2 ),  , ¬A1 ⇒   ⇒ ¬B1 ,  (¬∨L ) , ¬A2 ⇒  (¬∨R )  ⇒ ¬B2 ,  , ¬(A1 ∨ A2 ) ⇒   ⇒ ¬(B1 ∨ B2 ), 

Definition 3.3.1 A sequent  ⇒  is provable in Gt , denoted by t  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in Gt . Theorem 3.3.2 (Soundness theorem) For any sequent  ⇒ , if t  ⇒ , then |=t  ⇒ . Proof We prove that each axiom is valid and each deduction rule preserves the validity. (At ) For any assignment v, (i) assume that p, ¬p ∈ , then either v(p) = 0 or v(¬p) = 0; (ii) assume that q, ¬q ∈ , then either v(q) = 1 or v(¬q) = 1; (iii) l ∈  ∩ , then either v(l) = 0 or v(¬l) = 1. In each case, v |=t  ⇒ . (¬¬L ) Assume that for any assignment v, v |=t , A ⇒ . Then, for any assignment v, if v |= , ¬¬A then v |= A, and by induction assumption, v |= . (¬¬R ) Assume that for any assignment v, v |=t  ⇒ B, . Then, for any assignment v, if v |=  then by induction assumption, v |= B, , and v |= ¬¬B, , because v(¬¬B) = v(B). (∧L ) Assume that for any assignment v, v |=t , Ai ⇒ . Then, for any assignment v, if v |= , A1 ∧ A2 then v |= , Ai and by induction assumption, v |= . (∧R ) Assume that for any assignment v, v |=t  ⇒ B1 ,  and v |=  ⇒ B2 , . Then, for any assignment v, if v |=  then there are two cases: (i) v |= B1 or v |= B2 . By induction assumption, v |= ; (ii) v |= B1 and v |= B2 . Then, v |= B1 ∧ B2 , and hence, v |= B1 ∧ B2 , . (∨L ) Assume that for any assignment v, v |=t , A1 ⇒  and v |=t , A2 ⇒ . Then, for any assignment v, if v |= , A1 ∨ A2 then v |=  and v |= A1 ∨ A2 . There are two cases: If v |= A1 , then by induction assumption, v |= ; and if v |= A2 , then by induction assumption, v |= .

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(∨R ) Assume that for any assignment v, v |=t  ⇒ B1 , . Then, for any assignment v, if v |=  then by assumption, v |= B1 , . There are two cases: v |= B1 , then v |= B1 ∨ B2 , and hence, v |= B1 ∨ B2 , ; and v |= , and hence, v |=t B1 ∨ B2 , . Similar for other cases.   Theorem 3.3.3 (Completeness theorem) For any sequent  ⇒ , if |=t  ⇒  then t  ⇒ . Proof Given a sequent  ⇒ , we construct a tree T such that either (i) for each branch ξ of T , the sequent  ⇒  at the leaf of ξ is an axiom, or (ii) there is an assignment v such that v |=  ⇒ . T is constructed as follows: • the root of T is  ⇒ ; • for a node ξ , if the sequent  ⇒  at ξ is atomic then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following sequents: ⎧  ⎪ ⎪ 1 , A1 , A2 ⇒ 1 ⎪ ⎪ 1 ⇒ B1 , 1 ⎪ ⎪ ⎨   1 ⇒ B2 , 1 1 , A1 ⇒ 1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 , A2 ⇒ 1 ⎪ ⎩ 1 ⇒ B1 , B2 , 1 and

⎧  1 , A ⇒ 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ 1 ⇒ B, 1 ⎪ ⎪ 1 , ¬A1 ⇒ 1 ⎪ ⎪ ⎨ 1 , ¬A2 ⇒ 1 1 ⇒ ¬B1 , ¬B2 , 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ 1 , ¬A1 , ¬A2 ⇒ 1 ⎪ ⎪ 1 ⇒ ¬B1 , 1 ⎪ ⎪ ⎩ 1 ⇒ ¬B2 , 1

if 1 , A1 ∧ A2 ⇒ 1 ∈ ξ if 1 ⇒ B1 ∧ B2 , 1 ∈ ξ if 1 , A1 ∨ A2 ⇒ 1 ∈ ξ if 1 ⇒ B1 ∨ B2 , 1 ∈ ξ if 1 , ¬¬A ⇒ 1 ∈ ξ if 1 ⇒ ¬¬B, 1 ∈ ξ if 1 , ¬(A1 ∧ A2 ) ⇒ 1 ∈ ξ if 1 ⇒ ¬(B1 ∧ B2 ), 1 ∈ ξ if 1 , ¬(A1 ∨ A2 ) ⇒ 1 ∈ ξ if 1 ⇒ ¬(B1 ∨ B2 ), 1 ∈ ξ

Lemma 3.3.4 If for each branch ξ ⊆ T , there is a sequent  ⇒  ∈ ξ which is an axiom in Gt , then T is a proof tree of  ⇒ . Proof By the definition of T , T is a proof tree of  ⇒ .

 

Lemma 3.3.5 If there is a branch ξ ⊆ T such that each sequent  ⇒  ∈ ξ is not an axiom in Gt , then there is an assignment v such that v |=t  ⇒ . Proof Let ξ be a branch of T such that each sequent  ⇒  ∈ ξ is not an axiom in Gt .

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Let

 = {A ∈  :  ⇒  ∈ ξ }  = {B ∈  :  ⇒  ∈ ξ }.

Define an assignment v as follows:  v(p) =

1 if p ∈  or ¬p ∈  0 otherwise.

Then, v is well-defined and v |=t 0 ⇒ 0 , where 0 ⇒ 0 is a sequent at the leaf node of ξ . We prove by induction on nodes η of ξ that for each sequent  ⇒  at η, v |=t  ⇒  . Case  ⇒  = 2 , ¬¬A ⇒ 2 ∈ η. Then,  ⇒  has a child node ∈ ξ containing 2 , A ⇒ 2 ∈ η. By induction assumption, v |= 2 , A and v |= 2 , which implies that v |= 2 , ¬¬A and v |= 2 . Case  ⇒  = 2 ⇒ ¬¬B, 2 ∈ η. Then,  ⇒  has a child node ∈ ξ containing 2 ⇒ B, 2 ∈ η. By induction assumption, v |= 2 and v |= B, 2 , which implies that v |= 2 and v |= ¬¬B, 2 . Case  ⇒  = 2 , A1 ∧ A2 ⇒ 2 ∈ η. Then,  ⇒  has a child node ∈ ξ containing 2 , A1 ⇒ 2 ∈ η and 2 , A2 ⇒ 2 ∈ η. By induction assumption, v |= 2 , A1 ; v |= 2 , A2 and v |= 2 , i.e., v |= 2 , A1 ∧ A2 and v |= 2 . Case  ⇒  = 2 ⇒ B1 ∧ B2 , 2 ∈ η. Then,  ⇒  has a child node ∈ ξ containing 2 ⇒ Bi , 2 ∈ η. By induction assumption, v |= 2 , and v |= Bi , 2 , i.e., v |= 2 and v |= B1 ∧ B2 , 2 . Case  ⇒  = 2 , A1 ∨ A2 ⇒ 2 ∈ η. Then,  ⇒  has a child node ∈ ξ containing 2 , Ai ⇒ 2 ∈ η. By induction assumption, v |= 2 , Ai and v |= 2 . Hence, v |= 2 , A1 ∨ A2 and v |= 2 . Case  ⇒  = 2 ⇒ B1 ∨ B2 , 2 ∈ η. Then,  ⇒  has a child node ∈ ξ containing 2 ⇒ B1 , 2 ∈ η and 2 ⇒ B2 , 2 ∈ η. By induction assumption, v |= 2 , v |= B1 , 2 and v |= B2 , 2 . Hence, v |= 2 and v |= B1 ∨ B2 , 2 . Similar for other cases.  

3.3.2 Gentzen Deduction System Gf A sequent  ⇒  is f-valid, denoted by |=f  ⇒ , if for any assignment v, either v(A) = 1 for some A ∈ , or v(B) = 0 for some B ∈ . Gentzen deduction system Gf consists of the following axiom and deductions: • Axiom: (Af )

Ep(p, ¬p ∈ ) or Eq(q, ¬q ∈ ) or  ∩  = ∅ , ⇒

where ,  are sets of literals.

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• Deduction rules: 

 , A1 ⇒   ⇒ B1 ,  (∧L ) , A2 ⇒  (∧R )  ⇒ B2 ,  ,   A1 ∧ A2 ⇒   ⇒ B1 ∧ B2 ,  , A1 ⇒   ⇒ B1 ,  (∨L ) , A2 ⇒  (∨R )  ⇒ B2 ,  , A1 ∨ A2 ⇒   ⇒ B1 ∨ B2 ,  and

, A ⇒   ⇒ B,  (¬¬R ) ,   ¬¬A ⇒   ⇒ ¬¬B,  , ¬A1 ⇒   ⇒ ¬B1 ,  (¬∧L ) , ¬A2 ⇒  (¬∧R )  ⇒ ¬B2 ,  ,   ¬(A1 ∧ A2 ) ⇒   ⇒ ¬(B1 ∧ B2 ),  , ¬A1 ⇒   ⇒ ¬B1 ,  L R ⇒  , ¬A  ⇒ ¬B2 ,  (¬∨ ) (¬∨ ) 2 , ¬(A1 ∨ A2 ) ⇒   ⇒ ¬(B1 ∨ B2 ),  (¬¬L )

Definition 3.3.6 A sequent  ⇒  is provable in Gf , denoted by f  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in Gf . Theorem 3.3.7 (Soundness and completeness theorem) For any sequent  ⇒ , f  ⇒  iff |=f  ⇒ .

 

3.3.3 Gentzen Deduction System Gt A co-sequent  →  is Gt -valid, denoted by |=t  → , if there is an assignment v such that v |=  and v |= , where v |=  means that for each formula A ∈ , v(A) = 1; and v |=  means that for each formula B ∈ , v(B) = 0. Intuitively, if there is an assignment v such that v |= , A1 →  and v |= , A2 →  then for this assignment v, v |= , A1 ∧ A2 → ; and if there is an assignment v such that either v |=  → B1 ,  or v |=  → B2 ,  then for this assignment v, v |=  → B1 ∧ B2 , . If there is an assignment v such that either v |= , A1 →  or v |= , A2 →  then for this assignment v, v |= , A1 ∨ A2 → ; and if there is an assignment v such that v |=  → B1 ,  and v |=  → B2 ,  then for this assignment v, v |=  → B1 ∨ B2 , .

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Gentzen deduction system Gt consists of the following axiom and deductions: • Axiom:

con()&inval()& ∩  = ∅ ,  → 

where ,  are sets of literals. • Deduction rules: , A →   → B,  (¬¬L ) (¬¬R ) , ¬¬A →     → ¬¬B,  , A1 →   → B1 ,  (∧L ) , A2 →  (∧R )  → B2 ,  ,   A1 ∧ A2 →   → B1 ∧ B2 ,  , A1 →   → B1 ,  (∨L ) , A2 →  (∨R )  → B2 ,  , A1 ∨ A2 →   → B1 ∨ B2 ,  and



 , ¬A1 →   → ¬B1 ,  (¬∧L ) , ¬A2 →  (¬∧R )  → ¬B2 ,  ,   ¬(A1 ∧ A2 ) →   → ¬(B1 ∧ B2 ),  , ¬A1 →   → ¬B1 ,  (¬∨L ) , ¬A2 →  (¬∨R )  → ¬B2 ,  , ¬(A1 ∨ A2 ) →   → ¬(B1 ∨ B2 ), 

Definition 3.3.8 A co-sequent  →  is provable in Gt , denoted by t  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in Gt . Theorem 3.3.9 (Soundness theorem) For any co-sequent  → , if t  →  then |=t  → . Proof We prove that each axiom is valid and each deduction rule preserves the validity. Assume that con(), inval() and  ∩  = ∅. Then, we define an assignment v such that for any variable p, v(p) = 1 iff p ∈  or ¬p ∈ . Then, v |=  → . (¬¬L ) Assume that there is an assignment v such that v |= , A → . Then, for this very assignment v, v |= , ¬¬A → . (¬¬R ) Assume that there is an assignment v such that v |=  → B, . Then, for this very assignment v, v |=  → ¬¬B, .

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(∧L ) Assume that there is an assignment v such that v |= , A1 →  and v |= , A2 → . Then, for this very assignment v, v |= , A1 ∧ A2 → . (∧R ) Assume that there is an assignment v such that v |=  → B1 , . Then, for this very assignment v, v |=  → B1 ∧ B2 , . (¬∧L ) Assume that there is an assignment v such that v |= , ¬A1 → . Then, for this very assignment v, v |= , ¬(A1 ∧ A2 ) → . (¬∧R ) Assume that there is an assignment v such that v |=  → ¬B1 ,  and v |=  → ¬B2 , . Then, for this very assignment v, v |=  → ¬(B1 ∧ B2 ), . Similar for other cases.   Theorem 3.3.10 (Completeness theorem) For any co-sequent  → , if |=t  →  then t  → . Proof Given a co-sequent  → , we construct a tree T such that either (i) there is a branch ξ of T which is a proof of  → , or (ii) t  ⇒ . T is constructed as follows: • the root of T is  → ; • for a node ξ , if co-sequent  →  at ξ is literal then the node is a leaf; • otherwise, ξ has direct children nodes containing the following co-sequents: ⎧ 1 , A1 → 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  1 , A2 → 1 ⎪ ⎪ 1 → B1 , 1 ⎪ ⎪ ⎨   1 → B2 , 1 1 , A1 → 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , A2 → 1 ⎪ ⎪ 1 → B1 , 1 ⎪ ⎪ ⎩ 1 → B2 , 1 and

⎧ 1 , A → 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ 1 → B, 1 ⎪ ⎪ 1 , ¬A1 → 1 ⎪ ⎪ ⎪ ⎪ 1 , ¬A2 → 1 ⎪ ⎪ ⎨ 1 → ¬B1 , 1  ⎪ ⎪  1 → ¬B2 , 1 ⎪ ⎪ 1 , ¬A1 → 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , ¬A2 → 1 ⎪ ⎪ 1 → ¬B1 , 1 ⎪ ⎪ ⎩ 1 → ¬B2 , 1

if 1 , A1 ∧ A2 → 1 ∈ ξ if 1 → B1 ∧ B2 , 1 ∈ ξ if 1 , A1 ∨ A2 → 1 ∈ ξ if 1 → B1 ∨ B2 , 1 ∈ ξ

if 1 , ¬¬A → 1 ∈ ξ if 1 → ¬¬B, 1 ∈ ξ if 1 , ¬(A1 ∧ A2 ) → 1 ∈ ξ if 1 → ¬(B1 ∧ B2 ), 1 ∈ ξ if 1 , ¬(A1 ∨ A2 ) → 1 ∈ ξ if 1 → ¬(B1 ∨ B2 ), 1 ∈ ξ.

Lemma 3.3.11 If there is a branch ξ ⊆ T such that each co-sequent  →  at the leaf node of ξ is an axiom in Gt then ξ is a proof of  → .

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Proof By the definition of T , ξ is a proof of  → .

 

Lemma 3.3.12 If for each branch ξ ⊆ T , there is a co-sequent  →  at the leaf node of ξ which is not an axiom in Gt then T is a proof tree of sequent  ⇒  in Gt . Proof We prove by induction on nodes ξ of T that for some co-sequent  →  at ξ, t  ⇒  . Case ξ being a leaf node. By assumption, a co-sequent  →  ∈ ξ is not an axiom in Gt . Then,  ⇒  is an axiom in Gt . Case  →  = 2 , ¬¬A → 2 ∈ ξ. Then,  →  has a child node containing 2 , A → 2 . By induction assumption, t 2 , A ⇒ 2 , and by (¬¬L ), t 2 , ¬¬A ⇒ 2 . Case  →  = 2 , A1 ∧ A2 → 2 ∈ ξ. Then,  →  has a child node containing 2 , A1 , A2 → 2 . By induction assumption, t 2 , A1 , A2 ⇒ 2 , and by (∧L ), t 2 , A1 ∧ A2 ⇒ 2 . Case  →  = 2 , ¬(A1 ∧ A2 ) → 2 ∈ ξ. Then,  →  has children nodes containing 2 , ¬A1 → 2 and 2 , ¬A2 → 2 , respectively. By induction assumption, t 2 , ¬A1 ⇒ 2 , t 2 , ¬A2 ⇒ 2 , and by (¬∧L ), t 2 , ¬(A1 ∧ A2 ) ⇒ 2 . Similar for other cases.   We rewrite the proof of the completeness theorem for Gt as follows: Lemma 3.3.13 If for each branch ξ ⊆ T , there is a sequent  ⇒  ∈ ξ which is an axiom in Gt then T is a proof tree of  ⇒ . Proof By the definition of T , T is a proof tree of  ⇒ .

 

Lemma 3.3.14 If there is a branch ξ ⊆ T such that each sequent  ⇒  ∈ ξ is not an axiom in Gt then t  → . Proof Let ξ be a branch of T such that each sequent  ⇒  ∈ ξ is not an axiom in Gt . Let  = {A ∈  :  ⇒  ∈ ξ }  = {B ∈  :  ⇒  ∈ ξ }. Define an assignment v as follows:  v(p) =

1 if p ∈  or ¬p ∈  0 otherwise.

Then, v is well-defined and v |= 0 → 0 , where 0 ⇒ 0 is a sequent at the leaf node of ξ . That is, t 0 → 0 . We prove by induction on nodes η of ξ that for each sequent  ⇒  at η, t  →  .

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Case  ⇒  = 2 , ¬¬A ⇒ 2 ∈ η. Then,  ⇒  has a direct child node ∈ ξ containing 2 , A ⇒ 2 ∈ η. By induction assumption, t 2 , A → 2 , which implies that t 2 , ¬¬A → 2 . Case  ⇒  = 2 , A1 ∧ A2 ⇒ 2 ∈ η. Then,  ⇒  has a direct child node ∈ ξ containing 2 , A1 ⇒ 2 ∈ η and 2 , A2 ⇒ 2 ∈ η. By induction assumption, t 2 , A1 → 2 ; and t 2 , A2 → 2 , i.e., t 2 , A1 ∧ A2 → 2 . Case  ⇒  = 2 , A1 ∨ A2 ⇒ 2 ∈ η. Then,  ⇒  has a direct child node ∈ ξ containing 2 , Ai ⇒ 2 ∈ η. By induction assumption, t 2 , Ai → 2 . By (∨L ) in Gt , t 2 , A1 ∨ A2 → 2 . Similar for other cases.   There is a question: whether we can use the following tree to prove the completeness theorem of Gt : ⎧ 1 , A1 → 1 ⎪ ⎪ if 1 , A1 ∧ A2 → 1 ∈ ξ ⎪ ⎪ 1 , A2 → 1 ⎪ ⎪ ⎪ ⎪ 1 → B1 , 1 ⎪ ⎪ if 1 → B1 ∧ B2 , 1 ∈ ξ ⎨   1 → B2 , 1 1 , A1 → 1 ⎪ ⎪ if 1 , A1 ∨ A2 → 1 ∈ ξ ⎪ ⎪  ⎪ 1 , A2 → 1 ⎪  ⎪ ⎪ 1 → B1 , 1 ⎪ ⎪ if 1 → B1 ∨ B2 , 1 ∈ ξ ⎩ 1 → B2 , 1 and

⎧ 1 , A → 1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 → B, 1 ⎪ ⎪ ⎪ 1 , ¬A1 → 1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 , ¬A2 → 1 ⎪ ⎨ 1 → ¬B1 , 1 ⎪ ⎪  1 → ¬B2 , 1 ⎪ ⎪ 1 , ¬A1 → 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , ¬A2 → 1 ⎪ ⎪ 1 → ¬B1 , 1 ⎪ ⎪ ⎩ 1 → ¬B2 , 1

if 1 , ¬¬A → 1 ∈ ξ if 1 → ¬¬B, 1 ∈ ξ if 1 , ¬(A1 ∧ A2 ) → 1 ∈ ξ if 1 → ¬(B1 ∧ B2 ), 1 ∈ ξ if 1 , ¬(A1 ∨ A2 ) → 1 ∈ ξ if 1 → ¬(B1 ∨ B2 ), 1 ∈ ξ.

3.3.4 Gentzen Deduction System Gf A co-sequent  →  is Gf -valid, denoted by |=f  → , if there is an assignment v such that for each formula A ∈ , v(A) = 0; and for each formula B ∈ , v(B)=1.

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Gentzen deduction system Gf consists of the following axiom and deductions: • Axiom: (Af )

con()&inval()& ∩  = ∅ ,  → 

where ,  are sets of literals. • Deduction rules:   , A1 →   → B1 ,  (∧L ) , A2 →  (∧R )  → B2 ,  ,   A1 ∧ A2 →   → B1 ∧ B2 ,  , A1 →   → B1 ,  (∨L ) , A2 →  (∨R )  → B2 ,  , A1 ∨ A2 →   → B1 ∨ B2 ,  and

, A →   → B,  (¬¬R ) ,   ¬¬A →   → ¬¬B,  , ¬A1 →   → ¬B1 ,  (¬∧L ) , ¬A1 →  (¬∧R )  → ¬B2 ,  ,   ¬(A1 ∧ A2 ) →   → ¬(B1 ∧ B2 ),  , ¬A1 →   → ¬B1 ,  (¬∨L ) , ¬A2 →  (¬∨R )  → ¬B2 ,  , ¬(A1 ∨ A2 ) →   → ¬(B1 ∨ B2 ),  (¬¬L )

Definition 3.3.15 A co-sequent  →  is provable, denoted by f  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in Gf . Theorem 3.3.16 (Soundness and completeness theorem) For any co-sequent  → , f  →  iff |=f  → .  

3.4 R-Calculi for Sequents Let ∗ ∈ {t, f}. Given two theories ,  and formulas A ∈ , B ∈ , a reduction  ⇒ | (A, B) ⇒  ⇒  is R∗ -valid, denoted by |=∗  ⇒ |(A, B) ⇒  ⇒  , if 

 − {A} if |=∗  − {A} ⇒  otherwise;   − {B} if |=∗  ⇒  − {B}  =  otherwise;

 =

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85

Given two theories ,  and formulas A, B, a reduction  ⇒ |(A, B) ⇒  ⇒  is R∗ -valid, denoted by |=∗  ⇒ |(A, B) ⇒  ⇒  , if



 ∪ {A} if |=∗ , A ⇒   otherwise;   ∪ {B} if |=∗  ⇒ , B  =  otherwise;

 =

3.4.1 R-Calculus Rt Given two theories ,  and formulas A ∈ , B ∈ , a reduction  ⇒ |(A, B) ⇒  ⇒  is Rt -valid, denoted by |=t  ⇒ |(A, B) ⇒  ⇒  , if 

 − {A} if |=t  − {A} ⇒  otherwise;   − {B} if |=t  ⇒  − {B}  =  otherwise;  =

R-calculus Rt consists of the following axioms and deductions: • Axioms:

El = l(l , ¬l ∈  or l ∈  ∩ ) ,  ⇒ |(l, m) ⇒  − {l} ⇒ |m ∼ El = l(l , ¬l ∈  or l ∈  ∩ ) (At0 ) ,  ⇒ |(l, m) ⇒  ⇒ |m Em = m(m , ¬m ∈  or m ∈  ∩ ) (At− ) ,  ⇒ |m ⇒  ⇒  − {m} ∼ Em = m(m , ¬m ∈  or m ∈  ∩ ) (At0 ) ,  ⇒ |m ⇒  ⇒  (At− )

where l, m are literals and ,  are sets of literals. • Deduction rules:   ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B (∧L− )  ⇒ |(A2 , B) ⇒  − {A2 } ⇒ |B  ⇒ |(A1 ∧A2 , B) ⇒  − {A1 ∧ A2 } ⇒ |B  ⇒ |(A1 , B) ⇒  ⇒ |B (∧L0 )  ⇒ |(A2 , B) ⇒  ⇒ |B  ⇒ |(A1 ∧ A2 , B) ⇒  ⇒ |B   ⇒ |B1 ⇒  ⇒  − {B1 } (∧R− )  ⇒  − {B1 }|B2 ⇒  ⇒  − {B1 , B2 }  ⇒ |B1 ∧B2 ⇒  ⇒  − {B1 ∧ B2 }  ⇒ |B1 ⇒  ⇒  (∧R0 )  ⇒  − {B1 }|B2 ⇒  ⇒  − {B1 }  ⇒ |B1 ∧ B2 ⇒  ⇒ 

86

3 R-Calculi for Tableau/Gentzen Deduction Systems

and 

 ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 , A2 } ⇒ |B  ⇒ |(A1 ∨A2 , B) ⇒  − {A1 ∨ A2 } ⇒ |B  ⇒ |(A1 , B) ⇒  ⇒ |B (∨L0 )  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 } ⇒ |B  ⇒ |(A1 ∨ A2 , B) ⇒  ⇒ |B   ⇒ |B1 ⇒  ⇒  − {B1 } (∨R− )  ⇒ |B2 ⇒  ⇒  − {B2 }  ⇒ |B1 ∨B2 ⇒  ⇒  − {B1 ∨ B2 }  ⇒ |B1 ⇒  ⇒  R0 (∨ )  ⇒ |B2 ⇒  ⇒   ⇒ |B1 ∨ B2 ⇒  ⇒ 

(∨L− )

and  ⇒ |(A, B) ⇒  − {A} ⇒ |B  ⇒ |(¬¬A, B) ⇒  − {¬¬A} ⇒ |B  ⇒ |(A, B) ⇒  ⇒ |B (¬¬L0 )  ⇒ |(¬¬A, B) ⇒  ⇒ |B  ⇒ |B ⇒  ⇒  − {B} R− (¬¬ )  ⇒ |¬¬B ⇒  ⇒  − {¬¬B}  ⇒ |B ⇒  ⇒  (¬¬R0 )  ⇒ |¬¬B ⇒  ⇒  (¬¬L− )

and 

 ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B  − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 , ¬A2 } ⇒ |B  ⇒ |(¬(A1∧ A2 ), B) ⇒  − {¬(A1 ∧ A2 )} ⇒ |B  ⇒ |(¬A1 , B) ⇒  ⇒ |B (¬∧L0 )  − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 } ⇒ |B  ⇒ |(¬(A1 ∧ A2 ), B) ⇒  ⇒ |B   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } (¬∧R− )  ⇒ |¬B2 ⇒  ⇒  − {¬B2 }  ⇒ |¬(B1∧ B2 ) ⇒  ⇒  − {¬B1 ∧ ¬B2 }  ⇒ |¬B1 ⇒  ⇒  R0 ⇒ |¬B2 ⇒  ⇒   (¬∧ )  ⇒ |¬(B1 ∧ B2 ) ⇒  ⇒  (¬∧L− )

3.4 R-Calculi for Sequents

87

and 

 ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B  ⇒ |(¬A2 , B) ⇒  − {¬A2 } ⇒ |B  ⇒ |(¬(A1∨ A2 ), B) ⇒  − {¬(A1 ∨ A2 )} ⇒ |B  ⇒ |(¬A1 , B) ⇒  ⇒ |B (¬∨L0 )  ⇒ |(¬A2 , B) ⇒  ⇒ |B  ⇒ |(¬(A1 ∨ A2 ), B) ⇒  ⇒ |B   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } (¬∨R− )  ⇒  − {¬B1 }|¬B2 ⇒  ⇒  − {¬B1 , ¬B2 }  ⇒ |¬(B1∨ B2 ) ⇒  ⇒  − {¬(B1 ∨ B2 )}  ⇒ |¬B1 ⇒  ⇒  R0 (¬∨ )  ⇒  − {¬B1 }|¬B2 ⇒  ⇒  − {¬B1 }  ⇒ |¬(B1 ∨ B2 ) ⇒  ⇒ 

(¬∨L− )

Definition 3.4.1 A reduction δ =  ⇒ |(A, B) ⇒  ⇒  is provable in Rt , denoted by t δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in Rt . Theorem 3.4.2 (Soundness and completeness theorem) For any reduction  ⇒ |(A, B) ⇒  ⇒  , where A ∈ , B ∈ , t  ⇒ |(A, B) ⇒  ⇒  if and only if

|=t  ⇒ |(A, B) ⇒  ⇒  .

 

3.4.2 R-Calculus Rf Given two theories ,  and formulas A ∈ , B ∈ , a reduction  ⇒ |(A, B) ⇒  ⇒  is Rf -valid, denoted by |=f  ⇒ |(A, B) ⇒  ⇒  , if 

 − {A} if |=f  − {A} ⇒  otherwise;   − {B} if |=f  ⇒  − {B}  =  otherwise;  =

R-calculus Rf consists of the following axioms and deductions:

88

3 R-Calculi for Tableau/Gentzen Deduction Systems

• Axioms: El = l(l , ¬l ∈  or l ∈  ∩ ) ,  ⇒ |(l, m) ⇒  − {l} ⇒ |m ∼ El = l(l , ¬l ∈  or l ∈  ∩ ) (Af0 ) ,  ⇒ |(l, m) ⇒  ⇒ |m Em = m(m , ¬m ∈  or m ∈  ∩ ) (Af+ ) ,  ⇒ |m ⇒  ⇒  − {m} ∼ Em = m(m , ¬m ∈  or m ∈  ∩ ) , (Af0 )  ⇒ |m ⇒  ⇒  (Af+ )

where l, m are literals, ,  are sets of literals. • Deduction rules:   ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B (∧L− )  − {A1 } ⇒ |(A2 , B) ⇒  − {A1 , A2 } ⇒ |B  ⇒ |(A1 ∧A2 , B) ⇒  − {A1 ∧ A2 } ⇒ |B  ⇒ |(A1 , B) ⇒  ⇒ |B (∧L0 )  − {A1 } ⇒  |(A2 , B) ⇒  − {A1 } ⇒ |B  ⇒ |(A1 ∧ A2 , B) ⇒  ⇒ |B   ⇒ |B1 ⇒  ⇒  − {B1 } (∧R− )  ⇒ |B2 ⇒  ⇒  − {B2 }  ⇒ |B1 ∧B2 ⇒  ⇒  − {B1 ∧ B2 }  ⇒ |B1 ⇒  ⇒  R0 (∧ )  ⇒ |B2 ⇒  ⇒   ⇒ |B1 ∧ B2 ⇒  ⇒  and 

 ⇒ |(A1 , B) ⇒  − {A1 } ⇒ |B (∨ )  ⇒ |(A2 , B) ⇒  − {A2 } ⇒ |B  ⇒ |(A1 ∨A2 , B) ⇒  − {A1 ∨ A2 } ⇒ |B  ⇒ |(A1 , B) ⇒  ⇒ |B (∨L0 )  ⇒  |(A2 , B) ⇒  ⇒ |B  ⇒ |(A1 ∨ A2 , B) ⇒  ⇒ |B   ⇒ |B1 ⇒  ⇒  − {B1 } (∨R− )  ⇒  − {B1 }|B2 ⇒  ⇒  − {B1 , B2 }  ⇒ |B1 ∨B2 ⇒  ⇒  − {B1 ∨ B2 }  ⇒ |B1 ⇒  ⇒  R0 (∨ )  ⇒  − {B1 }|B2 ⇒  ⇒  − {B1 }  ⇒ |B1 ∨ B2 ⇒  ⇒  L−

3.4 R-Calculi for Sequents

and

89

 ⇒ |(A, B) ⇒  − {A} ⇒ |B  ⇒ |(¬¬A, B) ⇒  − {¬¬A} ⇒ |B  ⇒ |(A, B) ⇒  ⇒ |B (¬¬L0 )  ⇒ |(¬¬A, B) ⇒  ⇒ |B  ⇒ |B ⇒  ⇒  − {B} (¬¬R− )  ⇒ |¬¬B ⇒  ⇒  − {¬¬B}  ⇒ |B ⇒  ⇒  (¬¬R0 )  ⇒ |¬¬B ⇒  ⇒  (¬¬L− )

and 

 ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B  ⇒ |(¬A2 , B) ⇒  − {¬A2 } ⇒ |B  ⇒ |(¬(A1∧ A2 ), B) ⇒  − {¬(A1 ∧ A2 )} ⇒ |B  ⇒ |(¬A1 , B) ⇒  ⇒ |B (¬∧L0 )  ⇒  |(¬A2 , B) ⇒  ⇒ |B  ⇒ |(¬(A1 ∧ A2 ), B) ⇒  ⇒ |B   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } (¬∧R− )  ⇒  − {¬B1 }|¬B2 ⇒  ⇒  − {¬B1 , ¬B2 }  ⇒ |¬(B1∧ B2 ) ⇒  ⇒  − {¬(B1 ∧ B2 )}  ⇒ |¬B1 ⇒  ⇒  R0 (¬∧ )  ⇒  − {¬B1 }|¬B2 ⇒  ⇒  − {¬B1 }  ⇒ |¬(B1 ∧ B2 ) ⇒  ⇒  (¬∧L− )

and 

 ⇒ |(¬A1 , B) ⇒  − {¬A1 } ⇒ |B  − {¬A1 } ⇒ |(¬A2 , B) ⇒  − {¬A1 , ¬A2 } ⇒ |B  ⇒ |(¬(A1∨ A2 ), B) ⇒  − {¬(A1 ∨ A2 )} ⇒ |B  ⇒ |(¬A1 , B) ⇒  ⇒ |B (¬∨L0 )  − {¬A1 } ⇒  |(¬A2 , B) ⇒  − {¬A1 } ⇒ |B  ⇒ |(¬(A1 ∨ A2 ), B) ⇒  ⇒ |B   ⇒ |¬B1 ⇒  ⇒  − {¬B1 } (¬∨R− )  ⇒ |¬B2 ⇒  ⇒  − {¬B2 }  ⇒ |¬(B1∨ B2 ) ⇒  ⇒  − {¬(B1 ∨ B2 )}  ⇒ |¬B1 ⇒  ⇒  R0  ⇒ |¬B2 ⇒  ⇒  (¬∨ )  ⇒ |¬(B1 ∨ B2 ) ⇒  ⇒  (¬∨L− )

Definition 3.4.3 A reduction δ =  ⇒ |(A, B) ⇒  ⇒  is provable in Rf , denoted by f δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in Rf .

90

3 R-Calculi for Tableau/Gentzen Deduction Systems

Theorem 3.4.4 (Soundness and completeness theorem) For any reduction  ⇒ |(A, B) ⇒  ⇒  , f  ⇒ |(A, B) ⇒  ⇒  if and only if

|=f  ⇒ |(A, B) ⇒  ⇒  .

 

3.4.3 R-Calculus Rt Given two theories ,  and formulas A, B, a reduction  ⇒ |(A, B) ⇒  ⇒  is Rt -valid, denoted by |=t  ⇒ |(A, B) ⇒  ⇒  , if 

 ∪ {A} if |=t , A ⇒   otherwise;   ∪ {B} if |=t  ⇒ , B  =  otherwise;  =

R-calculus Rt consists of the following axioms and deductions: • Axioms: ¬l ∈ / &l ∈ / ¬l ∈  or l ∈  , (A0L , t )  → |(l, m) ⇒ , l → |m  → |(l, m) ⇒  → |m m∈ / &¬m ∈ / m ∈  or ¬m ∈  (A+R , (A0R , t ) t )  → |m ⇒  → , m  → |m ⇒  →  (A+L t )

where l, m are literals, and ,  are sets of literals. • Deduction rules:   → |(A1 , B) ⇒ , A1 → |B (∧L+ ) , A1 → |(A2 , B) ⇒ , A1 , A2 → |B  → |(A1 ∧A2 , B) ⇒ , A1 ∧ A2 → |B  → |(A1 , B) ⇒  → |B (∧L0 ) , A1 → |(A2 , B) ⇒ , A1 → |B  → |(A1 ∧ A2 , B) ⇒  → |B   → |B1 ⇒  → , B1 (∧R+ )  → |B2 ⇒  → , B2  → |B1 ∧B2 ⇒  → , B1 ∧ B2  → |B1 ⇒  →  R0 (∧ )  → |B2 ⇒  →   → |B1 ∧ B2 ⇒  → 

3.4 R-Calculi for Sequents

and



 → |(A1 , B) ⇒ , A1 → |B  → |(A2 , B) ⇒ , A2 → |B  → |(A1 ∨A2 , B) ⇒ , A1 ∨ A2 → |B  → |(A1 , B) ⇒  → |B (∨L0 )  → |(A2 , B) ⇒  → |B  → |(A1 ∨ A2 , B) ⇒  → |B   → |B1 ⇒  → , B1 (∨R− )  → , B1 |B2 ⇒  → , B1 , B2  → |B1 ∨B2 ⇒  → , B1 ∨ B2  → |B1 ⇒  →  R0 (∨ )  → , B1 |B2 ⇒  → , B1  → |B1 ∨ B2 ⇒  → 

(∨L+ )

and

 → |(A, B) ⇒ , A → |B  → |(¬¬A, B) ⇒ , ¬¬A → |B  → |(A, B) ⇒  → |B (¬¬L0 )  → |(¬¬A, B) ⇒  → |B  → |B ⇒  → , B R+ (¬¬ )  → |¬¬B ⇒  → , ¬¬B  → |B ⇒  →  (¬¬R0 )  → |¬¬B ⇒  → 

(¬¬L+ )

and



 → |(¬A1 , B) ⇒ , ¬A1 → |B  → |(¬A2 , B) ⇒ , ¬A2 → |B  → |(¬(A 1 ∧ A2 ), B) ⇒ , ¬(A1 ∧ A2 ) → |B  → |(¬A1 , B) ⇒  → |B (¬∧L0 )  → |(¬A2 , B) ⇒  → |B  → |(¬(A1 ∧ A2 ), B) ⇒  → |B   → |¬B1 ⇒  → , ¬B1 (¬∧R+ )  → , ¬B1 |¬B2 ⇒  → , ¬B1 , ¬B2  → |¬(B1∧ B2 ) ⇒  → , ¬(B1 ∧ B2 )  → |¬B1 ⇒  →  R0 (¬∧ )  → , ¬B1 |¬B2 ⇒  → , ¬B1  → |¬(B1 ∧ B2 ) ⇒  →  (¬∧L+ )

91

92

3 R-Calculi for Tableau/Gentzen Deduction Systems

and 

 → |(¬A1 , B) ⇒ , ¬A1 → |B , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 , ¬A2 → |B  → |(¬(A 1 ∨ A2 ), B) ⇒ , ¬(A1 ∨ A2 ) → |B  → |(¬A1 , B) ⇒  → |B (¬∨L0 ) , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 → |B  → |(¬(A1 ∨ A2 ), B) ⇒  → |B   → |¬B1 ⇒  → , ¬B1 (¬∨R+ )  → |¬B2 ⇒  → , ¬B2  → |¬(B1∨ B2 ) ⇒  → , ¬(B1 ∨ B2 )  → |¬B1 ⇒  →  R0 (¬∨ )  → |¬B2 ⇒  →   → |¬(B1 ∨ B2 ) ⇒  → 

(¬∨L+ )

Definition 3.4.5 A reduction δ =  → |(A, B) ⇒  →  is provable in Rt , denoted by t δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in Rt . Theorem 3.4.6 (Soundness and completeness theorem) For any reduction  → |(A, B) ⇒  →  , t  → |(A, B) ⇒  →  if and only if

|=t  → |(A, B) ⇒  →  .

Proof Here we give the proof of completeness part. Given a reduction  ⇒ |(A, B) ⇒ , A ⇒ |B, we construct a trees T such that either (i) T is a proof tree of  ⇒ |(A, B) ⇒ , A ⇒ |B; i.e., for each branch ξ of T , there is a reduction γ at the leaf of ξ which is an axiom of form (A+L t ), or (ii) there is a branch ξ ∈ T such that ξ is a proof of  ⇒ |(A, B) ⇒  ⇒ |B. T is constructed as follows: • the root of T is  ⇒ |(A, B) ⇒ , A ⇒ |B; • for a node ξ , if each reduction at ξ is of form  ⇒  |(l , B) ⇒ , l ⇒ |B then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following reductions:

3.4 R-Calculi for Sequents

⎧ ⎪ ⎪  1 ⇒ 1 |(A, B) ⇒ 1 , A ⇒ 1 |B ⎪ ⎪ 1 ⇒ 1 |(A1 , B) ⇒ 1 , A1 ⇒ 1 |B ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , A1 ⇒ 1 |(A2 , B) ⇒ 1 , A1 , A2 ⇒ 1 |B ⎪ ⎪ ⎨ 1 ⇒ 1 |(¬A1 , B) ⇒ 1 , ¬A1 ⇒ 1 |B  1 ⇒ 1 |(¬A2 , B) ⇒ 1 , ¬A2 ⇒ 1 |B ⎪ ⎪ 1 ⇒ 1 |(A1 , B) ⇒ 1 , A1 ⇒ 1 |B ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 ⇒ 1 |(A2 , B) ⇒ 1 , A2 ⇒ 1 |B ⎪ ⎪ 1 ⇒ 1 |(¬A1 , B) ⇒ 1 , ¬A1 ⇒ 1 |B ⎪ ⎪ ⎩ 1 , ¬A1 ⇒ 1 |(¬A2 , B) ⇒ 1 , ¬A1 , ¬A2 ⇒ 1 |B

93

if 1 (¬¬A) ∈ ξ if 1 (A1 ∧ A2 ) ∈ ξ if 1 (¬(A1 ∧ A2 )) ∈ ξ if 1 (A1 ∨ A2 ) ∈ ξ if 1 (¬(A1 ∨ A2 )) ∈ ξ

where 1 (A) = 1 ⇒ 1 |(A, B) ⇒ 1 , A ⇒ 1 |B. Lemma 3.4.7 If for each branch ξ ⊆ T , there is a reduction 1 ⇒ 1 | (A, B) ⇒ 1 , A ⇒ |B ∈ ξ which is an axiom in Rt then T is a proof tree of  ⇒ |(A, B) ⇒ , A ⇒ |B in Rt . Proof By the definition of T , T is a proof tree of  ⇒ |(A, B) ⇒ , A ⇒ |B.   Lemma 3.4.8 If there is a branch ξ ⊆ T such that each reduction 1 ⇒ 1 |(A , B) ⇒ 1 , A ⇒ |B ∈ ξ is not an axiom in Rt then ξ is a proof of  ⇒ |(A, B) ⇒  ⇒ |B in Rt . Proof Let ξ be a branch of T such that any reduction 1 ⇒ 1 |(A , B) ⇒ 1 , A ⇒ |B ∈ ξ is not an axiom in Rt . We proved by induction on node η ∈ ξ that for each reduction  ⇒ |(A , B) ⇒  ⇒ |B ∈ η, t  ⇒ |(A , B) ⇒  ⇒ |B. Case  ⇒  |(l, m) ⇒  , l ⇒  |m ∈ η is a leaf node of ξ. Then, ¬l∈ or l , and t  ⇒  |(l, m) ⇒  ⇒  |m. Case δ = 2 ⇒ 2 |(¬¬A, B) ⇒ 2 , ¬¬A ⇒ 2 |B ∈ η. Then, η has a child node ∈ ξ containing reduction 2 ⇒ 2 |(A, B) ⇒ 2 , A ⇒ 2 |B. By induction assumption, t 2 ⇒ 2 |(A, B) ⇒ 2 ⇒ 2 |B, and by (¬¬0 ), t 2 ⇒ 2 | (¬¬A, B) ⇒ 2 ⇒ 2 |B. Case δ = 2 ⇒ 2 |(A1 ∧ A2 , B) ⇒ 2 , A1 ∧ A2 ⇒ 2 |B ∈ η. Then, η has a child node ∈ ξ containing reduction 2 ⇒ 2 |(A1 , B) ⇒ 2 , A1 ⇒ 2 |B or 2 , A1 ⇒ 2 |(A2 , B) ⇒ 2 , A1 , A2 ⇒ 2 |B. By induction assumption, either t 2 ⇒ 2 |(A1 , B) ⇒ 2 ⇒ 2 |B, or t 2 , A1 ⇒ 2 |(A2 , B) ⇒ 2 , A1 ⇒ 2 |B. By (∧0 ), we have t 2 ⇒ 2 |(A1 ∧ A2 , B) ⇒ 2 ⇒ 2 |B. Case δ = 2 ⇒ 2 |(A1 ∨ A2 , B) ⇒ 2 ⇒ 2 |B ∈ η. Then, η has a child node ∈ ξ containing reductions 2 ⇒ 2 |(A1 , B) ⇒ 2 , A1 ⇒ 2 |B and 2 ⇒ 2 | (A2 , B) ⇒ 2 , A2 ⇒ 2 |B. By induction assumption, t 2 ⇒ 2 |(A1 , B) ⇒ 2 ⇒ 2 |B and t 2 ⇒ 2 |(A2 , B) ⇒ 2 ⇒ 2 |B. By (∨0 ), we have t 2 ⇒ 2 |(A1 ∨ A2 , B) ⇒ 2 ⇒ 2 |B. Similar for other cases.  

94

3 R-Calculi for Tableau/Gentzen Deduction Systems

Given a reduction  ⇒ |B ⇒  ⇒  , we construct a trees T such that either (i) T is a proof tree of  ⇒ |B ⇒  ⇒ , B; i.e., for each branch ξ of T , there is a reduction γ at the leaf of ξ which is an axiom of form (A+R t ), or (ii) there is a branch ξ ∈ T such that ξ is a proof of  ⇒ |B ⇒  ⇒ . T is constructed as follows: • the root of T is  ⇒ |B ⇒  ⇒ , B; • for a node ξ , if each reduction at ξ is of form  ⇒ |m ⇒  ⇒ , m then the node is a leaf; • otherwise, ξ has the direct children nodes containing the following reductions: ⎧  ⎪ ⎪ 1 ⇒ 1 |B ⇒ 1 ⇒ 1 , B ⎪ ⎪ 1 ⇒ 1 |B1 ⇒ 1 ⇒ 1 , B1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 ⇒ 1 |B2 ⇒ 1 ⇒ 1 , B2 ⎪ ⎪ ⎨ 1 ⇒ 1 |¬B1 ⇒ 1 ⇒ 1 , ¬B1  1 ⇒ 1 , ¬B1 |¬B2 ⇒ 1 ⇒ 1 , ¬B1 , ¬B2 ⎪ ⎪ ⎪ 1 ⇒ 1 |B1 ⇒ 1 ⇒ 1 , B1 ⎪ ⎪ ⎪ ⎪ ⎪  1 ⇒ 1 , B1 |B2 ⇒ 1 ⇒ 1 , B1 , B2 ⎪ ⎪ 1 ⇒ 1 |¬B1 ⇒ 1 ⇒ 1 , ¬B1 ⎪ ⎪ ⎩ 1 ⇒ 1 |¬B2 ⇒ 1 ⇒ 1 , ¬B2

if 1 (¬¬B) ∈ ξ if 1 (B1 ∧ B2 ) ∈ ξ if 1 (¬(B1 ∧ B2 )) ∈ ξ if 1 (B1 ∨ B2 ) ∈ ξ if 1 (¬(B1 ∨ B2 )) ∈ ξ

where 1 (B) = 1 ⇒ 1 |B ⇒ 1 ⇒ 1 , B. Lemma 3.4.9 If for each branch ξ ⊆ T , there is a reduction 1 ⇒ 1 |B ⇒ 1 ⇒ , B ∈ ξ which is an axiom in Rt then T is a proof tree of  ⇒ |B ⇒  ⇒ , B in Rt . Proof By the definition of T , T is a proof tree of  ⇒ |B ⇒  ⇒ , B.

 

Lemma 3.4.10 If there is a branch ξ ⊆ T such that reductions 1 ⇒ 1 |B ⇒ 1 ⇒ , B ∈ ξ are not axioms in Rt then ξ is a proof of  ⇒ |B ⇒  ⇒ . Proof Let ξ be a branch of T such that any reduction 1 ⇒ 1 |B ⇒ 1 ⇒ 1 , B ∈ ξ is not an axiom in Rt . We proved by induction on node η ∈ ξ that for each reduction  ⇒ |B ⇒  ⇒ , B ∈ η, t  ⇒ |B ⇒  ⇒ . Case  ⇒ |m ⇒  ⇒ , m ∈ η is a leaf node of ξ. Then, ¬m ∈  or m ∈  . t  ⇒ |m ⇒  ⇒ .

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Case δ = 2 ⇒ 2 |¬¬B ⇒ 2 ⇒ 2 , ¬¬B ∈ η. Then, η has a child node ∈ ξ containing reduction 2 ⇒ 2 |B ⇒ 2 ⇒ 2 , B. By induction assumption, t 2 ⇒ 2 |B ⇒ 2 ⇒ 2 , and by (¬¬0 ), t 2 ⇒ 2 |¬¬B ⇒ 2 ⇒ 2 . Case δ = 2 ⇒ 2 |B1 ∧ B2 ⇒ 2 ⇒ 2 , B1 ∧ B2 ∈ η. Then, η has a child node ∈ ξ containing reductions 2 ⇒ 2 |B1 ⇒ 2 ⇒ 2 , B1 and 2 ⇒ 2 |B2 ⇒ 2 ⇒ 2 , B2 . By induction assumption, t 2 ⇒ 2 |B1 ⇒ 2 ⇒ 2 , and t 2 ⇒ 2 |B2 ⇒ 2 ⇒ 2 . By (∧0 ), we have t 2 ⇒ 2 |B1 ∧ B2 ⇒ 2 ⇒ 2 . Case δ = 2 ⇒ 2 |B1 ∨ B2 ⇒ 2 ⇒ 2 , B1 ∨ B2 ∈ η. Then, η has a child node ∈ ξ containing reduction 2 ⇒2 |B1 ⇒ 2 ⇒ 2 , B1 or 2 ⇒ 2 , B1 |B2 ⇒ 2 ⇒ 2 , B1 , B2 . By induction assumption, either t 2 ⇒ 2 |B1 ⇒ 2 ⇒2 or t 2 ⇒2 , B1 |B2 ⇒ 2 ⇒ 2 , B1 . By (∨0 ), we have t 2 ⇒ 2 |B1 ∨ B2 ⇒ 2 ⇒ 2 . Similar for other cases.  

3.4.4 R-Calculus Rf Given two theories ,  and formulas A, B, a reduction  ⇒ |(A, B) ⇒  ⇒  is Rf -valid, denoted by |=f  ⇒ |(A, B) ⇒  ⇒  , if   ∪ {A} if |=f , A ⇒   =  otherwise;   ∪ {B} if |=f  ⇒ , B  =  otherwise; R-calculus Rf consists of the following axioms and deductions: • Axioms: ¬l ∈ / &l ∈ / ,  → |(l, m) ⇒ , l → |m ¬l ∈  or l ∈  , (A0L f )  → |(l, m) ⇒  → |m m∈ / &¬m ∈ / (A+R , f )  → |m ⇒  → , m m ∈  or ¬m ∈  , (A0R f )  → |m ⇒  → 

(A+L f )

where l, m are literals, and ,  are sets of literals.

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• Deduction rules: 

 → |(A1 , B) ⇒ , A1 → |B  → |(A2 , B) ⇒ , A2 → |B  → |(A1 ∧A2 , B) ⇒ , A1 ∧ A2 → |B  → |(A1 , B) ⇒  → |B (∧L0 )  → |(A2 , B) ⇒  → |B  → |(A1 ∧ A2 , B) ⇒  → |B   → |B1 ⇒  → , B1 (∧R+ )  → , B1 |B2 ⇒  → , B1 , B2  → |B1 ∧B2 ⇒  → , B1 ∧ B2  → |B1 ⇒  →  R0 (∧ )  → , B1 |B2 ⇒  → , B1  → |B1 ∧ B2 ⇒  → 

(∧L+ )

and



 → |(A1 , B) ⇒ , A1 → |B , A1 → |(A2 , B) ⇒ , A1 , A2 → |B  → |(A1 ∨A2 , B) ⇒ , A1 ∨ A2 → |B  → |(A1 , B) ⇒  → |B (∨L0 ) , A1 → |(A2 , B) ⇒ , A1 → |B  → |(A1 ∨ A2 , B) ⇒  → |B   → |B1 ⇒  → , B1 (∨R− )  → |B2 ⇒  → , B2  → |B1 ∨B2 ⇒  → , B1 ∨ B2  → |B1 ⇒  →  R0 (∨ )  → |B2 ⇒  →   → |B1 ∨ B2 ⇒  → 

(∨L+ )

and

 → |(A, B) ⇒ , A → |B  → |(¬¬A, B) ⇒ , ¬¬A → |B  → |(A, B) ⇒  → |B (¬¬L0 )  → |(¬¬A, B) ⇒  → |B  → |B ⇒  → , B (¬¬R+ )  → |¬¬B ⇒  → , ¬¬B  → |B ⇒  →  (¬¬R0 )  → |¬¬B ⇒  → 

(¬¬L+ )

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and 

 → |(¬A1 , B) ⇒ , ¬A1 → |B , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 , ¬A2 → |B  → |(¬(A 1 ∧ A2 ), B) ⇒ , ¬(A1 ∧ A2 ) → |B  → |(¬A1 , B) ⇒  → |B (¬∧L0 ) , ¬A1 → |(¬A2 , B) ⇒ , ¬A1 → |B  → |(¬(A1 ∧ A2 ), B) ⇒  → |B   → |¬B1 ⇒  → , ¬B1 (¬∧R+ )  → |¬B2 ⇒  → , ¬B2  → |¬(B1∧ B2 ) ⇒  → , ¬(B1 ∧ B2 )  → |¬B1 ⇒  →  R0 (¬∧ )  → |¬B2 ⇒  →   → |¬(B1 ∧ B2 ) ⇒  → 

(¬∧L+ )

and



 → |(¬A1 , B) ⇒ , ¬A1 → |B  → |(¬A2 , B) ⇒ , ¬A2 → |B  → |(¬(A 1 ∨ A2 ), B) ⇒ , ¬(A1 ∨ A2 ) → |B  → |(¬A1 , B) ⇒  → |B (¬∨L0 )  → |(¬A2 , B) ⇒  → |B  → |(¬(A1 ∨ A2 ), B) ⇒  → |B   → |¬B1 ⇒  → , ¬B1 (¬∨R+ )  → , ¬B1 |¬B2 ⇒  → , ¬B1 , ¬B2  → |¬(B1∨ B2 ) ⇒  → , ¬(B1 ∨ B2 )  → |¬B1 ⇒  →  R0 (¬∨ )  → , ¬B1 |¬B2 ⇒  → , ¬B1  → |¬(B1 ∨ B2 ) ⇒  →  (¬∨L+ )

Definition 3.4.11 A reduction δ =  → |(A, B) ⇒  →  is provable in Rf , denoted by f δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is an axiom or is deduced from the previous reductions by one of the deduction rules in Rf . Theorem 3.4.12 (Soundness and completeness theorem) For any reduction  → |(A, B) ⇒  →  , f  → |(A, B) ⇒  →  if and only if

|=f  → |(A, B) ⇒  →  .

 

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3.5 Conclusions Tableau proof systems: validity negation Tf : AvEA ∈ (v(A) = 0) Tt : EvAA ∈ (v(A) = 1) Tt : AvEB ∈ (v(B) = 1) Tf : EvAB ∈ (v(A) = 0) R-calculi: • for theories:

St : t t f S : f f St : t t Sf : f f

|B ⇒  − {B} iff t  − {B} |B ⇒  iff t  − {B} |A ⇒  − {A} iff f  − {A} |A ⇒  iff f  − {¬A} |B ⇒ , B iff   ¬B |B ⇒  iff   ¬B |A ⇒ , A iff   ¬A |A ⇒  iff   ¬A

• for sequents: validity negation validity negation

Gt Gt Gf Gf

   

⇒  iff Av(EA ∈ (v(A) = 0) or EB ∈ (v(B) = 1)) →  iff Ev(AA ∈ (v(A) = 1)&AB ∈ (v(B) = 0)) ⇒  iff Av(EA ∈ (v(A) = 1) or EB ∈ (v(B) = 0)) →  iff Ev(AA ∈ (v(A) = 0)&AB ∈ (v(B) = 1))

and for Rt , we have the following equivalences: t  ⇒ |(A, B) ⇒  − {A} ⇒  − {B} iff |=t  − {A} ⇒  − {B} t  ⇒ |(A, B) ⇒  − {A} ⇒  iff |=t  − {A} ⇒  t t  ⇒ |(A, B) ⇒  ⇒  − {B} ⎡ ifft |=  ⇒  − {B} |=  − {A} ⇒  − {B} t  ⇒ |(A, B) ⇒  ⇒  iff ⎣ |=t  − {A} ⇒  |=t  ⇒  − {B} and for Rf , we have the following equivalences: f  ⇒ |(A, B) ⇒  − {A} ⇒  − {B} iff |=f  − {A} ⇒  − {B} f  ⇒ |(A, B) ⇒  − {A} ⇒  iff |=f  − {A} ⇒  f f  ⇒ |(A, B) ⇒  ⇒  − {B} ⎡ ifff |=  ⇒  − {B} |=  − {A} ⇒  − {B} f  ⇒ |(A, B) ⇒  ⇒  iff ⎣ |=f  − {A} ⇒  |=f  ⇒  − {B}

References

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• co-sequents: for Rt , we have the following equivalences: t  → |(A, B) ⇒ , A → , B iff |=t , A → , B t  → |(A, B) ⇒ , A →  iff |=t , A →  t  → |(A, B) ⇒  → , B iff ⎡ |=t  → , B |=t  − {A} →  − {B} t  → |(A, B) ⇒  →  iff ⎣ |=t  − {A} →  |=t  →  − {B} and for Rf , we have the following equivalences: f  → |(A, B) ⇒ , A → , B iff |=f , A → , B f  → |(A, B) ⇒ , A →  iff |=f , A →  f  → |(A, B) ⇒  → , B iff ⎡ |=f  → , B |=f  − {A} →  − {B} f  → |(A, B) ⇒  →  iff ⎣ |=f  − {A} →  |=f  →  − {B}

References Cao, C., Sui, Y., Wang, Y.: The nonmonotonic propositional logics. Artif. Intell. Res. 5, 111–120 (2016) Clark, K.: Negation as failure. In: Readings in Nonmonotonic Reasoning, pp. 311–325. Morgan Kaufmann Publishers (1987) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Hähnle, R.: Advanced many-valued logics. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 297–395. Kluwer, Dordrecht (2001) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W.: Mathematical logic, foundations for information science. In: Progress in Computer Science and Applied Logic, vol. 25. Birkhäuser (2010) Malinowski, G.: Many-valued logic and its philosophy. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic, vol. 8. Elsevier, The Many Valued and Nonmonotonic Turn in Logic (2009) Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980) Takeuti, G.: Proof theory. In: Barwise, J. (ed.) Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn, vol. 2, pp. 249–295. Kluwer, Dordrecht (2001)

Chapter 4

R-Calculi RQ1 Q2 /RQ1 Q2

Let Q1 , Q2 ∈ {A, E}. We consider GQ1 Q2 -valid sequents and GQ1 Q2 -valid co-sequents, Gentzen deduction systems GQ1 Q2 , GQ1 Q2 , and corresponding R-calculi RQ1 Q2 , RQ1 Q2 . (1) Gentzen deduction systems (Li 2010; Takeuti and Barwise 1987): • GQ1 Q2 -validity: A sequent  ⇒  is GQ1 Q2 -valid, denoted by |=Q1 Q2  ⇒ , if for any assignment v, either Q1 A ∈ (v(A) = 0) or Q2 B ∈ (v(B) = 1). • GQ1 Q2 -validity: A co-sequent  →  is GQ1 Q2 -valid, denoted by |=Q1 Q2  → , if there is an assignment v such that Q1 A ∈ (v(A) = 1) and Q2 B ∈ (v(B) = 0). There are eight kinds of sequents and co-sequents, and sound and complete Gentzen deduction systems: GEE GEA GAE GAA sequents co-sequents GAA GAE GEA GEE . (2) R-calculi (Alchourrón et al. 1985; Cao et al. 2016; Darwiche and Pearl 1997; Fermé and Hansson 2011; Gärdenfors and Rott 1995; Ginsberg 1987; Li 2007; Reiter 1980): • RQ1 Q2 -validity: Given a sequent  ⇒  and pair (A, B) of formulas such that A ∈  and B ∈ , the result of  ⇒  GQ1 Q2 -revising (A, B) is sequent   ⇒  , denoted by |=Q1 Q2  ⇒ |(A, B) ⇒   ⇒  ,

© Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_4

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and we say that reduction  ⇒ |(A, B) ⇒   ⇒  is RQ1 Q2 -valid, where 

, A if |=Q1 Q2 , A ⇒   otherwise; , B if |=Q1 Q2   ⇒ , B  =  otherwise. 

 =

• RQ1 Q2 -validity: Given a sequent  →  and pair (A, B) of formulas, the result of  →  GQ1 Q2 -revising (A, B) is denoted by |=Q1 Q2  → |(A, B) ⇒   →  , and we say that reduction  → |(A, B) ⇒   →  is RQ1 Q2 -valid, if 

 ∪ {A} if Q1 Q2 , A →   otherwise;   ∪ {B} if Q1 Q2   → , B   =  otherwise.

 =

Correspondingly, there are eight kinds of sound and complete R-calculi for sequents and co-sequents: REE REA RAE RAA sequents co-sequents RAA RAE REA REE . There are eight basic forms of axioms: incon() or val() or  ∩  = ∅ ⇒ val() or  ⊆  (AAE ) ⇒ EA incon() or  ⊆  (A ) ⇒ AA  = ∅ or  = ∅ or  =  = {l} (A ) ⇒ (AEE )

and

con()&inval()& ∩  = ∅  →  inval()&   (AEA )  →  con()&   (AAE )  →   = ∅ = & =  = {l} (AEE )  → 

(AAA )

where  =  = {l} denotes that either  = {l} or  = {l}.

4.1 Gentzen Deduction Systems GQ1 Q2

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We have the following equivalences: Gt = GEE , Gf = −−; Gt = GAA , Gf = − − .

4.1 Gentzen Deduction Systems GQ1 Q2 Let Q1 , Q2 ∈ {A, E}. Definition 4.1.1 A sequent  ⇒  is GQ1 Q2 -valid, denoted by |=Q1 Q2  ⇒ , if for any assignment v, either Q1 A ∈ (v(A) = 0) or Q2 B ∈ (v(B) = 1).

4.1.1 Axioms Let ,  be sets of literals. We define incon() iff El(l, ¬l ∈ ) con() iff ∼ El(l, ¬l ∈ ) val() iff El(l, ¬l ∈ ) inval() iff ∼ El(l, ¬l ∈ ). Notice that incon() iff val() and con() iff inval(). Define v |=E  if EA ∈ (v(A) = 0) v |=E  if EB ∈ (v(B) = 1) v |=A  if AA ∈ (v(A) = 0) v |=A  if AB ∈ (v(B) = 1). Proposition 4.1.2 Let ,  be sets of literals.  ⇒  is GEE -valid if and only if incon() or val() or  ∩  = ∅. Proof Assume that incon() or val() or  ∩  = ∅. Then, (i) if there is a literal l such that l, ¬l ∈  then for any assignment v, either v(l) = 0, or v(¬l) = 0, and v |=E ; (ii) if there is a literal m such that m, ¬m ∈  then for any assignment v, either v(m) = 1, or v(¬m) = 1, and v |=E ; and (iii) if there is a literal l such that l ∈  ∩  then for any assignment v, either v(l) = 0, or v(l) = 1, and hence, either v |=E , or v |=E . Assume that con()&inval()& ∩  = ∅. We define an assignment v such that for any variable p,

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⎧ ⎨ 1 if p ∈  or ¬p ∈  v(p) = 0 if ¬p ∈  or p ∈  ⎩ 1 otherwise. Then, v is well-defined and v |=EE  ⇒ .



Hence, we have the following axiom: (AEE )

incon() or val() or  ∩  = ∅ ⇒

Proposition 4.1.3 Let ,  be sets of literals.  ⇒  is GEA -valid if and only if incon() or  ⊆ . Proof Assume that incon() or  ⊆ . Then, for any assignment v, either v(m) = 1 for every m ∈  (i.e., v |=A ), or v(m) = 0 for some m ∈  ⊆ , i.e., v |=E . Assume that incon()&  . Let l ∈  − . We define an assignment v such that for any variable p, ⎧ 1 if p ∈  ⎪ ⎪ ⎪ ⎪ ⎨ 0 if ¬p ∈  v(p) = 0 if p = l ⎪ ⎪ ⎪ 1 if p = ¬l ⎪ ⎩ 1 otherwise. Then, v is well-defined and v |=EA  ⇒ .



Hence, we have the following axiom: (AEA )

incon() or  ⊆  ⇒

Proposition 4.1.4 Let ,  be sets of literals.  ⇒  is GAE -valid if and only if val() or  ⊆ . Proof Assume that val() or  ⊆ . Then, for any assignment v, either v(l) = 0 for every l ∈  (i.e., v |=A ), or v(l) = 1 for some l ∈  ⊆ , i.e., v |=E . Assume that inval()&  . Let l ∈  − . We define an assignment v such that for any variable p, ⎧ 0 if p ∈  ⎪ ⎪ ⎪ ⎪ 1 ⎨ if ¬p ∈  v(p) = 1 if p = l ⎪ ⎪ 0 if p = ¬l ⎪ ⎪ ⎩ 0 otherwise. Then, v is well-defined and v |=AE  ⇒ .



4.1 Gentzen Deduction Systems GQ1 Q2

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Hence, we have the following axiom: (AAE )

val() or  ⊆  ⇒

Proposition 4.1.5 Let ,  be sets of literals.  ⇒  is GAA -valid if and only if  = ∅ or  = ∅ or  =  = {l}. Proof Assume that  = ∅ or  = ∅ or  =  = {l}. Then, (i) if  = ∅ then for any assignment v, v |=A ; (ii) if  = ∅ then for any assignment v, v |=A ; and (iii) if  =  = {l} then for any assignment v, either v(l) = 0 for every l ∈  (i.e., v |=A ), or v(m) = 1 for every m ∈ , i.e., v |=A . Assume that  = ∅ = &( = {l} or  = {l}). Let l ∈  and l = l  ∈  − . We define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v(p) = 0 ⎪ ⎪ ⎪1 ⎪ ⎩ 0

if p = l  if p = ¬l  if p = l if p = ¬l otherwise.

Then, v is well-defined and v |=AA  ⇒ . Similar for l = m ∈  − . Hence, we have the following axiom: (AAA )

 = ∅ or  = ∅ or  =  = {l} ⇒

4.1.2 Deduction Rules GL0 consists of the following deduction rules: , A ⇒  (¬¬L )  , ¬¬A ⇒   , A1 ⇒  , A1 ⇒  (∧L ) , A2 ⇒  (∨L ) , A2 ⇒  ,A1 ∧ A2 ⇒  ,A1 ∨ A2 ⇒  , ¬A1 ⇒  , ¬A1 ⇒  (¬∧L ) , ¬A2 ⇒  (¬∨L ) , ¬A2 ⇒  , ¬(A1 ∨ A2 ) ⇒  , ¬(A1 ∨ A2 ) ⇒ 



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GR1 consists of the following deduction rules:  ⇒ B,  (¬¬R )   ⇒ ¬¬B,    ⇒ B1 ,   ⇒ B1 ,  (∧R )  ⇒ B2 ,  (∨R )  ⇒ B2 ,   ⇒ B1 ∧ B2 ,   ⇒ B1 ∨ B2 ,   ⇒ ¬B1 ,   ⇒ ¬B1 ,  (¬∧R )  ⇒ ¬B2 ,  (¬∨R )  ⇒ ¬B2 ,   ⇒ ¬(B1 ∧ B2 ),   ⇒ ¬(B1 ∨ B2 ), 

4.1.3 Deduction Systems Let Q1 , Q2 ∈ {E, A} and GQ1 Q2 = AQ1 Q2 + GL0 + GR1 Definition 4.1.6 A sequent  ⇒  is provable in GQ1 Q2 , denoted by Q1 Q2  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in GQ1 Q2 . Theorem 4.1.7 (Soundness and completeness theorem) Let Q1 , Q2 ∈ {E, A}. For any sequent  ⇒ , Q1 Q2  ⇒  iff |=Q1 Q2  ⇒ .



4.2 R-Calculi RQ1 Q2 Let Q1 , Q2 ∈ {E, A}. Given a sequent  ⇒  and pair (A, B) of formulas such that A ∈  and B ∈ , the result of  ⇒  GQ1 Q2 -revising (A, B) is sequent   ⇒  , denoted by |=Q1 Q2  ⇒ |(A, B) ⇒   ⇒  , if 

, A if |=Q1 Q2 , A ⇒   otherwise; , B if |=Q1 Q2   ⇒ , B  =  otherwise. 

 =

4.2 R-Calculi RQ1 Q2

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4.2.1 Deduction Rules Let X =  ⇒  and X[A] =  ⇒ |(A, B) ⇒  − {A} ⇒ |B X(A) =  ⇒ |(A, B) ⇒ , A ⇒ |B X[B] =   ⇒ |B ⇒   ⇒  − {B} X(B) =   ⇒ |B ⇒   ⇒ , B. There are four basic sets of deduction rules: • SLE : where A ∈ . X|A ⇒ X X|A ⇒ X[A] (¬¬L− ) (¬¬L0 ) X|¬¬A ⇒ X X|¬¬A ⇒ X[¬¬A]   X|A1 ⇒ X X|A1 ⇒ X[A1 ] (∧L0 ) X|A2 ⇒ X (∧L− ) X|A2 ⇒ X[A2 ] X|A X|A  1 ∧ A2 ⇒ X  1 ∧ A2 ⇒ X[A1 ∧ A2 ] X|A1 ⇒ X X|A1 ⇒ X[A1 ] (∨L0 ) X[A1 ]|A2 ⇒ X[A1 ] (∨L− ) X[A1 ]|A2 ⇒ X[A1 , A2 ] X|A1 ∨ A2 ⇒ X X|A1 ∨ A2 ⇒ X[A1 ∨ A2 ] and



X|¬A1 ⇒ X X[¬A1 ]|¬A2 ⇒ X[¬A1 ] X|¬(A1 ∧ A2 ) ⇒ X X|¬A1 ⇒ X[¬A1 ] (¬∧L− ) X[¬A1 ]|¬A2 ⇒ X[¬A1 , ¬A2 ] ¬(A1 ∧ A2 ) ⇒ X[¬(A1 ∧ A2 )]  X|¬A1 ⇒ X (¬∨L0 ) X|¬A2 ⇒ X ¬(A1 ∨ A2 ) ⇒X X|¬A1 ⇒ X[¬A1 ] (¬∨L− ) X|¬A2 ⇒ X[¬A2 ] X|¬(A1 ∨ A2 ) ⇒ X[¬(A1 ∨ A2 )

(¬∧L0 )

A formula A1 ∧ A2 is extractable from , if either A1 or A2 is extractable from ; and A1 ∨ A2 is extractable from  if A1 is extractable from  and A2 is extractable from  − {A1 }. A formula A1 ∧ A2 is not extractable from  if both A1 and A2 are not extractable from ; and A1 ∨ A2 is not extractable from  if either A1 is not extractable from  or A2 is not extractable from  − {A1 }.

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• SRE : where B ∈  X|B ⇒ X X|B ⇒ X[B] (¬¬R− ) (¬¬R0 )  X|¬¬B ⇒ X  X|¬¬B ⇒ X[¬¬B] X|B1 ⇒ X X|B1 ⇒ X[B1 ] (∧R0 ) X[B1 ]|B2 ⇒ X[B1 ] (∧R− ) X[B1 ]|B2 ⇒ X[B1 , B2 ] X|B X|B  1 ∧ B2 ⇒ X  1 ∧ B2 ⇒ X[B1 ∧ B2 ] X|B1 ⇒ X X|B1 ⇒ X[B1 ] (∨R0 ) X|B2 ⇒ X (∨R− ) X|B2 ⇒ X[B2 ] X|B1 ∨ B2 ⇒ X X|B1 ∨ B2 ⇒ X[B1 ∨ B2 ] and



X|¬B1 ⇒ X X|¬B2 ⇒ X X|¬(B1 ∧ B2 ) ⇒ X X|¬B1 ⇒ X[¬B1 ] (¬∧R− ) X|¬B2 ⇒ X[¬B2 ] X|¬(B1 ∧ B2 ) ⇒ X[¬(B1 ∧ B2 )  X|¬B1 ⇒ X (¬∨R0 ) X[¬B1 ]|¬B2 ⇒ X[¬B1 ] X|¬(B1 ∨ B2 ) ⇒ X X|¬B1 ⇒ X[¬B1 ] (¬∨R− ) X[¬B1 ]|¬B2 ⇒ X[¬B1 , ¬B2 ] X|¬(B1 ∨ B2 ) ⇒ X[¬(B1 ∨ B2 )] (¬∧R0 )

A formula B1 ∧ B2 is extractable from , if B1 is extractable from  and B2 is extractable from  − {B1 }; and B1 ∨ B2 is extractable from  if either B1 or B2 is extractable from . A formula B1 ∧ B2 is not extractable from  if either B1 is not extractable from  or B2 are not extractable from  − {B1 }; and B1 ∨ B2 is not extractable from  if both B1 and B2 are not extractable from . • SLA : X|A ⇒ X X|A ⇒ X(A) (¬¬L− ) (¬¬L0 ) X|¬¬A ⇒ X X|¬¬A ⇒ X(¬¬A)   X|A1 ⇒ X X|A1 ⇒ X(A1 ) (∧L0 ) X|A2 ⇒ X (∧L− ) X|A2 ⇒ X(A2 ) X|A X|A  1 ∧ A2 ⇒ X  1 ∧ A2 ⇒ X(A1 ∧ A2 ) X|A1 ⇒ X X|A1 ⇒ X(A1 ) (∨L0 ) X(A1 )|A2 ⇒ X(A1 ) (∨L− ) X(A1 )|A2 ⇒ X(A1 , A2 ) X|A1 ∨ A2 ⇒ X X|A1 ∨ A2 ⇒ X(A1 ∨ A2 ) and

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X|¬A1 ⇒ X X(¬A1 )|¬A2 ⇒ X(¬A1 ) X|¬(A1 ∧ A2 ) ⇒ X X|¬A1 ⇒ X(¬A1 ) (¬∧L− ) X(¬A1 )|¬A2 ⇒ X(¬A1 , ¬A2 ) ¬(A1 ∧ A2 ) ⇒ X(¬(A1 ∧ A2 ))  X|¬A1 ⇒ X (¬∨L0 ) X|¬A2 ⇒ X ¬(A1 ∨ A2 ) ⇒X X|¬A1 ⇒ X(¬A1 ) (¬∨L− ) X|¬A2 ⇒ X(¬A2 ) X|¬(A1 ∨ A2 ) ⇒ X(¬(A1 ∨ A2 )

(¬∧L0 )

A formula A1 ∧ A2 is enumerable from , if either A1 or A2 is enumerable from ; and A1 ∨ A2 is enumerable from  if A1 is enumerable from  and A2 is enumerable from , A1 . A formula A1 ∧ A2 is not enumerable from  if both A1 and A2 are not enumerable from ; and A1 ∨ A2 is not enumerable from  if either A1 is not enumerable from  or A2 is not enumerable from , A1 . • SRA : X|B ⇒ X X|B ⇒ X(B) (¬¬R− ) (¬¬R0 ) X|¬¬B ⇒ X X|¬¬B ⇒ X(¬¬B)   X|B1 ⇒ X X|B1 ⇒ X(B1 ) (∧R0 ) X(B1 )|B2 ⇒ X(B1 ) (∧R− ) X(B1 )|B2 ⇒ X(B1 , B2 ) X|B X|B  1 ∧ B2 ⇒ X  1 ∧ B2 ⇒ X(B1 ∧ B2 ) X|B1 ⇒ X X|B1 ⇒ X(B1 ) (∨R0 ) X|B2 ⇒ X (∨R− ) X|B2 ⇒ X(B2 ) X|B1 ∨ B2 ⇒ X X|B1 ∨ B2 ⇒ X(B1 ∨ B2 ) and



X|¬B1 ⇒ X X|¬B2 ⇒ X X|¬(B1 ∧ B2 ) ⇒ X X|¬B1 ⇒ X(¬B1 ) R X|¬B (¬∧− ) 2 ⇒ X(¬B2 ) X|¬(B1 ∧ B2 ) ⇒ X(¬(B1 ∧ B2 )  X|¬B1 ⇒ X (¬∨R0 ) X(¬B1 )|¬B2 ⇒ X(¬B1 ) X|¬(B1 ∨ B2 ) ⇒ X X|¬B1 ⇒ X(¬B1 ) (¬∨R− ) X(¬B1 )|¬B2 ⇒ X(¬B1 , ¬B2 ) X|¬(B1 ∨ B2 ) ⇒ X(¬(B1 ∨ B2 )) (¬∧R0 )

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A formula B1 ∧ B2 is enumerable from , if B1 is enumerable from  and B2 is enumerable from , B1 ; and B1 ∨ B2 is enumerable from  if either B1 or B2 is enumerable from . A formula B1 ∧ B2 is not enumerable from  if either B1 is not enumerable from  or B2 are not enumerable from , B1 ; and B1 ∨ B2 is not enumerable from  if both B1 and B2 are not enumerable from .

4.2.2 Axioms If a reduction  ⇒ |(l, m) ⇒   ⇒  is GQ1 Q2 -valid for literals l, m, then the reduction is called an axiom, where   ,  are sets of literals. We use deduction rules in GQ1 Q2 to reduce  ⇒  into literal sets. Let ,  be sets of literals and l, m be literals such that l ∈  and m ∈ . (AQ1 Q2 )  ⇒ |(l, m) ⇒   ⇒  

if

 ± l if Q1 Q2  ± l ⇒  otherwise   ± m if Q1 Q2   ⇒  ± m  =  otherwise 

 =



where ±1 l =

∪{l} if Q1 = A ±2 m = −{l} if Q1 = E,



∪{m} if Q2 = A −{m} if Q2 = E.

In detail, we have the following axioms: 

El  = l(l  , ¬l  ∈ ) El  = l(l  ∈ &l  ∈ )  ⇒ |(l, m)⇒  − {l} ⇒ |m ∼ El  = l(l  , ¬l  ∈ ) EE (A0L ) ∼ El  = l(l  ∈ &l  ∈ )  ⇒ |(l, m) ⇒  ⇒ |m  Em = m(m , ¬m ∈ ) Em = m(m ∈   &m ∈ ) (AEE −R )   ⇒ |m ⇒   ⇒  − {m} ∼ Em = m(m , ¬m ∈ ) EE (A0R ) ∼ Em = m(m ∈   &m ∈ )   ⇒ |m ⇒   ⇒  (AEE −L )

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where l ∈  and m ∈ , and   El = l(l  , ¬l  ∈ ) EA (A−L )  ⊆  − {l}  ⇒ |(l, m)⇒  − {l} ⇒ |m ∼ El  = l(l  , ¬l  ∈ ) EA (A0L )    − {l}  ⇒ |(l, m) ⇒  ⇒ |m   ∪ {m} ⊆  (AEA +R )   ⇒ |m ⇒   ⇒ , m  ∪ {m}    (AEA 0R )   ⇒ |m ⇒   ⇒  where l ∈ , and  ∪ {l} ⊆   ⇒ |(l, m) ⇒ , l ⇒ |m  ∪ {l}   (AAE 0L )  ⇒ |(l, m) ⇒  ⇒ |m  Em = m(m , ¬m ∈ ) AE (A−R )   ⊆  − {m}   ⇒ |m ⇒   ⇒  − {m} ∼ Em = m(m , ¬m ∈ ) AE (A+R )     − {m}   ⇒ |m ⇒   ⇒  (AAE +L )

where m ∈ , and l∈  ⇒ |(l, m) ⇒ , l ⇒ |m l∈ / (AAA 0L )  ⇒ |(l, m) ⇒  ⇒ |m m∈ (AAA +R )   ⇒ |m ⇒   ⇒ , m m∈ / (AAA 0R )   ⇒ |m ⇒   ⇒ 

(AAA +L )

Proposition 4.2.1 Let  ⇒  be literal.  ⇒ |(A, B) ⇒   ⇒  is RQ1 Q2 -valid if and only if  ⇒ |(A, B) ⇒   ⇒  is the conclusion of an axiom. 

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4.2.3 Deduction Systems RQ1 Q2 Define RQ1 Q2 = AQ1 Q2 + SLQ1 + SRQ2 . Definition 4.2.2 A reduction δ =  ⇒ |(A, B) ⇒   ⇒  is provable in RQ1 Q2 , denoted by Q1 Q2 δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in RQ1 Q2 . Theorem 4.2.3 (Soundness and completeness theorem) For any Q1 , Q2 ∈ {A, E} and reduction δ =  ⇒ |(A, B) ⇒   ⇒  , δ is RQ1 Q2 -valid if and only if δ is provable in RQ1 Q2 . That is, Q1 Q2 δ iff |=Q1 Q2 δ. Proof Here we give the proof of completeness part for Q1 = Q2 = E. Given a reduction  ⇒ |(A, B) ⇒  ± {A} ⇒ |B, we construct a trees T such that either (i) T is a proof tree of  ⇒ |(A, B) ⇒  ± {A} ⇒ |B, i.e., for each branch ξ of T , there is a reduction γ at the leaf of ξ which is an axiom of form (At− ), or (ii) there is a branch ξ ∈ T such that ξ is a proof of  ⇒ |(A, B) ⇒  ⇒ |B. T is constructed as follows: • the root of T is  ⇒ |(A, B) ⇒  ± {A} ⇒ |B; • for a node ξ , if each reduction at ξ is of form   ⇒  |(l  , B) ⇒  − {l  } ⇒ |B then the node is a leaf; • otherwise, ξ has the direct child containing the following reductions: ⎧  ⎪ 1 ⇒ 1 |(A, B) ⇒ 1 − {A} ⇒ 1 |B ⎪ ⎪ ⎪ 1 ⇒ 1 |(A1 , B) ⇒ 1 − {A1 } ⇒ 1 |B ⎪ ⎪ ⎪ 1 ⇒ 1 |(A2 , B) ⇒ 1 − {A2 } ⇒ 1 |B ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⇒ 1 |(¬A1 , B) ⇒ 1 − {¬A1 } ⇒ 1 |B  1 − {¬A1 } ⇒ 1 |(¬A2 , B) ⇒ 1 − {¬A1 , ¬A2 } ⇒ 1 |B ⎪ ⎪ ⎪ 1 ⇒ 1 |(A1 , B) ⇒ 1 − {A1 } ⇒ 1 |B ⎪ ⎪ ⎪ ⎪ ⎪  1 − {A1 } ⇒ 1 |(A2 , B) ⇒ 1 − {A1 , A2 } ⇒ 1 |B ⎪ ⎪ ⎪ ⎩ 1 ⇒ 1 |(¬A1 , B) ⇒ 1 − {¬A1 } ⇒ 1 |B 1 ⇒ 1 |(¬A2 , B) ⇒ 1 − {¬A2 } ⇒ 1 |B

if 1 (¬¬A, B) ∈ ξ if 1 (A1 ∧ A2 , B) ∈ ξ if 1 (¬(A1 ∧ A2 ), B) ∈ ξ if 1 (A1 ∨ A2 , B) ∈ ξ if 1 (¬(A1 ∨ A2 ), B) ∈ ξ

where 1 (A, B) = 1 ⇒ 1 |(A, B) ⇒ 1 − {A} ⇒ 1 |B. Lemma 4.2.4 If for each branch ξ ⊆ T , there is a reduction 1 ⇒ 1 |(A, B) ⇒ 1 − {A} ⇒ |B ∈ ξ which is the conclusion of an axiom in REE then T is a proof tree of  ⇒ |(A, B) ⇒  − {A} ⇒ |B in REE . Proof By the definition of T , T is a proof tree of  ⇒ |(A, B) ⇒  − {A} ⇒ |B. 

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Lemma 4.2.5 If there is a branch ξ ⊆ T such that each reduction 1 ⇒ 1 | (A , B) ⇒ 1 − {A } ⇒ |B ∈ ξ is not the conclusion of an axioms in REE then ξ is a proof of  ⇒ |(A, B) ⇒  ⇒ |B. Proof Let ξ be a branch of T such that each reduction 1 ⇒ 1 |(A , B) ⇒ 1 − {A } ⇒ |B ∈ ξ is not the conclusion of an axiom in REE . We proved by induction on node η ∈ ξ that for each reduction   ⇒ |(A , B) ⇒   ⇒ |B ∈ η, t   ⇒ |(A , B) ⇒   ⇒ |B. Case   ⇒  |(l, m) ⇒   − {l} ⇒  |m ∈ η is a leaf node of ξ. Then, there is no literal l  = l such that l  , ¬l  ∈   or l  ∈   ∩  . t   ⇒  |(l, m) ⇒   ⇒  |m. Case δ = 2 ⇒ 2 |(¬¬A, B) ⇒ 2 − {¬¬A} ⇒ 2 |B ∈ η. Then, η has a child node ∈ ξ containing reduction 2 ⇒ 2 |(A, B) ⇒ 2 − {A} ⇒ 2 |B. By induction assumption, t 2 ⇒ 2 |(A, B) ⇒ 2 ⇒ 2 |B, and by (¬¬0 ), t 2 ⇒ 2 |(¬¬ A, B) ⇒ 2 ⇒ 2 |B. Case δ = 2 ⇒ 2 |(A1 ∧ A2 , B) ⇒ 2 − {A1 ∧ A2 } ⇒ 2 |B ∈ η. Then, η has a child node ∈ ξ containing reductions 2 ⇒ 2 |(A1 , B) ⇒ 2 − {A1 } ⇒ 2 |B and 2 ⇒ 2 |(A2 , B) ⇒ 2 − {A2 } ⇒ 2 |B. By induction assumption, t 2 ⇒ 2 |(A1 , B) ⇒ 2 ⇒ 2 |B, and t 2 ⇒ 2 |(A2 , B) ⇒ 2 ⇒ 2 |B. By (∧0 ), we have t 2 ⇒ 2 |(A1 ∧ A2 , B) ⇒ 2 ⇒ 2 |B. Case δ = 2 ⇒ 2 |(A1 ∨ A2 , B) ⇒ 2 ⇒ 2 |B ∈ η. Then, η has a child node ∈ ξ containing reduction either 2 ⇒ 2 |(A1 , B) ⇒ 2 − {A1 } ⇒ 2 |B or 2 − {A1 } ⇒ 2 |(A2 , B) ⇒ 2 − {A1 , A2 } ⇒ 2 |B. By induction assumption, either t 2 ⇒ 2 |(A1 , B) ⇒ 2 ⇒ 2 |B or t 2 − {A1 } ⇒ 2 |(A2 , B) ⇒ 2 − {A1 } ⇒ 2 |B. By (∨0 ), we have t 2 ⇒ 2 |(A1 ∨ A2 , B) ⇒ 2 ⇒ 2 |B. Similar for other cases.  Given a reduction  ⇒ |B ⇒  ⇒  , we construct a trees T  such that either (i) T  is a proof tree of  ⇒ |B ⇒  ⇒  − {B}, i.e., for each branch ξ of T  , there is a reduction γ at the leaf of ξ which is an axiom of form (At− ). (ii) There is a branch ξ ∈ T  such that ξ is a proof of  ⇒ |B ⇒  ⇒ . T  is constructed as follows: • The root of T  is  ⇒ |B ⇒  ⇒  − {B}. • For a node ξ , if each reduction at ξ is of form  ⇒ |m ⇒  ⇒  − {m} then the node is a leaf. • Otherwise, ξ has the direct child containing the following reductions:

114

⎧ ⎪ ⎪ 1 ⇒ 1 |B ⇒ 1 ⇒ 1 − {B} ⎪ ⎪ 1 ⇒ 1 |B1 ⇒ 1 ⇒ 1 − {B1 } ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 ⇒ 1 − {B1 }|B2 ⇒ 1 ⇒ 1 − {B1 , B2 } ⎪ ⎪ ⎨ 1 ⇒ 1 |¬B1 ⇒ 1 ⇒ 1 − {¬B1 }  1 ⇒ 1 |¬B2 ⇒ 1 ⇒ 1 − {¬B2 } ⎪ ⎪ 1 ⇒ 1 |B1 ⇒ 1 ⇒ 1 − {B1 } ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 ⇒ 1 |B2 ⇒ 1 ⇒ 1 − {B2 } ⎪ ⎪ 1 ⇒ 1 |¬B1 ⇒ 1 ⇒ 1 − {¬B1 } ⎪ ⎪ ⎩ 1 ⇒ 1 − {¬B1 }|¬B2 ⇒ 1 ⇒ 1 − {¬B1 , ¬B2 }

4 R-Calculi RQ1 Q2 /RQ1 Q2

if 1 (¬¬B) ∈ ξ if 1 (B1 ∧ B2 ) ∈ ξ if 1 (¬(B1 ∧ B2 )) ∈ ξ if 1 (B1 ∨ B2 ) ∈ ξ if 1 (¬(B1 ∨ B2 )) ∈ ξ

where 1 (B) = 1 ⇒ 1 |B ⇒ 1 ⇒ 1 − {B}. Lemma 4.2.6 If for each branch ξ ⊆ T  , there is a reduction 1 ⇒ 1 |B ⇒ 1 ⇒  − {B } ∈ ξ which is the conclusion of an axiom in REE then T  is a proof tree of  ⇒ |B ⇒  ⇒  − {B} in REE . Proof By the definition of T  , T  is a proof tree of  ⇒ |B ⇒  ⇒  − {B} in  REE . Lemma 4.2.7 If there is a branch ξ ⊆ T  such that each reduction 1 ⇒ 1 |B ⇒ 1 ⇒  − {B } ∈ ξ is not the conclusion of an axiom in REE then ξ is a proof of  ⇒ |B ⇒  ⇒ . Proof Let ξ be a branch of T  such that any reduction 1 ⇒ 1 |B ⇒ 1 ⇒ 1 − {B } ∈ ξ is not an axiom in REE . We proved by induction on node η ∈ ξ that for each reduction   ⇒ |B ⇒   ⇒  − {B } ∈ η, t   ⇒ |B ⇒   ⇒ . Case   ⇒ |m ⇒   ⇒  − {m} ∈ η is a leaf node of ξ. Then, there is no literal  m = m such that m , ¬m ∈  or m ∈   ∩ . t   ⇒ |m ⇒   ⇒ . Case 2 ⇒ 2 |¬¬B ⇒ 2 ⇒ 2 − {¬¬B} ∈ η. Then, η has a child node ∈ ξ containing reduction 2 ⇒ 2 |B ⇒ 2 ⇒ 2 − {B}. By induction assumption, t 2 ⇒ 2 |B ⇒ 2 ⇒ 2 , and by (¬¬0 ), t 2 ⇒ 2 |¬¬B ⇒ 2 ⇒ 2 . Case δ = 2 ⇒ 2 |B1 ∧ B2 ⇒ 2 ⇒ 2 − {B1 ∧ B2 } ∈ η. Then, η has a child node ∈ ξ containing reduction either 2 ⇒ 2 |B1 ⇒ 2 ⇒ 2 − {B1 } or 2 ⇒ 2 − {B1 }|B2 ⇒ 2 ⇒ 2 − {B1 , B2 }. By induction assumption, either t 2 ⇒ 2 |B1 ⇒ 2 ⇒ 2 , or t 2 ⇒ 2 − {B1 }|B2 ⇒ 2 ⇒ 2 − {B1 }. By (∧0 ), we have t 2 ⇒ 2 |B1 ∧ B2 ⇒ 2 ⇒ 2 . Case δ = 2 ⇒ 2 |B1 ∨ B2 ⇒ 2 ⇒ 2 − {B1 ∨ B2 } ∈ η. Then, η has a child node ∈ ξ containing reductions 2 ⇒ 2 |B1 ⇒ 2 ⇒ 2 − {B1 } and 2 ⇒ 2 | B2 ⇒ 2 ⇒ 2 − {B2 }. By induction assumption, t 2 ⇒ 2 |B1 ⇒ 2 ⇒ 2 and t 2 ⇒ 2 |B2 ⇒ 2 ⇒ 2 . By (∨0 ), we have t 2 ⇒ 2 |B1 ∨ B2 ⇒ 2 ⇒ 2 . Similar for other cases. 

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115

4.3 Gentzen Deduction Systems GQ1 Q2 Definition 4.3.1 A co-sequent  →  is GQ1 Q2 -valid, denoted by |=Q1 Q2  → , if there is an assignment v such that Q1 A ∈ (v(A) = 1) and Q2 B ∈ (v(B) = 0).

4.3.1 Axioms Lemma 4.3.2 Given two sets ,  of literals, |=AA  →  if and only if con()& inval()& ∩  = ∅.  Hence, we have the following axiom: (AAA )

con()&inval()& ∩  = ∅  → 

Proposition 4.3.3 Let ,  be sets of literals. |=EA  →  if and only if inval()&   .  Hence, we have the following axiom: (AEA )

inval()&    → 

Proposition 4.3.4 Let ,  be sets of literals. |=AE  →  if and only if con()&   .  Hence, we have the following axiom: (AAE )

con()&    → 

Proposition 4.3.5 Let ,  be sets of literals. |=EE  →  if and only if  = ∅ = & =  = {l} for any literal l.  Hence, we have the following axiom: (AEE )

 = ∅ = & =  = {l}  → 

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4.3.2 Deduction Rules GL1 consists of the following deduction rules: , A →  (¬¬L ) ,  ¬¬A →   , A1 →  , A1 →  (∧L ) , A2 →  (∨L ) , A2 →  ,A1 ∧ A2 →  ,A1 ∨ A2 →  , ¬A1 →  , ¬A1 →  (¬∧L ) , ¬A2 →  (¬∨L ) , ¬A2 →  , ¬(A1 ∨ A2 ) →  , ¬(A1 ∨ A2 ) →  GR0 consists of the following deduction rules:  → B,  (¬¬R )   → ¬¬B,    → B1 ,   → B1 ,  (∧R )  → B2 ,  (∨R )  → B2 ,   → B1 ∧ B2 ,   → B1 ∨ B2 ,   → ¬B1 ,   → ¬B1 ,  (¬∧R )  → ¬B2 ,  (¬∨R )  → ¬B2 ,   → ¬(B1 ∧ B2 ),   → ¬(B1 ∨ B2 ), 

4.3.3 Deduction Systems For Q1 , Q2 ∈ {A, E}, let GQ1 Q2 = AQ1 Q2 + GL1 + GR0 . Definition 4.3.6 A co-sequent  →  is provable in GQ1 Q2 , denoted by Q1 Q2  → , if there is a sequence {1 → 1 , . . . , n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in GQ1 Q2 . Theorem 4.3.7 (Soundness and completeness theorem) Let Q1 , Q2 ∈ {E, A}. For any co-sequent  → , Q1 Q2  →  if and only if |=Q1 Q2  → .



4.4 R-Calculi RQ1 Q2

117

4.4 R-Calculi RQ1 Q2 Let Q1 , Q2 ∈ {E, A}. Given a co-sequent  →  and pair (A, B) of formulas, the result of  →  GQ1 Q2 -revising (A, B) is denoted by |=Q1 Q2  → |(A, B) ⇒   →  , 

if

 ± {A} if Q1 Q2  ± {A} →   otherwise;   ± {B} if Q1 Q2   →  ± {B}  =  otherwise, 

 =

where



∪{l} if Q1 = A −{l} if Q1 = E and l ∈ , ∪{m} if Q2 = A ±2 m = −{m} if Q2 = E and m ∈ .

±1 l =

For example, let Q1 = A, Q2 = E and B ∈ . Then, 

, A if AE , A →   otherwise;   − {B} if AE   →  − {B}  =  otherwise. 

 =

4.4.1 Axioms A reduction  → |(A, B) ⇒   →  is reduced to literals for A and B, that is,   ,  are sets of literals. We use deduction rules in GQ1 Q2 to reduce  →  into literal ones, say   →  . Let ,  be sets of literals and l, m be literals. (AQ1 Q2 )  ⇒ |(l, m) ⇒   ⇒  if



 ± l if Q1 Q2  ± l ⇒  otherwise   ± m if Q1 Q2   ⇒  ± m  =  otherwise. 

 =

In detail, we have the following axioms: Let ,  be sets of literals.

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¬l ∈ / l∈ /  → |(l, m)⇒ , l → |m ¬l ∈  l∈ (A0L ) AA  → |(l, m) ⇒  → |m  m∈ /  ¬m ∈ / (A+R AA )   → |m ⇒  → , m m ∈  0R (AAA ) ¬m ∈    → |m ⇒   → 

(A+L AA )

and

, l    → |(l, m) ⇒  − {l} → |m , l ⊆  (A0L EA )   → |(l, m) ⇒   → |m ¬m ∈ / (A+R EA )    → |m ⇒    →  − {m} ¬m ∈  (A0R EA )    → |m ⇒    → 

(A−L EA )

where l ∈ , and ¬l ∈ /  → |(l, m) ⇒ , l → |m ¬l ∈  (A0L AA )   → |(l, m) ⇒   → |m  − {m}   (A−R AA )    → |m ⇒    →  − {m}  − {m} ⊆  (A0R AA )    → |m ⇒    →  (A+L AE )

where m ∈ , and  − {l} = {l  } =   → |(l, m) ⇒  − {l} → |m  − {l} = {l  } =  (A0L EE )   → |(l, m) ⇒   → |m   = {l  } =  − {m} (A−R EE )    → |m ⇒    →  − {m}   = {l  } =  − {m} (A0R EE )    → |m ⇒    → 

(A−L EE )

where l ∈  and m ∈ .

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Proposition 4.4.1 Let   →  be literal.  → |(A, B) ⇒   →  is RQ1 Q2  valid if and only if  → |(A, B) ⇒   →  is the conclusion of an axiom.

4.4.2 Deduction Rules Let X =  →  and X[A] =  → |(A, B) ⇒  − {A} → |B X(A) =  → |(A, B) ⇒ , A → |B X[B] =   → |B ⇒   →  − {B} X(B) =   → |B ⇒   → , B. • SLA :

and

X|A ⇒ X(A) X|A ⇒ X (¬¬L0 ) (¬¬L+ ) X|¬¬A ⇒ X(¬¬A ) X|¬¬A ⇒X 1   X|A1 ⇒ X(A1 ) X|A1 ⇒ X (∧L+ ) X(A1 )|A2 ⇒ X(A1 , A2 ) (∧L0 ) X(A1 )|A2 ⇒ X(A1 ) X|A X|A  1 ∧ A2 ⇒ X(A1 ∧ A2 )  1 ∧ A2 ⇒ X X|A1 ⇒ X(A1 ) X|A1 ⇒ X (∨L+ ) X|A2 ⇒ X(A2 ) (∨L0 ) X|A2 ⇒ X X|A1 ∨ A2 ⇒ X(A1 ∨ A2 ) X|A1 ∨ A2 ⇒ X 

X|¬A1 ⇒ X(¬A1 ) X|¬A2 ⇒ X(¬A2 ) X|¬(A1 ∧ A2 )⇒ X(¬(A1 ∧ A2 )) X|¬A1 ⇒ X (¬∧L0 ) X|¬A2 ⇒ X X|¬(A1 ∧ A2 ) ⇒ X  X|¬A1 ⇒ X(¬A1 ) (¬∨L+ ) X(¬A1 )|¬A2 ⇒ X(¬A1 , ¬A2 ) X|¬(A1 ∨ A2 )⇒ X(¬(A1 ∨ A2 )) X|¬A1 ⇒ X (¬∨L0 ) X(¬A1 )|¬A2 ⇒ X(¬A1 ) X|¬(A1 ∨ A2 ) ⇒ X (¬∧L+ )

A formula A1 ∧ A2 is enumerable into , if either A1 or A2 is enumerable into ; and A1 ∨ A2 is enumerable into  if A1 is enumerable into  and A2 is enumerable into  ∪ {A1 }.

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A1 ∧ A2 is not enumerable into , if both A1 and A2 are not enumerable into ; and A1 ∨ A2 is not enumerable into  if either A1 is not enumerable into  or A2 is not enumerable into  ∪ {A1 }. • SRA : X|B ⇒ X(B) X|B ⇒ X (¬¬R+ ) (¬¬R0 )  X|¬¬B ⇒ X(¬¬B)  X|¬¬B ⇒ X X|B1 ⇒ X(B1 ) X|B1 ⇒ X (∧R+ ) X|B2 ⇒ X(B2 ) (∧R0 ) X|B2 ⇒ X X|B X|B  1 ∧ B2 ⇒ X(B1 ∧ B2 )  1 ∧ B2 ⇒ X X|B1 ⇒ X(B1 ) X|B1 ⇒ X (∨R+ ) X(B1 )|B2 ⇒ X(B1 , B2 ) (∨R0 ) X(B1 )|B2 ⇒ X(B1 ) X|B1 ∨ B2 ⇒ X(B1 ∨ B2 ) X|B1 ∨ B2 ⇒ X and 

X|¬B1 ⇒ X(¬B1 ) X(¬B1 )|¬B2 ⇒ X(¬B1 , ¬B2 ) X|¬(B1 ∧ B2 )⇒ X(¬(B1 ∧ B2 )) X|¬B1 ⇒ X (¬∧R0 ) X(¬B1 )|¬B2 ⇒ X(¬B1 ) X|¬(B1 ∧ B2 ) ⇒ X  X|¬B1 ⇒ X(¬B1 ) (¬∨R+ ) X|¬B2 ⇒ X(¬B2 ) X|¬(B1 ∨ B2 )⇒ X¬(B1 ∨ B2 )) X|¬B1 ⇒ X (¬∨R0 ) X|¬B2 ⇒ X X|¬(B1 ∨ B2 ) ⇒ X (¬∧R+ )

A formula B1 ∧ B2 is enumerable into , if B1 or B2 is enumerable into ; and B1 ∨ B2 is enumerable into  if B1 is enumerable into  and B2 is enumerable into  ∪ {B1 }. Formula B1 ∧ B2 is not enumerable into , if both B1 and B2 are not enumerable into ; and B1 ∨ B2 is not enumerable into  if either B1 is not enumerable into , or B2 is not enumerable into  ∪ {B1 }.

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• SLE : X|A ⇒ X[A] X|¬¬A ⇒ X[¬¬A1 ] X|A ⇒ X (¬¬L0 ) X|¬¬A ⇒X  X|A1 ⇒ X[A1 ] (∧L+ ) X[A1 ]|A2 ⇒ X[A1 , A2 ] X|A1 ∧ A2 ⇒ X[A1 ∧ A2 ] X|A1 ⇒ X (∧L0 ) X[A1 ]|A2 ⇒ X[A1 ] X|A1 ∧ A2 ⇒ X  X|A1 ⇒ X[A1 ] (∨L+ ) X|A2 ⇒ X[A2 ] X|A1 ∨ A2 ⇒ X[A1 ∨ A2 ] X|A1 ⇒ X (∨L0 ) X|A2 ⇒ X X|A1 ∨ A2 ⇒ X (¬¬L+ )

and 

X|¬A1 ⇒ X[¬A1 ] X|¬A2 ⇒ X[¬A2 ] X|¬(A1 ∧ A2 )⇒ X[¬(A1 ∧ A2 )] X|¬A1 ⇒ X L (¬∧0 ) X|¬A2 ⇒ X X|¬(A1 ∧ A2 ) ⇒ X  X|¬A1 ⇒ X[¬A1 ] (¬∨L+ ) X[¬A1 ]|¬A2 ⇒ X[¬A1 , ¬A2 ] X|¬(A1 ∨ A2 )⇒ X[¬(A1 ∨ A2 )] X|¬A1 ⇒ X (¬∨L0 ) X[¬A1 ]|¬A2 ⇒ X[¬A1 ] X|¬(A1 ∨ A2 ) ⇒ X (¬∧L+ )

A formula A1 ∧ A2 is extractable into , if either A1 or A2 is extractable into ; and A1 ∨ A2 is extractable into  if A1 is extractable into  and A2 is extractable into  ∪ {A1 }. A1 ∧ A2 is not extractable into , if both A1 and A2 are not extractable into ; and A1 ∨ A2 is not extractable into  if either A1 is not extractable into  or A2 is not extractable into  ∪ {A1 }.

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• SRE : X|B ⇒ X[B] X|¬¬B ⇒ X[¬¬B] X|B ⇒ X (¬¬R0 ) X|¬¬B ⇒X  X|B1 ⇒ X[B1 ] (∧R+ ) X|B2 ⇒ X[B2 ] X|B1 ∧ B2 ⇒ X[B1 ∧ B2 ] X|B1 ⇒ X (∧R0 ) X|B2 ⇒ X X|B1 ∧ B2 ⇒ X  X|B1 ⇒ X[B1 ] (∨R+ ) X[B1 ]|B2 ⇒ X[B1 , B2 ] X|B1 ∨ B2 ⇒ X[B1 ∨ B2 ] X|B1 ⇒ X (∨R0 ) X[B1 ]|B2 ⇒ X[B1 ] X|B1 ∨ B2 ⇒ X

(¬¬R+ )

and 

X|¬B1 ⇒ X[¬B1 ] X[¬B1 ]|¬B2 ⇒ X[¬B1 , ¬B2 ] X|¬(B1 ∧ B2 )⇒ X[¬(B1 ∧ B2 )] X|¬B1 ⇒ X R X[¬B (¬∧0 ) 1 ]|¬B2 ⇒ X[¬B1 ] X|¬(B1 ∧ B2 ) ⇒ X  X|¬B1 ⇒ X[¬B1 ] (¬∨R+ ) X|¬B2 ⇒ X[¬B2 ] X|¬(B1 ∨ B2 )⇒ X¬(B1 ∨ B2 )] X|¬B1 ⇒ X (¬∨R0 ) X|¬B2 ⇒ X X|¬(B1 ∨ B2 ) ⇒ X (¬∧R+ )

A formula B1 ∧ B2 is extractable into , if B1 or B2 is extractable into ; and B1 ∨ B2 is extractable into  if B1 is extractable into  and B2 is extractable into  ∪ {B1 }. Formula B1 ∧ B2 is not extractable into , if both B1 and B2 are not extractable into ; and B1 ∨ B2 is not extractable into  if either B1 is not extractable into , or B2 is not extractable into  ∪ {B1 }.

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4.4.3 Deduction Systems RQ1 Q2 For Q1 , Q2 ∈ {A, E}, define RQ1 Q2 = AQ1 Q2 + SLQ1 + SRQ2 . Theorem 4.4.2 (Soundness and completeness theorem) For any Q1 , Q2 ∈ {A, E} and reduction δ =  → |(A, B) ⇒   →  , δ is RQ1 Q2 -valid if and only if δ is provable in RQ1 Q2 . That is, Q1 Q2 δ iff |=Q1 Q2 δ. 

4.5 Conclusions It is true that

|=Q1 Q2  ⇒  iff |=Q1 Q2  →  Q1 Q2  ⇒  iff Q1 Q2  → 

and

|=Q1 Q2  ⇒ |(A, B) ⇒   ⇒  iff |=Q1 Q2  → |(A, B) ⇒   →  Q1 Q2  ⇒ |(A, B) ⇒   ⇒  iff Q1 Q2  → |(A, B) ⇒   →  , where E = A and A = E. There are eight Gentzen deduction systems: Av GEE GEA GAE GAA Qv GAA GAE GEA GEE where

GQ1 Q2 = AQ1 Q2 + GL0 + GR1 GQ1 Q2 = AQ1 Q2 + GL1 + GR0 GL0 ≡ GL0 , GL1 ≡ GL1 , GR0 ≡ GR0 , GR1 ≡ GR1 .

Moreover, GQ1 Q2 /GQ1 Q2 is monotonic in  if and only if Q1 = E; and nonmonotonic in  if and only if Q2 = A. Correspondingly there are eight R-calculi: Av REE REA RAE RAA Ev RAA RAE REA REE

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where

RQ1 Q2 = AQ1 Q2 + RLQ1 + RRQ2 RQ1 Q2 = AQ1 Q2 + RLQ1 + RRQ2 ; RLQ1 ≡ RRQ1 , RRQ2 ≡ RLQ2 .

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symbolic Logic 50, 510–530 (1985) Cao, C., Sui, Y., Wang, Y.: The nonmonotonic propositional logics. Artif. Intell. Res. 5, 111–120 (2016) Darwiche, A., Pearl, J.: On the logic of iterated belief revision. Artif. Intell. 89, 1–29 (1997) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Gärdenfors, P., Rott, H.: Belief revision. In: Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp. 35–132. Oxford University Press, Oxford (1995) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W.: Mathematical logic, foundations for information science. In: Progress in Computer Science and Applied Logic, vol. 25. Birkhäuser (2010) Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980) Takeuti, G.: Proof theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987)

Chapter 5

R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j

We consider sequents of form G Q 1 i Q 2 j and co-sequents of form G Q 1 i Q 2 j (Li 2010; Takeuti and Barwise 1987), and corresponding R-calculi R Q 1 i Q 2 j and R Q 1 i Q 2 j (Li 2007), where Q 1 , Q 2 ∈ A, E and i, j ∈ 0, 1. Let Q 1 , Q 2 ∈ A, E and i, j ∈ 0, 1. • G Q 1 i Q 2 j : A sequent  ⇒  is G Q 1 i Q 2 j -valid, denoted by |= Q 1 i Q 2 j  ⇒ , if for any assignment v, either Q 1 A ∈ (v(A) = i) or Q 2 B ∈ (v(B) = j). • G Q 1 i Q 2 j : A co-sequent  →  is G Q 1 i Q 2 j -valid, denoted by |= Q 1 i Q 2 j  → , if there is an assignment v such that Q 1 A ∈ (v(A) = i) and Q 2 B ∈ (v(B) = j). There are 32 kinds of sequents and co-sequents: sequents GE0E0 GE0E1 GE1E0 GE1E1 co-sequents GA1A1 GA1A0 GA0A1 GA0A0

GE0A0 GE0A1 GE1A0 GE1A1 GA1E1 GA1E0 GA0E1 GA0E0

GA0E0 GA0E1 GA1E0 GA1E1 GE1A1 GE1A0 GE0A1 GE0A0

GA0A0 GA0A1 GA1A0 GA1A1 GE1E1 GE1E0 GE0E1 GE0E0

where GE0E1 = GEE , GE0A1 = GEA , GA0E1 = GAE , GA0A1 = GAA . • R Q 1 i Q 2 j : Given a sequent  ⇒  and pair (A, B) of formulas, the result of  ⇒  G Q 1 i Q 2 j -revising (A, B) is   ⇒  , denoted by |= Q 1 i Q 2 j  ⇒ |(A, B) ⇒   ⇒  , © Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_5

125

5 R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j

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where ⎧  ±1 A ⇒  ±2 B if |= Q 1 i Q 2 j  ±1 A ⇒  ±2 B ⎪ ⎪ ⎨ if |= Q 1 i Q 2 j  ±1 A ⇒   ±1 A ⇒    ⇒  = B if |= Q 1 i Q 2 j  ⇒  ±2 B  ⇒  ± ⎪ 2 ⎪ ⎩ ⇒ otherwise, where  ±1 A =

∪{A} if Q 1 = A −{A} if Q 1 = E,

 ±2 B =

∪{B} if Q 2 = A −{B} if Q 2 = E.

• R Q 1 i Q 2 j : Given a co-sequent  →  and pair (A, B), the result of  →  G Q 1 i Q 2 j -revising (A, B) is   →  , denoted by |= Q 1 i Q 2 j  → |(A, B) ⇒   →  , where ⎧  ±1 A →  ±2 B if  ±1 A →  ±2 BisG Q 1 i Q 2 j −valid ⎪ ⎪ ⎨ otherwise, if  ±1 A → isG Q 1 i Q 2 j −valid  ±1 A →    →  = B otherwise, if  →  ±2 BisG Q 1 i Q 2 j −valid   →  ± ⎪ 2 ⎪ ⎩  →  otherwise; Correspondingly, there are 32 kinds of R-calculi: sequents RE0E0 RE0E1 RE1E0 RE1E1 co-sequents RA1A1 RA1A0 RA0A1 RA0A0

RE0A0 RE0A1 RE1A0 RE1A1 RA1E1 RA1E0 RA0E1 RA0E0

RA0E0 RA0E1 RA1E0 RA1E1 RE1A1 RE1A0 RE0A1 RE0A0

RA0A0 RA0A1 RA1A0 RA1A1 RE1E1 RE1E0 RE0E1 RE0E0

where RE0E1 = REE , RE0A1 = REA , RA0E1 = RAE , RA0A1 = RAA . We have the following equivalences: GEE = GE0E1 , GAA = GA0A1 ; GAA = GA1A0 , GEE = GE1E0 . About the monotonicity (Cao et al. 2016; Ginsberg 1987; Reiter 1980), we have the following conclusions: • G Q 1 i Q 2 j is monotonic in  iff Q 1 = E, and monotonic in  iff Q 2 = E; • G Q 1 i Q 2 j is nonmonotonic in  iff Q 1 = A, and nonmonotonic in  iff Q 2 = A;

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• R Q 1 i Q 2 j (+/−) is monotonic in  iff Q 1 = E, and monotonic in  iff Q 2 = E; and • R Q 1 i Q 2 j (+/−) is nonmonotonic in  iff Q 1 = E, and nonmonotonic in  iff Q 2 = E.

5.1 Gentzen Deduction Systems GE0∗ /GE1∗ Let ∗ ∈ {A1, A0, E1, E0}. Definition 5.1.1 A sequent  ⇒  is GE0∗ -valid, denoted by |=E0∗  ⇒ , if for any assignment v, either v(A) = 0 for some A ∈  or ⎧ v(B) = 0 for someB ∈  if ∗ = E0 ⎪ ⎪ ⎨ v(B) = 0 for everyB ∈  if ∗ = A0 v(B) = 1 for someB ∈  if ∗ = E1 ⎪ ⎪ ⎩ v(B) = 1 for everyB ∈  if ∗ = A1 Definition 5.1.2 A co-sequent  →  is GA1∗ -valid, denoted by |=A1∗  → , if there is an assignment v such that v(A) = 1 for every A ∈ , and ⎧ v(B) = 1 for every B ∈  if ∗ = A1 ⎪ ⎪ ⎨ v(B) = 1 for some B ∈  if ∗ = E1 v(B) = 0 for every B ∈  if ∗ = A0 ⎪ ⎪ ⎩ v(B) = 0 for some B ∈  if ∗ = E0

5.1.1 Axioms Proposition 5.1.3 Let ,  be sets of literals.  ⇒  is GE0E1 -valid if and only if incon() or val() or  ∩  = ∅. Proof Because  ⇒  is GE0E1 -valid iff  ⇒  is GEE -valid.



Hence, we have the following axioms: incon() or val() or d ∩  = ∅ ⇒ con()&inval()& ∩  = ∅ (AA1A0 )  →  (AE0E1 )

Proposition 5.1.4 Let ,  be sets of literals.  ⇒  is GE0E0 -valid if and only if incon() or val() or  ∩ ¬ = ∅.

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Proof Assume that incon() or val() or  ∩ ¬ = ∅, and  ∩ ¬ = ∅. Then, if incon() or val() then for any assignment v, v |=E  or v |=E ; and if  ∩ ¬ = ∅ then there is a literal l ∈  ∩ ¬, and for any assignment v, either v(l) = 0 or v(l) = 1. If v(l) = 0 then v |=E ; and if v(l) = 1 then v(¬l) = 0, ¬l ∈ , v |=E . Assume that con()&inval()& ∩ ¬ = ∅. We define an assignment v such that for any variable p, ⎧ ⎨ 1 if p ∈  or ¬ p ∈  v( p) = 0 if ¬ p ∈  or p ∈  ⎩ 1 otherwise. Then, v is well-defined and v |=E0E0  ⇒ .



Hence, we have the following axioms: incon() or val() or  ∩ ¬ = ∅ ⇒ con()&inval()& ∩ ¬ = ∅ (AA1A1 )  →  (AE0E0 )

Proposition 5.1.5 Let ,  be sets of literals. |=E0A1  ⇒  if and only if  ⊆  or incon(). Proof Assume that  ⊆ . Then, for any assignment v, either v |=A , or there is a formula B ∈  such that v(B) = 0, and by assumption, B ∈ , i.e., v |=E . Hence, v |=E0A1  ⇒ . Conversely, assume that    and con(). There is a literal l ∈  − . Define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v( p) = 0 ⎪ ⎪ 1 ⎪ ⎪ ⎩ 0

if p ∈  if ¬ p ∈  if p = l if p = ¬l otherwise.

Then, v |=A , v |=E , and v |=A1E0  → . Hence, we have the following axioms:  ⊆  or incon() ⇒   &con() (AA1E0 )  →  (AE0A1 )



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Proposition 5.1.6 Let ,  be sets of literals. |=E0A0  ⇒  if and only if ¬ ⊆  or incon(). Proof Assume that ¬ ⊆ . Then, for any assignment v, either v |=E , or there is a formula B ∈  such that v(B) = 1, and by assumption, ¬B ∈  and v(¬B) = 0, i.e., v |=A . Hence, v |=E0A0  ⇒ . Conversely, assume that ¬   and con(). There is a literal l ∈ ¬ − . Define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v( p) = 0 ⎪ ⎪ 1 ⎪ ⎪ ⎩ 0

if p ∈  if ¬ p ∈  if p = l if p = ¬l otherwise.

Then, v |=A , v |=E , and v |=A1E0  → . Hence, we have the following axioms: ¬ ⊆  or incon() ⇒ ¬  &con() (AA1E0 )  →  (AE0A1 )

5.1.2 Deduction Rules There are four kinds of deduction rules. G L0 consists of the following deduction rules: , A ⇒  (¬¬ L ) ,  ¬¬A ⇒   , A1 ⇒  , A1 ⇒  (∧ L ) , A2 ⇒  (∨ L ) , A2 ⇒  ,A1 ∧ A2 ⇒  ,A1 ∨ A2 ⇒  , ¬A1 ⇒  , ¬A1 ⇒  (¬∧ L ) , ¬A2 ⇒  (¬∨ L ) , ¬A2 ⇒  , ¬(A1 ∨ A2 ) ⇒  , ¬(A1 ∨ A2 ) ⇒  G R0 consists of the following deduction rules:



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 ⇒ B,  (¬¬ R )    ⇒ ¬¬B,   ⇒ B1 ,   ⇒ B1 ,  (∧ R )  ⇒ B2 ,  (∨ R )  ⇒ B2 ,   ⇒ B1 ∧ B2 ,   ⇒ B1 ∨ B2 ,   ⇒ ¬B1 ,   ⇒ ¬B1 ,  (¬∧ R )  ⇒ ¬B2 ,  (¬∨ R )  ⇒ ¬B2 ,   ⇒ ¬(B1 ∧ B2 ),   ⇒ ¬(B1 ∨ B2 ),  G R1 consists of the following deduction rules:  ⇒ B,  (¬¬ R )    ⇒ ¬¬B,   ⇒ B1 ,   ⇒ B1 ,  (∧ R )  ⇒ B2 ,  (∨ R )  ⇒ B2 ,   ⇒ B1 ∧ B2 ,   ⇒ B1 ∨ B2 ,   ⇒ ¬B1 ,   ⇒ ¬B1 ,  (¬∧ R )  ⇒ ¬B2 ,  (¬∨ R )  ⇒ ¬B2 ,   ⇒ ¬(B1 ∧ B2 ),   ⇒ ¬(B1 ∨ B2 ),  G L1 consists of the following deduction rules: , A →  (¬¬ L ) ,  ¬¬A →   , A1 →  , A1 →  (∧ L ) , A2 →  (∨ L ) , A2 →  ,A1 ∧ A2 →  ,A1 ∨ A2 →  , ¬A1 →  , ¬A1 →  (¬∧ L ) , ¬A2 →  (¬∨ L ) , ¬A2 →  , ¬(A1 ∨ A2 ) →  , ¬(A1 ∨ A2 ) →  G R0 consists of the following deduction rules:  → B,  (¬¬ R )   ⇒ ¬¬B,    → B1 ,   → B1 ,  (∧ R )  → B2 ,  (∨ R )  → B2 ,   → B1 ∧ B2 ,   → B1 ∨ B2 ,   → ¬B1 ,   → ¬B1 ,  (¬∧ R )  → ¬B2 ,  (¬∨ R )  → ¬B2 ,   → ¬(B1 ∧ B2 ),   → ¬(B1 ∨ B2 ),  G R1 consists of the following deduction rules:

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 → B,  (¬¬ R )   ⇒ ¬¬B,    → B1 ,   → B1 ,  (∧ R )  → B2 ,  (∨ R )  → B2 ,   → B1 ∧ B2 ,   → B1 ∨ B2 ,   → ¬B1 ,   → ¬B1 ,  (¬∧ R )  → ¬B2 ,  (¬∨ R )  → ¬B2 ,   → ¬(B1 ∧ B2 ),   → ¬(B1 ∨ B2 ),  It is easy to see that G L1 = G L1 G L0 = G L0 G R1 = G R1 G R0 = G R0 .

5.1.3 Deduction Systems Let GE0E0 = AE0E0 + G L0 + G R0 GE0E1 = AE0E1 + G L0 + G R1 GE0A0 = AE0A0 + G L0 + G R0 GE0A1 = AE0A1 + G L0 + G R1 and GA1E0 = AA1E0 + G L1 + G R0 GA1E1 = AA1E1 + G L1 + G R1 GA1A0 = AA1A0 + G L1 + G R0 GA1A1 = AA1A1 + G L1 + G R1 Let ∗ ∈ {E0, E1, A0, A1}. Definition 5.1.7 A sequent  ⇒  is provable in GE0∗ , denoted by E0∗  ⇒ , if there is a sequence {1 ⇒ 1 , ..., n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in GE0∗ . Definition 5.1.8 A co-sequent  →  is provable in GE0∗ , denoted by E0∗  →  if there is a sequence {1 → 1 , ..., n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in GE0∗ . Theorem 5.1.9 (Soundness and completeness theorem) For any sequent  ⇒  and co-sequent  → ,

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E0∗  ⇒ iff |=A1∗  ⇒  A1∗  → iff |=A1∗  → .

Notice that GE0∗ GA1∗

⎧ A1 ⎪ ⎪ negation ⎨ A0 GA1∗ , where ∗ = ⎪ E0 E0∗ ⎪ G , ⎩ E1

if ∗ = E0 if ∗ = E1 if ∗ = A1 if ∗ = A0



5.1.4 Monotonicity of GE0∗ and GA1∗ Definition 5.1.10 A deduction system X is monotonic in  if for any formula sets ,   and , X  ⇒ &  ⊇ imply X   ⇒ . X is monotonic in  if for any formula sets ,  and  , X  ⇒ & ⊇ imply X  ⇒  . Theorem 5.1.11 (Monotonicity theorem) GE0Ei is monotonic in both  and , that is, for any formula sets ,   ,  and  ,  ⊆   & E0Ei  ⇒ imply E0Ei   ⇒ ;  ⊆  & E0Ei  ⇒ imply E0Ei  ⇒  ; and GE0Ai is monotonic in  and nonmonotonic in , that is, for any formula sets ,   ,  and  ,  ⊆   & E0Ai  ⇒ imply E0Ai   ⇒ ;  ⊆  & E0Ai  ⇒ may not imply E0Ai  ⇒  . Definition 5.1.12 A deduction system X is nonmonotonic in  if for any formula sets ,   and , X  → &  ⊇ may not imply X   → . X is nonmonotonic in  if for any formula sets ,  and  , X  → & ⊇ may not imply X  →  . Theorem 5.1.13 (Nonmonotonicity theorem) (i) GA1Ei is nonmonotonic in  and and monotonic in , that is, for any formula sets ,   ,  and  ,

5.1 Gentzen Deduction Systems GE0∗ /GE1∗

133

 ⊆   & GA1Ei  → may not imply GA1Ei   → ;  ⊆  & GA1Ei  → imply GA1Ei  →  . (ii) GA1Ai is nonmonotonic in both  and , that is, for any formula sets ,   ,  and  ,  ⊆   & GA1Ai  → may not imply GA1Ai   → ;  ⊆  & GA1Ai  → may not imply GA1Ai  →  . Proof We prove that the axiom is nonmonotonic and each deduction rule preserves the monotonicity. Assume that con(), inval() and  ∩  = ∅. There is a superset   ⊇  such that   is inconsistent; and there is a superset  ⊇  such that  is inconsistent; and there are supersets   ⊇  and  ⊇  such that   ∩  = ∅. Hence, GA1Ai is nonmonotonic in both  and . To show that (∧ L ) preserves the monotonicity of , assume that , A1 →  and , A1 →  are monotonic with respect to . By (∧ L ), from , A1 →  and , A2 → , we infer , A1 ∧ A2 → . Then, for any   ⊇ ,   , A1 →  and , A2 →  follows; and by (∧ L ), from   , A1 →  and   , A2 → , we infer   , A1 ∧ A2 → . Hence, , A1 ∧ A2 →  implies   , A1 ∧ A2 → , that is, , A1 ∧ A2 →  is monotonic with respect to . To show that (∧ L ) preserves the nonmonotonicity of , assume that , A1 →  and , A2 →  are nonmonotonic with respect to . By (∧ L ), from , A1 →  and , A2 → , we infer , A1 ∧ A2 → . Then, for some   ⊇ , , A1 → may not imply  , A1 → ; , A2 → may not imply  , A2 → ; and by (∧ L ), , A1 ∧ A2 →  may not imply   , A1 ∧ A2 → , that is, , A1 ∧ A2 →  is nonmonotonic with respect to . To show that (∧ L ) preserves the monotonicity of , assume that , A1 →  and , A2 →  are monotonic with respect to . By (∧ L ), from , A1 , →  and , A2 → , we infer , A1 ∧ A2 → . Then, for any  ⊇ , , A1 →  and , A2 →  follow; and by (∧ L ), from , A1 →  and , A2 →  , we infer , A1 ∧ A2 →  . Hence, , A1 ∧ A2 →  implies , A1 ∧ A2 →  , that is, , A1 ∧ A2 →  is monotonic with respect to . To show that (∧ L ) preserves the nonmonotonicity of , assume that , A1 →  and , A2 →  are nonmonotonic with respect to  . By (∧ L ), from , A1 →  and , A2 → , we infer , A1 ∧ A2 → . Then, for some  ⊇ , , A1 → may not imply, A1 →  ; , A2 → may not imply, A2 →  ; and by (∧ L ), , A1 ∧ A2 →  may not imply , A1 ∧ A2 →  , that is, , A1 ∧ A2 →  is nonmonotonic with respect to .

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Similar to show that other deduction rules preserves the monotonicity and nonmonotonicity with respect to  and .

Generally we have the following table for the monotonicity of Getzen deduction systems GE0∗ /GA1∗ : System GE0E1 GA1A0 GE0E0 GA1A1 GE0A1 GA1E0 GE0A0 GA1E1

Axiom mono  incon() or val() or  ∩  = ∅ Y con()&inval()& ∩  = ∅ N incon() or val() or  ∩ ¬ = ∅ Y con()&inval()& ∩ ¬ = ∅ N  ⊆  or incon() Y   &con() N ¬ ⊆  or incon() Y ¬  &con() N

mono  Y N Y N N Y N Y.

5.2 Gentzen Deduction Systems G Q 1 i Q 2 j Definition 5.2.1 A sequent  ⇒  is G Q 1 i Q 2 j -valid, denoted by |= Q 1 i Q 2 j  ⇒ , if for any assignment v, either ⎧ v(A) = 1 for every A ∈  ⎪ ⎪ ⎨ v(A) = 0 for every A ∈  v(A) = 1 for someA ∈  ⎪ ⎪ ⎩ v(A) = 0 for someA ∈ 

if if if if

Q1i Q1i Q1i Q1i

= A1 = A0 = E1 = E0

⎧ v(B) = 1 for everyB ∈  if ⎪ ⎪ ⎨ v(B) = 0 for everyB ∈  if v(B) = 1 for someB ∈  if ⎪ ⎪ ⎩ v(B) = 0 for someB ∈  if

Q2 j Q2 j Q2 j Q2 j

= A1 = A0 = E1 = E0

or

5.2.1 Axioms The axioms are classified into four classes: E0Q 2 j, E1Q 2 j, A0Q 2 j, A1Q 2 j. • E0Q 2 j: Proposition 5.2.2 Let ,  be sets of literals.  ⇒  is GE0E0 -valid if and only if incon() or val() or  ∩ ¬ = ∅. Proof Same as Proposition 5.1.5.



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Hence, we have the following axioms: (AE0E0 )

incon() or val() or  ∩ ¬ = ∅ ⇒

Proposition 5.2.3 Let ,  be sets of literals.  ⇒  is GE0E1 -valid if and only if incon() or val() or  ∩  = ∅. Proof  ⇒  is GE0E1 -valid iff  ⇒  is GEE -valid.



Hence, we have the following axiom: (AE0E1 )

incon()orval() or  ∩  = ∅ ⇒

Proposition 5.2.4 Let ,  be sets of literals.  ⇒  is GE0A1 -valid if and only if  ⊆  or incon(). Proof  ⇒  is GE0A1 -valid iff  ⇒  is GEA -valid.



Hence, we have the following axiom: (AE0A1 )

 ⊆  or incon() ⇒

Proposition 5.2.5 Let ,  be sets of literals.  ⇒  is GE0A0 -valid if and only if ¬ ⊆  or incon().



Proof Same as Proposition 5.1.8. Hence, we have the following axiom: (AE0A0 )

¬ ⊆  or incon() ⇒

• E1Q 2 j: Proposition 5.2.6 Let ,  be sets of literals.  ⇒  is GE1E1 -valid if and only if incon() or val() or  ∩ ¬ = ∅. Proof  ⇒  is GE1E1 -valid iff ¬ ⇒ ¬ is GE0E0 -valid.



Hence, we have the following axiom: (AE1E1 )

incon() or val() or  ∩ ¬ = ∅ ⇒

Proposition 5.2.7 Let ,  be sets of literals.  ⇒  is GE1E0 -valid if and only if incon() or val() or  ∩  = ∅.

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Proof  ⇒  is GE1E0 -valid iff ¬ ⇒ ¬ is GE0E1 -valid.



Hence, we have the following axiom: (AE1E0 )

incon() or val() or  ∩  = ∅ ⇒

Proposition 5.2.8 Let ,  be sets of literals. |=E1A1  ⇒  if and only if ¬ ⊆  or incon(). Proof Assume that ¬ ⊆ . Then, for any assignment v, either v |=A , or there is a formula B ∈  such that v(B) = 0, and by assumption, ¬B ∈  and v(¬B) = 1, i.e., v |=E . Hence, v |=E1A1  ⇒ . Conversely, assume that ¬   and con(). There is a literal l ∈ ¬ − . Define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v( p) = 1 ⎪ ⎪ ⎪0 ⎪ ⎩ 0

if p ∈  if ¬ p ∈  if p = l if p = ¬l otherwise.

Then, v |=A , v |=E , and v |=A1E0  → .



Hence, we have the following axioms: (AE1A1 )

¬ ⊆  or incon() ⇒

Proposition 5.2.9 Let ,  be sets of literals. |=E1A0  ⇒  if and only if  ⊆  or incon(). Proof Assume that  ⊆ . Then, for any assignment v, either v |= , or there is a formula B ∈  such that v(B) = 1, and by assumption, B ∈ , i.e., v |= . Hence, v |=E1A0  ⇒ . Conversely, assume that    and con(). There is a literal l ∈  − . Define an assignment v such that for any variable p, ⎧ ⎪ ⎪0 ⎪ ⎪ ⎨1 v( p) = 1 ⎪ ⎪ ⎪0 ⎪ ⎩ 0

if p ∈  if ¬ p ∈  if p = l if p = ¬l otherwise.

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137

Then, v |=  and v |=A0E1  → .



Hence, we have the following axioms: (AE1A0 )

 ⊆  or incon() ⇒

• A0Q 2 j: Proposition 5.2.10 Let ,  be sets of literals.  ⇒  is GA0E0 -valid if and only if val() or ¬ ⊆ . Proof Assume that ¬ ⊆  or val(). Then, if val() then for any assignment v, v |=E ; and if ¬ ⊆  then for any assignment v, either v(l) = 0 for every literal l ∈  (hence, v |=A ) or v(l) = 1 for some l ∈ , and ¬l ∈ , i.e., v(¬l) = 0 and hence, v |=E , that is, |=A0E0  ⇒ . Conversely, assume that ¬   and inval(). There is a literal l ∈ ¬ − . Define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v( p) = 0 ⎪ ⎪ 1 ⎪ ⎪ ⎩ 0

if p ∈  if ¬ p ∈  if p = l if p = ¬l otherwise.

Then, v |=E , v |=A , and v |=E1A1  → .



Hence, we have the following axioms: (AA0E0 )

val() or ¬ ⊆  ⇒

Proposition 5.2.11 Let ,  be sets of literals.  ⇒  is GA0E1 -valid if and only if val() or  ⊆ . Proof Assume that  ⊆  or val(). Then, if val() then for any assignment v, v |=E ; and if  ⊆  then for any assignment v, either v(l) = 0 for every literal l ∈  (hence, v |=A ) or v(l) = 1 for some l ∈  ⊆ , v |=E , that is, |=A0E1  ⇒ . Conversely, assume that    and inval(). There is a literal l ∈  − . Define an assignment v such that for any variable p, ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨1 v( p) = 1 ⎪ ⎪ 0 ⎪ ⎪ ⎩ 0

if p ∈  if ¬ p ∈  if p = l if ¬ p = l otherwise.

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Then, v |=E , v |=A , and v |=E1A0  → .



Hence, we have the following axioms: (AA0E1 )

 ⊆  or val() ⇒

Proposition 5.2.12 Given two sets ,  of literals, |=A0A0  ⇒  if and only if  = ∅ or  = ∅ or  = {l} = ¬. Proof Assume that  = ∅ or  = ∅ or  = {l} = ¬. Then, for any assignment v, either v(l) = 0 for every l ∈ , or v(m) = 0 for every m ∈ . Assume that  = ∅ = &( = {l} or {l} = ¬). Let l = l  ∈  −  and l ∈ . We define an assignment v such that for any variable p, ⎧ ⎪ ⎪1 ⎪ ⎪ ⎨0 v( p) = 1 ⎪ ⎪ 0 ⎪ ⎪ ⎩ 1

if p = l  if p = ¬l  if p = l if p = ¬l otherwise.

Then, v is well-defined and v |=E1E1  → . Similar for l ∈  − .



Hence, we have the following axioms: (AA0A0 )

 = ∅ or  = ∅ or  = {l} = ¬ ⇒

Proposition 5.2.13 Given two sets ,  of literals, |=A0A1  ⇒  if and only if  = ∅ or  = ∅ or  = {l} = . Proof |=A0A1  ⇒  iff |=AA  ⇒ . Hence, we have the following axioms: (AA0A1 ) • A1Q 2 j:

 = ∅ or  = ∅ or  = {l} =  ⇒



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139

Proposition 5.2.14 Let ,  be sets of literals.  ⇒  is GA1E0 -valid if and only if val() or  ⊆ . Proof Assume that  ⊆  or val(). Then, if val() then for any assignment v, v |=E ; and if  ⊆  then for any assignment v, either v(l) = 1 for every literal l ∈  or v(l) = 0 for some l ∈  ⊆ , that is, |=A1E0  ⇒ . Conversely, assume that    and inval(). There is a literal l ∈  − . Define an assignment v such that for any variable p, ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨1 v( p) = 1 ⎪ ⎪ 0 ⎪ ⎪ ⎩ 1

if p = l if p = ¬l if p ∈  if ¬ p ∈  otherwise.

Then, v |=E , v |=A , and v |=E0A1  → .



Hence, we have the following axioms: (AA1E0 )

val() or  ⊆  ⇒

Proposition 5.2.15 Let ,  be sets of literals.  ⇒  is GA1E1 -valid if and only if val() or ¬ ⊆ . Proof Assume that ¬ ⊆  or val(). Then, if val() then for any assignment v, v |=E ; and if ¬ ⊆  then for any assignment v, either v(l) = 1 for every literal l ∈  (hence, v |=A ), or v(l) = 0 for some l ∈ , i.e., v(¬l) = 1. Then, ¬l ∈  and v |=E . Hence, |=A1E1  ⇒ . Conversely, assume that ¬   and inval(). There is a literal l ∈ ¬ − . Define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v( p) = 0 ⎪ ⎪ ⎪1 ⎪ ⎩ 1

if p = l if p = ¬l if p ∈  if ¬ p ∈  otherwise.

Then, v |=E , v |=A , and v |=E0A0  → . Hence, we have the following axioms: (AA1E1 )

¬ ⊆  or val() ⇒



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Proposition 5.2.16 Given two sets ,  of literals, |=A1A0  ⇒  if and only if  = ∅ or  = ∅ or  = {l} = . Proof Assume that  = ∅ or  = ∅ or  = {l} = . Then, for any assignment v, either v(l) = 1 for every l ∈ , or v(m) = 0 for every m ∈ . Assume that  = ∅ = &( = {l} = ). Let l  ∈  −  and l  = l ∈ . We define an assignment v such that for any variable p, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎨0 v( p) = 0 ⎪ ⎪ 1 ⎪ ⎪ ⎩ 1

if p = l if p = ¬l if p = l  if p = ¬l  otherwise.

Then, v is well-defined and v |=E0E1  → . Similar for l ∈  − .



Hence, we have the following axioms: (AA1A0 )

 = ∅ or  = ∅ or  = {l} =  ⇒

Proposition 5.2.17 Given two sets ,  of literals, |=A1A1  ⇒  if and only if  = ∅ or  = ∅ or  = {l} = ¬. Proof Assume that  = ∅ or  = ∅ or  = {l} = ¬. Then, for any assignment v, either v(l) = 1 for every l ∈ , or v(m) = 1 for every m ∈ . Assume that  = ∅ = &( = {l} = ¬). Let l = l  ∈  − ¬ and l ∈ . We define an assignment v such that for any variable p, ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨1 v( p) = 0 ⎪ ⎪ ⎪1 ⎪ ⎩ 1

if p = l if p = ¬l if p = l  if p = ¬l  otherwise.

Then, v is well-defined and v |=A1A1  ⇒ . Similar for ¬l = ¬l  ∈  − . Hence, we have the following axioms: (AA1A1 )

 = ∅ or  = ∅ or  = {l} = ¬ ⇒



5.2 Gentzen Deduction Systems G Q 1 i Q 2 j

5.2.2 Deduction Rules G L Q 1 0 consists of the following deduction rules: , A ⇒  (¬¬ L )  , ¬¬A ⇒   , A1 ⇒  , A1 ⇒  (∧ L ) , A2 ⇒  (∨ L ) , A2 ⇒  ,A1 ∧ A2 ⇒  ,A1 ∨ A2 ⇒  , ¬A1 ⇒  , ¬A1 ⇒  (¬∨ L ) , ¬A2 ⇒  (¬∧ L ) , ¬A2 ⇒  , ¬(A1 ∨ A2 ) ⇒  , ¬(A1 ∨ A2 ) ⇒ 

G L Q 1 1 consists of the following deduction rules: , A ⇒  (¬¬ L ) ,  ¬¬A ⇒   , A1 ⇒  , A1 ⇒  (∧ L ) , A2 ⇒  (∨ L ) , A2 ⇒  ,A1 ∧ A2 ⇒  ,A1 ∨ A2 ⇒  , ¬A1 ⇒  , ¬A1 ⇒  (¬∧ L ) , ¬A2 ⇒  (¬∨ L ) , ¬A2 ⇒  , ¬(A1 ∨ A2 ) ⇒  , ¬(A1 ∨ A2 ) ⇒ 

G R Q 2 1 consists of the following deduction rules:  ⇒ B,  (¬¬ R )   ⇒ ¬¬B,    ⇒ B1 ,   ⇒ B1 ,  (∨ R )  ⇒ B2 ,  (∧ R )  ⇒ B2 ,   ⇒ B1 ∧ B2 ,   ⇒ B1 ∨ B2 ,   ⇒ ¬B1 ,   ⇒ ¬B1 ,  (¬∧ R )  ⇒ ¬B2 ,  (¬∨ R )  ⇒ ¬B2 ,   ⇒ ¬(B1 ∧ B2 ),   ⇒ ¬(B1 ∨ B2 ), 

G R Q 2 0 consists of the following deduction rules:  ⇒ B,  (¬¬ R )   ⇒ ¬¬B,    ⇒ B1 ,   ⇒ B1 ,  (∨ R )  ⇒ B2 ,  (∧ R )  ⇒ B2 ,   ⇒ B1 ∧ B2 ,   ⇒ B1 ∨ B2 ,   ⇒ ¬B1 ,   ⇒ ¬B1 ,  (¬∨ R )  ⇒ ¬B2 ,  (¬∧ R )  ⇒ ¬B2 ,   ⇒ ¬(B1 ∧ B2 ),   ⇒ ¬(B1 ∨ B2 ), 

141

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5.2.3 Deduction Systems Let Q 1 , Q 2 ∈ {E, A} and i, j ∈ {0, 1}. Define G Q1i Q2 j = AQ1i Q2 j + GL Q1i + G R Q2 j . Definition 5.2.18 A sequent  ⇒  is provable in G Q 1 i Q 2 j , denoted by  Q 1 i Q 2 j  ⇒ , if there is a sequence {1 ⇒ 1 , ..., n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or is deduced from the previous sequents by one of the deduction rules in G Q 1 i Q 2 j . Theorem 5.2.19 (Soundness and completeness theorem) For any sequent  ⇒ ,  Q 1 i Q 2 j  ⇒  iff |= Q 1 i Q 2 j  ⇒ .

Theorem 5.2.20 (Monotonicity theorem) G Q 1 i Q 2 j is monotonic in  if and only if Q 1 = E; and nonmonotonic in  if and only if Q 2 = A.

5.3 R-Calculi R Q 1 i Q 2 j Let Q 1 , Q 2 ∈ E, A and i, j ∈ 0, 1. Given a sequent  ⇒ Q 1 i Q 2 j  and pair (A, B) of formulas, the result of  ⇒  G Q 1 i Q 2 j -revising (A, B) is denoted by |= Q 1 i Q 2 j  ⇒ |(A, B) ⇒   ⇒  , where ⎧ Q1i Q2 j ⎪ ⎪  ±1 A ⇒  ±2 B if |= Q i Q j  ±1 A ⇒  ±2 B ⎨ if |= 1 2  ±1 A ⇒   ±1 A ⇒    ⇒  = if |= Q 1 i Q 2 j  ⇒  ±2 B  ⇒  ±2 B ⎪ ⎪ ⎩ ⇒ otherwise, where  ±1 A =

∪{A} if Q 1 = A −{A} if Q 1 = E,

 ±2 B =

∪{B} if Q 2 = A −{B} if Q 2 = E.

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143

5.3.1 Axioms We reorder the axioms in Gentzen deduction systems as follows: incon() or val() or  ∩  = ∅ ⇒ incon() or val() or  ∩  = ∅ (AE1E0 ) ⇒ incon() or val() or  ∩ ¬ = ∅ (AE0E0 ) ⇒ E1E1 incon() or val() or  ∩ ¬ = ∅ (A ) ⇒

(AE0E1 )

and  ⊆  or incon() ⇒  ⊆  or incon() (AE1A0 ) ⇒ ¬ ⊆  or incon() (AE0A0 ) ⇒ E1A1 ¬ ⊆  or incon() ) (A ⇒

(AE0A1 )

and  ⊆  or val() ⇒ val() or  ⊆  (AA1E0 ) ⇒ val() or ¬ ⊆  (AA0E0 ) ⇒ A1E1 ¬ ⊆  or val() ) (A ⇒ (AA0E1 )

and  = ∅ or  = ∅ or  ⇒  = ∅ or  = ∅ or  (AA1A0 ) ⇒ A0A0  = ∅ or  = ∅ or  (A ) ⇒ A1A1  = ∅ or  = ∅ or  (A ) ⇒ (AA0A1 )

= {l} =  = {l} =  = {l} = ¬ = {l} = ¬

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Correspondingly we have the following axioms for R-calculi: • Q 1 i Q 2 j = E0E1/E1E0: 

El  = l(l  , ¬l  ∈ ) El  = l(l  ∈ &l  ∈ )  ⇒ |(l, m) ⇒  − {l} ⇒ |m  ∼ El  = l(l  , ¬l  ∈ ) Q1i Q2 j ) ∼ El  = l(l  ∈ &l  ∈ ) (A0L  ⇒ |(l, m) ⇒  ⇒ |m  Em  = m(m  , ¬m  ∈ ) Q1i Q2 j ) Em  = m(m  ∈   &m  ∈ ) (A−R   ⇒ |m ⇒   ⇒  − {m}  ∼ Em  = m(m  , ¬m  ∈ ) Q iQ j (A0R1 2 ) ∼ Em  = m(m  ∈   &m  ∈ )   ⇒ |m ⇒   ⇒  Q iQ j (A−L1 2 )

where l ∈  and m ∈ . • Q 1 i Q 2 j = E0E0/E1E1: 

El  = l(l  , ¬l  ∈ ) El  = l(l  ∈ &¬l  ∈ )  ⇒ |(l, m) ⇒  − {l} ⇒ |m  ∼ El  = l(l  , ¬l  ∈ ) Q1i Q2 j ) ∼ El  = l(l  ∈ &¬l  ∈ ) (A0L  ⇒ |(l, m) ⇒  ⇒ |m  Em  = m(m  , ¬m  ∈ ) Q iQ j (A−R1 2 ) Em  = m(m  ∈   &¬m  ∈ )   ⇒ |m ⇒   ⇒  − {m}  ∼ Em  = m(m  , ¬m  ∈ ) Q1i Q2 j ) ∼ Em  = m(m  ∈   &¬m  ∈ ) (A0R   ⇒ |m ⇒   ⇒  Q iQ j (A−L1 2 )

where l ∈  and m ∈ . • Q 1 i Q 2 j = E0A1/E1A0: 

El  = l(l  , ¬l  ∈ )  ⊆  − {l}  m) ⇒  − {l} ⇒ |m  ⇒ |(l, ∼ El  = l(l  , ¬l  ∈ ) Q iQ j (A0L1 2 )    − {l}  ⇒ |(l, m) ⇒  ⇒ |m m ∈  Q iQ j (A−R1 2 )   ⇒ |m ⇒   ⇒ , m /  Q iQ j m ∈ (A0R1 2 )   ⇒ |m ⇒   ⇒  Q iQ j (A−L1 2 )

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145

where l ∈ . • Q 1 i Q 2 j = E0A0/E1A1: 

El  = l(l  , ¬l  ∈ ) ¬ ⊆  − {l}  m) ⇒  − {l} ⇒ |m  ⇒ |(l, ∼ El  = l(l  , ¬l  ∈ ) Q iQ j (A0L1 2 ) ¬   − {l}  ⇒ |(l, m) ⇒  ⇒ |m  Q 1 i Q 2 j ¬m ∈  )  (A−R  ⇒ |m ⇒   ⇒ , m /  Q i Q j ¬m ∈ (A0R1 2 )   ⇒ |m ⇒   ⇒  Q iQ j (A−L1 2 )

where l ∈ . • Q 1 i Q 2 j = A0E1/A1E0: l∈  ⇒ |(l, m) ⇒ , l ⇒ |m / Q1i Q2 j l ∈ ) (A0L   ⇒ |(l, m)  ⇒   ⇒ |m Em = m(m , ¬m ∈ ) Q iQ j (A−R1 2 )   ⊆  − {m}   ⇒   ⇒  − {m}  ⇒ |m  ∼ Em = m(m  , ¬m  ∈ ) Q1i Q2 j (A0R )     − {m}   ⇒ |m ⇒   ⇒  Q i Q2 j

(A−L1

)

where m ∈ . • Q 1 i Q 2 j = A0E0/A1E1:

¬l ∈   ⇒ |(l, m) ⇒ , l ⇒ |m / Q i Q j ¬l ∈ (A0L1 2 )  ⇒ |(l, m) ⇒  ⇒ |m  Em  = m(m  , ¬m  ∈ ) Q iQ j (A−R1 2 ) ¬  ⊆  − {m}   ⇒   ⇒  − {m}  ⇒ |m ∼ Em  = m(m  , ¬m  ∈ ) Q iQ j (A0R1 2 ) ¬    − {m}   ⇒ |m ⇒   ⇒  Q i Q2 j

(A−L1

)

5 R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j

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where m ∈ . • Q 1 i Q 2 j = A0A1/A1A0:

l∈  ⇒ |(l, m) ⇒ , l ⇒ |m / Q1i Q2 j l ∈ ) (A0L  ⇒ |(l, m) ⇒  ⇒ |m Q iQ j m ∈  (A+R1 2 )   ⇒ |m ⇒   ⇒ , m / Q iQ j m ∈ (A0R1 2 )   ⇒ |m ⇒   ⇒  Q i Q2 j

(A+L1

)

• Q 1 i Q 2 j = A0A0/A1A1:

¬l ∈   ⇒ |(l, m) ⇒ , l ⇒ |m / Q i Q j ¬l ∈ (A0L1 2 )  ⇒ |(l, m) ⇒  ⇒ |m  Q i Q j ¬m ∈  (A+R1 2 )   ⇒ |m ⇒   ⇒ , m /  Q i Q j ¬m ∈ (A0R1 2 )   ⇒ |m ⇒   ⇒  Q i Q2 j

(A+L1

)

Theorem 5.3.1 Let  ⇒  be literal and l, m be literals.  ⇒ |(l, m) ⇒   ⇒  is R Q 1 i Q 2 j -valid iff  ⇒ |(l, m) ⇒   ⇒  is the conclusion of an axiom.



5.3.2 Deduction Rules Let X =  ⇒  and X(A) = , A ⇒  X(B) =  ⇒ B,  X[A] =  − {A} ⇒  X[B] =  ⇒  − {B}. There are four basic sets of deduction rules: SLQ1 0 , SLQ1 1 , SRQ2 0 , SRQ2 1 , which are same as in the last chapter.

5.4 Gentzen Deduction Systems G Q 1 i Q 2 j

147

5.3.3 R-Calculi Define R Q1i Q2 j = AQ1i Q2 j + SL Q1i + S R Q2 j . Definition 5.3.2 A reduction δ =  ⇒ |(A, B) ⇒   ⇒  is provable in R Q 1 i Q 2 j , denoted by  Q 1 i Q 2 j δ, if there is a sequence {δ1 , ..., δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in R Q 1 i Q 2 j . Theorem 5.3.3 (Soundness and completeness theorem) For any reduction δ =  → |(A, B) ⇒   →  , δ is R Q 1 i Q 2 j -valid if and only if δ is provable in R Q 1 i Q 2 j . That is,  Q 1 i Q 2 j δiff |= Q 1 i Q 2 j δ.

Theorem 5.3.4 (Nonmonotonicity theorem) R Q 1 i Q 2 j is nonmonotonic in  and in . Proof R Q 1 i Q 2 j is composed of two parts: the enumeration/elimination part (denoted Q iQ j Q iQ j Q iQ j by R± 1 2 ) and the zero (doing nothing) part (denoted by R0 1 2 ). Then, R± 1 2 Q iQ j is nonmonotonic in  and in ; and R0 1 2 is monotonic in  and in .



5.4 Gentzen Deduction Systems G Q 1 i Q 2 j Definition 5.4.1 A co-sequent  →  is G Q 1 i Q 2 j -valid, denoted by |= Q 1 i Q 2 j  → , if there is an assignment v such that ⎧ v(A) = 1 for every A ∈  ⎪ ⎪ ⎨ v(A) = 0 for every A ∈  v(A) = 1 for some A ∈  ⎪ ⎪ ⎩ v(A) = 0 for some A ∈ 

if if if if

Q 1i Q 1i Q 1i Q 1i

= A1 = A0 = E1 = E0

⎧ v(B) = 1 for some B ∈  if ⎪ ⎪ ⎨ v(B) = 0 for some B ∈  if v(B) = 1 for every B ∈  if ⎪ ⎪ ⎩ v(B) = 0 for every B ∈  if

Q2 j Q2 j Q2 j Q2 j

= A1 = A0 = E1 = E0

and

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5.4.1 Axioms The axioms are classified into the following four classes: • A1Q 2 j: Lemma 5.4.2 Given two sets ,  of literals, |=A1A0  →  if and only if con()& inval()& ∩  = ∅.

Hence, we have the following axiom: (AA1A0 )

con()&inval()& ∩  = ∅  → 

Lemma 5.4.3 Given two sets ,  of literals, |=A1E1  →  if and only if con()& inval()& ∩ ¬ = ∅.

Hence, we have the following axiom: (AA1A1 )

con()&inval()& ∩ ¬ = ∅  → 

Lemma 5.4.4 Let ,  be sets of literals. |=A1E0  ⇒  if and only if   &con(). Hence, we have the following axioms: (AA1E0 )

  &con()  → 

Lemma 5.4.5 Let ,  be sets of literals. |=A1E1  →  if and only if ¬  &con().



Hence, we have the following axioms: (AA1E1 )

¬  &con()  → 

• A0Q 2 j: Lemma 5.4.6 Given two sets ,  of literals, |=A0A0  →  if and only if con()& inval()& ∩ ¬ = ∅.

Hence, we have the following axiom: (AA0A0 )

con()&inval()& ∩ ¬ = ∅  → 

5.4 Gentzen Deduction Systems G Q 1 i Q 2 j

149

Lemma 5.4.7 Given two sets ,  of literals, |=A0A1  →  if and only if con()& inval()& ∩  = ∅.

Hence, we have the following axiom: (AA0A1 )

con()&inval()& ∩  = ∅  → 

Lemma 5.4.8 Let ,  be sets of literals. |=A0E0  ⇒  if and only if ¬  &con(). Hence, we have the following axioms: ¬  &con()  → 

(AA0E0 )

Lemma 5.4.9 Let ,  be sets of literals. |=A0E1  →  if and only if   &con().

Hence, we have the following axioms: (AA0E1 )

  &con()  → 

• E1Q 2 j: Lemma 5.4.10 Let ,  be sets of literals.  ⇒  is GE1A0 -valid if and only if inval()&¬  . Hence, we have the following axioms: (AE1A0 )

inval()&¬    → 

Lemma 5.4.11 Let ,  be sets of literals.  ⇒  is GE1A1 -valid if and only if inval()&  . Hence, we have the following axioms: (AE1A1 )

  &inval()  → 

Lemma 5.4.12 Given two sets ,  of literals, |=E1E0  →  if and only if  = ∅& = ∅&( = {l} = ).

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Hence, we have the following axioms: (AE1E0 )

 = ∅& = ∅&( =  = {l})  → 

Lemma 5.4.13 Given two sets ,  of literals, |=E1E1  →  if and only if  = ∅& = ∅&( = {l} or  = {¬l}). Hence, we have the following axioms: (AE1E1 )

 = ∅& = ∅&( = {l} or  = {¬l})  → 

• E0Q 2 j: Lemma 5.4.14 Let ,  be sets of literals.  ⇒  is GE0A0 -valid if and only if inval()&  . Hence, we have the following axioms: (AE0A0 )

inval()&    → 

Lemma 5.4.15 Let ,  be sets of literals.  ⇒  is GE0A1 -valid if and only if inval()&¬  . Hence, we have the following axioms: (AE0A1 )

¬  &inval()  → 

Lemma 5.4.16 Given two sets ,  of literals, |=E0E0  →  if and only if  = ∅& = ∅&( = {l} or {¬l} = ). Hence, we have the following axioms: (AE0E0 )

 = ∅& = ∅&( = {l} or  = {¬l})  → 

Lemma 5.4.17 Given two sets ,  of literals, |=E0E1  →  if and only if  = ∅& = ∅&( = {l} or  = {l}). Hence, we have the following axioms: (AE0E1 )

 = ∅& = ∅&( = {l} or  = {l})  → 

Theorem 5.4.18 Let  →  be literal.  →  is G Q 1 i Q 2 j -valid if and only if the

precondition in (A Q 1 i Q 2 j ) holds.

5.4 Gentzen Deduction Systems G Q 1 i Q 2 j

5.4.2 Deduction Rules G L0 consists of the following deduction rules: , A →  (¬¬ L ) ,  ¬¬A →   , A1 →  , A1 →  (∧ L ) , A2 →  (∨ L ) , A2 →  ,A1 ∧ A2 →  ,A1 ∨ A2 →  , ¬A1 →  , ¬A1 →  (¬∧ L ) , ¬A2 →  (¬∨ L ) , ¬A2 →  , ¬(A1 ∨ A2 ) →  , ¬(A1 ∨ A2 ) →  G L1 consists of the following deduction rules: , A →  (¬¬ L )  , ¬¬A →   , A1 →  , A1 →  (∧ L ) , A2 →  (∨ L ) , A2 →  ,A1 ∧ A2 →  ,A1 ∨ A2 →  , ¬A1 →  , ¬A1 →  (¬∧ L ) , ¬A2 →  (¬∨ L ) , ¬A2 →  , ¬(A1 ∨ A2 ) →  , ¬(A1 ∨ A2 ) →  G R1 consists of the following deduction rules:  → B,  (¬¬ R )   → ¬¬B,    → B1 ,   → B1 ,  (∧ R )  → B2 ,  (∨ R )  → B2 ,   → B1 ∧ B2 ,   → B1 ∨ B2 ,   → ¬B1 ,   → ¬B1 ,  R R ,    → ¬B  → ¬B2 ,  (¬∧ ) (¬∨ ) 2  → ¬(B1 ∧ B2 ),   → ¬(B1 ∨ B2 ),  G R0 consists of the following deduction rules:  → B,  (¬¬ R )   → ¬¬B,    → B1 ,   → B1 ,  (∧ R )  → B2 ,  (∨ R )  → B2 ,   → B1 ∧ B2 ,   → B1 ∨ B2 ,   → ¬B1 ,   → ¬B1 ,  (¬∧ R )  → ¬B2 ,  (¬∨ R )  → ¬B2 ,   → ¬(B1 ∧ B2 ),   → ¬(B1 ∨ B2 ), 

151

5 R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j

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Then, we have the following equalities: G L1 = G L1 G L0 = G L0 G R1 = G R1 G R0 = G R0 .

5.4.3 Deduction Systems Let Q 1 , Q 2 ∈ {E, A} and i, j ∈ {0, 1}. Define G Q 1 i Q 2 j = A Q 1 i Q 2 j + G Li + G R j . Definition 5.4.19 A co-sequent  →  is provable in G Q 1 i Q 2 j , denoted by  Q 1 i Q 2 j  → , if there is a sequence {1 → 1 , ..., n → n } of co-sequents such that n → n =  → , and for each 1 ≤ i ≤ n, i → i is an axiom or is deduced from the previous co-sequents by one of the deduction rules in G Q 1 i Q 2 j . Theorem 5.4.20 (Soundness and completeness theorem) For any co-sequent  → ,  Q 1 i Q 2 j  →  iff |= Q 1 i Q 2 j  → .

Theorem 5.4.21 (Nonmonotonicity theorem) G Q 1 i Q 2 j is monotonic in  if and only if Q 1 = E; and nonmonotonic in  if and only if Q 2 = A.

5.5 R-Calculi R Q 1 i Q 2 j Let Q 1 , Q 2 ∈ E, A and i, j ∈ 0, 1. Given a co-sequent  →  and pair (A, B) of formulas, the result of  →  G Q 1 i Q 2 j -revising (A, B) is denoted by |= Q 1 i Q 2 j  → |(A, B) ⇒   →  , where   →  = ⎧  ±1 A →  ±2 B if  ±1 A →  ±2 B is G Q 1 i Q 2 j -valid ⎪ ⎪ ⎨ otherwise, if  ±1 A →  is G Q 1 i Q 2 j -valid  ±1 A →  B otherwise, if  →  ±2 B is G Q 1 i Q 2 j -valid   →  ± ⎪ 2 ⎪ ⎩  →  otherwise; where  ±1 A =

∪{A} if Q 1 = A −{A} if Q 1 = E,

 ±2 B =

∪{B} if Q 2 = A −{B} if Q 2 = E.

5.5 R-Calculi R Q 1 i Q 2 j

153

5.5.1 Axioms Hence, we have the following axioms for R-calculi: • Q 1 i Q 2 j = A0A1/A1A0: 

¬l ∈ / l∈ /   → |(l, m)  ⇒ , l  → |m ¬l ∈  l∈ (A0L Q1i Q2 j )   → |(l, m) ⇒   → |m  m∈ / / (A+R Q 1 i Q 2 j ) ¬m ∈    → |m ⇒     → , m m∈ ¬m ∈  ) (A0R Q1i Q2 j    → |m ⇒    →  (A+L Q1i Q2 j )

• Q 1 i Q 2 j = A0A0/A1A1: 

¬l ∈ / ¬l ∈ /   → |(l, m)  ⇒ , l  → |m ¬l ∈  ¬l ∈  ) (A0L Q1i Q2 j   → |(l, m) ⇒   → |m  ¬m ∈ / / (A+R Q 1 i Q 2 j ) ¬m ∈    → |m ⇒     → , m ¬m ∈  ¬m ∈  ) (A0R Q1i Q2 j    → |m ⇒    →  (A+L Q1i Q2 j )

• Q 1 i Q 2 j = A0E1/A1E0: ¬l ∈ /   → |(l, m) ⇒ , l  → |m ¬l ∈  (A0L Q 1 i Q 2 j )   → |(l, m) ⇒   → |m ( − {m})    (A−R ) Q 1 i Q 2 j    → |m ⇒    →  − {m} ( − {m}) ⊆   (A0R Q 1 i Q 2 j )    → |m ⇒    →  (A+L Q1i Q2 j )

5 R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j

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where m ∈ . • Q 1 i Q 2 j = A0E0/A1E1:

¬l ∈ /  → |(l, m) ⇒ , l → |m ¬l ∈  (A0L Q 1 i Q 2 j )   → |(l, m) ⇒   → |m ¬( − {m})   (A−R Q 1 i Q 2 j )    → |m ⇒    →  − {m} ¬( − {m}) ⊆  (A0R Q 1 i Q 2 j )    → |m ⇒    →  (A+L Q1i Q2 j )

where m ∈ . • Q 1 i Q 2 j = E0A1/E1A0:

¬( − {l})    → |(l, m) ⇒  − {l} → |m ¬( − {l}) ⊆  (A0L Q 1 i Q 2 j )   → |(l, m) ⇒   → |m  ¬m ∈ / ¬  , m (A+R ) Q1i Q2 j   → |m ⇒  → , m ¬m ∈  ¬ ⊆ , m (A0R ) Q1i Q2 j   → |m ⇒   →  (A−L Q1i Q2 j )

where l ∈ . • Q 1 i Q 2 j = E1A1/E0A0:

( − {l})    → |(l, m) ⇒  − {l} → |m ( − {l}) ⊆  (A0L Q 1 i Q 2 j )   → |(l, m) ⇒   → |m  ¬m ∈ /   , m (A+R Q1i Q2 j )   → |m ⇒  → , m ¬m ∈   ⊆ , m (A0R ) Q1i Q2 j   → |m ⇒   →  (A−L Q1i Q2 j )

5.5 R-Calculi R Q 1 i Q 2 j

155

where l ∈ . • Q 1 i Q 2 j = E0E1/E1E0: 

( − {l}) = {l  }  = {l  }  → |(l, m) ⇒  − {l} → |m ( − {l}) =  = {l  } (A0L Q 1 i Q 2 j )   → |(l, m) ⇒   → |m  ( − {m}) = {m  } −R (A Q 1 i Q 2 j )  = {m  }   → |m ⇒   →  − {m} ( − {m}) =  = {m  } (A0R Q 1 i Q 2 j )    → |m ⇒    →  (A−L Q1i Q2 j )

where l ∈  and m ∈ . • Q 1 i Q 2 j = E0E0/E1E1: 

( − {l}) = {l  } ¬ = {l  }  → |(l, m) ⇒  − {l} → |m ( − {l}) = ¬ = {l  } (A0L Q 1 i Q 2 j )   → |(l, m) ⇒   → |m  ¬( − {m}) = {m  } −R (A Q 1 i Q 2 j )  = {m  }   → |m ⇒   →  − {m} ¬( − {m}) =  = {m  } ) (A0R Q 1 i Q 2 j    → |m ⇒    →  (A−L Q1i Q2 j )

where l ∈  and m ∈ . Theorem 5.5.1 Let  →  be literal.  → |(l, m) ⇒   →  is R Q 1 i Q 2 j -valid if and only if  → |(l, m) ⇒   →  is a conclusion of some axiom (A Q 1 i Q 2 j )x , where x ∈ {+L , −L , 0L , +R, −R, 0R}.



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5.5.2 Deduction Rules Let X =  →  and X(A) = , A →  X(B) =  → B,  X[A] =  − {A} →  X[B] =  →  − {B}. Define SLQ1 i = SLQ1 i (⇒ / →) SRQ2 j = SRQ2 j (⇒ / →).

5.5.3 R-Calculi Define R Q 1 i Q 2 j = A Q 1 i Q 2 j + SLQ1 i + SRQ2 j . Definition 5.5.2 A reduction  → |(A, B) ⇒   →  is provable in R Q 1 i Q 2 j , denoted by  Q 1 i Q 2 j  → |(A, B) ⇒   →  , if there is a sequence {δ1 , ..., δn } of reductions such that δn =  → |(A, B) ⇒   →  , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in R Q 1 i Q 2 j . Theorem 5.5.3 (Soundness and completeness theorem) For any Q 1 , Q 2 ∈ {A, E}, i, j ∈ {0, 1} and reduction δ =  → |(A, B) ⇒   →  , δ is R Q 1 i Q 2 j -valid if and only if δ is provable in R Q 1 i Q 2 j . That is,  Q 1 i Q 2 j δiff |= Q 1 i Q 2 j δ.

Theorem 5.5.4 (Nonmonotonicity theorem) R Q 1 i Q 2 j is composed of two parts: the enumeration/elimination part (denoted by R± Q 1 i Q 2 j ) and the zero part (denoted by is nonmonotonic in  and in ; and R0Q 1 i Q 2 j is monotonic R0Q 1 i Q 2 j ) Then, R± Q1i Q2 j in  and in .



5.6 Conclusions

157

5.6 Conclusions There are 16 Gentzen deduction systems: Av GEiE j GEiA j GAiE j GAiA j Ev GAiA j GAiE j GEiA j GEiE j where G Q 1 i Q 2 j = A Q 1 i Q 2 j + G Li + G R j G Q 1 i Q 2 j = A Q 1 i Q 2 j + G Li + G R j G Li ≡ G Li GR j ≡ GR j . Moreover, G Q 1 i Q 2 j /G Q 1 i Q 2 j is monotonic in  if and only if Q 1 = E; and nonmonotonic in  if and only if Q 2 = A. Correspondingly there are 16 R-calculi: Av REiE j REiA j RAiE j RAiA j Ev RAiA j RAiE j REiA j REiE j where R Q 1 i Q 2 j = A Q 1 i Q 2 j + SLQ1 i + SRQ2 j R Q 1 i Q 2 j = A Q 1 i Q 2 j + SLQ1 i + SRQ2 j ; SLQ1 i ≡ SLQ1 i SRQ2 j ≡ SRQ2 j . EL AL Moreover, R+ , R− are nonmonotonic in ; R0EL , R0AL are monotonic in ; ER AR R− , R+ are nonmonotonic in ; and R0ER , R0AR are monotonic in , where EL AL /R− is the set of deduction rules in REL /RAL with mark +/−, respectively, R+ EL AL ER AR with mark 0. Similar for R− , R+ and R0ER , R0AR . and R+ /R−

Appendix: List of all the Gentzen deduction systems G Q 1 i Q 2 j /G Q 1 i Q 2 j Let Gi j = G Li + G R j Gi j = G Li + G R j

5 R-Calculi R Q 1 i Q 2 j /R Q 1 i Q 2 j

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Then we have the following list of all the axioms for G Q 1 i Q 2 j and G Q 1 i Q 2 j : m m

system

 ∩  = ∅ or incon() or val() Y ⇒  ∩  = ∅&con()&inval() N = G10 +  → 

GE0E1 = G01 +

Y

GA1A0

N

 ⊆  or incon() ⇒   &con() = G10 +  → 

GE0A1 = G01 +

Y

N

GA1E0

N

Y

where m denotes monotonic in , and m denotes monotonic in , and m m

system

 ∩ ¬ = ∅ or incon() or val(¬) Y ⇒  ∩ ¬ = ∅&con()&inval(¬) N = G11 +  → 

GE0E0 = G00 +

Y

GA1A1

N

¬ ⊆  or incon() ⇒ ¬  &con() = G11 +  → 

GE0A0 = G10 +

Y

N

GA1E1

N

Y

and system  ∩ ¬ = ∅ or incon() or val() ⇒  ∩ ¬ = ∅&con()&inval() GA0A0 = G00 +  →  ¬ ⊆  or incon() GE1A1 = G11 + ⇒ ¬  &con() GA0E0 = G00 +  →  GE1E1 = G11 +

m m Y

Y

N

N

Y

N

N

Y

5.6 Conclusions

159

and m m

system

 ∩  = ∅ or incon() or val() Y ⇒  ∩  = ∅&con()&inval(¬) = G01 + N  → 

GE1E0 = G10 +

Y

GA0A1

N

 ⊆  or incon() ⇒   &con() = G01 +  → 

GE1A0 = G10 +

Y

N

GA0E1

N

Y

and m m

system  ⊆  or val() ⇒   &con() = G10 +  → 

GA0E1 = G01 +

N

Y

GE1A0

Y

N

 =  = {l} or  = ∅ or  = ∅ N ⇒ ( = ∅ = &( = {l} or  = {l}) Y = G10 +  → 

GA0A1 = G01 +

N

GE1E0

Y

and m m

system ¬ ⊆  or val(¬) ⇒ ¬  &inval(¬) = G11 +  → 

GA0E0 = G00 +

N

Y

GE1A1

Y

N

 = ¬ = {l} or  = ∅ or  = ∅ N ⇒ ( = {l} or ¬ = {l})& = ∅ = ¬ Y = G11 +  → 

GA0A0 = G00 +

N

GE1E1

Y

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160

and m m

system ¬ ⊆  or val() ⇒ ¬  &inval() = G00 +  → 

GA1E1 = G11 +

N

Y

GE0A0

Y

N

 = ¬ = {l} or ¬ = ∅ or  = ∅ N ⇒ ( = {l} or ¬ = {l})&¬ = ∅ =  Y = G00 +  → 

GA1A1 = G11 +

N

GE0E0

Y

and m m

system  ⊆  or val(¬) ⇒   &inval(¬) = G01 +  → 

GA1E0 = G10 +

N

Y

GE0A1

Y

N

¬ = ¬ = {l} or ¬ = ∅ or ¬ = ∅ N ⇒ (¬ = {l} or ¬ = {l})&¬ = ∅ =  Y = G01 +  → 

GA1A0 = G10 +

N

GE0E1

Y

References Cao, C., Sui, Y., Wang, Y.: The nonmonotonic propositional logics. Artif. Intell. Res. 5, 111–120 (2016) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W.: Mathematical logic, foundations for information science. In: Progress in Computer Science and Applied Logic, vol.25, Birkhäuser (2010) Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980) Takeuti, G.: Proof Theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987)

Chapter 6

R-Calculi: RY1 Q 1 i Y2 Q 2 j /RY1 Q 1 i Y2 Q 2 j

Let Q 1 , Q 2 ∈ {A, E}, i, j ∈ {0, 1}, and Y1 , Y2 ∈ {R, Q, P}. Let X =  ⇒  and X ∈ (A, B). A reduction X|(A, B) ⇒ X is RY1 Q 1 iY2 Q 2 j -valid (Li 2010; Takeuti and Barwise 1987), denoted by |=Y1 Q 1 iY2 Q 2 j X|(A, B) ⇒ X , if ⎧ X{A}{B} if X{A} and X{A}{B} are GY1 Q 1 Y2 Q 2 − valid ⎪ ⎪ ⎨ X{A} if X{A} is GY1 Q 1 Y2 Q 2 − valid and X{A}{B} is not X = X{B} if X{A} is not GY1 Q 1 Y2 Q 2 − valid and X{B} is ⎪ ⎪ ⎩ X otherwise where 

(A) if Q 1 = A [A] if Q 1 = Eand A ∈   (B) if Q 2 = Eand B ∈  {B} = [B] if Q 2 = A.

{A} =

Dually, a reduction X|X ⇒ X is RY1 Q 1 iY2 Q 2 j -valid (Alchourrón et al. 1985; Darwiche and Pearl 1997; Fermé and Hansson 2011; Gärdenfors and Rott 1995), denoted by |=Y1 Q 1 iY2 Q 2 j X|X ⇒ X , if ⎧ ⎨  ± A → |B if |=Y1 Q 1 iY2 Q 2 j  ± A → |B X =   →  ± B if |=Y1 Q 1 iY2 Q 2 j   →  ± B ⎩  otherwise. © Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_6

161

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We consider the following R-calculi (Li 2007): RY1 Q 1 iY2 Q 2 j QY1 Q 1 iY2 Q 2 j PY1 Q 1 iY2 Q 2 j About the monotonicity (Cao et al. 2016; Reiter 1980), we have the following conclusions: • GY1 Q 1 iY2 Q 2 j /GY1 Q 1 iY2 Q 2 j is monotonic in  iff Q 1 = E, and monotonic in  iff Q 2 = E; and Y Q iY Q j Y Q iY Q j • R0 1 1 2 2 /RY01 Q 1 iY2 Q 2 j is monotonic in  and in ; and R±1 1 2 2 /RY±1 Q 1 iY2 Q 2 j is nonmonotonic in  iff Q 1 = A, and nonmonotonic in  iff Q 2 = A.

6.1 Variant R-Calculi By choosing different minimal changes, there are following R-calculi: Rt Rf Rt Rf

Qt Qf Qt Qf

Pt Pf Pt Pf .

We will give R-calculi Rt , Qt , Pt , respectively.

6.1.1 R-Calculus Rt Definition 6.1.1 Given any theory  and a formula γ , a theory , γ  is a subset  minimal (⊆-minimal) change of  by γ , denoted by |=R t |γ ⇒ , γ , if (i) γ = γ    or γ = λ; (ii) , γ is consistent, and (iii) for any set  with , γ ⊂  ⊆ , γ ,  is inconsistent. R-calculus Rt consists of the following axioms and deduction rules: • Axioms:

(At+ ) where l is a literal.

  ¬l  ¬l (At0 ) |l ⇒ , l |l ⇒ 

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• Deduction rules:

|A1 ⇒ , A1 (¬¬+ )  |¬¬A1 ⇒ , ¬¬A1 |A1 ⇒ , A1 (∧+ ) , A1 |A2 ⇒ , A1 , A2 |A  1 ∧ A2 ⇒ , A1 ∧ A2 |A1 ⇒ , A1 (∨+ ) |A2 ⇒ , A2 |A  1 ∨ A2 ⇒ , A1 ∨ A2 |¬A1 ⇒ , ¬A1 (¬∧+ ) |¬A2 ⇒ , ¬A2 |¬(A 1 ∧ A2 ) ⇒ , ¬(A1 ∧ A2 )  |¬A1 ⇒ , ¬A1 (¬∨+ ) , ¬A1 |¬A2 ⇒ , ¬A1 , ¬A2 |¬(A1 ∨ A2 ) ⇒ , ¬(A1 ∨ A2 )

|A1 ⇒  (¬¬0 )  |¬¬A1 ⇒  |A1 ⇒  (∧0 ) , A1 |A2 ⇒ , A1 |A  1 ∧ A2 ⇒  |A1 ⇒  (∨0 ) |A2 ⇒  |A  1 ∨ A2 ⇒  |¬A1 ⇒  (¬∧0 ) |¬A2 ⇒  |¬(A 1 ∧ A2 ) ⇒   |¬A1 ⇒  (¬∨0 ) , ¬A1 |¬A2 ⇒ , ¬A1 |¬(A1 ∨ A2 ) ⇒ 

Definition 6.1.2 A reduction δ = |A ⇒ , C is provable in Rt , denoted by t |A ⇒ , C, if there is a sequence {δ1 , . . . , δm } of reductions such that δm = δ, and for each i < m, δi+1 is an axiom or is deduced from the previous statements by a deduction rule in Rt . Theorem 6.1.3 (Soundness and completeness theorem) For any consistent formula set  and formula A, if |A ⇒ , A is provable in Rt then  ∪ {A} is consistent, i.e., t |A ⇒ , A implies |=t |A ⇒ , A; and if |A ⇒  is provable in Rt then  ∪ {A} is inconsistent, i.e., t |A ⇒  implies |=t |A ⇒ . 

Rt is composed of two parts: Rt0 (consisting of axiom and deduction rules with 0) and Rt+ (consisting of axiom and deduction rules with +). Then, Rt0 is monotonic in , and Rt+ is nonmonotonic in .

6.1.2 R-Calculus Rf Definition 6.1.4 Given any theory  and a formula γ , a theory , γ  is a subset  minimal (⊆-minimal) change of  by γ , denoted by |=R f |γ ⇒ , γ , if (i) γ = γ

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or γ  = λ; (ii) , γ  is f-consistent, and (iii) for any set  with , γ  ⊂  ⊆ , γ ,  is f-inconsistent. R-calculus Rf consists of the following axioms and deduction rules: • Axioms:

(At+ )

 f ¬l  f ¬l (At0 ) |l ⇒ , l |l ⇒ 

where l is a literal. • Deduction rules:

|A1 ⇒ , A1 (¬¬+ ) |¬¬A 1 ⇒ , ¬¬A1  |A1 ⇒ , A1 (∧+ ) |A2 ⇒ , A2 |A  1 ∧ A2 ⇒ , A1 ∧ A2 |A1 ⇒ , A1 (∨+ ) , A1 |A2 ⇒ , A1 , A2 |A  1 ∨ A2 ⇒ , A1 ∨ A2 |¬A1 ⇒ , ¬A1 (¬∧+ ) , ¬A1 |¬A2 ⇒ , ¬A1 , ¬A2 |¬(A 1 ∧ A2 ) ⇒ , ¬(A1 ∧ A2 )  |¬A1 ⇒ , ¬A1 (¬∨+ ) |¬A2 ⇒ , ¬A2 |¬(A1 ∨ A2 ) ⇒ , ¬(A1 ∨ A2 )

|A1 ⇒  (¬¬0 ) |¬¬A 1 ⇒  |A1 ⇒  (∧0 ) |A2 ⇒  |A  1 ∧ A2 ⇒  |A1 ⇒  (∨0 ) , A1 |A2 ⇒ , A1 |A  1 ∨ A2 ⇒  |¬A1 ⇒  (¬∧0 ) , ¬A1 |¬A2 ⇒ , ¬A1 |¬(A 1 ∧ A2 ) ⇒   |¬A1 ⇒  (¬∨0 ) |¬A2 ⇒  |¬(A1 ∨ A2 ) ⇒ 

Definition 6.1.5 A reduction δ = |A ⇒ , C is provable in Rf , denoted by f |A ⇒ , C, if there is a sequence {δ1 , . . . , δm } of reductions such that δm = δ, and for each i < m, δi+1 is an axiom or is deduced from the previous statements by a deduction rule in Rf . Theorem 6.1.6 (Soundness and completeness theorem) For any consistent formula set  and formula A, f |A ⇒   iff |=f |A ⇒   . 

Rf is composed of two parts: Rf0 (consisting of axiom and deduction rules with 0) and Rf+ (consisting of axiom and deduction rules with +). Then, Rf0 is monotonic in , and Rf+ is nonmonotonic in .

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6.1.3 R-Calculus Qt Definition 6.1.7 A theory , γ  is a -minimal change of  by γ , denoted by |=Q t |γ ⇒ , γ  , if (i) γ   γ , (ii)  ∪ {γ  } is consistent, and (iii) for any theory  with , γ  ≺   , γ ,  is inconsistent. R-calculus Qt consists of the following axioms and deduction rules: • Axioms:

(At+ )

  ¬A |A ⇒ , A

(At0 )

 ¬l |l ⇒ , λ

• Deduction rules: 

(∧)

|A1 ⇒ , C1 , C1 |A2 ⇒ , C1 , C2 |A1 ∧ A2 ⇒ , C1 ∧ C2

⎧ |A1 ⇒ , C1  = λ ⎪ ⎪ ⎪ ⎪ ⎨ |A1 ⇒   |A2 ⇒ , C2  = λ (∨) ⎪ ⎪ ⎪ |A1 ⇒  ⎪ ⎩ |A2 ⇒  |A1 ∨ A2 ⇒ , C1 ∨ C2

⎧ |¬A1 ⇒ , ¬C1  = λ ⎪ ⎪  ⎪ ⎪  ⎨ |¬A1 ⇒  |¬A1 ⇒ , ¬C1 |¬A ⇒ , ¬C  = λ 2 2  (¬∨) , ¬C1 |¬A2 ⇒ , ¬C1 , ¬C2 (¬∧) ⎪ ⎪ |¬A1 ⇒  ⎪ ⎪ ⎩ |¬(A1 ∨ A2 ) ⇒ , ¬(C1 ∨ C2 ) |¬A2 ⇒  |¬(A1 ∧ A2 ) ⇒ , ¬(C1 ∧ C2 )

where ⎧ ⎨ C1 ∨ A2 if C1 = λ C1 ∨ C2 = A1 ∨ C2 if C1 = λ and C2 = λ ⎩ λ otherwise and ⎧ ⎨ ¬(C1 ∧ A2 if C1 = λ ¬(C1 ∧ C2 ) = ¬(A1 ∧ C2 ) if C1 = λ and C2 = λ ⎩ λ otherwise. Definition 6.1.8 A reduction δ = |A ⇒ , C is provable in Qt , denoted by t |A ⇒ , C, if there is a sequence {δ1 , . . . , δm } of reductions such that δm = δ, and for each i < m, δi+1 is an axiom or is deduced from previous reductions by a deduction rule in Qt .

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Theorem 6.1.9 For any formula set  and formula A, there is a formula C such 

that C  A and |A ⇒ , C is provable in Qt . Theorem 6.1.10 For any formula set  and formula A, if |A ⇒ , C is provable in Qt then C  A is a -minimal of  by A. 

Qt is composed of two parts: Q0t (axiom with 0 and deduction rules) and Q+ t (axiom with + and deduction rules). Then, Q0t is monotonic in , and Q+ t is nonmonotonic in .

6.1.4 R-Calculus Qf Definition 6.1.11 A theory , γ  is a -minimal change of  by γ , denoted by  |=Q f |γ ⇒ , γ , if (i) γ   γ , (ii)  ∪ {γ  } is f-consistent, and (iii) for any theory  with , γ  ≺   , γ ,  is f-inconsistent. R-calculus Qf consists of the following axioms and deduction rules: • Axioms:

(Af+ )

  ¬A |A ⇒ , A

(Af0 )

 ¬l |l ⇒ , λ

• Deduction rules: ⎧ |A1 ⇒ , C1 = λ ⎪ ⎪ ⎪ ⎪ ⎨ |A1 ⇒   |A2 ⇒ , C2 = λ (∧) ⎪ ⎪ |A1 ⇒  ⎪ ⎪ ⎩ |A2 ⇒  |A1 ∧ A2 ⇒ , C1 ∧ C2

 (∨)

|A1 ⇒ , C1 , C1 |A2 ⇒ , C1 , C2 |A1 ∨ A2 ⇒ , C1 ∨ C2

⎧ |¬A ⎪ 1 ⇒ , ¬C 1  = λ ⎪  ⎪ ⎪  ⎨ |¬A1 ⇒  |¬A1 ⇒ , ¬C1  |¬A2 ⇒ , ¬C2 = λ (¬∧) , ¬C1 |¬A2 ⇒ , ¬C1 , ¬C2 (¬∨) ⎪ ⎪ |¬A1 ⇒  ⎪ ⎪ |¬(A1 ∧ A2 ) ⇒ , ¬(C1 ∧ C2 ) ⎩ |¬A2 ⇒  |¬(A1 ∨ A2 ) ⇒ , ¬(C1 ∨ C2 )

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where ⎧ ⎨ C1 ∧ A2 if C1 = λ C1 ∧ C2 = A1 ∧ C2 if C1 = λ and C2 = λ ⎩ λ otherwise and ⎧ ⎨ ¬(C1 ∨ A2 if C1 = λ ¬(C1 ∨ C2 ) = ¬(A1 ∨ C2 ) if C1 = λ and C2 = λ ⎩ λ otherwise. Definition 6.1.12 A reduction δ = |A ⇒ , C is provable in Qf , denoted by Q f |A ⇒ , C, if there is a sequence {δ1 , . . . , δm } of reductions such that δm = δ, and for each i < m, δi+1 is an axiom or is deduced from previous reductions by a deduction rule in Qf . Theorem 6.1.13 (Soundness and completeness theorem) For any formula set  and formula A, Q Q f |A ⇒ , C iff |=f |A ⇒ , C.



Qf is composed of two parts: Q0f (axiom with 0 and deduction rules) and Q+ f (axiom with + and deduction rules). Then, Q0f is monotonic in , and Q+ f is nonmonotonic in .

6.1.5 R-Calculus Pt Definition 6.1.14 Given a theory  and a formula δ, theory , δ  is a deductionbased minimal (  -minimal) change of δ by , denoted by |=Pt |δ ⇒ , δ  , if (i)  ∪ {δ  } is consistent; (ii) δ   δ, and (iii) for any theory  with , δ     , δ, either  , δ  and , δ  , or  is inconsistent. R-calculus Pt consists of the following axioms and deduction rules: • Axiom:

(A+ P)

  ¬l |l ⇒ , l

(A0P )

 ¬l |l ⇒ 

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• Deduction rules:

(¬¬)

|A1 ⇒ , C1 |¬¬A1 ⇒ , ¬¬C1

 |A1 ⇒ , C1 |A1 ⇒ , C1 (∧) (∨) |A2 ⇒ , C2 |A1 ∧ A2 ⇒ , C1 |A2 |A1 ∨ A2 ⇒ , C1 ∨ C2  |¬A1 ⇒ , ¬C1 |¬A1 ⇒ , ¬C1 (¬∧) |¬A2 ⇒ , ¬C2 (¬∨) |¬(A1 ∨ A2 ) ⇒ , ¬C1 |¬A2 |¬(A1 ∧ A2 ) ⇒ , ¬(C1 ∧ C2 ) where if C is consistent then λ∨C ≡C ∨λ≡C λ ∧ C ≡ C ∧ λ ≡ C; , λ ≡  and if C is inconsistent then λ∨C ≡C ∨λ≡λ λ ∧ C ≡ C ∧ λ ≡ λ. Theorem 6.1.15 For any consistent set  of formulas and formula A in conjunctive normal form, there is a formula C such that (1) |A ⇒ , C is provable in Pt ; (2) C  A, and (3)  ∪ {C} is consistent, and for any D with C ≺ D  A, either , C D and , D C, or  ∪ {D} is inconsistent. Theorem 6.1.16 For any consistent set  of formulas and formula A in conjunctive normal form, if t |A ⇒ , C then (1) C  A, and (2)  ∪ {C} is consistent, and (3) for any D with C ≺ D  A, either , C D and , D C, or  ∪ {D} is inconsistent. 

Pt is composed of two parts: Pt0 (axiom with 0 and deduction rules) and Pt+ (axiom with + and deduction rules). Then, Pt0 is monotonic in , and Pt+ is nonmonotonic in .

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6.1.6 R-Calculus Pf Definition 6.1.17 Given a theory  and a formula δ, theory , δ  is a deductionbased minimal (  -minimal) change of δ by , denoted by |=Pf |δ ⇒ , δ  , if (i)  ∪ {δ  } is f-consistent; (ii) δ   δ, and (iii) for any theory  with , δ     , δ, either  , δ  and , δ  , or  is f-inconsistent. R-calculus Pf consists of the following axioms and deduction rules: • Axiom:

(A+ f)

  ¬l |l ⇒ , l

(A0f )

 ¬l |l ⇒ 

• Deduction rules: |A1 ⇒ , C1 (¬¬)  |¬¬A1 ⇒ , ¬¬C1 |A1 ⇒ , C1 (∧) |A2 ⇒ , C2 |A1 ∧ A2 ⇒ , C1 ∧ C2

|A1 ⇒ , C1 |A1 ∨ A2 ⇒ , C1 |A2  |¬A1 ⇒ , ¬C1 |¬A1 ⇒ , ¬C1 (¬∧) (¬∨) |¬A2 ⇒ , ¬C2 |¬(A1 ∧ A2 ) ⇒ , ¬C1 |¬A2 |¬(A1 ∨ A2 ) ⇒ , ¬(C1 ∨ C2 ) (∨)

Theorem 6.1.18 For any consistent set  of formulas and formula A in disjunctive normal form, Pf |A ⇒ , C iff |=Pf |A ⇒ , C.



Pf is composed of two parts: Pf0 (axiom with 0 and deduction rules) and Pf+ (axiom with + and deduction rules). Then, Pf0 is monotonic in , and Pf+ is nonmonotonic in .

6.2 R-Calculi RY1 Q 1 i Y2 Q 2 j Let Q 1 , Q 2 ∈ {E, A} and Y1 , Y2 ∈ {S, T, P}. Given a sequent  ⇒  and pair (A, B) of formulas, the result of  ⇒  GY1 Q 1 iY2 Q 2 j -revising (A, B) is denoted by |=Y1 Q 1 iY2 Q 2 j  ⇒ |(A, B) ⇒   ⇒  , where

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⎧ Y1 Q 1 iY2 Q 2 j -valid ⎪ ⎪  ±1 A ⇒  ±2 B if  ±1 A ⇒  ±2 B is G ⎨ otherwise, if  ±1 A ⇒  is GY1 Q 1 iY2 Q 2 j -valid  ±1 A ⇒     | = otherwise, if  ⇒  ±2 B is GY1 Q 1 iY2 Q 2 j -valid  ⇒  ±2 B ⎪ ⎪ ⎩ ⇒ otherwise, where  ±1 A =

∪{A} if Q 1 = A −{A} if Q 1 = E,

 ±2 B =

∪{B} if Q 2 = A −{B} if Q 2 = E.

6.2.1 Axioms Because |=Y1 Q 1 iY2 Q 2 j  ⇒  iff |= Q 1 i Q 2 j  ⇒ , axioms for RY1 Q 1 iY2 Q 2 j are same as those for R Q 1 i Q 2 j . Proposition 6.2.1 Let  ⇒  be literal. |=Y1 Q 1 iY2 Q 2 j  ⇒ |(l, m) ⇒   ⇒  if 

and only if |= Q 1 i Q 2 j  ⇒ |(l, m) ⇒   ⇒  .

6.2.2 Deduction Rules There are three basic sets R, Q, P of deduction rules: ♦ Deduction class R : • R L0 : X|A ⇒ X(A) L ) (¬¬+  X|¬¬A ⇒ X(¬¬A1 ) X|A1 ⇒ X(A1 ) L ) X|A2 ⇒ X(A2 ) (∧+ X|A  1 ∧ A2 ⇒ X(A1 ∧ A2 ) X|A1 ⇒ X(A1 ) L (∨+ ) X(A1 )|A2 ⇒ X(A1 , A2 ) X|A  1 ∨ A2 ⇒ X(A1 ∨ A2 ) X|¬A1 ⇒ X(¬A1 ) L ) X(¬A1 )|¬A2 ⇒ X(¬A1 , ¬A2 ) (¬∧+ X|¬(A 1 ∧ A2 ) ⇒ X(¬(A1 ∧ A2 ))  X|¬A1 ⇒ X(¬A1 ) L (¬∨+ ) X|¬A2 ⇒ X(¬A2 ) X|¬(A1 ∨ A2 ) ⇒ X(¬(A1 ∨ A2 ))

X|A ⇒ X (¬¬0L )  X|¬¬A ⇒ X X|A1 ⇒ X (∧0L ) X|A2 ⇒ X X|A  1 ∧ A2 ⇒ X X|A1 ⇒ X (∨0L ) X(A1 )|A2 ⇒ X(A1 ) X|A  1 ∨ A2 ⇒ X X|¬A1 ⇒ X (¬∧0L ) X(¬A1 )|¬A2 ⇒ X(¬A1 ) X|¬(A 1 ∧ A2 ) ⇒ X  X|¬A1 ⇒ X (¬∨0L ) X|¬A2 ⇒ X X|¬(A1 ∨ A2 ) ⇒ X

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• R R1 :

X|B ⇒ X(B) (¬¬+R )  X|¬¬B ⇒ X(¬¬B) X|B1 ⇒ X(B1 ) (∧+R ) X(B1 )|B2 ⇒ X(B1 , B2 ) X|B  1 ∧ B2 ⇒ X(B1 ∧ B2 ) X|B1 ⇒ X(B1 ) (∨+R ) X|B2 ⇒ X(B2 ) X|B  1 ∨ B2 ⇒ X(B1 ∨ B2 ) X|¬B1 ⇒ X(¬B1 ) (¬∧+R ) X|¬B2 ⇒ X(¬B2 ) X|¬(B 1 ∧ B2 ) ⇒ X(¬(B1 ∧ B2 ))  X|¬B1 ⇒ X(¬B1 ) (¬∨+R ) X(¬B1 )|¬B2 ⇒ X(¬B1 , ¬B2 ) X|¬(B1 ∨ B2 ) ⇒ X(¬(B1 ∨ B2 ))

X|B ⇒ X (¬¬0R )  X|¬¬B ⇒ X X|B1 ⇒ X (∧0R ) X(B1 )|B2 ⇒ X(B1 ) X|B  1 ∧ B2 ⇒ X X|B1 ⇒ X (∨0R ) X|B2 ⇒ X X|B  1 ∨ B2 ⇒ X X|¬B1 ⇒ X (¬∧0R ) X|¬B2 ⇒ X X|¬(B 1 ∧ B2 ) ⇒ X  X|¬B1 ⇒ X (¬∨0R ) X(¬B1 )|¬B2 ⇒ X(¬B1 ) X|¬(B1 ∨ B2 ) ⇒ X

• R L1 : where A ∈ .

X|A ⇒ X (¬¬0L ) X|¬¬A ⇒X  X|A1 ⇒ X (∧0L ) X[A1 ]|A2 ⇒ X[A1 ] X|A  1 ∧ A2 ⇒ X X|A1 ⇒ X (∨0L ) X|A2 ⇒ X X|A  1 ∨ A2 ⇒ X X|¬A1 ⇒ X (¬∧0L ) X|¬A2 ⇒ X X|¬(A 1 ∧ A2 ) ⇒ X  X|¬A1 |X ⇒ X (¬∨0L ) X[¬A1 ]|¬A2 ⇒ X[¬A1 ] X|¬(A1 ∨ A2 ) ⇒ X • R R0 : where B ∈ 

X|A ⇒ X[A] L (¬¬− ) X|¬¬A ⇒ X[¬¬A]  X|A1 ⇒ X[A1 ] L ) X[A1 ]|A2 ⇒ X[A1 , A2 ] (∧− X|A  1 ∧ A2 ⇒ X[A1 ∧ A2 ] X|A1 ⇒ X[A1 ] L ) X|A2 ⇒ X[A2 ] (∨− X|A  1 ∨ A2 ⇒ X[A1 ∨ A2 ] X|¬A1 ⇒ X[¬A1 ] L ) X|¬A2 ⇒ X[¬A2 ] (¬∧− X|¬(A 1 ∧ A2 ) ⇒ X[¬(A1 ∧ A2 )]  X|¬A1 ⇒ X[¬A1 ] L ) X[¬A1 ]|¬A2 ⇒ X[¬A1 , ¬A2 ] (¬∨− X|¬(A1 ∨ A2 ) ⇒ X[¬(A1 ∨ A2 )

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X|B ⇒ X (¬¬0R )  X|¬¬B ⇒ X X|B1 ⇒ X R X|B (∧0 ) 2 ⇒X X|B  1 ∧ B2 ⇒ X X|B1 ⇒ X (∨0R ) X|B2 ⇒ X X|B  1 ∨ B2 ⇒ X X|¬B1 ⇒ X (¬∧0R ) X|¬B2 ⇒ X X|¬(B 1 ∧ B2 ) ⇒ X  X|¬B1 ⇒ X (¬∨0R ) X|¬B2 ⇒ X X|¬(B1 ∨ B2 ) ⇒ X

X|B ⇒ X[B] (¬¬−R )  X|¬¬B ⇒ X[¬¬B] X|B1 ⇒ X[B1 ] R (∧− ) X[B1 ]|B2 ⇒ X[B1 , B2 ] X|B  1 ∧ B2 ⇒ X[B1 ∧ B2 ] X|B1 ⇒ X[B1 ] (∨−R ) X|B2 ⇒ X[B2 ] X|B  1 ∨ B2 ⇒ X[B1 ∨ B2 ] X|¬B1 ⇒ X[¬B1 ] (¬∧−R ) X|¬B2 ⇒ X[¬B2 ] X|¬(B 1 ∧ B2 ) ⇒ X[¬(B1 ∧ B2 )  X|¬B1 ⇒ X[¬B1 ] (¬∨−R ) X|¬B2 ⇒ X[¬B2 ] X|¬(B1 ∨ B2 ) ⇒ X[¬(B1 ∨ B2 )]

♦ Deduction class Q : • Q L0 : X|A ⇒ X(A ) (¬¬ L ) ⇒ X(¬¬A ) ⎧X|¬¬A  X|A ⇒ X(C1 ) ⎪ 1 ⎪ ⎪ ⎪  = λ C ⎪ 1 ⎪ ⎡ ⎪ ⎪  ⎨ X|A1 ⇒ X X|A1 ⇒ X(C1 ) ⎣ X|A2 ⇒ X(C2 ) L X(C ) (∧ L ) ⎪ (∨ 1 )|A2 ⇒ X(C 1 , C 2 ) ⎪ ⎪ ⎪  C2 = λ X|A ∨ A2 ⇒ X(C1 ∨ C2 ) ⎪ 1 ⎪ X|A1 ⇒ X ⎪ ⎪ ⎩ X|A2 ⇒ X X|A1 ∧ A2 ⇒ X(C1 ∧ C2 ) and 

(¬∧ L )

X|¬A1 ⇒ X(¬C1 ) X(¬C1 )|¬A2 ⇒ X(¬C1 , ¬C2 ) X|¬(A1 ∧ A2 )⎧⇒  X(¬(C1 ∧ C2 )) X|¬A1 ⇒ X(¬C1 ) ⎪ ⎪ ⎪ ⎪ C ⎪ ⎪ ⎡ 1 = λ ⎪ ⎪ ⎨ X|¬A1 ⇒ X ⎣ X|¬A2 ⇒ X(¬C2 ) (¬∨ L ) ⎪ ⎪ ⎪ ⎪  C2 = λ ⎪ ⎪ X|¬A1 ⇒ X ⎪ ⎪ ⎩ X|¬A2 ⇒ X X|¬(A1 ∨ A2 ) ⇒ X(¬(C1 ∨ C2 ))

6.2 R-Calculi RY1 Q 1 iY2 Q 2 j

where ⎧ ⎨ C1 ∧ A2 if C1 = λ C1 ∧ C2 = A1 ∧ C2 if C1 = λ and C2 = λ ⎩ λ⎧ otherwise ¬(C ⎨ 1 ∨ A2 ) if C 1  = λ   ¬(A ¬(C1 ∨ C2 ) = 1 ∨ C 2 ) if C 1 = λ and C 2  = λ ⎩ λ otherwise • Q R1 :

(¬¬ R )

X|B ⇒ X(C) X|¬¬B ⇒ X(¬¬C)

⎧ X|B1 ⇒ X(D1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪  ⎨ X|B1 ⇒ X X|B1 ⇒ X(D1 ) ⎣ X|B2 ⇒ X(D2 ) (∧ R ) X(D1 )|B2 ⇒ X(D1 , D2 ) (∨ R ) ⎪ ⎪ ⎪ ⎪  D2  = λ X|B1 ∧ B2 ⇒ X(D1 ∧ D2 ) ⎪ ⎪ X|B1 ⇒ X ⎪ ⎪ ⎩ X|B2 ⇒ X X|B1 ∨ B2 ⇒ X(D1 ∨ D2 ) and ⎧ X|¬B1 ⇒ X(¬D1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪ ⎨ X|¬B1 ⇒ X ⎣ X|¬B2 ⇒ X(¬D2 ) (¬∧ R ) ⎪ ⎪ ⎪ ⎪  D2  = λ ⎪ ⎪ X|¬B1 ⇒ X ⎪ ⎪ ⎩ X|¬B2 ⇒ X X|¬(B1 ∧ B2 )⇒ X(¬(D1 ∧ D2 )) X|¬B1 ⇒ X(¬D1 ) (¬∨ R ) X(¬D1 )|¬B2 ⇒ X(¬D1 , ¬D2 ) X|¬(B1 ∨ B2 ) ⇒ X¬(D1 ∨ D2 )) where ⎧ ⎨ D1 ∨ B2 if D1 = λ D1 ∨ D2 = B1 ∨ D2 if D1 = λ and D2 = λ ⎩ λ⎧ otherwise ⎨ ¬(D1 ∧ B2 ) if D1 = λ ¬(D1 ∧ D2 ) = ¬(B1 ∧ D2 ) if D1 = λ and D2 = λ ⎩ λ otherwise

173

6 R-Calculi: RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j

174

• Q L1 : where A ∈ . X|A ⇒ X[C] (¬¬ L ) X|¬¬A ⇒ X[¬¬C] ⎧ X|A ⇒ X[C1 ] ⎪ 1 ⎪ ⎪ ⎪  = λ C ⎪ 1 ⎪ ⎡ ⎪ ⎪  ⎨ X|A1 ⇒ X X|A1 ⇒ X[C1 ] ⎣ X|A2 ⇒ X[C2 ] L X[C1 ]|A2 ⇒ X[C1 , C2 ] ) (∧ L ) ⎪ (∨ ⎪ ⎪ ⎪  C2 = λ X|A ⎪ 1 ∨ A2 ⇒ X[C 1 ∨ C 2 ] ⎪ X|A1 ⇒ X ⎪ ⎪ ⎩ X|A2 ⇒ X X|A1 ∧ A2 ⇒ X[C1 ∧ C2 ] and 

(¬∧ L )

X|¬A1 ⇒ X[¬C1 ] X[¬C1 ]|¬A2 ⇒ X[¬C1 , ¬C2 ] X|¬(A1 ∧ A2 )⎧⇒  X[¬(C1 ∧ C2 )] X|¬A1 ⇒ X[¬C1 ] ⎪ ⎪ ⎪ ⎪ C ⎪ ⎪ ⎡ 1 = λ ⎪ ⎪ ⎨ X|¬A1 ⇒ X ⎣ X|¬A2 ⇒ X[¬C2 ] (¬∨ L ) ⎪ ⎪ ⎪ ⎪  C2 = λ ⎪ ⎪ X|¬A1 ⇒ X ⎪ ⎪ ⎩ X|¬A2 ⇒ X X|¬(A1 ∨ A2 ) ⇒ X[¬(C1 ∨ C2 )]

where ⎧ ⎨ C1 ∨ A1 if C1 = λ C1 ∧ C2 = A1 ∨ C1 if C1 = λ and C2 = λ ⎩ λ⎧ otherwise; ⎨ ¬(C1 ∨ A1 ) if C1 = λ ¬(C1 ∧ C2 ) = ¬(A1 ∨ C2 ) if C1 = λ and C2 = λ ⎩ λ otherwise; • Q R0 : where B ∈ 

6.2 R-Calculi RY1 Q 1 iY2 Q 2 j

(¬¬ R )

X|B ⇒ X[D] X|¬¬B ⇒ X[¬¬D]

⎧ X|B1 ⇒ X[D1 ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪  ⎨ X|B1 ⇒ X X|B1 ⇒ X[D1 ] ⎣ X|B2 ⇒ X[D2 ] (∧ R ) X[D1 ]|B2 ⇒ X[D1 , D2 ] (∨ R ) ⎪ ⎪ ⎪ ⎪  D2  = λ X|B1 ∧ B2 ⇒ X[D1 ∧ D2 ] ⎪ ⎪ X|B1 ⇒ X ⎪ ⎪ ⎩ X|B2 ⇒ X X|B1 ∨ B2 |X ⇒ X[D1 ∨ D2 ]

and ⎧ X|¬B1 ⇒ X[¬D1 ] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪ ⎨ X|¬B1 ⇒ X ⎣ X|¬B2 ⇒ X[¬D2 ] (¬∧ R ) ⎪ ⎪ ⎪  D2  = λ ⎪ ⎪ ⎪ ⎪ X|¬B1 ⇒ X ⎪ ⎩ X|¬B2 ⇒ X X|¬(B1 ∧ B2 )⇒ X[¬(D1 ∧ D2 )] X|¬B1 ⇒ X[¬D1 ] (¬∨ R ) X[¬D1 ]|¬B2 ⇒ X[¬D1 , ¬D2 ] X|¬(B1 ∨ B2 ) ⇒ X[¬(D1 ∨ D2 )] where ⎧ ⎨ D1 ∨ B1 if D1 = λ D1 ∨ D2 = B1 ∨ D1 if D1 = λ and D2 = λ ⎩ λ⎧ otherwise; ⎨ ¬(D1 ∨ B2 ) if D1 = λ ¬(D1 ∨ D2 ) = ¬(B1 ∨ D2 ) if D1 = λ and D2 = λ ⎩ λ otherwise; ♦ Deduction class P : • P L0 : X|A ⇒ X(C) (¬¬ L ) X|¬¬A ⇒ X(¬¬C)   X|A1 ⇒ X(C1 ) X|A1 ⇒ X(C1 ) (∧ L ) X|A2 ⇒ X(C2 ) (∨ L ) X(C1 )|A2 ⇒ X(C1 , C2 ) X|A1 ∧ A2 ⇒ X(C1 ∧ C2 ) X|A1 ∨ A2 ⇒ X(C1 ∨ C2 )

175

6 R-Calculi: RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j

176

and 

(¬∧ L )

X|¬A1 ⇒ X(¬C1 ) X(¬C1 )|¬A2 ⇒ X(¬C1 , ¬C2 ) X|¬(A1 ∧ A2 )⇒ X(¬(C1 ∧ C2 )) X|¬A1 ⇒ X(¬C1 ) (¬∨ L ) X|¬A2 ⇒ X(¬C2 ) X|¬(A1 ∨ A2 ) ⇒ X(¬(C1 ∨ C2 ))

• P R1 : X|B ⇒ X(D) (¬¬ R )  X|¬¬B ⇒ X(¬¬D)  X|B1 ⇒ X(D1 ) X|B1 ⇒ X(D1 ) R R )|B ⇒ X(D , D ) X(D (∧ ) (∨ ) X|B2 ⇒ X(D2 ) 1 2 1 2 X|B1 ∧ B2 ⇒ X(D1 ∧ D2 ) X|B1 ∨ B2 ⇒ X(D1 ∨ D2 ) and 

(¬∧ R )

X|¬B1 ⇒ X(¬D1 ) X|¬B2 ⇒ X(¬D2 ) X|¬(B1 ∧ B2 )⇒ X(¬(D1 ∧ D2 )) X|¬B1 ⇒ X(¬D1 ) (¬∨ R ) X(¬D1 )|¬B2 ⇒ X(¬D1 , ¬D2 ) X|¬(B1 ∨ B2 ) ⇒ X(¬(D1 ∨ D2 ))

• P L1 : where A ∈ . X|A ⇒ X[C] (¬¬ L )  X|¬¬A ⇒ X[¬¬C]  X|A1 ⇒ X[C1 ] X|A1 ⇒ X[C1 ] (∧ L ) X|A2 ⇒ X[C2 ] (∨ L ) X[C1 ]|A2 ⇒ X[C1 , C2 ] X|A1 ∧ A2 ⇒ X[C1 ∧ C2 ] X|A1 ∨ A2 ⇒ X[C1 ∨ C2 ] and 

(¬∧ L )

X|¬A1 ⇒ X[¬C1 ] X[¬C1 ]|¬A2 ⇒ X[¬C1 , ¬C2 ] X|¬(A1 ∧ A2 )⇒ X[¬(C1 ∧ C2 )] X|¬A1 ⇒ X[¬C1 ] (¬∨ L ) X|¬A2 ⇒ X[¬C2 ] X|¬(A1 ∨ A2 ) ⇒ X[¬(C1 ∨ C2 ]

6.2 R-Calculi RY1 Q 1 iY2 Q 2 j

177

• P R0 : where B ∈  X|B ⇒ X[D] (¬¬ R ) X|¬¬B ⇒ X[¬¬D]   X|B1 ⇒ X[D1 ] X|B1 ⇒ X[D1 ] (∧ R ) X[D1 ]|B2 ⇒ X[D1 , D2 ] (∨ R ) X|B2 ⇒ X[D2 ] X|B1 ∧ B2 ⇒ X[D1 ∧ D2 ] X|B1 ∨ B2 ⇒ X[D1 ∨ D2 ] and 

(¬∧ R )

X|¬B1 ⇒ X[¬D1 ] X|¬B2 ⇒ X[¬D2 ] X|¬(B1 ∧ B2 )⇒ X[¬(D1 ∧ D2 )] X|¬B1 ⇒ X[¬D1 ] (¬∨ R ) X[¬D1 ]|¬B1 ⇒ X[¬D1 , ¬D2 ] X|¬(B1 ∨ B2 ) ⇒ X[¬(D1 ∨ D2 )]

Notice that Y L0 , Y R1 are for enumerating elements into , and Y L1 , Y R0 are for not enumerating elements into .

6.2.3 R-Calculi For Y ∈ {R, Q, P}, there are eight R-calculi Y LE0 Y LA0 Y LE1 Y LA1 Y RE0 Y RA0 Y RE1 Y RA1 and there are four given R-calculi: R L0 R R1 R L1 R R0 . We define Y RE0 = Y R0 Y LE0 = Y L0 LA0 L0 Y = Y ((A)/[A]) Y RA0 = Y R1 ((B)/[B]) LE1 = Y L1 ([A]/(A)) Y RE1 = Y R1 Y LA1 Y = Y L1 Y RA1 = Y R1 ((B)/[B]). Define R Q2 j

RY1 Q 1 iY2 Q 2 j = A Q 1 i Q 2 j + Y1L Q 1 i + Y2

.

6 R-Calculi: RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j

178

Definition 6.2.2 A reduction δ =  ⇒ |(A, B) ⇒   ⇒  is provable in RY1 Q 1 iY2 Q 2 j , denoted by Y1 Q 1 iY2 Q 2 j δ, if there is a sequence {δ1 , . . . , δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in RY1 Q 1 iY2 Q 2 j . Theorem 6.2.3 (Soundness and completeness theorem) For any Q 1 , Q 2 ∈ {A, E}, i, j ∈ {0, 1}, Y1 , Y2 ∈ {R, Q} and reduction δ =  → |(A, B) ⇒   →Y1 Q 1 iY2 Q 2 j  , δ is RY1 Q 1 iY2 Q 2 j -valid if and only if δ is provable in RY1 Q 1 iY2 Q 2 j . That is, |=Y1 Q 1 iY2 Q 2 j δ iff Y1 Q 1 iY2 Q 2 j δ.



Theorem 6.2.4 (Soundness and completeness theorem) For any Q 1 , Q 2 ∈ {A, E}, i, j ∈ {0, 1} and reduction δ =  → |(A, B) ⇒   →PQ 1 iPQ 2 j  , where A is in conjunctive normal form and B is in disjunctive normal form, δ is RPQ 1 iPQ 2 j -valid if and only if δ is provable in RPQ 1 iPQ 2 j . That is, |=PQ 1 iPQ 2 j δ iff PQ 1 iPQ 2 j δ.



Y Q 1 iY2 Q 2 j

and R0 1

Y Q iY2 Q 2 j

is monotonic in  and in 

RY1 Q 1 iY2 Q 2 j is decomposed into two parts: R±1

Theorem 6.2.5 (Nonmonotonicity theorem) R0 1 1 Y Q iY Q j ; and R±1 1 2 2 is nonmonotonic in  and in .

Y Q 1 iY2 Q 2 j

.

6.3 R-Calculi RY1 Q 1 i Y2 Q 2 j Let Q 1 , Q 2 ∈ {E, A}, i, j ∈ {0, 1} and Y1 , Y2 ∈ {R, Q, P}. Given a co-sequent  →  and pair (A, B) of formulas, the result of  →  GY1 Q 1 iY2 Q 2 j -revising (A, B) is denoted by |=Y1 Q 1 iY2 Q 2 j  → |(A, B) ⇒   →  , if   →  = ⎧ , {C} → , {D} if , {C} → , {D} is GY1 Q 1 iY2 Q 2 j -valid ⎪ ⎪ ⎨ , {C} →  if , {C} →  is GY1 Q 1 iY2 Q 2 j -valid   → , {D} if  → , {D} is GY1 Q 1 iY2 Q 2 j -valid ⎪ ⎪ ⎩  →  otherwise.

6.3 R-Calculi RY1 Q 1 iY2 Q 2 j

179

6.3.1 Axioms Proposition 6.3.1 Let  →  be literal. |=Y1 Q 1 iY2 Q 2 j  → |(l, m) ⇒   →  if and only if |= Q 1 i Q 2 j  → |(l, m) ⇒   → 



6.3.2 Deduction Rules We define Y RE0 = Y R0 (⇒ / →) Y LE0 = Y L0 (⇒ / →) L0 Y LA0 = Y (⇒ / →)((A)/[A]) Y RA0 = Y R1 (⇒ / →)((B)/[B]) Y LE1 = Y L1 (⇒ / →)([A]/(A)) Y RE1 = Y R1 (⇒ / →) Y RA1 = Y R1 (⇒ / →)((B)/[B]). Y LA1 = Y L1 (⇒ / →)

6.3.3 R-Calculi Define R Q2 j

RY1 Q 1 iY2 Q 2 j = A Q 1 i Q 2 j + Y1L Q 1 i + Y2

.

Definition 6.3.2 A reduction  → |(A, B) ⇒   →  is provable in RY1 Q 1 iY2 Q 2 j , denoted by Y1 Q 1 iY2 Q 2 j  → |(A, B) ⇒   →  , if there is a sequence {δ1 , . . . , δn } of reductions such that δn =  → |(A, B) ⇒   →  , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in RY1 Q 1 iY2 Q 2 j . Theorem 6.3.3 (Soundness and completeness theorem) For any Q 1 , Q 2 ∈ {A, E}, i, j ∈ {0, 1}, Y1 , Y2 ∈ {R, Q} and reduction δ =  → |(A, B) ⇒   →  , δ is RY1 Q 1 iY2 Q 2 j -valid if and only if δ is provable in RY1 Q 1 iY2 Q 2 j . That is, |=Y1 Q 1 iY2 Q 2 j δ iff Y1 Q 1 iY2 Q 2 j δ.



6 R-Calculi: RY1 Q 1 iY2 Q 2 j /RY1 Q 1 iY2 Q 2 j

180

Theorem 6.3.4 (Soundness and completeness theorem) For any Q 1 , Q 2 ∈ {A, E}, i, j ∈ {0, 1} and reduction δ =  → |(A, B) ⇒   →  , where A is in disjunctive normal form and B is in conjunctive normal form, δ is RPQ 1 iPQ 2 j -valid if and only if δ is provable in RPQ 1 iPQ 2 j . That is, |=PQ 1 iPQ 2 j δ iff PQ 1 iPQ 2 j δ. RY1 Q 1 iY2 Q 2 j is decomposed into two parts:



RY±1 Q 1 iY2 Q 2 j

and RY01 Q 1 iY2 Q 2 j .

Theorem 6.3.5 (Nonmonotonicity theorem) RY01 Q 1 iY2 Q 2 j is monotonic in  and in

 ; and RY±1 Q 1 iY2 Q 2 j is nonmonotonic in  and in .

6.4 Conclusions There are the following sound and complete Gentzen deduction systems GY1 Q 1 iY2 Q 2 j = A Q 1 i Q 2 j + G Li + G R j GY1 Q 1 iY2 Q 2 j = A Q 1 i Q 2 j + G Li + G R j . Moreover, GY1 Q 1 iY2 Q 2 j /GY1 Q 1 iY2 Q 2 j is monotonic in  if and only if Q 1 = E; and nonmonotonic in  if and only if Q 2 = A. Correspondingly, there are the following sound and complete R-calculi: RQ j

RY1 Q 1 iY2 Q 2 j = A Q 1 i Q 2 j + Y1L Q 1 i + Y2 2 RQ j RY1 Q 1 iY2 Q 2 j = A Q 1 i Q 2 j + Y1L Q 1 i + Y2 2 , Y Q iY Q j

Y Q 1 iY2 Q 2 j

and R0 1 1 2 2 /RY01 Q 1 iY2 Q 2 j is monotonic in  and in ; and R±1 is nonmonotonic in  and in .

/RY±1 Q 1 iY2 Q 2 j

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symbol. Log. 50, 510–530 (1985) Cao, C., Sui, Y., Wang, Y.: The nonmonotonic propositional logics. Artif. Intell. Res. 5, 111–120 (2016) Darwiche, A., Pearl, J.: On the logic of iterated belief revision. Artif. Intell. 89, 1–29 (1997) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philosoph. Log. 40, 295–331 (2011) Gärdenfors, P., Rott, H.: Belief revision. In: Handbook of logic in artificial intelligence and logic programming, vol. 4, pp. 35-132. Oxford University Press (1995) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W.: Mathematical logic, foundations for information science. In: Progress in Computer Science and Applied Logic, vol. 25, Birkhäuser (2010)

References

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Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980) Takeuti, G.: Proof Theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987)

Chapter 7

R-Calculi for Supersequents

A sequent  ⇒  is valid if for any assignment v, v satisfying each formula in  implies v satisfying some formula in , equivalently, either v satisfies the negation of some formula in  or v satisfies some formula in . Correspondingly, there are four kinds of tableau proof systems in propositional logic: A multisequent | is valid (Avron 1991; Baaz and Zach 2000) if for any assignment v, either v satisfies some formula in  or v does not satisfy some formula in . In L3 -valued propositional logic, a multisequent is of form ||, which is valid (Bochvar 1938; Fitting 1991; Gottwald 2001; Hähnle 2001) if for any assignment v, either some formula in  has truth-value t, or some formula B ∈  has truth-value m, or some formula C in  has truth-value f. In B2 -valued propositional logic, a sequent  ⇒  is equivalent to multisequent | (Li 2010). A supersequent δ is of form | ⇒ |, where , , ,  are sets of formulas. δ is valid if for any assignment v, both each formula in  has truth-value 1 and each formula in  has truth-value 0 imply either some formula in  has truth-value 1 or some formula in  has truth-value 0. In this paper, we will give a sound and complete Gentzen deduction system G+ for supersequents, from which four sound and complete Gentzen deduction systems for sequents are deduced (Łukasiewicz 1970; Malinowski 2009; Urquhart 2001; Wronski 1987): • Gft : a sequent  ⇒  is valid if for any assignment v, each formula A ∈  having truth-value 1 implies some formula C ∈  having truth-value 1; • Gff : a sequent  ⇒  is valid if for any assignment v, each formula A ∈  having truth-value 1 implies some formula D ∈  having truth-value 0; • Gtt : a sequent  ⇒  is valid if for any assignment v, each formula B ∈  having truth-value 0 implies some formula C ∈  having truth-value 1; • Gtf : a sequent  ⇒  is valid if for any assignment v, each formula B ∈  having truth-value 0 implies some formula D ∈  having truth-value 0.

© Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_7

183

184

7 R-Calculi for Supersequents

Conversely, a supersequent | ⇒ | is equivalent to sequents , ¬ ⇒ , ¬; , ¬ ⇒ ¬, ; ¬,  ⇒ , ¬; ¬,  ⇒ ¬, . respectively. That is, | ⇒ | is provable in G+ if and only if , ¬ ⇒ , ¬is provable inGft ; , ¬ ⇒ ¬, is provable inGff ; ¬,  ⇒ , ¬is provable inGtt ; ¬,  ⇒ ¬, is provable inGtf . Also, G+ is taken as a combination of four tableau proof systems: • G Lf :  is provable if and only if for any assignment v, there exists a formula A ∈  such that v(A) = 0; • G Lt :  is provable if and only if for any assignment v, there exists a formula B ∈  such that v(B) = 1; • G Rt :  is provable if and only if for any assignment v, there is a formula C ∈  such that v(C) = 1; • G Rf :  is provable if and only if for any assignment v, there is a formula D ∈  such that v(D) = 0. | ⇒ |

G

HH  H

GL

HH  H

GR

H

|



H

HH 

G Lf

G Lf

G Rt

G Rf

|

multisequents

H

⇒ ⇒

Gft , Gff , Gtt , Gtf

P  @PPP  P @



hybridsequents

PP @ PP P @

sequents

 



⇒ ⇒

⇒ ⇒

semi-sequents

Therefore, a deduction system G+ for supersequents is decomposed into two deduction systems G L and G R for multisequents; between G L and G R , there are four deduction systems Gft , Gff , Gtt , Gtf for sequents, which are decomposed into four tableau proof systems G Lt , G Lf , G Rt , G Rf for theories. Correspondingly, there are R-calculi R+ , R− for supersequents and R-calculi Rft , Rff , Rtt , Rtf

7.1 Supersequents

185

for sequents, and Rtf , Rtt , Rff , Rft for co-sequents. We will give the following deduction systems and R-calculi: supersequents +

sequents ft

G G−

G Gtf

Gff Gtt

Gtt Gtf Gff Gft

R+ R−

Rft Rff Rtf Rtt

Rtt Rtf Rff Rft

7.1 Supersequents A supersequent δ is a quadruple (, , , ) of formula sets, denoted by | ⇒ |. And a supersequent δ is satisfied in v, denoted by v |= δ, if v |= | implies v |= |, where • v |= | if for each formula A∈, v(A) = 1; and for each formula B ∈ , v(A) = 0; • v |= | if either for some formula C ∈ , v(C) = 1; or for some formula D ∈ , v(D) = 0. A supersequent δ is valid, denoted by |= δ, if for any assignment v, v |= δ. Proposition 7.1.1 For any supersequent | ⇒ | and any assignment v, (i) v |= | iff v |= , ¬; and (ii) v |= | iff v |= , ¬. Therefore, |⇒| is G+ -valid iff , ¬⇒, ¬ is Gft -valid, iff , ¬ ⇒ ¬,  is Gff -valid; iff ¬,  ⇒ , ¬ is Gtt -valid; and iff ¬,  ⇒ ¬,  is Gtf -valid.

7.1.1 Supersequent Gentzen Deduction System G+ Supersequent Gentzen deduction system G+ consists of the following axioms and deduction rules: • Axiom:

incon( ∪ ¬) or val( ∪ ¬) (A+ ) or ( ∪ ¬) ∩ ( ∪ ¬) = ∅ | ⇒ |

186

7 R-Calculi for Supersequents

where , , ,  are sets of literals. • Deduction rules: , A| ⇒ | |B,  ⇒ | (¬¬ B ) , ¬¬A| ⇒ | |¬¬B,  ⇒ | | ⇒ , C| | ⇒ |D,  (¬¬C ) (¬¬ D ) | ⇒ , ¬¬C| | ⇒ |¬¬D, 

(¬¬ A )



and (∧ A ) (∧C ) (∨ A ) (∨C )

, A1 | ⇒ | , A2 | ⇒ | ,  A1 ∧ A2 | ⇒ | | ⇒ , C1 | | ⇒ , C2 | | ⇒ , C1 ∧ C2 |  , A1 | ⇒ | , A2 | ⇒ | ,  A1 ∨ A2 | ⇒ | | ⇒ , C1 | | ⇒ , C2 | | ⇒ , C1 ∨ C2 |



|B1 ,  ⇒ | |B2 ,  ⇒ | |B  1 ∧ B2 ,  ⇒ | | ⇒ |D1 ,  (∧ D ) | ⇒ |D2 ,  | ⇒ |D1 ∧ D2 ,   |B1 ,  ⇒ | (∨ B ) |B2 ,  ⇒ | |B  1 ∨ B2 ,  ⇒ | | ⇒ |D1 ,  (∨ D ) | ⇒ |D2 ,  | ⇒ |D1 ∨ D2 ,  (∧ B )

and 

(¬∧ ) A

(¬∧C ) (¬∨ A ) (¬∧C )

, ¬A1 | ⇒ | , ¬A2 | ⇒ | ,  ¬(A1 ∧ A2 )| ⇒ | | ⇒ , ¬C1 | | ⇒ , ¬C2 | |  ⇒ , ¬(C1 ∧ C2 )| , ¬A1 | ⇒ | , ¬A2 | ⇒ | ,  ¬(A1 ∨ A2 )| ⇒ | | ⇒ , ¬C1 | | ⇒ , ¬C2 | | ⇒ , ¬(C1 ∨ C2 )|



(¬∧ ) B

(¬∧ D ) (¬∨ B ) (¬∨ D )

|¬B1 ,  ⇒ | |¬B2 ,  ⇒ | |¬(B 1 ∧ B2 ),  ⇒ |  | ⇒ |¬D1 ,  | ⇒ |¬D2 ,  | ⇒ |¬(D1 ∧ D2 ),   |¬B1 ,  ⇒ | |¬B2 ,  ⇒ | |¬(B 1 ∨ B2 ),  ⇒ |  | ⇒ |¬D1 ,  | ⇒ |¬D2 ,  | ⇒ |¬(D1 ∨ D2 ), 

Definition 7.1.2 A supersequent | ⇒ | is provable in G+ , denoted by + | ⇒ |, if there is a sequence {1 |1 ⇒ 1 |1 , . . . , n |n ⇒ n |n } of supersequents such that n |n ⇒ n |n = | ⇒ |, and for each 1 ≤ i ≤ n, i |i ⇒ i |i is either an axiom or deduced from the previous supersequents by one of the deduction rules in G+ . Theorem 7.1.3 (Soundness theorem) For any supersequent δ = | ⇒ |, if

+ δ then |=+ δ.

7.1 Supersequents

187

Proof We prove that each axiom is valid and each deduction rule preserves validity. (A+ ) By Proposition 7.1.1, literal supersequent | ⇒ | is G+ -valid iff , ¬ ⇒ , ¬ is Gft -valid, iff either  ∪ ¬ is inconsistent or  ∪ ¬ is valid, or ( ∪ ¬) ∩ ( ∪ ¬) = ∅. Hence, for any assignment v, v |= | ⇒ |. (¬¬ A ) Assume that for any assignment v, v |= , A| ⇒ |. For any assignment v, assume that v |= , ¬¬A| then v(A) = 1 = v(¬¬A), v |= , A|. By induction assumption, v |= |, and hence, v |= , ¬¬A| ⇒ |. (∧ A ) Assume that for any assignment v, either v |= , A1 | ⇒ |, or v |= , A2 | ⇒ |. For any assignment v, if v |= , A1 ∧ A2 | then v |= | and v(A1 ∧ A2 ) = 1, i.e., v(A1 ) = v(A2 ) = 1. By induction assumption, each case implies v |= |. (∧ B ) Assume that for any assignment v, v |= |B1 ,  ⇒ |, v |= |B2 ,  ⇒ |. For any assignment v, if v |= |B1 ∧ B2 ,  then v |= | and v(B1 ∧ B2 ) = 0, i.e., either v(B1 ) = 0 or v(B2 ) = 0. By induction assumption, each case implies v |= |. (∧C ) Assume that for any assignment v, v |= | ⇒ , C1 |, v |= | ⇒ , C2 |. For any assignment v, if v |= | then v |= , C1 | and v |= , C2 |, i.e., either v |= |, or v(C1 ) = v(C2 ) = 1. Hence, v |= , C1 ∧ C2 |. (∧ A ) Assume that for any assignment v, either v |= | ⇒ |D1 , , or v |= | ⇒ |D2 , . For any assignment v, if v |= | then either v |= |D1 , , or v |= |D2 , , i.e., either v |= |, or either v(D1 ) = 0 or v(D2 ) = 0. Hence, v |= |D1 ∧ D2 , . Similar for other deduction rules.

Theorem 7.1.4 (Completeness theorem) For any supersequent δ = | ⇒ |, if |= δ then δ. Proof Given a supersequent δ = | ⇒ |, we construct a tree T such that either (i) for each branch ξ of T, some supersequent δ  at the leaf of ξ is an axiom, or (ii) there is an assignment v such that v |= δ. T is constructed as follows: • the root of T is δ; • for a node ξ , if the supersequent δ  at ξ is literal then the node is a leaf;

188

7 R-Calculi for Supersequents

• otherwise, ξ has the direct children nodes containing the following supersequents: ⎧ 1 , A|1 ⇒ 1 |1 ⎪ ⎪ ⎨ 1 |B, 1 ⇒ 1 |1  ⎪ 1 |1 ⇒ 1 , C|1 ⎪ ⎩ 1 |1 ⇒ 1 |D, 1 and

if 1 , ¬¬A|1 ⇒ 1 |1 ∈ ξ if 1 |¬¬B, 1 ⇒ 1 |1 ∈ ξ if 1 |1 ⇒ 1 , ¬¬C|1 ∈ ξ if 1 |1 ⇒ 1 |¬¬D, 1 ∈ ξ

⎧  ⎪ ⎪ 1 , A1 , A2 |1 ⇒ 1 |1 ⎪ ⎪ 1 |B1 , 1 ⇒ 1 |1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 |B2 , 1 ⇒ 1 |1 ⎪ ⎪ ⎪  ⎪ 1 |1 ⇒ 1 , C 1 |1 ⎪ ⎪ ⎪  ⎪ 1 |1 ⇒ 1 , C 2 |1 ⎪ ⎨ |  1 1 ⇒ 1 |D1 , D2 , 1  , A1 |1 ⇒ 1 |1  ⎪ 1 ⎪ ⎪ ⎪  , A2 |1 ⇒ 1 |1 ⎪ 1 ⎪ ⎪ ⎪  |B , B2 , 1 ⇒ 1 |1 ⎪ 1 1 ⎪ ⎪ ⎪ | ⇒ 1 , C1 , C2 |1  ⎪ 1 1 ⎪  ⎪ ⎪ |  ⎪ 1 1 ⇒ 1 |D1 , 1 ⎪ ⎩ 1 |1 ⇒ 1 |D2 , 1

if 1 , A1 ∧ A2 |1 ⇒ 1 |1 ∈ ξ if 1 |B1 ∧ B2 , 1 ⇒ 1 |1 ∈ ξ if 1 |1 ⇒ 1 , C1 ∧ C2 |1 ∈ ξ if 1 |1 ⇒ 1 |D1 ∧ D2 1 ∈ ξ if 1 , A1 ∨ A2 |1 ⇒ 1 |1 ∈ ξ if 1 |B1 ∨ B2 , 1 ⇒ 1 |1 ∈ ξ if 1 |1 ⇒ 1 , C1 ∨ C2 |1 ∈ ξ if 1 |1 ⇒ 1 |D1 ∨ D2 1 ∈ ξ

and ⎧ 1 , ¬A1 |1 ⇒ 1 |1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 , ¬A2 |1 ⇒ 1 |1 ⎪ ⎪ ⎪ |¬B  ⎪ 1 1 , ¬B2 , 1 ⇒ 1 |1 ⎪ ⎪ ⎪ | ⇒ 1 , ¬C1 , ¬C2 |1  ⎪ 1 1 ⎪  ⎪ ⎪ |  ⎪ 1 1 ⇒ 1 |¬D1 , 1 ⎪ ⎨ 1 |1 ⇒ 1 |¬D2 , 1  ⎪ ⎪ 1 , ¬A1 , ¬A2 |1 ⇒ 1 |1 ⎪ ⎪ 1 |¬B1 , 1 ⇒ 1 |1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 |¬B2 , 1 ⇒ 1 |1 ⎪ ⎪ 1 |1 ⇒ 1 , ¬C1 |1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 |1 ⇒ 1 , ¬C 2 |1 ⎪ ⎩ 1 |1 ⇒ 1 |¬D1 , ¬D2 , 1

if 1 , ¬(A1 ∧ A2 )|1 ⇒ 1 |1 ∈ ξ if 1 |¬(B1 ∧ B2 ), 1 ⇒ 1 |1 ∈ ξ if 1 |1 ⇒ 1 , ¬(C1 ∧ C2 )|1 ∈ ξ if 1 |1 ⇒ 1 |¬(D1 ∧ D2 ), 1 ∈ ξ if 1 , ¬(A1 ∨ A2 )|1 ⇒ 1 |1 ∈ ξ if 1 |¬(B1 ∨ B2 ), 1 ⇒ 1 |1 ∈ ξ if 1 |1 ⇒ 1 , ¬(C1 ∨ C2 )|1 ∈ ξ if 1 |1 ⇒ 1 |¬(D1 ∨ D2 ), 1 ∈ ξ

Lemma 7.1.5 If for each branch ξ ⊆ T , there is a supersequent δ  ∈ ξ which is an axiom in G+ then T is a proof of δ. Proof By the definition of T, T is a proof tree of δ in G+ .



Lemma 7.1.6 If there is a branch ξ ⊆ T such that each supersequent δ  ∈ ξ is not an axiom in G+ then there is an assignment v such that v |= δ.

7.2 Reduction of Supersequents to Sequents

189

Proof Let γ be the set of all the literal supersequents δ  in T which is not an axiom. By Proposition 7.1.1, there is an assignment v such that v |= δ  , where δ  is at the leaf node of ξ . We proved by induction on nodes η of ξ that each supersequent δ  ∈ ξ is not satisfied by v. Case δ  =   , ¬¬A| ⇒   | ∈ η. Then, η has a direct child node ∈ ξ containing   , A| ⇒   | . By induction assumption, v |=   , A| ⇒   | , i.e., v(A) = 1 = v(¬¬A), and v |=   , ¬¬A| ⇒   | . Case δ  =   , A1 ∧ A2 | ⇒   | ∈ η. Then, η has a direct child node ∈ ξ containing   , A1 , A2 | ⇒   | . By induction assumption, v |=   , A1 , A2 | ⇒   | , i.e., v(A1 ) = 1 = v(A2 ), which imply v(A1 ∧ A2 ) = 1. Hence, v |=   , A1 ∧ A2 | ⇒   | . Case δ  =   |B1 ∧ B2 ,  ⇒   | ∈ η. Then, η has a direct child node ∈ ξ containing one of the following supersequents:   |B1 ,  ⇒   | ;   |B2 ,  ⇒   | . say,   |B1 ,  ⇒   | . By induction assumption, v |=   |B1 ,  ⇒   | , i.e., v |=   | ⇒   | and v(B1 ) = 0. Hence, v(B1 ∧ B2 ) = 0, and v |=   |B1 ∧ B2 ,  ⇒   | ; . Case δ  =   | ⇒   , C1 ∧ C2 | ∈ η. Then, η has a direct child node ∈ ξ containing one of supersequents:   | ⇒   , C1 | ;   | ⇒   , C2 | , say   | ⇒   , C2 | . By induction assumption, v |=   | ⇒   , C2 | ; i.e., v |=   | ⇒   | , and v(C2 ) = 0. Hence, v(C1 ∧ C2 ) = f, and v |=   | ⇒   , C1 ∧ C2 | . Case δ  =   | ⇒   |D1 ∧ D2 ,  ∈ η. Then, η has a direct child node ∈ ξ containing   | ⇒   |D1 , D2 ,  . By induction assumption, v |=   | ⇒   |D1 , D2 ,  , i.e., either v(D1 ) = 0 or v(D2 ) = 0, which imply v(D1 ∧ D2 ) = 0. Hence, v |=   | ⇒   |D1 , D2 ,  . Similar for other cases.



7.2 Reduction of Supersequents to Sequents A supersequent supersequent | ⇒ | is reduced into four kinds of sequents: ft :  ⇒ ,  =  = ∅ ff :  ⇒ ,  =  = ∅ tt :  ⇒ ,  =  = ∅ tf :  ⇒ ,  =  = ∅.

190

7 R-Calculi for Supersequents

7.2.1 Gentzen Deduction System Gft  ⇒  is Gft -valid, denoted by |=ft  ⇒ , if for any assignment v, v |=  implies v |= , where • v |=  if for each formula A ∈ , v(A) = 1; and • v |=  if for some formula C ∈ , v(C) = 1. Gentzen deduction system Gft consists of the following axioms and deduction rules: • Axiom: ⎧ ⎨ incon() val() ft (A ) ⎩  ∩  = ∅ ⇒ where ,  are sets of literals. • Deduction rules: , A ⇒   ⇒ , C (¬¬C ) (¬¬ A )  , ¬¬A ⇒    ⇒ , ¬¬C , A1 ⇒   ⇒ , C1 (∧ A ) , A2 ⇒  (∧C )  ⇒ , C2 ,   A1 ∧ A2 ⇒   ⇒ , C1 ∧ C2 , A1 ⇒   ⇒ , C1 (∨ A ) , A2 ⇒  (∨C )  ⇒ , C2 , A1 ∨ A2 ⇒   ⇒ , C1 ∨ C2 and



 , ¬A1 ⇒   ⇒ , ¬C1 A C ⇒   ⇒ , ¬C2 , ¬A (¬∧ ) (¬∧ ) 2 ,   ¬(A1 ∧ A2 ) ⇒   ⇒ , ¬(C1 ∧ C2 ) , ¬A1 ⇒   ⇒ , ¬C1 (¬∨ A ) , ¬A2 ⇒  (¬∧C )  ⇒ , ¬C2 , ¬(A1 ∨ A2 ) ⇒   ⇒ , ¬(C1 ∨ C2 )

Theorem 7.2.1 (Soundness and completeness theorem) For any sequent  ⇒ , ft  ⇒  if and only if |=ft  ⇒ .

7.2.2 Gentzen Deduction System Gff  ⇒  is Gff -valid, denoted by |=ff  ⇒ , if for any assignment v, v |=  implies v |= , where

7.3 R-Calculus R+

191

• v |=  if for each formula A ∈ , v(A) = 1; and • v |=  if for some formula D ∈ , v(D) = 0. Gentzen deduction system Gff consists of the following axioms and deduction rules: • Axiom: ⎧ ⎨ incon() val() (Aff ) ⎩  ∩ ¬ = ∅ ⇒ where ,  are sets of literals. • Deduction rules: , A ⇒   ⇒ D,  (¬¬ D ) (¬¬ A ) , ¬¬A ⇒     ⇒ ¬¬D,  , A1 ⇒   ⇒ D1 ,  (∧ A ) , A2 ⇒  (∧ D )  ⇒ D2 ,  ,   A1 ∧ A2 ⇒   ⇒ D1 ∧ D2 ,  , A1 ⇒   ⇒ D1 ,  (∨ A ) , A2 ⇒  (∨ D )  ⇒ D2 ,  , A1 ∨ A2 ⇒   ⇒ D1 ∨ D2 ,  and



 , ¬A1 ⇒   ⇒ ¬D1 ,  (¬∧ A ) , ¬A2 ⇒  (¬∧ D )  ⇒ ¬D2 ,  ,   ¬(A1 ∧ A2 ) ⇒   ⇒ ¬(D1 ∧ D2 ),  , ¬A1 ⇒   ⇒ ¬D1 ,  (¬∨ A ) , ¬A2 ⇒  (¬∨ D )  ⇒ ¬D2 ,  , ¬(A1 ∨ A2 ) ⇒   ⇒ ¬(D1 ∨ D2 ), 

Theorem 7.2.2 (Soundness and completeness theorem) For any sequent  ⇒ , ff  ⇒  if and only if |=ff  ⇒ .

7.3 R-Calculus R+ Given a quadruple (A, B, C, D) with A ∈ , B ∈ , C ∈  and D ∈ , we use | ⇒ | to revise (A, B, C, D) and obtain   | ⇒   | , denoted by | ⇒ ||(A, B, C, D) ⇒   | ⇒   | , if   | ⇒   | = ⎧ ⎨  − {A}| − {B} ⇒  − {C}| − {D} if |=+  − {A}| − {B} ⇒  − {C}| − {D} ⎩ | ⇒ | otherwise.

192

7 R-Calculi for Supersequents

We denote X = | ⇒ | X((A, B, C, D)) =  − {A}| − {B} ⇒  − {C}| − {D}. R-calculus R+ consists of the following axioms and deduction rules: • Axioms:

⎡

¬l ∈ /  l ∈ /  ⎢l∈ /  ∈ / ¬l ⎢  ⎢l∈ /  ∈ / ¬l ⎢  ⎢ ¬l ∈ /  ∈ /  l ⎢  ∈ / ¬m ∈ /  m (A− ) ⎢ ⎢  ⎢m∈ ∈ / ¬m /  ⎢  ⎣ ¬m ∈ / / m ∈ / ¬m  ∈ m∈ /   X|(l, l , m, m ) ⇒ X((l, l  , m, m  )) ⎧  ¬l ∈  l ∈  ⎪ ⎪ ⎪ ⎪ l∈ ¬l  ∈  ⎪ ⎪ ⎪ ⎪ l∈ ¬l  ∈  ⎪ ⎪ ⎨ ¬l ∈  l  ∈  (A0 ) ⎪ ⎪ ¬m ∈  m  ∈  ⎪ ⎪ ¬m  ∈  ⎪m∈ ⎪ ⎪ ⎪ ¬m ∈  m  ∈  ⎪ ⎪ ⎩ ¬m  ∈  m∈ X|(l, l  , m, m  ) ⇒ X

where X is literal and l, l  , m, m  are literals. • Deduction rules: X|(A, B, C, D) ⇒ X((A, B, C, D)) X|(¬¬A, B, C, D) ⇒ X((¬¬A, B, C, D)) X|(A, B, C, D) ⇒ X (¬¬0A ) X|(¬¬A, B, C, D) ⇒ X X|(A, B, C, D) ⇒ X((A, B, C, D)) (¬¬−B ) X|(A, ¬¬B, C, D) ⇒ X((A, ¬¬B, C, D)) X|(A, B, C, D) ⇒ X (¬¬0B ) X|(A, ¬¬B, C, D) ⇒ X X|(A, B, C, D) ⇒ X((A, B, C, D)) (¬¬C− ) X|(A, B, ¬¬C, D) ⇒ X((A, B, ¬¬C, D)) X|(A, B, C, D) ⇒ X (¬¬C0 ) X|(A, B, ¬¬C, D) ⇒ X X|(A, B, C, D) ⇒ X((A, B, C, D)) (¬¬−D ) X|(A, B, C, ¬¬D) ⇒ X((A, B, C, ¬¬D)) X|(A, B, C, D) ⇒ X (¬¬0D ) X|(A, B, C, ¬¬D) ⇒ X (¬¬−A )

7.3 R-Calculus R+

and 

X|(A1 , B, C, D) ⇒ X((A1 , B, C, D)) X|(A2 , B, C, D) ⇒ X((A2 , B, C, D)) X|(A1 ∧ A2 , B,  C, D) ⇒ X((A1 ∧ A2 , B, C, D)) X|(A1 , B, C, D) ⇒ X (∧0A ) X|(A2 , B, C, D) ⇒ X X|(A1 ∧ A2 , B, C, D) ⇒ X  X|(A, B1 , C, D) ⇒ X((A, B1 , C, D)) (∧−B ) X((A, B1 , C, D))|(λ, B2 , λ, λ) ⇒ X((A, B1 , C, D))((λ, B2 , λ, λ)) X|(A, B1 ∧ B2, C, D) ⇒ X((A, B1 ∧ B2 , C, D)) X|(A, B1 , C, D) ⇒ X (∧0B ) X((A, B1 , C, D))|(λ, B2 , λ, λ) ⇒ X((A, B1 , C, D)) X|(A, B1 ∧ B2 , C, D) ⇒ X (∧−A )

and 

X|(A, B, C1 , D) ⇒ X((A, B, C1 , D)) X((A, B, C1 , D))|(λ, λ, C2 , λ) ⇒ X((A, B, C1 , D))((λ, λ, C2 , λ)) X|(A, B, C1 ∧C2 , D) ⇒ X((A, B, C1 ∧ C2 , D)) X|(A, B, C1 , D) ⇒ X (∧C0 ) X((A, B, C1 , D))|(λ, λ, C2 , λ) ⇒ X((A, B, C1 , D)) X|(A, B, C1 ∧ C2 , D) ⇒ X  X|(A, B, C, D1 ) ⇒ X((A, B, C, D1 )) (∧−D ) X|(A, B, C, D2 ) ⇒ X((A, B, C, D2 )) X|(A, B, C, D1 ∧ D2 ) ⇒ X((A, B, C, D1 ∧ D2 )) X|(A, B, C, D1 ) ⇒ X (∧0D ) X|(A, B, C, D2 ) ⇒ X X|(A, B, C, D1 ∧ D2 ) ⇒ X

(∧C− )

and 

X|(A1 , B, C, D) ⇒ X((A1 , B, C, D)) X((A1 , B, C, D))|(A2 , λ, λ, λ) ⇒ X((A1 , B, C, D))((A2 , λ, λ, λ)) X(A1 ∨ A2 , B,C, D) ⇒ X((A1 ∨ A2 , B, C, D)) X|(A1 , B, C, D) ⇒ X (∨0A ) X((A1 , B, C, D))|(A2 , λ, λ, λ) ⇒ X((A1 , B, C, D)) X|(A1 ∨ A2 , B, C, D) ⇒ X  X|(A, B1 , C, D) ⇒ X((A, B1 , C, D)) (∨−B ) X|(A, B2 , C, D) ⇒ X((A, B2 , C, D)) X|(A, B1 ∨ B2, C, D) ⇒ X((A, B1 ∨ B2 , C, D)) X|(A, B1 , C, D) ⇒ X (∨0B ) X|(A, B2 , C, D) ⇒ X X|(A, B1 ∨ B2 , C, D) ⇒ X (∨−A )

193

194

7 R-Calculi for Supersequents

and 

X|(A, B, C1 , D) ⇒ X((A, B, C1 , D)) X|(A, B, C2 , D) ⇒ X((A, B, C2 , D)) X|(A, B, C1 ∨C2 , D) ⇒ X((A, B, C1 ∨ C2 , D)) X|(A, B, C1 , D) ⇒ X (∨C0 ) X|(A, B, C2 , D) ⇒ X X(A, B, C1 ∨ C2 , D) ⇒ X  X|(A, B, C, D1 ) ⇒ X((A, B, C, D1 )) (∨−D ) X((λ, λ, λ, D1 ))|(A, B, C, D2 ) ⇒ X((A, B, C, D1 ))((λ, λ, λ, D2 )) X(A, B, C, D1∨ D2 ) ⇒ X((A, B, C, D1 ∨ D2 )) X|(A, B, C, D1 ) ⇒ X (∨0D ) X((A, B, C, D1 ))|(λ, λ, λ, D2 ) ⇒ X((A, B, C, D1 )) X(A, B, C, D1 ∨ D2 ) ⇒ X

(∨C− )

and 

X|(¬A1 , B, C, D) ⇒ X((¬A1 , B, C, D)) X((¬A1 , B, C, D))|(¬A2 , λ, λ, λ) ⇒ X((¬A1 , B, C, D))((¬A2 , λ, λ, λ)) X|(¬(A1 ∧ A2 ), B, C, D) ⇒ X((¬(A1 ∧ A2 ), B, C, D)) X|(¬A1 , B, C, D) ⇒ X (¬∧0A ) X((¬A1 , B, C, D))|(¬A2 , λ, λ, λ) ⇒ X((¬A1 , B, C, D)) X|(¬(A1 ∧ A2 ), B, C, D) ⇒ X  X|(A, ¬B1 , C, D) ⇒ X((A, ¬B1 , C, D)) B ) X|(A, ¬B , C, D) ⇒ X((A, ¬B , C, D)) (¬∧− 2 2 X|(A, ¬(B1 ∧ B2 ), C, D) ⇒ X((A, ¬(B1 ∧ B2 ), C, D)) X|(A, ¬B1 , C, D) ⇒ X (¬∧0B ) X|(A, ¬B2 , C, D) ⇒ X X|(A, ¬(B1 ∧ B2 ), C, D) ⇒ X A) (¬∧−

and 

X|(A, B, ¬C1 , D) ⇒ X((A, B, ¬C1 , D)) X|(A, B, ¬C2 , D) ⇒ X((A, B, ¬C2 , D)) X|(A, B, ¬(C1 ∧ C2 ), D) ⇒ X((A, B, ¬(C1 ∧ C2 ), D)) X|(A, B, ¬C1 , D) ⇒ X (¬∧C 0 ) X|(A, B, ¬C 2 , D) ⇒ X X|(A, B, ¬(C1 ∧ C2 ), D) ⇒ X  X|(A, B, C, ¬D1 ) ⇒ X((A, B, C, ¬D1 )) D ) X((A, B, C, ¬D ))|(λ, λ, λ, ¬D ) ⇒ X((A, B, C, ¬D ))((λ, λ, λ, ¬D )) (¬∧− 1 2 1 2 X|(A, B, C, ¬(D ∧ D )) ⇒ X((A, B, C, ¬(D ∧ D ))) 1 2 1 2  X|(A, B, C, ¬D1 ) ⇒ X D (¬∧0 ) X((A, B, C, ¬D1 ))|(λ, λ, λ, ¬D2 ) ⇒ X((A, B, C, ¬D1 )) X|(A, B, C, ¬(D1 ∧ D2 )) ⇒ X (¬∧C −)

7.3 R-Calculus R+

195

and 

X|(¬A1 , B, C, D) ⇒ X((¬A1 , B, C, D)) X|(¬A2 , B, C, D) ⇒ X((¬A2 , B, C, D)) X|(¬(A1 ∨ A2 ), B, C, D) ⇒ X((¬(A1 ∨ A2 ), B, C, D)) X|(¬A1 , B, C, D) ⇒ X (¬∨0A ) X|(¬A2 , B, C, D) ⇒ X X|(¬(A1 ∨ A2 ), B, C, D) ⇒ X  X|(A, ¬B1 , C, D) ⇒ X((A, ¬B1 , C, D)) B ) X((A, ¬B , C, D))|(λ, ¬B , λ, λ) ⇒ X((A, ¬B , C, D))((λ, ¬B , λ, λ)) (¬∨− 1 2 1 2 X|(A, ¬(B1 ∨ B2 ), C, D) ⇒ X((A, ¬(B1 ∨ B2 ), C, D)) X|(A, ¬B1 , C, D) ⇒ X (¬∨0B ) X((A, ¬B1 , C, D))|(λ, ¬B2 , λ, λ) ⇒ X((A, ¬B1 , C, D)) X|(A, ¬(B1 ∨ B2 ), C, D) ⇒ X A) (¬∨−

and 

X|(A, B, ¬C1 , D) ⇒ X((A, B, ¬C1 , D)) X((A, B, ¬C1 , D))|(λ, λ, ¬C2 , λ) ⇒ X((A, B, ¬C1 , D))((λ, λ, ¬C2 , λ)) X|(A, B, ¬(C1  ∨ C2 ), D) ⇒ X((A, B, ¬(C1 ∨ C2 ), D)) X|(A, B, ¬C1 , D) ⇒ X X((A, B, ¬C1 , D))|(λ, λ, ¬C2 , λ) ⇒ X((A, B, ¬C1 , D)) ) (¬∨C 0 X|(A, B, ¬(C1 ∨ C2 ), D) ⇒ X  X|(A, B, C, ¬D1 ) ⇒ X((A, B, C, ¬D1 )) D ) X|(A, B, C, ¬D ) ⇒ X((A, B, C, ¬D )) (¬∨− 2 2 X|(A, B, C, ¬(D  1 ∨ D2 )) ⇒ X((A, B, C, ¬(D1 ∨ D2 ))) X|(A, B, C, ¬D1 ) ⇒ X (¬∨0D ) X|(A, B, C, ¬D2 ) ⇒ X X|(A, B, C, ¬(D1 ∨ D2 )) ⇒ X (¬∨C −)

Definition 7.3.1 Given a supersequent X and a quadruple X with X ∈ X, a reduction X|X ⇒ X is provable in R+ , denoted by + X|X ⇒ X , if there is a sequence {δ1 , . . . , δn } of reductions such that δn = X|X ⇒ X , and for each 1 ≤ i ≤ n, δi is an axiom or deduced from the previous reductions by one of the deduction rules in R+ . Theorem 7.3.2 (Soundness and completeness theorem) For any reduction X|X ⇒ X with X ∈ X,

+ X|X ⇒ X if and only if |=+ X|X ⇒ X .



7.3.1 Rft Given a pair (A, C) of formulas such that A ∈  and C ∈ , we use  ⇒  to revise (A, C) and obtain a sequence   ⇒   , denoted by

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7 R-Calculi for Supersequents

|=ft  ⇒ |(A, C) ⇒   ⇒   , if   ⇒   = 

 − {A} ⇒  − {C} if |=ft  − {A} ⇒  − {C} ⇒ otherwise.

Gentzen deduction system Gft consists of the following axioms and deduction rules: • Axioms: ⎡

¬l ∈ / ⎢l ∈ /  ⎢ ⎣ m ∈ /  (Aft ) (Att − 0 ) ¬m ∈ /  ⇒ |(l, m) ⇒  − {l} ⇒  − {m}

⎧ ¬l ∈  ⎪ ⎪ ⎨ l∈ m∈ ⎪ ⎪ ⎩ ¬m ∈   ⇒ |(l, m) ⇒  ⇒ 

where ,  are sets of literals, and l, m are literals such that l ∈  and m ∈ . • Deduction rules:  ⇒ |(A, C) ⇒  − {A} ⇒  − {C}  ⇒ |(¬¬A, C) ⇒  − {¬¬A} ⇒  − {C}  ⇒ |(A, C) ⇒  ⇒  (¬¬0A )  ⇒ |(¬¬A, C) ⇒  ⇒   ⇒ |(A, C) ⇒  − {A} ⇒  − {C} (¬¬C− )  ⇒ |(A, ¬¬C) ⇒  − {A} ⇒  − {¬¬C}  ⇒ |(A, C) ⇒  ⇒  (¬¬0A )  ⇒ |(A, ¬¬C) ⇒  ⇒  (¬¬−A )

and 

 ⇒ |(A1 , C) ⇒  − {A1 } ⇒  − {C}  ⇒ |(A2 , C) ⇒  − {A2 } ⇒  − {C}  ⇒ |(A1 ∧A2 , C) ⇒  − {A1 ∧ A2 } ⇒  − {C}  ⇒ |(A1 , C) ⇒  ⇒  (∧0A )  ⇒ |(A2 , C) ⇒  ⇒   ⇒ |(A1 ∧ A2 , C) ⇒  ⇒    ⇒ |(A, C1 ) ⇒  − {A} ⇒  − {C1 } (∧C− )  − {A} ⇒  − {C1 }|(λ, C2 ) ⇒  − {A} ⇒  − {C1 , C2 }  ⇒ |(A, C 1 ∧ C 2 ) ⇒  − {A} ⇒  − {C 1 ∧ C 2 }  ⇒ |(A, C1 ) ⇒  ⇒  (∧0A )  − {A} ⇒  − {C1 }|(λ, C2 ) ⇒  − {A} ⇒  − {C1 }  ⇒ |(A1 ∧ A2 , C) ⇒  ⇒  (∧−A )

7.3 R-Calculus R+

197

and 

 ⇒ |(A1 , C) ⇒  − {A1 } ⇒  − {C}  − {A1 } ⇒  − {C}|(A2 , λ) ⇒  − {A1 , A2 } ⇒  − {C}  ⇒ |(A1 ∨A2 , C) ⇒  − {A1 ∨ A2 } ⇒  − {C}  ⇒ |(A1 , C) ⇒  ⇒  (∨0A )  − {A1 } ⇒  − {C}|(A2 , λ) ⇒  − {A1 } ⇒  − {C}  ⇒ |(A1 ∨ A2 , C) ⇒  ⇒    ⇒ |(A, C1 ) ⇒  − {A} ⇒  − {C1 } (∨C− )  ⇒ |(A, C2 ) ⇒  − {A} ⇒  − {C2 }  ⇒ |(A, C 1 ∨ C 2 ) ⇒  − {A} ⇒  − {C 1 ∨ C 2 }  ⇒ |(A, C1 ) ⇒  ⇒  (∨0A )  ⇒ |(A, C2 ) ⇒  ⇒   ⇒ |(A, C1 ∨ C2 ) ⇒  ⇒  (∨−A )

and 

 ⇒ |(¬A1 , C) ⇒  − {¬A1 } ⇒  − {C}  − {¬A1 } ⇒  − {C}|(¬A2 , λ) ⇒  − {¬A1 , ¬A2 } ⇒  − {C}  ⇒ |(¬(A1 ∧  A2 ), C) ⇒  − {¬(A1 ∧ A2 )} ⇒  − {C}  ⇒ |(¬A1 , C) ⇒  ⇒  (¬∧0A )  − {¬A1 } ⇒  − {C}|(¬A2 , λ) ⇒  − {¬A1 } ⇒  − {C}  ⇒ |(¬(A1 ∧ A2 ), C) ⇒  ⇒    ⇒ |(A, ¬C1 ) ⇒  − {A} ⇒  − {¬C1 } (¬∧C − )  ⇒ |(A, ¬C 2 ) ⇒  − {A} ⇒  − {¬C 2 }  ⇒ |(A, ¬(C1 ∧ C2 )) ⇒  − {A} ⇒  − {¬(C1 ∧ C2 )}  ⇒ |(A, ¬C1 ) ⇒  ⇒   ⇒ |(A, ¬C2 ) ⇒  ⇒  ) (¬∧C 0  ⇒ |(A, ¬(C1 ∧ C2 )) ⇒  ⇒  A) (¬∧−

and 

 ⇒ |(¬A1 , C) ⇒  − {¬A1 } ⇒  − {C}  ⇒ |(¬A2 , C) ⇒  − {¬A2 } ⇒  − {C}  ⇒ |(¬(A1 ∨  A2 ), C) ⇒  − {¬(A1 ∨ A2 )} ⇒  − {C}  ⇒ |(¬A1 , C) ⇒  ⇒  (¬∨0A )  ⇒ |(¬A2 , C) ⇒  ⇒   ⇒ |(¬(A1 ∨ A2 ), C) ⇒  ⇒    ⇒ |(A, ¬C1 ) ⇒  − {A} ⇒  − {¬C1 } (¬∨C − )  − {A} ⇒  − {¬C 1 }|(λ, ¬C 2 ) ⇒  − {A} ⇒  − {¬C 1 , ¬C 2 }  ⇒ |(A, ¬(C1 ∨ C2 )) ⇒  − {A} ⇒  − {¬(C1 ∨ C2 )}  ⇒ |(A, ¬C1 ) ⇒  ⇒   − {A} ⇒  − {¬C1 }|(λ, ¬C2 ) ⇒  − {A} ⇒  − {¬C1 } ) (¬∨C 0  ⇒ |(A, ¬(C1 ∨ C2 )) ⇒  ⇒  A) (¬∨−

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7 R-Calculi for Supersequents

Theorem 7.3.3 (Soundness and completeness theorem) For any sequent  ⇒  and pair (A, C) with A ∈  and C ∈ ,

ft  ⇒ |(A, C) ⇒   ⇒   if and only if

|=ft  ⇒ |(A, C) ⇒   ⇒   ,

where   = | − {A} and  = | − {C}.



7.3.2 Rff Given a pair (A, D) of formulas such that A ∈  and D ∈ , we use  ⇒  to revise (A, D) and obtain a sequence   ⇒  , denoted by |=ff  ⇒ |(A, D) ⇒   ⇒  , if 



 ⇒ =



 − {A} ⇒  − {D} if |=ff  − {A} ⇒  − {D} ⇒ otherwise.

Gentzen deduction system Gff consists of the following axioms and deduction rules: • Axioms: ⎡

¬l ∈ / ⎢ ¬l ∈ /  ⎢ ⎣ ¬m ∈ /  (Aff ) (Aff − 0 ) ¬m ∈ /  ⇒ |(l, m) ⇒  − {l} ⇒  − {m}

⎧ ¬l ∈  ⎪ ⎪ ⎨ ¬l ∈  ¬m ∈  ⎪ ⎪ ⎩ ¬m ∈   ⇒ |(l, m) ⇒  ⇒ 

where ,  are sets of literals, and l, m are literals such that l ∈  and m ∈ . • Deduction rules:  ⇒ |(A, D) ⇒  − {A} ⇒  − {D}  ⇒ |(¬¬A, D) ⇒  − {¬¬A} ⇒  − {D}  ⇒ |(A, D) ⇒  ⇒  (¬¬0A )  ⇒ |(¬¬A, D) ⇒  ⇒   ⇒ |(A, D) ⇒  − {A} ⇒  − {D} (¬¬−D )  ⇒ |(A, ¬¬D) ⇒  − {A} ⇒  − {¬¬D}  ⇒ |(A, D) ⇒  ⇒  (¬¬0A )  ⇒ |(A, ¬¬D) ⇒  ⇒  (¬¬−A )

7.3 R-Calculus R+

199



and

 ⇒ |(A1 , D) ⇒  − {A1 } ⇒  − {D}  ⇒ |(A2 , D) ⇒  − {A2 } ⇒  − {D}  ⇒ |(A1 ∧A2 , D) ⇒  − {A1 ∧ A2 } ⇒  − {D}  ⇒ |(A1 , D) ⇒  ⇒  (∧0A )  ⇒ |(A2 , D) ⇒  ⇒   ⇒ |(A1 ∧ A2 , D) ⇒  ⇒    ⇒ |(A, D1 ) ⇒  − {A} ⇒  − {D1 } (∧−D )  ⇒ |(A, D2 ) ⇒  − {A} ⇒  − {D2 }  ⇒ |(A, D1 ∧ D2 ) ⇒  − {A} ⇒  − {D1 ∧ D2 }  ⇒ |(A, D1 ) ⇒  ⇒  (∧0A )  ⇒ |(A, D2 ) ⇒  ⇒   ⇒ |(A, D1 ∧ D2 ) ⇒  ⇒  (∧−A )

and 

 ⇒ |(A1 , D) ⇒  − {A1 } ⇒  − {D}  − {A1 } ⇒  − {D}|(A2 , λ) ⇒  − {A1 , A2 } ⇒  − {D}  ⇒ |(A1 ∨A2 , D) ⇒  − {A1 ∨ A2 } ⇒  − {D}  ⇒ |(A1 , D) ⇒  ⇒  (∨0A )  − {A1 } ⇒  − {D}|(A2 , λ) ⇒  − {A1 } ⇒  − {D}  ⇒ |(A1 ∨ A2 , D) ⇒  ⇒    ⇒ |(A, D1 ) ⇒  − {A} ⇒  − {D1 } (∨−D )  − {A} ⇒  − {D1 }|(λ, D2 ) ⇒  − {A} ⇒  − {D1 , D2 }  ⇒ |(A, D1 ∨ D2 ) ⇒  − {A} ⇒  − {D1 ∨ D2 }  ⇒ |(A, D1 ) ⇒  ⇒  (∨0A )  − {A} ⇒  − {D1 }|(λ, D2 ) ⇒  − {A} ⇒  − {D1 }  ⇒ |(A1 ∨ A2 , D) ⇒  ⇒  (∨−A )

and 

 ⇒ |(¬A1 , D) ⇒  − {¬A1 } ⇒  − {D}  − {¬A1 } ⇒  − {D}|(¬A2 , λ) ⇒  − {¬A1 , ¬A2 } ⇒  − {D}  ⇒ |(¬(A1  ∧ A2 ), D) ⇒  − {¬(A1 ∧ A2 )} ⇒  − {D}  ⇒ |(¬A1 , D) ⇒  ⇒  (¬∧0A )  − {¬A1 } ⇒  − {D}|(¬A2 , λ) ⇒  − {¬A1 } ⇒  − {D}  ⇒ |(¬(A1 ∧ A2 ), D) ⇒  ⇒    ⇒ |(A, ¬D1 ) ⇒  − {A} ⇒  − {¬D1 } D )  − {A} ⇒  − {¬D }|(λ, ¬D ) ⇒  − {A} ⇒  − {¬D , ¬D } (¬∧− 1 2 1 2  ⇒ |(A, ¬(D ∧ D )) ⇒  − {A} ⇒  − {¬(D ∧ D )} 1 2 1 2   ⇒ |(A, ¬D1 ) ⇒  ⇒  D (¬∧0 )  − {A} ⇒  − {¬D1 }|(λ, ¬D2 ) ⇒  − {A} ⇒  − {¬D1 }  ⇒ |(A, ¬(D1 ∧ D2 )) ⇒  ⇒  A) (¬∧−

200

7 R-Calculi for Supersequents

and 

 ⇒ |(¬A1 , D) ⇒  − {¬A1 } ⇒  − {D}  ⇒ |(¬A2 , D) ⇒  − {¬A2 } ⇒  − {D}  ⇒ |(¬(A1∨ A2 ), D) ⇒  − {¬(A1 ∨ A2 )} ⇒  − {D}  ⇒ |(¬A1 , D) ⇒  ⇒  (¬∨0A )  ⇒ |(¬A2 , D) ⇒  ⇒   ⇒ |(¬(A1 ∨ A2 ), D) ⇒  ⇒    ⇒ |(A, ¬D1 ) ⇒  − {A} ⇒  − {¬D1 } (¬∨−D )  ⇒ |(A, ¬D2 ) ⇒  − {A} ⇒  − {¬D2 }  ⇒ |(A, ¬(D  1 ∨ D2 )) ⇒  − {A} ⇒  − {¬(D1 ∨ D2 )}  ⇒ |(A, ¬D1 ) ⇒  ⇒  (¬∨0D )  ⇒ |(A, ¬D2 ) ⇒  ⇒   ⇒ |(A, ¬(D1 ∨ D2 )) ⇒  ⇒ 

(¬∨−A )

Theorem 7.3.4 (Soundness and completeness theorem) For any sequent  ⇒  and pair (A, D) with A ∈  and D ∈ ,

ff  ⇒ |(A, D) ⇒   ⇒  if and only if

|=ff  ⇒ |(A, D) ⇒   ⇒  ,

where   = | − {A} and   = | − {D}.



7.4 Gentzen Deduction System G− A co-supersequent | → | is valid, denoted by |=− | → |, if there is an assignment v such that v(A) = 1 for each A ∈ ; v(B) = 0 for each B ∈ ; v(C) = 0 for each C ∈ , and v(D) = 1 for each D ∈ . | ⇒ | f|t ⇒ t|f | → | t|f → f|t Co-supersequent Gentzen deduction system G− consists of the following axioms and deduction rules: • Axiom:

con( ∪ ¬)&inval( ∪ ¬) (A− ) ( ∪ ¬) ∩ ( ∪ ¬) = ∅ | → |

where , , ,  are sets of literals.

7.4 Gentzen Deduction System G−

201

• Deduction rules: , A| → | |B,  → | (¬¬ B ) , ¬¬A| → | |¬¬B,  → | | → , C| | → |D,  (¬¬C ) (¬¬ D ) | → , ¬¬C| | → |¬¬D, 

(¬¬ A )



and (∧ ) A

(∧C ) (∨ A ) (∨C )

, A1 | → | , A2 | → | ,  A1 ∧ A2 | → | | → , C1 | | → , C2 | | → , C1 ∧ C2 |  , A1 | → | , A2 | → | ,  A1 ∨ A2 | → | | → , C1 | | → , C2 | | → , C1 ∨ C2 |



(∧ ) B

(∧ D ) (∨ B ) (∨ D )

|B1 ,  → | |B2 ,  → | |B  1 ∧ B2 ,  → | | → |D1 ,  | → |D2 ,  | → |D1 ∧ D2 ,   |B1 ,  → | |B2 ,  → | |B  1 ∨ B2 ,  → | | → |D1 ,  | → |D2 ,  | → |D1 ∨ D2 , 

and 

(¬∧ A ) (¬∧C ) (¬∨ A ) (¬∧C )

, ¬A1 | → | , ¬A2 | → | ,  ¬(A1 ∧ A2 )| → | | → , ¬C1 | | → , ¬C1 | | → , ¬(C1 ∧ C2 )|  , ¬A1 | → | , ¬A2 | → | ,  ¬(A1 ∨ A2 )| → | | → , ¬C1 | | → , ¬C2 | | → , ¬(C1 ∨ C2 )|



|¬B1 ,  → | |¬B2 ,  → | |¬(B 1 ∧ B2 ),   → |  | → |¬D1 ,  (¬∧ D ) | → |¬D2 ,  | → |¬(D1 ∧ D2 ),   |¬B1 ,  → | (¬∨ B ) |¬B2 ,  → | |¬(B 1 ∨ B2 ),   → |  | → |¬D1 ,  (¬∨ D ) | → |¬D2 ,  | → |¬(D1 ∨ D2 ),  (¬∧ B )

Definition 7.4.1 A co-supersequent | → | is provable in G− , denoted by

− | → |, if there is a sequence {1 |1 → 1 |1 , . . . , n |n → n |n } of co-supersequents such that n |n → n |n = | → |, and for each 1 ≤ i ≤ n, i |i → i |i is deduced from the previous co-supersequents by one of the deduction rules in G− . Theorem 7.4.2 (Soundness and completeness theorem) For any co-supersequent δ = | → |,

− δ iff |=− δ.



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7 R-Calculi for Supersequents

7.5 Reduction of Supersequents to Sequents A co-supersequent | → | is reduced into four kinds of co-sequents: tf :  → ,  =  = ∅ tt :  → ,  =  = ∅ ff :  → ,  =  = ∅ ft :  → ,  =  = ∅.

7.5.1 Gentzen Deduction System Gff A co-sequent  →  is Gff -valid, denoted by |=ff  → , if there is an assignment v such that for every formula B ∈ , v(A) = 0, and for every formula C ∈ , v(C) = 0. Gentzen deduction system Gff consists of the following axiom and deduction rules: • Axiom:

⎡

con() ⎣ inval() (Aff ) ¬ ∩  = ∅  → 

where ,  are sets of literals. • Deduction rules: B,  →   → , C (¬¬C ) (¬¬ B ) ¬¬B,   →     → , ¬¬C B1 ,  →   → , C1 (∧ B ) B2 ,  →  (∧C )  → , C2  B1 ∧ B2 ,  →   → , C1 ∧ C2 B1 ,  →   → , C1 (∨ B ) B2 ,  →  (∨C )  → , C2 B1 ∨ B2 ,  →   → , C1 ∨ C2 and



 ¬B1 ,  →   → , ¬C1 (¬∧ B ) ¬B2 ,  →  (¬∧C )  → , ¬C2 ¬(B   1 ∧ B2 ),  →   → , ¬(C1 ∧ C2 ) ¬B1 ,  →   → , ¬C1 (¬∨ B ) ¬B2 ,  →  (¬∧C )  → , ¬C2 ¬(B1 ∨ B2 ),  →   → , ¬(C1 ∨ C2 )

7.5 Reduction of Supersequents to Sequents

203

Theorem 7.5.1 (Soundness and completeness theorem) For any co-sequent  →

, ff  →  if and only if |=ff  → .

7.5.2 Gentzen Deduction System Gft A co-sequent  →  is Gft -valid, denoted by |=ft  → , if there is an assignment v such that for every formula B ∈ , v(B) = 1, and for every formula D ∈ , v(D) = 0. Gentzen deduction system Gft consists of the following axiom and deduction rules: • Axiom:

⎡

con() ⎣ inval() (Aft ) ∩=∅  → 

where ,  are sets of literals. • Deduction rules: B,  →   → D,  (¬¬ D ) (¬¬ B )  ¬¬B,  →    → ¬¬D,  B1 ,  →    → D1 ,  (∧ B ) B2 ,  →  (∧ D )  → D2 ,   B1 ∧ B2 ,  →    → D1 ∧ D2 ,  B1 ,  →    → D1 ,  (∨ B ) B2 ,  →  (∨ D )  → D2 ,  B1 ∨ B2 ,  →    → D1 ∨ D2 ,  and



 ¬B1 ,  →   → ¬D1 ,  B D ,   →  ¬B  → ¬D2 ,  (¬∧ ) (¬∧ ) 2 ¬(B   1 ∧ B2 ),  →   → ¬(D1 ∧ D2 ),  ¬B1 ,  →   → ¬D1 ,  (¬∨ B ) ¬B2 ,  →  (¬∨ D )  → ¬D2 ,  ¬(B1 ∨ B2 ),  →   → ¬(D1 ∨ D2 ), 

Theorem 7.5.2 (Soundness and completeness theorem) For any co-sequent  → ,

ft  →  iff |=ft  → .



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7 R-Calculi for Supersequents

7.6 R-Calculus R− Given a quadruple (A, B, C, D), we use | → | to revise (A, B, C, D) and obtain   | →   | , denoted by |=− | → | ⇒   | →   | , if   | →   | = 

, A|, B → , C|, D if , A|, B → , C|, D is valid | → | otherwise.

We denote

X = | → | X[X ] = , A|, B → , C|, D.

R-calculus R− consists of the following axioms and deduction rules: • Axioms:

⎡

¬l ∈ /  l ∈ /  ⎢l∈ /  ∈ / ¬l ⎢  ⎢l∈ /  ∈ / ¬l ⎢  ⎢ ¬l ∈ /  ∈ /  l ⎢  ∈ / m ∈ /  ¬m (A+ ) ⎢ ⎢  ⎢ ¬m ∈ / / m ∈ ⎢ ⎣ ¬m ∈ / /  m ∈ / ¬m  ∈ m∈ / X|(l, l  , m, m ⎧ ) ⇒ X[(l, l  , m, m  )] ¬l ∈  l  ∈  ⎪ ⎪ ⎪ ⎪ l∈ ¬l  ∈  ⎪ ⎪ ⎪ ⎪ l∈ ¬l  ∈  ⎪ ⎪ ⎨ ¬l ∈  l  ∈  m∈ ¬m  ∈  (A0 ) ⎪ ⎪ ⎪ ⎪ ¬m ∈  m  ∈  ⎪ ⎪ ⎪ ⎪ ¬m ∈  m  ∈  ⎪ ⎪ ⎩ ¬m  ∈  m∈   X|(l, l , m, m ) ⇒ X

where X is literal and l, l  , m, m  are literals. • Deduction rules:

7.6 R-Calculus R−

205

X|(A, B, C, D) ⇒ X[(A, B, C, D)] X|(¬¬A, B, C, D) ⇒ X[(¬¬A, B, C, D)] X|(A, B, C, D) ⇒ X (¬¬0A ) X|(¬¬A, B, C, D) ⇒ X X|(A, B, C, D) ⇒ X[(A, B, C, D)] B (¬¬+ ) X|(A, ¬¬B, C, D) ⇒ X[(A, ¬¬B, C, D)] X|(A, B, C, D) ⇒ X (¬¬0B ) X|(A, ¬¬B, C, D) ⇒ X A) (¬¬+

and

X|(A, B, C, D) ⇒ X[(A, B, C, D)] X|(A, B, ¬¬C, D) ⇒ X[(A, B, ¬¬C, D)] X|(A, B, C, D) ⇒ X (¬¬C 0 ) X|(A, B, ¬¬C, D) ⇒ X D ) X|(A, B, C, D) ⇒ X[(A, B, C, D)] (¬¬+ X|(A, B, C, ¬¬D) ⇒ X[(A, B, C, ¬¬D)] X|(A, B, C, D) ⇒ X (¬¬0D ) X|(A, B, C, ¬¬D) ⇒ X

(¬¬C +)

and 

X|(A1 , B, C, D) ⇒ X[(A1 , B, C, D)] X[(A1 , B, C, D)]|(A2 , B, C, D) ⇒ X[(A1 , B, C, D)][(A2 , B, C, D)] X|(A1 ∧ A2 , B,C, D) ⇒ X[(A1 ∧ A2 , B, C, D)] X|(A1 , B, C, D) ⇒ X (∧0A ) X[(A1 , B, C, D)]|(A2 , B, C, D) ⇒ X[(A1 , B, C, D)] X|(A1 ∧ A2 , B, C, D) ⇒ X  X|(A, B1 , C, D) ⇒ X[(A, B1 , C, D)] B ) X|(A, B , C, D) ⇒ X((A, B , C, D)] (∧+ 2 2 X|(A, B1 ∧ B2 ,C, D) ⇒ X[(A, B1 ∧ B2 , C, D)] X|(A, B1 , C, D) ⇒ X (∧0B ) X|(A, B2 , C, D) ⇒ X X|(A, B1 ∧ B2 , C, D) ⇒ X A) (∧+

and 

X|(A, B, C1 , D) ⇒ X[(A, B, C1 , D)] X|(A, B, C2 , D) ⇒ X((A, B, C2 , D)] X|(A, B, C1 ∧ C  2 , D) ⇒ X[(A, B, C1 ∧ C2 , D)] X|(A, B, C1 , D) ⇒ X (∧C 0 ) X|(A, B, C 2 , D) ⇒ X X|(A, B, C1 ∧ C2 , D) ⇒ X  X|(A, B, C, D1 ) ⇒ X[(A, B, C, D1 )] D ) X[(A, B, C, D )]|(A, B, C, D ) ⇒ X[(A, B, C, D )]((A, B, C, D )] (∧+ 1 2 1 2 X|(A, B, C, D1∧ D2 ) ⇒ X[(A, B, C, D1 ∧ D2 )] X|(A, B, C, D1 ) ⇒ X (∧0D ) X[(A, B, C, D1 )]|(A, B, C, D2 ) ⇒ X[(A, B, C, D1 )] X|(A, B, C, D1 ∧ D2 ) ⇒ X (∧C +)

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7 R-Calculi for Supersequents

and 

X|(A1 , B, C, D) ⇒ X[(A1 , B, C, D)] X|(A2 , B, C, D) ⇒ X[(A2 , B, C, D)] X(A1 ∨ A2 , B, C, D) ⇒ X[(A1 ∨ A2 , B, C, D)] X|(A1 , B, C, D) ⇒ X (∨0A ) X|(A2 , B, C, D) ⇒ X X(A1 ∨ A2 , B, C, D) ⇒ X  X|(A, B1 , C, D) ⇒ X[(A, B1 , C, D)] (∨+B ) X[(A, B1 , C, D)]|(A, B2 , C, D) ⇒ X[(A, B1 , C, D)][(A, B2 , C, D)] X(A, B1 ∨ B2 ,C, D) ⇒ X[(A, B1 ∨ B2 , C, D)] X|(A, B1 , C, D) ⇒ X (∨0B ) X[(A, B1 , C, D)]|(A, B2 , C, D) ⇒ X[(A, B1 , C, D)] X(A, B1 ∨ B2 , C, D) ⇒ X (∨+A )

and 

X|(A, B, C1 , D) ⇒ X[(A, B, C1 , D)] X[(A, B, C1 , D)]|(A, B, C2 , D) ⇒ X[(A, B, C1 , D)][(A, B, C2 , D)] X(A, B, C1 ∨ C2 , D) ⇒ X[(A, B, C1 ∨ C2 , D)] X|(A, B, C1 , D) ⇒ X (∨C0 ) X[(A, B, C1 , D)]|(A, B, C2 , D) ⇒ X[(A, B, C1 , D)] X(A, B, C1 ∨ C2 , D) ⇒ X  X|(A, B, C, D1 ) ⇒ X[(A, B, C, D1 )] (∨+D ) X|(A, B, C, D2 ) ⇒ X[(A, B, C, D2 )] X(A, B, C, D1∨ D2 ) ⇒ X[(A, B, C, D1 ∨ D2 )] X|(A, B, C, D1 ) ⇒ X (∨0D ) X|(A, B, C, D2 ) ⇒ X X(A, B, C, D1 ∨ D2 ) ⇒ X

(∨C+ )

and 

X|(¬A1 , B, C, D) ⇒ X[(¬A1 , B, C, D)] X|(¬A2 , B, C, D) ⇒ X[(¬A2 , B, C, D)] X|(¬(A1 ∧ A2 ), B, C, D) ⇒ X[(¬(A1 ∧ A2 ), B, C, D)] X|(¬A1 , B, C, D) ⇒ X (¬∧0A ) X|(¬A2 , B, C, D) ⇒ X X|(¬(A1 ∧ A2 ), B, C, D) ⇒ X  X|(A, ¬B1 , C, D) ⇒ X[(A, ¬B1 , C, D)] B ) X[(A, ¬B , C, D)]|(A, ¬B , C, D) ⇒ X[(A, ¬B , C, D)][(A, ¬B , C, D)] (¬∧+ 1 2 1 2 X|(A, ¬(B1 ∧ B2 ), C, D) ⇒ X[(A, ¬(B1 ∧ B2 ), C, D)] X|(A, ¬B1 , C, D) ⇒ X (¬∧0B ) X[(A, ¬B1 , C, D)]|(A, ¬B2 , C, D) ⇒ X[(A, ¬B1 , C, D)] X|(A, ¬(B1 ∧ B2 ), C, D) ⇒ X A) (¬∧+

7.6 R-Calculus R−

207

and 

X|(A, B, ¬C1 , D) ⇒ X[(A, B, ¬C1 , D)] X[(A, B, ¬C1 , D)]|(A, B, ¬C2 , D) ⇒ X[(A, B, ¬C1 , D)][(A, B, ¬C2 , D)] X|(A, B, ¬(C1  ∧ C2 ), D) ⇒ X[(A, B, ¬(C1 ∧ C2 ), D)] X|(A, B, ¬C1 , D) ⇒ X (¬∧C 0 ) X[(A, B, ¬C 1 , D)]|(A, B, ¬C 2 , D) ⇒ X[(A, B, ¬C 1 , D)] X|(A, B, ¬(C1 ∧ C2 ), D) ⇒ X  X|(A, B, C, ¬D1 ) ⇒ X[(A, B, C, ¬D1 )] D ) X|(A, B, C, ¬D ) ⇒ X[(A, B, C, ¬D )] (¬∧+ 2 2 X|(A, B, C, ¬(D ∧ D )] ⇒ X[(A, B, C, ¬(D1 ∧ D2 )]) 1 2  X|(A, B, C, ¬D1 ) ⇒ X (¬∧0D ) X|(A, B, C, ¬D2 ) ⇒ X X|(A, B, C, ¬(D1 ∧ D2 )] ⇒ X (¬∧C +)

and 

X|(¬A1 , B, C, D) ⇒ X[(¬A1 , B, C, D)] X[(¬A1 , B, C, D)]|(¬A2 , B, C, D) ⇒ X[(¬A1 , B, C, D)][(¬A2 , B, C, D)] X|(¬(A1 ∨ A2 ), B, C, D) ⇒ X[(¬(A1 ∨ A2 ), B, C, D)] X|(¬A1 , B, C, D) ⇒ X (¬∨0A ) X[(¬A1 , B, C, D)]|(¬A2 , B, C, D) ⇒ X[(¬A1 , B, C, D)] X|(¬(A1 ∨ A2 ), B, C, D) ⇒ X  X|(A, ¬B1 , C, D) ⇒ X[(A, ¬B1 , C, D)] B ) X|(A, ¬B , C, D) ⇒ X[(A, ¬B , C, D)] (¬∨+ 2 2 X|(A, ¬(B1 ∨ B2 ), C, D) ⇒ X[(A, ¬(B1 ∨ B2 ), C, D)] X|(A, ¬B1 , C, D) ⇒ X (¬∨0B ) X|(A, ¬B2 , C, D) ⇒ X X|(A, ¬(B1 ∨ B2 ), C, D) ⇒ X A) (¬∨+

and 

X|(A, B, ¬C1 , D) ⇒ X[(A, B, ¬C1 , D)] X|(A, B, ¬C2 , D) ⇒ X[(A, B, ¬C2 , D)] X|(A, B, ¬(C1 ∨ C2 ), D) ⇒ X[(A, B, ¬(C1 ∨ C2 ), D)] X|(A, B, ¬C1 , D) ⇒ X (¬∨C 0 ) X|(A, B, ¬C 2 , D) ⇒ X X|(A, B, ¬(C1 ∨ C2 ), D) ⇒ X  X|(A, B, C, ¬D1 ) ⇒ X[(A, B, C, ¬D1 )] D ) X[(A, B, C, ¬D )]|(A, B, C, ¬D ) ⇒ X[(A, B, C, ¬D )][(A, B, C, ¬D )] (¬∨+ 1 2 1 2 X|(A, B, C, ¬(D ∨ D )] ⇒ X[(A, B, C, ¬(D ∨ D )]) 1 2 1 2  X|(A, B, C, ¬D1 ) ⇒ X D (¬∨0 ) X[(A, B, C, ¬D1 )]|(A, B, C, ¬D2 ) ⇒ X[(A, B, C, ¬D1 )] X|(A, B, C, ¬(D1 ∨ D2 )] ⇒ X (¬∨C +)

Definition 7.6.1 Given a supersequent X and a quadruple X, a reduction X|X ⇒ X is provable in R− , denoted by − X|X ⇒ X , if there is a sequence {δ1 , . . . , δn } of

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7 R-Calculi for Supersequents

reductions such that δn = X|X ⇒ X , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in R− . Theorem 7.6.2 (Soundness and completeness theorem) For any reduction X|X ⇒ X ,

− X|X ⇒ X if and only if |=− X|X ⇒ X .



7.6.1 Rff Given a pair (B, C) of formulas, we use  ⇒ff  to revise (B, C) and obtain a sequence  →   , denoted by |=ff  → |(B, C) ⇒  →   , if  →   = 

, B → , C if , B → , C is Gff -valid  →  otherwise.

R-calculus Rff consists of the following axioms and deduction rules: • Axioms: ⎡

¬l ∈ / ⎢l ∈ /  ⎢ ⎣ m ∈ /  (A+ ) (A0ff ) ff ¬m ∈ /  → |(l, m) ⇒ , l → , m

⎧ ¬l ∈  ⎪ ⎪ ⎨ l∈ m∈ ⎪ ⎪ ⎩ ¬m ∈   → |(l, m) ⇒  → 

where ,  are sets of literals, and l, m are literals. • Deduction rules:  → |(B, C) ⇒ , B → , C  → |(¬¬B, C) ⇒ , ¬¬B → , C  → |(B, C) ⇒  →  (¬¬0B )  → |(¬¬B, C) ⇒  →   → |(B, C) ⇒ , B → , C (¬¬C+ )  → |(B, ¬¬C) ⇒ , B → , ¬¬C  → |(B, C) ⇒  →  (¬¬C0 )  → |(B, ¬¬C) ⇒  →  (¬¬+B )

7.6 R-Calculus R−

209



and

 → |(B1 , C) ⇒ , B1 → , C  → |(B2 , C) ⇒ , B2 → , C  → |(B1 ∧ B2 , C) ⇒ , B1 ∧ B2 → , C  → |(B1 , C) ⇒  →  (∧0B )  → |(B2 , C) ⇒  →   → |(B1 ∧ B2 , C) ⇒  →    → |(B, C1 ) ⇒ , B → , C1 (∧C+ )  → |(B, C2 ) ⇒ , B → , C2  → |(B, C1 ∧ C2 ) ⇒ , B → , C1 ∧ C2  → |(B, C1 ) ⇒  →  (∧0B )  → |(B, C2 ) ⇒  →   → |(B, C1 ∧ C2 ) ⇒  → 

(∧+B )

and



 → |(B1 , C) ⇒ , B1 → , C , B1 → , C|(B2 , C) ⇒ , B1 , B2 → , C  → |(B1 ∨B2 , C) ⇒ , B1 ∨ B2 → , C  → |(B1 , C) ⇒  →  (∨0B ) , B1 → , C|(B2 , C) ⇒ , B1 → , C  → |(B1 ∨ B2 , C) ⇒  →    → |(B, C1 ) ⇒ , B → , C1 (∨C+ ) , B → , C1 |(B, C2 ) ⇒ , B → , C1 , C2  → |(B, C1 ∨ C2 ) ⇒ , B → , C1 ∨ C2  → |(B, C1 ) ⇒  →  (∨0B ) , B → , C1 |(B, C2 ) ⇒ , B → , C1  → |(B, C1 ∨ C2 ) ⇒  →  (∨+B )

and 

 → |(¬B1 , C) ⇒ , ¬B1 → , C , ¬B1 → , C|(¬B2 , C) ⇒ , ¬B1 , ¬B2 → , C  → |(¬(B 1 ∧ B2 ), C) ⇒ , ¬(B1 ∧ B2 )  → , C  → |(¬B1 , C) ⇒  →  (¬∧0B ) , ¬B1 → , C|(¬B2 , C) ⇒ , ¬B1 → , C  → |(¬(B1 ∧ B2 ), C) ⇒  →    → |(B, ¬C1 ) ⇒ , B → , ¬C1 (¬∧C+ ) , B → , ¬C1 |(B, ¬C2 ) ⇒ , B → , ¬C1 , ¬C2  → |(B, ¬(C  1 ∧ C2 )) ⇒ , B → , ¬(C1 ∧ C2 )  → |(B, ¬C1 ) ⇒  →  (¬∧C0 ) , B → , ¬C1 |(B, ¬C2 ) ⇒ , B → , ¬C1  → |(B, ¬(C1 ∧ C2 )) ⇒  →  (¬∧+B )

210

7 R-Calculi for Supersequents



and

 → |(¬B1 , C) ⇒ , ¬B1 → , C  → |(¬B2 , C) ⇒ , ¬B2 → , C  → |(¬(B1 ∨ B2 ), C) ⇒ , ¬(B1 ∨ B2 ) → , C  → |(¬B1 , C) ⇒  →  (¬∨0B )  → |(¬B2 , C) ⇒  →   → |(¬(B1 ∨ B2 ), C) ⇒  →    → |(B, ¬C1 ) ⇒ , B → , ¬C1 (¬∧C+ )  → |(B, ¬C2 ) ⇒ , B → , ¬C2  → |(B, ¬(C  1 ∨ C2 )) ⇒ , B → , ¬(C1 ∨ C2 )  → |(B, ¬C1 ) ⇒  →  (¬∨C0 )  → |(B, ¬C2 ) ⇒  →   → |(B, ¬(C1 ∨ C2 )) ⇒  → 

(¬∨+B )

Theorem 7.6.3 (Soundness and completeness theorem) For any sequent  →  and pair (B, C) with B ∈  and C ∈ ,

ff  → |(B, C) ⇒  →   if and only if

|=ff  → |(B, C) ⇒  →   ,

where  = |, B and   = |, C.



7.6.2 Rft Given a pair (B, D) of formulas, we use  →  to revise (B, D) and obtain a sequence  →  , denoted by |=ft  → |(B, D) ⇒  →  , 

, B → , D if , B → , DisGft −valid  →  otherwise. R-calculus Rft consists of the following axioms and deduction rules: 



if  →  = • Axioms: ⎡

¬l ∈ / ⎢ ¬l ∈ /  ⎢ ⎣ ¬m ∈ /  (A+ ) (A0ft ) ft ¬m ∈ /  → |(l, m) ⇒ , l → , m

⎧ ¬l ∈  ⎪ ⎪ ⎨ ¬l ∈  ¬m ∈  ⎪ ⎪ ⎩ ¬m ∈   → |(l, m) ⇒  → 

where ,  are sets of literals, and l, m are literals.

7.6 R-Calculus R−

• Deduction rules:  → |(B, D) ⇒ , B → , D  → |(¬¬B, D) ⇒ , ¬¬B → , D  → |(B, D) ⇒  →  (¬¬0B )  → |(¬¬B, D) ⇒  →   → |(B, D) ⇒ , B → , D (¬¬+D )  → |(B, ¬¬D) ⇒ , B → , ¬¬D  → |(B, D) ⇒  →  (¬¬0B )  → |(B, ¬¬D) ⇒  → 

(¬¬+B )

and 

 → |(B1 , D) ⇒ , B1 → , D  → |(B2 , D) ⇒ , B2 → , D  → |(B1 ∧ B2 , D) ⇒ , B1 ∧ B2 → , D  → |(B1 , D) ⇒  →  (∧0B )  → |(B2 , D) ⇒  →   → |(B1 ∧ B2 , D) ⇒  →    → |(B, D1 ) ⇒ , B → , D1 (∧+D ) , B → , D1 |(B, D2 ) ⇒ , B → , D1 , D2  → |(B, D1 ∧ D2 ) ⇒ , B → , D1 ∧ D2  → |(B, D1 ) ⇒  →  (∧0B ) , B → , D1 |(B, D2 ) ⇒ , B → , D1  → |(B1 ∧ B2 , D) ⇒  →  (∧+B )

and



 → |(B1 , D) ⇒ , B1 → , D , B1 → , D|(B2 , D) ⇒ , B1 , B2 → , D  → |(B1 ∨B2 , D) ⇒ , B1 ∨ B2 → , D  → |(B1 , D) ⇒  →  (∨0B ) , B1 → , D|(B2 , D) ⇒ , B1 → , D  → |(B1 ∨ B2 , D) ⇒  →    → |(B, D1 ) ⇒ , B → , D1 (∨+D )  → |(B, D2 ) ⇒ , B → , D2  → |(B, D1 ∨ D2 ) ⇒ , B → , D1 ∨ D2  → |(B, D1 ) ⇒  →  (∨0B )  → |(B, D2 ) ⇒  →   → |(B, D1 ∨ D2 ) ⇒  →  (∨+B )

211

212

7 R-Calculi for Supersequents

and 

 → |(¬B1 , D) ⇒ , ¬B1 → , D , ¬B1 → , D|(¬B2 , D) ⇒ , ¬B1 , ¬B2 → , D  → |(¬(B 1 ∧ B2 ), D) ⇒ , ¬(B1 ∧ B2 )  → , D  → |(¬B1 , D) ⇒  →  (¬∧0B ) , ¬B1 → , D|(¬B2 , D) ⇒ , ¬B1 → , D  → |(¬(B1 ∧ B2 ), D) ⇒  →    → |(B, ¬D1 ) ⇒ , B → , ¬D1 (¬∧+D )  → |(B, ¬D2 ) ⇒ , B → , ¬D2  → |(B, ¬(D  1 ∧ D2 )) ⇒ , B → , ¬(D1 ∧ D2 )  → |(B, ¬D1 ) ⇒  →  (¬∧0D )  → |(B, ¬D2 ) ⇒  →   → |(B, ¬(D1 ∧ D2 )) ⇒  → 

(¬∧+B )

and 

 → |(¬B1 , D) ⇒ , ¬B1 → , D  → |(¬B2 , D) ⇒ , ¬B2 → , D  → |(¬(B1 ∨ B2 ), D) ⇒ , ¬(B1 ∨ B2 ) → , D  → |(¬B1 , D) ⇒  →  (¬∨0B )  → |(¬B2 , D) ⇒  →   → |(¬(B1 ∨ B2 ), D) ⇒  →    → |(B, ¬D1 ) ⇒ , B → , ¬D1 (¬∨+D ) , B → , ¬D1 |(B, ¬D2 ) ⇒ , B → , ¬D1 , ¬D2  → |(B, ¬(D  1 ∨ D2 )) ⇒ , B → , ¬(D1 ∨ D2 )  → |(B, ¬D1 ) ⇒  →  (¬∨0D ) , B → , ¬D1 |(B, ¬D2 ) ⇒ , B → , ¬D1  → |(B, ¬(D1 ∨ D2 )) ⇒  →  (¬∨+B )

Theorem 7.6.4 (Soundness and completeness theorem) For any sequent  →  and pair (B, D) with B ∈  and D ∈ ,

ft  → |(B, D) ⇒  →  if and only if

|=ft  → |(B, D) ⇒  →  ,

where  = |, B and  = |, D.



7.7 Conclusions

213

7.7 Conclusions A Gentzen deduction system is a combination of two semi-sequent deduction systems. Therefore, in each logic, there are a Gentzen deduction system and two semisequent deduction systems (tableau proof systems). From these deduction systems, we find that the validity/satisfiability determines the axioms; and the truth-values determine one or two sequents/semi-sequents in the premise of a deduction rule. Gentzen deduction system G+ for supersequent | ⇒ | is equivalent to one of the following Gentzen deduction systems for sequents: • • • •

Gft Gff Gtt Gtf

for sequent , ¬ ⇒ , ¬; for sequent , ¬ ⇒ ¬, ; for sequent ¬,  ⇒ , ¬; for sequent ¬,  ⇒ ¬, .

Here, G+ for supersequent | ⇒ | is equivalent to Gtt for sequent , ¬ ⇒ , ¬, that is, | ⇒ | is provable in G+ iff , ¬ ⇒ , ¬ is provable in Gtt . Correspondingly, R-calculi R+ for supersequent | ⇒ | is equivalent to one of the following R-calculi for sequents: • • • •

Rft Rff Rtt Rtf

for sequent , ¬ ⇒ , ¬; for sequent , ¬ ⇒ ¬, ; for sequent ¬,  ⇒ , ¬; for sequent ¬,  ⇒ ¬, .

Dually, there is Gentzen deduction system G− for co-supersequents | → | is equivalent to one of the following Gentzen deduction systems for co-sequents: • • • •

Gtf Gtt Gff Gft

for co-sequent , ¬ → , ¬; for co-sequent , ¬ → ¬, ; for co-sequent ¬,  → , ¬; for co-sequent ¬,  → ¬, .

Here, G− for co-supersequent | → | is equivalent to Gtf for co-sequent , ¬ → , ¬, that is, | → | is provable in G− iff , ¬ → , ¬ is provable in Gtf . Correspondingly, R-calculi R− for co-supersequent | → | is equivalent to one of the following R-calculi for sequents: • • • •

Rtf Rtt Rff Rft

for co-sequent , ¬ → , ¬; for co-sequent , ¬ → ¬, ; for co-sequent ¬,  → , ¬; for co-sequent ¬,  → ¬, .

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References Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Artif. Intell. 4, 225–248 (1991) Baaz, M., Zach, R.: Hypersequent and cut-elimination for intuitionistic fuzzy logic. In: Clote, P.G., Schwichtenberg, H., (eds.), Computer Science Logic, Proceedings of the CSL’2000, LNCS 1862, pp. 178–201. Springer Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Log. 2, 87–112 (1938) Fitting, M.C.: Many-valued modal logics (I,II), Fundamenta Informaticae, 15(1991), 235–254; 17(1992), 55–73 Gottwald, S.: A treatise on many-valued logics. Studies in Logic and Computation, vol. 9. Research Studies Press Ltd., Baldock (2001) Hähnle, R.: Advanced many-valued logics. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 297–395. Kluwer, Dordrecht (2001) Li, W.: Mathematical logic, foundations for information science. Progress in Computer Science and Applied Logic, vol.25, Birkhäuser (2010) Łukasiewicz, J.: Selected Works. In: Borkowski, L. (ed.) North-Holland and Warsaw: PWN, Amsterdam (1970) Malinowski, G.: Many-valued Logic and its Philosophy. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic, vol. 8. The Many Valued and Nonmonotonic Turn in Logic, Elsevier (2009) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner F. (eds.), Handbook of Philosophical Logic, vol. 2 (2d edn), pp. 249–295 Kluwer, Dordrecht (2001) Wronski, A.: Remarks on a survey article on many valued logic by A. Urquhart, StudiaLogica 46, 275–278 (1987)

Chapter 8

R-Calculi for -Propositional Logic

By taking ¬ as a logical connective, in traditional Getzen deduction system, we have the following deduction rules (¬ L )

 ⇒ A,  , ¬A ⇒ 

(¬ R )

, B ⇒   ⇒ ¬B, 

to eliminate ¬ from the conclusion. We take ¬ as a meta-logical connective, instead of as a logical connective, and obtain Gentzen deduction system Gt , where we use (¬¬ L )/(¬¬ R ) instead of (¬ L )/(¬ R ). Such a deduction system is used as the one for multivalued logic (Bolc and Borowik 2003). Similarly we take → as a meta-logical connective, denoted by , and obtain a t for . Gentzen deduction system G t t , Qt , P for -propositional logic, where In this chapter, we give R-calculi R t t t R , Q , P are sound and complete with respect to ⊆-minimal change, -minimal change and  -minimal change (Li and Sui 2014, 2013; Li et al. 2015), respectively. By the following equivalences and inclusions: for any formulas A1 , A2 , B1 , B2 , A1 ∧ A2  B ⇐ A1  B ∨ A2  B A1 ∨ A2  B ≡ A1  B ∧ A2  B A  B1 ∧ B2 ≡ A  B1 ∧ A  B2 A  B1 ∨ B2 ⇐ A  B1 ∨ A  B2 ;

© Science Press 2023 W. Li and Y. Sui, R-Calculus, IV: Propositional Logic, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-19-8633-8_8

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and A1 ∧ A2   B ⇒ A1   B ∧ A2   B A1 ∨ A2   B ≡ A1   B ∨ A2   B A  B1 ∧ B2 ≡ A  B1 ∨ A  B2 A  B1 ∨ B2 ⇒ A  B1 ∧ A  B2 . t t , Qt , P for -propositional logic. we will give R-calculi R We will give the following deduction systems and R-calculi:

sequents theories t G Gt

Tt T t

Tf T f

and sequents t t R , Qt , P   Rt , Qt , Pt

theories t t f f R , Qt , P , R , Qf , P      Rt , Qt , Pt , Rf , Qf , Pf

8.1 -Propositional Logic -Propositional logic are similar to logic programs by the correspondence given in the following. • literals correspond to statements l  l ; , . . . , l1n  l1n ⇒ • clauses l1 , . . . , lm ← l1 , . . . , ln correspond to sequents l11  l11  l21 , . . . , l2m  l2m , l21 where l1 , . . . , lm , l1 , . . . , ln are literals and

l11 , l11 , . . . , l1n , l1n ; are literals in propol21 , l21 , . . . , l2m , l2m

sitional logic. t for -proposiThis section gives basic definitions and a deduction system G t t t tional logic, based on which R-calculi R , Q , P for -propositional logic will be built.

8.1.1 Basic Definitions The logical language of -propositional logic contains the following symbols: • atomic propositional variables: p0 , p1 , . . . ; • formula constructors: ¬, ∨, ∧; and

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• formula connectives: ,  . Formulas are defined as follows: A ::= p|¬A|A1 ∧ A2 |A1 ∨ A2 . Statements are defined as follows: δ:: = A  B|A  B, where A, B are formulas. δ is literal (atomic) if A and B are literals (atoms). Let v be an assignment. A statement δ is satisfied in v, denoted by v |= δ, if 

v(A) = 1 ⇒ v(B) = 1 if δ = A  B v(A) = 1&v(B) = 0 if δ = A  B.

δ is valid, denoted by |= δ, if 

Av(v |= δ) if δ = A  B Ev(v |= δ) if δ = A  B.

Let  be a set of literal statements. A literal statement l1  l2 is deduced from l1  l and l  l2 . Define  ∗ be the transitive closure of , and for each l1  l2 ∈  ∗ we say that l1  l2 is deduced from , denoted by   l1  l2 . There are two deduction rules for transitivity: l1  l2 l  l2 (tran) l2  l3 (neg) 1 , ¬l2  ¬l1 l1  l3 

¬ p if l = p p if l = ¬ p A set  of statements is valid, denoted by |=  ⇒, if for any assignment v, there is a statement A  B/A  B ∈  such that v |= A  B/v |= A  B. where ¬l =

Proposition 8.1.1 For any set  of literal statements, |=  ⇒ if and only if there is a statement l1  l2 ∈  such that   l1  l2 . Proof Because l11  l12 , . . . , ln1  ln2 ; l11  l12 , . . . , ln 1  ln 2 ⇒

is Tf -valid iff

l11  l12 , . . . , ln1  ln2 ⇒ l11  l12 , . . . , ln 1  ln 2

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t is G -valid iff there is a literal statement l  l such that

l11  l12 , . . . , ln1  ln2  lm 2  l  l l11  l12 , . . . , ln 1  ln 2  l  l iff there is a statement l  l ∈  such that   l  l .

 

Proposition 8.1.2 For any nonempty set  of literal statements, |=⇒  if and only   if there is a statement m 1  m 2 ∈  such that   m 1  m 2 .

8.2 Tableau Proof Systems A sequent ⇒  is Tt -valid, denoted by |=t ⇒ , if for any assignment v, there is a statement δ ∈  such that v |= δ. A sequent  ⇒ is Tf -valid, denoted by |=f  ⇒, if for any assignment v, there is a statement γ ∈  such that v |= γ .  A co-sequent →  is T f -valid, denoted by |=f  → , if there is an assignment v such that for any statement δ ∈ , v |= δ.  A co-sequent  → is T t -valid, denoted by |=t   →, if there is an assignment v such that for any statement γ ∈ , v |= γ .

8.2.1 Tt We say that a sequent ⇒  is Tt -valid, denoted by |=t ⇒ , if for any assignment v, there is a statement δ ∈  such that v |= δ. Tableau proof system Tt consists of the following axiom and deduction rules: • Axiom: (At )

Em 1  m 2 (m 1  m 2 ∈ &  m 1  m 2 ) ⇒

where  is a set of literal statements. • Deduction rules: 

 ⇒ C1  D,  ⇒ C  D1 ,  (+∧ R R ) ⇒ C  D2 ,  (+∧ R L ) ⇒ C2  D,  ⇒ ⇒  C1 ∧ C2  D,   C  D1 ∧ D2 ,  ⇒ C1  D,  ⇒ C   D1 ,  (−∧ R L ) ⇒ C2  D,  (−∧ R R ) ⇒ C  D2 ,  ⇒ C1 ∧ C2  D,  ⇒ C   D1 ∧ D2 , 

8.2 Tableau Proof Systems

219



 ⇒ C1  D,  ⇒ C  D1 ,  (+∨ R L ) ⇒ C2  D,  (+∨ R R ) ⇒ C  D2 ,  ⇒ ⇒  C1 ∨ C2  D,   C  D1 ∨ D2 ,  ⇒ C1  D,  ⇒ C   D1 ,  (−∨ R L ) ⇒ C2  D,  (−∨ R R ) ⇒ C  D2 ,  ⇒ C1 ∨ C2  D,  ⇒ C   D1 ∨ D2 , 

and

and 

 ⇒ ¬C1  D,  ⇒ C  ¬D1 ,  (+¬∧ R L ) ⇒ ¬C2  D,  (+¬∧ R R ) ⇒ C  ¬D2 ,  ⇒ ⇒  ¬(C1 ∧ C2 )  D,   C  ¬(D1 ∧ D2 ),  ⇒ ¬C1  D,  ⇒ C  ¬D1 ,  (−¬∧ R L ) ⇒ ¬C2  D,  (−¬∧ R R ) ⇒ C  ¬D2 ,  ⇒ ¬(C1 ∧ C2 )  D,  ⇒ C  ¬(D1 ∧ D2 ),  and 

 ⇒ ¬C1  D,  ⇒ C  ¬D1 ,  (+¬∨ R L ) ⇒ ¬C2  D,  (+¬∨ R R ) ⇒ C  ¬D2 ,  ⇒ ⇒  ¬(C1 ∨ C2 )  D,   C  ¬(D1 ∨ D2 ),  ⇒ ¬C1  D,  ⇒ C  ¬D1 ,  (−¬∨ R L ) ⇒ ¬C2  D,  (−¬∨ R R ) ⇒ C  ¬D2 ,  ⇒ ¬(C1 ∨ C2 )  D,  ⇒ C  ¬(D1 ∨ D2 ),  Definition 8.2.1 A sequent ⇒  is provable in Tt , denoted by t ⇒ , if there is a sequence {⇒ 1 , . . . , ⇒ n } of sequents such that n = , and for each 1 ≤ i ≤ n, ⇒ i is an axiom or is deduced from the previous sequents by one deduction rule in Tt . Theorem 8.2.2 (Soundness and completeness theorem) For any sequent  ⇒, t  ⇒ if and only if |=t  ⇒ .

 

8.2.2 Tf We say that a sequent  ⇒ is valid, denoted by |=f  ⇒, if for any assignment v, there is a statement γ ∈  such that v |= γ . Tableau proof system Tf consists of the following axiom and deduction rules: • Axiom: (Af )

El1  l2 (l1  l2 ∈ &  l1  l2 ) ⇒

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where  is a set of literal statements. • Deduction rules: 

 , A1  B ⇒ , A  B1 ⇒ (+∧ L L ) , A2  B ⇒ (+∧ L R ) , A  B2 ⇒ , ,  A1 ∧ A2  B ⇒  A  B1 ∧ B2 ⇒ , A1  B ⇒ , A  B1 ⇒ (−∧ L L ) , A2  B ⇒ (−∧ L R ) , A  B2 ⇒ , A1 ∧ A2  B ⇒ , A  B1 ∧ B2 ⇒ and



 , A1  B ⇒ , A  B1 ⇒ (+∨ L L ) , A2  B ⇒ (+∨ L R ) , A  B2 ⇒ , ,  A1 ∨ A2  B ⇒  A  B1 ∨ B2 ⇒ , A1  B ⇒ , A  B1 ⇒ (−∨ L L ) , A2  B ⇒ (−∨ L R ) , A  B2 ⇒ , A1 ∨ A2  B ⇒ , A  B1 ∨ B2 ⇒

and 

 , ¬A1  B ⇒ , A  ¬B1 ⇒ (+¬∧ L L ) , ¬A2  B ⇒ (+¬∧ L R ) , A  ¬B2 ⇒ , ,  ¬(A1 ∧ A2 )  B ⇒  A  ¬(B1 ∧ B2 ) ⇒ , ¬A1  B ⇒ , A  ¬B1 ⇒ (−¬∧ L L ) , ¬A2  B ⇒ (−¬∧ L R ) , A  ¬B2 ⇒ , ¬(A1 ∧ A2 )  B ⇒ , A  ¬(B1 ∧ B2 ) ⇒ and 

 , ¬A1  B ⇒ , A  ¬B1 ⇒ (+¬∨ L L ) , ¬A2  B ⇒ (+¬∨ L R ) , A  ¬B2 ⇒ ,  ¬(A1 ∨ A2 )  B ⇒  , A  ¬(B1 ∨ B2 ) ⇒ , ¬A1  B ⇒ , A  ¬B1 ⇒ (−¬∨ L L ) , ¬A2  B ⇒ (−∨ L R ) , A  ¬B2 ⇒ , ¬(A1 ∨ A2 )  B ⇒ , A  ¬(B1 ∨ B2 ) ⇒ Definition 8.2.3 A sequent  ⇒ is provable, denoted by f  ⇒, if there is a sequence {1 ⇒, . . . , n ⇒} of sequents such that n = , and for each 1 ≤ i ≤ n, i ⇒ is an axiom or is deduced from the previous sequents by one deduction rule in Tf . Theorem 8.2.4 (Soundness and completeness theorem) For any sequent  ⇒, f  ⇒ iff |=f  ⇒ .

 

∗ 8.3 R-Calculi R

221

 8.2.3 Tableau Proof Systems T t /Tf  Dually we have nonmonotonic tableau proof systems T t , Tf . ∗ ∗ ∗ Let T consist of two parts A and T (−). Then,  f T t = At + σ (T (−)),   Tf = Af + σ (Tt (−)),  where σ is a mapping mapping deduction rules in Tt /Tf into T t /Tf , such that

σ (⇒ ) =  → σ (C) = A σ (D) = B σ (x) = x,

σ ( ⇒) =→  σ (A) = C, σ (B) = D,

for any other symbols x, and ∼ El1  l2 (l1  l2 ∈ &  l1  l2 )  → ∼ Em 1  m 2 (m 1  m 2 ∈ &  m 1  m 2 ) (A t ) →  (A t )

where ,  are sets of literal statements. Theorem 8.2.5 (Soundness and completeness theorem) (i) For any co-sequent →,   t   → iff |=t   → .

(ii) For any co-sequent → ,   f  →  iff |=f  → .

 

∗ 8.3 R-Calculi R t f In this section we will R-calculi R and R .

t 8.3.1 R-Calculus R

Given  and a statement δ ∈ , we use  to revise δ and obtain  , denoted by |=t |δ ⇒  , if

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 =



 − {δ} if |=t ⇒  − {δ}  otherwise.

t R-calculus R consists of the following axioms and deduction rules:

• Axioms: (At− )

Eδ = δ(  δ & ∼ δ ∈ ) Aδ = δ(  δ or ∼ δ ∈ / ) (At0 ) |δ ⇒  − {δ} |δ ⇒ 

where  ∪ {δ} is a set of literal statements, and  ∼δ=

A  B if δ = A  B A  B if δ = A  B.

• Deduction rules: 

|C1  D ⇒  − {C1  D} |C2  D ⇒  − {C2  D} |C1 ∧ C2 D ⇒  − {C1 ∧ C2  D} |C1  D ⇒  (+∧0L ) |C2  D ⇒  |C1 ∧ C2  D ⇒   |C  D1 ⇒  − {C  D1 } (+∧−R )  − {C  D1 }|C  D2 ⇒  − {C  D1 , C  D2 } |C  D1 ∧ D2 ⇒  − {C  D1 ∧ D2 } |C  D1 ⇒  (+∧0R )  − {C  D1 }|C  D2 ⇒  − {C  D1 } |C  D1 ∧ D2 ⇒ 

L ) (+∧−

and 

|C1  D ⇒  − {C1  D}  − {C1  D}|C2  D ⇒  − {C1  D, C2  D} |C1 ∧ C2 D ⇒  − {C1 ∧ C2  D} |C1  D ⇒  (−∧0L )  − {C1  D}|C2  D ⇒  − {C1  D} |C1 ∧ C2  D ⇒   |C  D1 ⇒  − {C  D1 } (−∧−R ) |C  D2 ⇒  − {C  D2 } |C  D1 ∧ D2 ⇒  − {C  D1 ∧ D2 } |C  D1 ⇒  (−∧0R ) |C  D2 ⇒  |C  D1 ∧ D2 ⇒ 

L (−∧− )

∗ 8.3 R-Calculi R

and 

|C1  D ⇒  − {C1  D}  − {C1  D}|C2  D ⇒  − {C1  D, C2  D} |C1 ∨ C2 D ⇒  − {C1 ∨ C2  D} |C1  D ⇒  (+∨0L )  − {C1  D}|C2  D ⇒  − {C1  D} |C1 ∨ C2  D ⇒   |C  D1 ⇒  − {C1  D} (+∨−R ) |C  D2 ⇒  − {C2  D} |C  D1 ∨ D2 ⇒  − {C1 ∨ C2  D} |C  D1 ⇒  (+∨0R ) |C  D2 ⇒  |C  D1 ∨ D2 ⇒ 

L (+∨− )

and 

|C1  D ⇒  − {C1  D} |C2  D ⇒  − {C2  D} |C1 ∨ C2 D ⇒  − {C1 ∨ C2  D} |C1  D ⇒  (−∨0L ) |C2  D ⇒  |C1 ∨ C2  D ⇒   |C  D1 ⇒  − {C  D1 } (−∨−R )  − {C  D1 }|C  D2 ⇒  − {C  D1 , C  D2 } |C  D1 ∨ D2 ⇒  − {C  D1 ∨ D2 } |C  D1 ⇒  (−∨0R )  − {C  D1 }|C  D2 ⇒  − {C  D1 } |C  D1 ∨ D2 ⇒  L (−∨− )

and 

|¬C1  D ⇒  − {¬C1  D}  − {¬C1  D}|¬C2  D ⇒  − {¬C1  D, ¬C2  D} |¬(C1 ∧ C2 ) D ⇒  − {¬(C1 ∧ C2 )  D} |¬C1  D ⇒  (+¬∧0L )  − {¬C1  D}|¬C2  D ⇒  − {¬C1  D} |¬(C1 ∧ C2 )  D ⇒   |C  ¬D1 ⇒  − {C  ¬D1 } (+¬∧−R ) |C  ¬D2 ⇒  − {C  ¬D2 } |C  ¬(D1∧ D2 ) ⇒  − {C  ¬(D1 ∧ D2 )} |C  ¬D1 ⇒  (+¬∧0R ) |C  ¬D2 ⇒  |C  ¬(D1 ∧ D2 ) ⇒ 

L (+¬∧− )

223

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and 

|¬C1  D ⇒  − {¬C1  D} |¬C2  D ⇒  − {¬C2  D} |¬(C1 ∧ C2 ) D ⇒  − {¬(C1 ∧ C2 )  D} |¬C1  D ⇒  (−¬∧0L ) |¬C2  D ⇒  |¬(C1 ∧ C2 )  D ⇒   |C  ¬D1 ⇒  − {C  ¬D1 } (−¬∧−R )  − {C  ¬D1 }|C  ¬D2 ⇒  − {C  ¬D1 , C  ¬D2 } |C  ¬(D1∧ D2 ) ⇒  − {C  ¬(D1 ∧ D2 )} |C  ¬D1 ⇒  (−¬∧0R )  − {C  ¬D1 }|C  ¬D2 ⇒  − {C  ¬D1 |C  ¬(D1 ∧ D2 ) ⇒  L (−¬∧− )

and 

|¬C1  D ⇒  − {¬C1  D} |¬C2  D ⇒  − {¬C2  D} |¬(C1 ∨ C2 ) D ⇒  − {¬(C1 ∨ C2 )  D} |¬C1  D ⇒  (+¬∨0L ) |¬C2  D ⇒  |¬(C1 ∨ C2 )  D ⇒   |C  ¬D1 ⇒  − {C  ¬D1 } (+¬∨−R )  − {C  ¬D1 }|C  ¬D2 ⇒  − {C  ¬D1 , C  ¬D2 } |C  ¬(D1∨ D2 ) ⇒  − {C  ¬(D1 ∨ D2 )} |C  ¬D1 ⇒  (+¬∨0R )  − {C  ¬D1 }|C  ¬D2 ⇒  − {C  ¬D1 } |C  ¬(D1 ∨ D2 ) ⇒  L (+¬∨− )

and 

|¬C1  D ⇒  − {¬C1  D}  − {¬C1  D}|¬C2  D ⇒  − {¬C1  D, ¬C2  D} |¬(C1 ∨ C2 ) D ⇒  − {¬(C1 ∨ C2 )  D} |¬C1  D ⇒  (−¬∨0L )  − {¬C1  D}|¬C2  D ⇒  − {¬C1  D} |¬(C1 ∨ C2 )  D ⇒   |C  ¬D1 ⇒  − {C  ¬D1 } (−¬∨−R ) |C  ¬D2 ⇒  − {C  ¬D2 } |C  ¬(D1∨ D2 ) ⇒  − {C  ¬(D1 ∨ D2 )} |C  ¬D1 ⇒  (−¬∨0R ) |C  ¬D2 ⇒  |C  ¬(D1 ∨ D2 ) ⇒ 

L (−¬∨− )

∗ 8.3 R-Calculi R

225

Definition 8.3.1 A reduction θ = |δ ⇒  is provable, denoted by t θ, if there is a sequence {θ1 , . . . , θn } of reductions such that θn = θ, and for each 1 ≤ i ≤ n, θi t is an axiom or is deduced from the previous reductions by one deduction rule in R . Theorem 8.3.2 (Soundness and completeness theorem) For any reduction θ = |δ ⇒  ,   t θ iff |=t θ.

f 8.3.2 R-Calculus R

Given  and a statement γ ∈ , we use  to revise γ and obtain  , denoted by |=f |γ ⇒  , if   − {γ } if |=f ⇒  − {γ }  =  otherwise. f R-calculus R consists of the following axioms and deduction rules:

• Axioms: (Af− )

Eγ = γ (∼ γ ∈ &  γ ) ∼ Eγ = γ (∼ γ ∈ &  γ ) (Af0 ) |γ ⇒  − {γ } |γ ⇒ 

where  is a set of literal statements. • Deduction rules: 

|A1  B ⇒  − {A1  B}  − {A1  B}|A2  B ⇒  − {A1  B, A2  B} |A1 ∧ A2 B ⇒  − {A1 ∧ A2  B} |A1  B ⇒  (+∧0L )  − {A1  B}|A2  B ⇒  − {A1  B} |A1 ∧ A2  B ⇒   |A  B1 ⇒  − {A  B1 } (+∧−R ) |A  B2 ⇒  − {A  B2 } |A  B1 ∧ B 2 ⇒  − {A  B1 ∧ B2 } |A  B1 ⇒  (+∧0R ) |A  B2 ⇒  |A  B1 ∧ B2 ⇒  L ) (+∧−

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and 

|A1  B ⇒  − {A1  B} |A2  B ⇒  − {A2  B} |A1 ∧ A2 B ⇒  − {A1 ∧ A2  B} |A1  B ⇒  (−∧0L ) |A2  B ⇒  |A1 ∧ A2  B ⇒   |A  B1 ⇒  − {A  B1 } (−∧−R )  − {A  B1 }|A  B2 ⇒  − {A  B1 , A  B2 } |A  B1 ∧ B 2 ⇒  − {A  B1 ∧ B2 } |A  B1 ⇒  (−∧0R )  − {A  B1 }|A  B2 ⇒  − {A  B1 } |A  B1 ∧ B2 ⇒ 

L (−∧− )

and 

|A1  B ⇒  − {A1  B} |A2  B ⇒  − {A1  B} |A1 ∨ A2 B ⇒  − {A1 ∨ A2  B} |A1  B ⇒  (+∨0L ) |A2  B ⇒  |A1 ∨ A2  B ⇒   |A  B1 ⇒  − {A  B1 } (+∨−R )  − {A  B1 }|A  B2 ⇒  − {A  B1 , A  B2 } |A  B1 ∨ B 2 ⇒  − {A  B1 ∨ B2 } |A  B1 ⇒  (+∨0R )  − {A  B1 }|A  B2 ⇒  − {A  B1 } |A  B1 ∨ B2 ⇒ 

L (+∨− )

and 

|A1  B ⇒  − {A1  B}  − {A1  B}|A2  B ⇒  − {A1  B, A2  B} |A1 ∨ A2 B ⇒  − {A1 ∨ A2  B} |A1  B ⇒  (−∨0L )  − {A1  B}|A2  B ⇒  − {A1  B} |A1 ∨ A2  B ⇒   |A  B1 ⇒  − {A  B1 } (−∨−R ) |A  B2 ⇒  − {A  B2 } |A  B1 ∨ B 2 ⇒  − {A  B1 ∨ B2 } |A  B1 ⇒  (−∨0R ) |A  B2 ⇒  |A  B1 ∨ B2 ⇒  L (−∨− )

∗ 8.3 R-Calculi R

and 

|¬A1  B ⇒  − {¬A1  B} |¬A2  B ⇒  − {¬A2  B} |¬(A1 ∧ A2 ) B ⇒  − {¬(A1 ∧ A2 )  B} |¬A1  B ⇒  (+¬∧0L ) |¬A2  B ⇒  |¬(A1 ∧ A2 )  B ⇒   |A  ¬B1 ⇒  − {A  ¬B1 } (+¬∧−R )  − {A  ¬B1 }|A  ¬B2 ⇒  − {A  ¬B1 , A  ¬B2 } |A  ¬(B1 ∧  B2 ) ⇒  − {A  ¬(B1 ∧ B2 )} |A  ¬B1 ⇒  (+¬∧0R )  − {A  ¬B1 }|A  ¬B2 ⇒  − {A  ¬B1 } |A  ¬(B1 ∧ B2 ) ⇒ 

L (+¬∧− )

and 

|¬A1  B ⇒  − {¬A1  B}  − {¬A1  B}|¬A2  B ⇒  − {¬A1  B, ¬A2  B} |¬(A1 ∧ A2 ) B ⇒  − {¬(A1 ∧ A2 )  B} |¬A1  B ⇒  (−¬∧0L )  − {¬A1  B}|¬A2  B ⇒  − {¬A1  B} |¬(A1 ∧ A2 )  B ⇒   |A  ¬B1 ⇒  − {A  ¬B1 } (−¬∧−R ) |A  ¬B2 ⇒  − {A  ¬B2 } |A  ¬(B1  ∧ B2 ) ⇒  − {A  ¬(B1 ∧ B2 )} |A  ¬B1 ⇒  (−¬∧0R ) |A  ¬B2 ⇒  |A  ¬(B1 ∧ B2 ) ⇒  L (−¬∧− )

and 

|¬A1  B ⇒  − {¬A1  B}  − {¬A1  B}|¬A2  B ⇒  − {¬A1  B, ¬A2  B} |¬(A1 ∨ A2 ) B ⇒  − {¬(A1 ∨ A2 )  B} |¬A1  B ⇒  (+¬∨0L )  − {¬A1  B}|¬A2  B ⇒  − {¬A1  B} |¬(A1 ∨ A2 )  B ⇒   |A  ¬B1 ⇒  − {A  ¬B1 } (+¬∨−R ) |A  ¬B2 ⇒  − {A  ¬B2 } |A  ¬(B1 ∨  B2 ) ⇒  − {A  ¬(B1 ∨ B2 )} |A  ¬B1 ⇒  (+¬∨0R ) |A  ¬B2 ⇒  |A  ¬(B1 ∨ B2 ) ⇒  L (+¬∨− )

227

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228

and 

|¬A1  B ⇒  − {¬A1  B} |¬A2  B ⇒  − {¬A2  B} |¬(A1 ∨ A2 ) B ⇒  − {¬(A1 ∨ A2 )  B} |¬A1  B ⇒  (−¬∨0L ) |¬A2  B ⇒  |¬(A1 ∨ A2 )  B ⇒   |A  ¬B1 ⇒  − {A  ¬B1 } (−∨−R )  − {A  ¬B1 }|A  ¬B2 ⇒  − {A  ¬B1 , A  ¬B2 } |A  ¬(B1  ∨ B2 ) ⇒  − {A  ¬(B1 ∨ B2 )} |A  ¬B1 ⇒  (−∨0R )  − {A  ¬B1 }|A  ¬B2 ⇒  − {A  ¬B1 } |A  ¬(B1 ∨ B2 ) ⇒ 

L (−¬∨− )

f Definition 8.3.3 A reduction θ = |γ ⇒  is provable in R , denoted by f θ, if there is a sequence {θ1 , . . . , θn } of reductions such that θn = θ, and for each 1 ≤ i ≤ n, θi is an axiom or is deduced from the previous reductions by one deduction f . rule in R

Theorem 8.3.4 (Soundness and completeness theorem) For any reduction θ = |γ ⇒  ,   f θ iff |=f θ.

8.4 Other Minimal Changes Definition 8.4.1 Given a formula A, a formula B is a subformula of A, denoted by B ≤ A, if either A = B, or (i) if A = ¬A1 then B ≤ A1 ; (ii) if A = A1 ∨ A2 or A1 ∧ A2 then either B ≤ A1 or B ≤ A2 . For example, let A = ( p ∨ q) ∧ (r ∨ s). Then, p ∨ q, r ∨ s ≤ A; and p ∧ r, q ∧ r, p ∧ (r ∨ s)  A. t f , R preserves the ⊆-minimal change, that is, if It is clear that the R-calculi R |γ ⇒  then  is a ⊆-minimal subtheory of , γ , which is maximal consistent with , that is, for any theory  with  ⊇  ⊃  ,  is inconsistent with .

8.4 Other Minimal Changes

229

Definition 8.4.2 Given a formula A[B1 , . . . , Bn ], where [B1 ] is an occurrence of B1 in A, a formula B = A[λ, . . . , λ] = A[B1 /λ, . . . , Bn /λ], where the occurrence Bi is replaced by the empty formula λ, is called a pseudo-subformula of A, denoted by B  A. For example, let A = ( p ∨ q) ∧ (r ∨ s). Then, p ∨ q, r ∨ s, p ∧ r, q ∧ r, p ∧ (r ∨ s)  A. There are two other minimal changes, and we will consider: (i) Pseudo-subformula-minimal (-minimal) change, where  is the pseudosubformula relation, just as the subformula relation ≤, where a statement γ is a pseudo-substatement of γ if eliminating some substatements in γ results in γ. (ii) Deduction-based minimal ( -minimal) change, where a theory  is a  minimal change of  (denoted by |=P |γ ⇒  ), if    − {γ } is consistent with γ , and for any theory  with      − {γ } either γ ,    and γ ,   , or  is inconsistent with γ . t We will give R-calculi Qt and P which are sound and complete for -minimal change and  -minimal change, respectively. In fact, there should be following R-calculi t R f R Rt Rf

Qt Qf Q t Q f

t P f P Pt Pf

 and we will give three R-calculi Rt , Q t and Pt .

8.4.1 R-Calculus Rt Given  and a statement γ ∈ , we use  to revise γ and obtain  , denoted by |= t |γ ⇒  , if  , γ if |= t , γ  →  =  otherwise. R-calculus Rt consists of the following axioms and deduction rules: • Axioms: (A+ t )

Eγ = γ (  γ & ∼ γ ∈ ) Aγ = γ (  γ or ∼ γ ∈ / ) ) (A0 t |γ ⇒ , γ |γ ⇒ 

where  ∪ {γ } is a set of literal statements.

8 R-Calculi for -Propositional Logic

230

• Deduction rules: 

|A1  B ⇒ , A1  B |A2  B ⇒ , A2  B |A1 ∧ A2 B ⇒ , A1 ∧ A2  B |A1  B ⇒  (+∧0L ) |A2  B ⇒  |A1 ∧ A2  B ⇒   |A  B1 ⇒ , A  B1 (+∧−R ) , A  B1 |A  B2 ⇒ , A  B1 , A  B2 |A  B1 ∧ B 2 ⇒ , A  B1 ∧ B2 |A  B1 ⇒  (+∧0R ) , A  B1 |A  B2 ⇒ , A  B1 |A  B1 ∧ B2 ⇒ , A  B1 ∧ B2

L (+∧− )

and



|A1  B ⇒ , A1  B , A1  B|A2  B ⇒ , A1  B, A2  B |A1 ∧ A2 B ⇒ , A1 ∧ A2  B |A1  B ⇒  (−∧0L ) , A1  B|A2  B ⇒ , A1  B |A1 ∧ A2  B ⇒ , A1 ∧ A2  B  |A  B1 ⇒ , A  B1 (−∧−R ) |A  B2 ⇒ , A  B2 |A  B1 ∧ B 2 ⇒ , A  B1 ∧ B2 |A  B1 ⇒  (−∧0R ) |A  B2 ⇒  |A  B1 ∧ B2 ⇒ 

L (−∧− )

and



|A1  B ⇒ , A1  B , A1  B|A2  B ⇒ , A1  B, A2  B |A1 ∨ A2 B ⇒ , A1 ∨ A2  B |A1  B ⇒  (+∨0L ) , A1  B|A2  B ⇒ , A1  B |A1 ∨ A2  B ⇒   |A  B1 ⇒ , A  B1 (+∨−R ) |A  B2 ⇒ , A  B2 |A  B1 ∨ B 2 ⇒ , A  B1 ∨ B2 |A  B1 ⇒  (+∨0R ) |A  B2 ⇒  |A  B1 ∨ B2 ⇒ 

L (+∨− )

8.4 Other Minimal Changes

and



|A1  B ⇒ , A1  B |A2  B ⇒ , A2  B |A1 ∨ A2 B ⇒ , A1 ∨ A2  B |A1  B ⇒  (−∨0L ) |A2  B ⇒  |A1 ∨ A2  B ⇒   |A  B1 ⇒ , A  B1 (−∨−R ) , A  B1 |A  B2 ⇒ , A  B1 , A  B2 |A  B1 ∨ B 2 ⇒ , A  B1 ∨ B2 |A  B1 ⇒  (−∨0R ) , A  B1 |A  B2 ⇒ , A  B1 |A  B1 ∨ B2 ⇒ 

L (−∨− )

and 

|¬A1  B ⇒ , ¬A1  B , ¬A1  B|¬A2  B ⇒ , ¬A1  B, ¬A2  B |¬(A1 ∧ A2 ) B ⇒ , ¬(A1 ∧ A2 )  B |¬A1  B ⇒  (+¬∧0L ) , ¬A1  B|¬A2  B ⇒ , ¬A1  B |¬(A1 ∧ A2 )  B ⇒   |A  ¬B1 ⇒ , A  ¬B1 (+¬∧−R ) |A  ¬B2 ⇒ , A  ¬B2 |A  ¬(B1 ∧  B2 ) ⇒ , A  ¬(B1 ∧ B2 ) |A  ¬B1 ⇒  (+¬∧0R ) |A  ¬B2 ⇒  |A  ¬(B1 ∧ B2 ) ⇒  L (+¬∧− )

and 

|¬A1  B ⇒ , ¬A1  B |¬A2  B ⇒ , ¬A2  B |¬(A1 ∧ A2 ) B ⇒ , ¬(A1 ∧ A2 )  B |¬A1  B ⇒  (−¬∧0L ) |¬A2  B ⇒  |¬(A1 ∧ A2 )  B ⇒   |A  ¬B1 ⇒ , A  ¬B1 (−¬∧−R ) , A  ¬B1 |A  ¬B2 ⇒ , A  ¬B1 , A  ¬B2 |A  ¬(B1  ∧ B2 ) ⇒ , A  ¬(B1 ∧ B2 ) |A  ¬B1 ⇒  (−¬∧0R ) , A  ¬B1 |A  ¬B2 ⇒ , A  ¬B1 |A  ¬(B1 ∧ B2 ) ⇒ 

L (−¬∧− )

231

8 R-Calculi for -Propositional Logic

232

and 

|¬A1  B ⇒ , ¬A1  B |¬A2  B ⇒ , ¬A2  B |¬(A1 ∨ A2 ) B ⇒ , ¬(A1 ∨ A2 )  B |¬A1  B ⇒  (+¬∨0L ) |¬A2  B ⇒  |¬(A1 ∨ A2 )  B ⇒   |A  ¬B1 ⇒ , A  ¬B1 (+¬∨−R ) , A  ¬B1 |A  ¬B2 ⇒ , A  ¬B1 , A  ¬B2 |A  ¬(B1 ∨  B2 ) ⇒ , A  ¬(B1 ∨ B2 ) |A  ¬B1 ⇒  (+¬∨0R ) , A  ¬B1 |A  ¬B2 ⇒ , A  ¬B1 |A  ¬(B1 ∨ B2 ) ⇒ 

L (+¬∨− )

and 

|¬A1  B ⇒ , ¬A1  B , ¬A1  B|¬A2  B ⇒ , ¬A1  B, ¬A2  B |¬(A1 ∨ A2 ) B ⇒ , ¬(A1 ∨ A2 )  B |¬A1  B ⇒  (−¬∨0L ) , ¬A1  B|¬A2  B ⇒ , ¬A1  B |¬(A1 ∨ A2 )  B ⇒   |A  ¬B1 ⇒ , A  ¬B1 (−¬∨−R ) |A  ¬B2 ⇒ , A  ¬B2 |A  ¬(B1  ∨ B2 ) ⇒ , A  ¬(B1 ∨ B2 ) |A  ¬B1 ⇒  (−¬∨0R ) |A  ¬B2 ⇒  |A  ¬(B1 ∨ B2 ) ⇒  L (−¬∨− )

Definition 8.4.3 A reduction θ = |γ ⇒  is provable in Rt , denoted by  t θ, if there is a sequence {θ1 , . . . , θn } of reductions such that θn = θ, and for each 1 ≤ i ≤ n, θi is an axiom or is deduced from the previous reductions by one deduction rule in Rt . Theorem 8.4.4 (Soundness and completeness theorem) For any reduction θ = |γ ⇒  ,     t θ iff |=t θ.

8.4.2 R-Calculus Q t Given  and a statement γ , we use  to revise γ and obtain  , denoted by |= t |γ ⇒  , if

8.4 Other Minimal Changes

233

 =



, γ if |= t , γ  →  otherwise.

R-calculus Q t consists of the following axioms and deduction rules: • Axioms: (A +t )

 ∼ γ  ∼ γ ) (A 0t |γ ⇒ , γ |γ ⇒ 

where , γ is a set of literal statements. • Deduction rules: ⎧ |A1  B ⇒ , C1  B ⎪ ⎪ ⎪ ⎪ C ⎪ ⎪ ⎡ 1 = λ ⎪ ⎪ ⎨ |A1  B ⇒  ⎣ |A2  B ⇒ , C2  B (+∧ L ) ⎪ ⎪ ⎪ ⎪  C2 = λ ⎪ ⎪ |A1  B ⇒  ⎪ ⎪ ⎩ |A2  B ⇒  |A1  B ⇒ , γ |A  B1 ⇒ , A  D1 (+∧ R ) , A  D1 |A  B2 ⇒ , A  D1 , A  D2 |A  B1 ∧ B2 ⇒ , A  D1 ∧ D2 ⎧ ⎨ C1 ∧ A2  B if C1 = λ where γ = A1 ∧ C2  B if C1 = λ = C2 and ⎩ λ otherwise; |A1  B ⇒ , C1  B (−∧ L ) , C1  B|A2  B ⇒ , C1  B, C2  B |A1 ∧ A2  ⎧B⇒ , C1 ∧ C2  B} |A  B1 ⇒ , A  D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪ ⎨ |A  B1 ⇒  ⎣ |A  B2 ⇒ , A  D2 R (−∧ ) ⎪ ⎪ ⎪ ⎪  D2  = λ ⎪ ⎪ |A  B1 ⇒  ⎪ ⎪ ⎩ |A  B2 ⇒  |A  B1 ∧ B2 ⇒ , γ ⎧ ⎨ A  D1 ∧ B2 if D1 = λ where γ = A  B1 ∧ D2 if D1 = λ = D2 and ⎩ λ otherwise;

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|A1  B ⇒ , C1  B (+∨ L ) , C1  B|A2  B ⇒ , C1  B, C2  B |A1 ∨ A2  ⎧ B ⇒ , C1 ∨ C2  B} |A  B1 ⇒ , A  D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪ ⎨ |A  B1 ⇒  ⎣ |A  B2 ⇒ , A  D2 (+∨ R ) ⎪ ⎪ ⎪ ⎪  D2  = λ ⎪ ⎪ |A  B1 ⇒  ⎪ ⎪ ⎩ |A  B2 ⇒  |A  B1 ∨ B2 ⇒ , γ ⎧ ⎨ A  D1 ∨ B2 if D1 = λ where γ = A  B1 ∨ D2 if D1 = λ = D2 and ⎩ λ otherwise; ⎧ |A1  B ⇒ , C1  B ⎪ ⎪ ⎪ ⎪ C ⎪ ⎪ ⎡ 1 = λ ⎪ ⎪ ⎨ |A1  B ⇒  ⎣ |A2  B ⇒ , C2  B (−∨ L ) ⎪ ⎪ ⎪ ⎪  C2 = λ ⎪ ⎪ |A1  B ⇒  ⎪ ⎪ ⎩ |A2  B ⇒  |A1 ∨ A2  B ⇒ , γ |A  B1 ⇒ , A  D1 (−∨ R ) , A  D1 |A  B2 ⇒ , A  D1 , A  D2 |A  B1 ∨ B2 ⇒ , A  D1 ∨ D2 } where γ

⎧ ⎨ C1 ∨ A2  B if C1 = λ = A1 ∨ C2  B if C1 = λ = C2 and ⎩ λ otherwise; |¬A1  B ⇒ , ¬C1  B (+¬∧ L ) , ¬C1  B|¬A2  B ⇒ , ¬C1  B, ¬C2  B |¬(A1 ∧ A2 )⎧ B ⇒ , ¬(C1 ∧ C2 )  B} |A  ¬B1 ⇒ , A  ¬D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪ ⎨ |A  ¬B1 ⇒  ⎣ |A  ¬B2 ⇒ , A  ¬D2 (+¬∧ R ) ⎪ ⎪ ⎪ ⎪  D2  = λ ⎪ ⎪ |A  ¬B1 ⇒  ⎪ ⎪ ⎩ |A  ¬B2 ⇒  |A  ¬(B1 ∧ B2 ) ⇒ , γ (4)

8.4 Other Minimal Changes

where γ (4)

⎧ ⎨ A  ¬(D1 ∨ B2 ) if D1 = λ = A  ¬(B1 ∨ D2 ) if D1 = λ = D2 and ⎩ λ otherwise;

⎧ |¬A1  B ⇒ , ¬C1  B ⎪ ⎪ ⎪ ⎪ C ⎪ 1 = λ ⎪ ⎪⎡ ⎪ ⎨ |¬A1  B ⇒  ⎣ |¬A2  B ⇒ , ¬C2  B (−¬∧ L ) ⎪ ⎪ ⎪ ⎪  C2 = λ ⎪ ⎪ |¬A1  B ⇒  ⎪ ⎪ ⎩ |¬A2  B ⇒  |¬(A1 ∧ A2 )  B ⇒ , γ (5) |A  ¬B1 ⇒ , A  ¬D1 (−¬∧ R ) , A  ¬D1 |A  ¬B2 ⇒ , A  ¬D2 |A  ¬(B1 ∧ B2 ) ⇒ , A  ¬(D1 ∧ D2 )} where γ (5)

⎧ ⎨ ¬(C1 ∧ A2 )  B if C1 = λ = ¬(A1 ∧ C2 )  B if C1 = λ = C2 and ⎩ λ otherwise;

⎧ |¬A1  B ⇒ , ¬C1  B ⎪ ⎪ ⎪ ⎪ C ⎪ ⎪ ⎡ 1 = λ ⎪ ⎪ ⎨ |¬A1  B ⇒  ⎣ |¬A2  B ⇒ , ¬C2  B (+¬∨ L ) ⎪ ⎪ ⎪  C2 = λ ⎪ ⎪ ⎪ ⎪ |¬A1  B ⇒  ⎪ ⎩ |¬A2  B ⇒  |¬(A1 ∨ A2 )  B ⇒ , γ (6) |A  ¬B1 ⇒ , A  ¬D1 (+¬∨ R ) , A  ¬D1 |A  ¬B2 ⇒ , A  ¬D1 , A  ¬D2 |A  ¬(B1 ∨ B2 ) ⇒ , A  ¬(D1 ∨ D2 )} where γ (6)

⎧ ⎨ ¬(C1 ∨ A2 )  B if C1 = λ = ¬(A1 ∨ C2 )  B if C1 = λ = C2 and ⎩ λ otherwise;

235

8 R-Calculi for -Propositional Logic

236

|¬A1  B ⇒ , ¬C1  B (−¬∨ L ) , ¬C1  B|¬A2  B ⇒ , ¬C1  B, ¬C2  B |¬(A1 ∨ A⎧  B ⇒ , ¬(C1 ∨ C2 )  B} 2)  |A  ¬B1 ⇒ , A  ¬D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1  = λ ⎪ ⎪ ⎨ |A  ¬B1 ⇒  ⎣ |A  ¬B2 ⇒ , A  ¬D2 (−∨ R ) ⎪ ⎪ ⎪ ⎪  D2  = λ ⎪ ⎪ |A  ¬B1 ⇒  ⎪ ⎪ ⎩ |A  ¬B2 ⇒  |A  ¬(B1 ∨ B2 ) ⇒ , γ (7) where γ (7)

⎧ ⎨ A  ¬(D1 ∨ B2 ) if D1 = λ = A  ¬(B1 ∨ D2 ) if D1 = λ = D2 ⎩ λ otherwise.

 Definition 8.4.5 A reduction θ = |γ ⇒  is provable in Q t , denoted by t θ, if there is a sequence {θ1 , . . . , θn } of reductions such that θn = θ, and for each 1 ≤ i ≤ n, θi is an axiom or is deduced from the previous reductions by one deduction rule in Q t .

Theorem 8.4.6 (Soundness and completeness theorem) For any reduction θ = |γ ⇒  ,     t θ iff |=t θ.

8.4.3 R-Calculus Pt A statement A  B is in conjunctive normal form if A is in disjunctive normal form and B is in conjunctive normal form; and A  B is in disjunctive normal form if A is in conjunctive normal form and B is in disjunctive normal form. Given  and a statement γ , we use  to revise γ and obtain , γ , denoted by |= t |γ ⇒ , γ , if  γ if |= t , γ  → γ = λ otherwise. R-calculus Pt consists of the following axioms and deduction rules: • Axioms: (A +t )

¬γ ∈ / &  ¬γ |γ ⇒ , γ

where , γ is a set of literal statements.

(A 0t )

¬γ ∈  or   ¬γ |γ ⇒ 

8.4 Other Minimal Changes

• Deduction rules: |A1  B ⇒ , C1  B (+∧ L ) |A2  B ⇒ , C2  B |A1 ∧ A2  B ⇒ , C1 ∧ C2  B |A  B1 ⇒ , A  D1 (+∧ R ) , A  D1 |A  B2 ⇒ , A  D1 , A  D2 |A  B1 ∧ B2 ⇒ , A  D1 ∧ D2 |A1  B ⇒ , C1  B (−∧ L ) , C1  B|A2  B ⇒ , C1  B, C2  B |A1 ∧ A2  B ⇒ , C1 ∧ C2  B} |A  B1 ⇒ , A  D1 (−∧ R ) |A  B2 ⇒ , A  D2 |A  B1 ∧ B2 ⇒ , A  D1 ∧ D2 } and |A1  B ⇒ , C1  B (+∨ L ) , C1  B|A2  B ⇒ , C1  B, C2  B |A1 ∨ A2  B ⇒ , C1 ∨ C2  B} |A  B1 ⇒ , A  D1 (+∨ R ) |A  B2 ⇒ , A  D2 |A  B1 ∨ B2 ⇒ , A  B1 ∨ D2 } |A1  B ⇒ , C1  B (−∨ L ) |A2  B ⇒ , C2  B |A1 ∨ A2  B ⇒ , C1 ∨ C2  B} |A  B1 ⇒ , A  D1 (−∨ R ) , A  D1 |A  B2 ⇒ , A  D1 , A  D2 |A  B1 ∨ B2 ⇒ , A  D1 ∨ D2 } and |¬A1  B ⇒ , ¬C1  B (+¬∧ L ) , ¬C1  B|¬A2  B ⇒ , ¬C1  B, ¬C2  B |¬(A1 ∧ A2 )  B ⇒ , ¬(C1 ∧ C2 )  B} |A  ¬B1 ⇒ , A  ¬D1 (+¬∧ R ) |A  ¬B2 ⇒ , A  ¬D2 |A  ¬(B1 ∧ B2 ) ⇒ , A  ¬(D1 ∧ D2 )} |¬A1  B ⇒ , ¬C1  B (−¬∧ L ) |¬A2  B ⇒ , ¬C2  B |¬(A1 ∧ A2 )  B ⇒ , ¬(C1 ∧ C2 )  B} |A  ¬B1 ⇒ , A  ¬D1 (−¬∧ R ) , A  ¬D1 |A  ¬B2 ⇒ , A  ¬D2 |A  ¬(B1 ∧ B2 ) ⇒ , A  ¬(D1 ∧ D2 )}

237

8 R-Calculi for -Propositional Logic

238

and |¬A1  B ⇒ , ¬C1  B (+¬∨ L ) |¬A2  B ⇒ , ¬C2  B |¬(A1 ∨ A2 )  B ⇒ , ¬(C1 ∨ C2 )  B} |A  ¬B1 ⇒ , A  ¬D1 (+¬∨ R ) , A  ¬D1 |A  ¬B2 ⇒ , A  ¬D1 , A  ¬D2 |A  ¬(B1 ∨ B2 ) ⇒ , A  ¬(D1 ∨ D2 )} |¬A1  B ⇒ , ¬C1  B (−¬∨ L ) , ¬C1  B|¬A2  B ⇒ , ¬C1  B, ¬C2  B |¬(A1 ∨ A2 )  B ⇒ , ¬(C1 ∨ C2 )  B} |A  ¬B1 ⇒ , A  ¬D1 (−∨ R ) |A  ¬B2 ⇒ , A  ¬D2 |A  ¬(B1 ∨ B2 ) ⇒ , A  ¬(D1 ∨ D2 )} Definition 8.4.7 A reduction θ = |γ ⇒  is provable in Pt , denoted by  t θ, if there is a sequence {θ1 , . . . , θn } of reductions such that θn = θ, and for each 1 ≤ i ≤ n, θi is an axiom or is deduced from the previous reductions by one deduction rule in Pt . Theorem 8.4.8 (Soundness and completeness theorem) For any reduction θ = |γ ⇒  , where γ is in conjunctive normal form,   t θ iff |=t θ.

 

8.5 Gentzen Deduction Systems There are four Gentzen deduction systems t f G G  Gt Gf . t We will give the full description of G , and others follow from it.

8.5.1 Sequents Given two sets ,  of statements, let  ⇒  be a sequent, and an assignment v satisfies  ⇒ , denoted by v |=  ⇒ , if v satisfying every statement γ ∈  implies v satisfying some δ ∈ . t -valid, denoted by |=t  ⇒ , if for any assignment v, v |=  ⇒  is G ⇒. A statement l1  l2 is called a literal statement if l1 , l2 ::= p|¬ p.

8.5 Gentzen Deduction Systems

239

Proposition 8.5.1 For any formulas A1 , A2 , B1 , B2 , A1 ∧ A2  B ⇐ A1  B∨A2  B A1 ∨ A2  B ≡ A1  B∧A2  B A  B1 ∧ B2 ≡ A  B1 ∧A  B2 A  B1 ∨ B2 ⇐ A  B1 ∨A  B2 ; and A1 ∧ A2   B ⇒ A1   B∧A2   B A1 ∨ A2   B ≡ A1   B∨A2   B A   B1 ∧ B2 ≡ A   B1 ∨A   B2 A   B1 ∨ B2 ⇒ A   B1 ∧A   B2 .

  For example, |=⇒ l1  l2 , l3  l4 iff |=⇒ ¬l1 , l2 , ¬l3 , l4 , iff ¬l1 l2 = l1 ¬l3 = l1 l4 = l1 l2 ¬l1 = ¬l2 ¬l3 = ¬l2 l4 = ¬l2 ¬l3 ¬l1 = l3 l2 = l3 l4 = l3 l4 ¬l1 = ¬l4 l2 = ¬l4 ¬l3 = ¬l4

⇒ l1  l2 , l3  l4 ⇒ l1  l1 , l3  l4 ⇒ l1  l2 , ¬l1  l4 ⇒ l1  l2 , l3  l1 ⇒ l2  l2 , l3  l4 ⇒ l1  l2 , l2  l4 ⇒ l1  l2 , l3  ¬l2 ⇒ ¬l3  l2 , l3  l4 ⇒ l1  l3 , l3  l4 ⇒ l1  l2 , l3  l3 ⇒ l4  l2 , l3  l4 ⇒ l1  ¬l4 , ¬l3  l4 ⇒ l1  l2 , l4  l4

Proposition 8.5.2 |= l1  l2 ⇒ l3  l4 if and only if ¬l1 , l2 ∈ {¬l3 , l4 }. Proof For any literals l1 , l2 , l3 , l4 ,

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240

|= l1  l2 ⇒ l3  l4 iff |= ¬l1 ⇒ ¬l3 , l4 & |= l2 ⇒ ¬l3 , l4 iff ¬l1 ∈ {¬l3 , l4 }&l2 ∈ {¬l3 , l4 } iff (¬l1 = ¬l3 &l2 = ¬l3 )or (¬l1 = ¬l3 &l2 = l4 ) or (¬l1 = l4 &l2 = ¬l3 )or (¬l1 = l4 &l2 = l4 ) iff (l1 = l3 &l2 = ¬l3 )or (l1 = l3 &l2 = l4 ) or (¬l1 = l4 &l2 = ¬l3 )or (¬l1 = l4 &l2 = l4 ) ⎧ l  ¬l3 ⇒ l3  l4 ⎪ ⎪ ⎨ 3 l3  l4 ⇒ l3  l4 iff l1  l2 ⇒ l3  l4 = ⎪ ¬l4  ¬l3 ⇒ l3  l4 ⎪ ⎩ ¬l4  l4 ⇒ l3  l4

Proposition 8.5.3 |= l1  l2 , l3  l4 ⇒ l1  l2 , l3  l4 iff

 

{¬l1 , ¬l3 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅&{¬l1 , l4 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅ &{l2 , ¬l3 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅&{l2 , l4 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅. Proof |= l1  l2 , l3  l4 ⇒ l1  l2 , l3  l4 iff |= ¬l1 , ¬l3 ⇒ l1  l2 , l3  l4 & |= ¬l1 , l4 ⇒ l1  l2 , l3  l4 & |= l2 , ¬l3 ⇒ l1  l2 , l3  l4 & |= l2 , l4 ⇒ l1  l2 , l3  l4 , iff

{¬l1 , ¬l3 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅&{¬l1 , l4 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅ &{l2 , ¬l3 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅&{l2 , l4 } ∩ {¬l1 , l2 , ¬l3 l4 } = ∅.  

Proposition 8.5.4 Let ,  be sets of literal statements such that  = {l11  l12 , . . . , ln1  ln2 ; l11  l12 , . . . , ln 1  ln 2 }  = {l11  l12 , . . . , lm1  lm2 ; l11  l12 , . . . , lm 1  lm 2 }.

Then, |=  ⇒  if and only if for any f : {1, . . . , n}  {1, 2} and g : {1, . . . , m }  {1, 2} either (i) El(l, ¬l ∈ σ f ()), or (ii) El(l, ¬l ∈ τg ()), or (iii) σ f () ∩ τg () = ∅, where , ¬l12 , . . . , ln 1 , ¬ln 2 } σ f () = {¬ f (1)l1 f (1) , . . . , ¬ f (n)ln f (n) ; l11 g(1) τg () = {l11 , ¬l12 , . . . , lm1 , ¬lm2 ; ¬ l1g(1) , . . . , ¬g(m )lm g(m ) },

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Proof Because  ⇒   l12 , . . . , ln 1  ln 2 iff l11  l12 , . . . , ln1  ln2 ; l11 ⇒ l11  l12 , . . . , lm1  lm2 ; l11  l12 , . . . , lm 1  lm 2 iff ¬l11 ∨ l12 , . . . , ¬ln1 ∨ ln2 ; l11 ∧ ¬l12 , . . . , ln 1 ∧ ¬ln 2 ⇒ ¬l11 ∨ l12 , . . . , ¬lm1 ∨ lm2 ; l11 ∧ ¬l12 , . . . , lm 1 ∧ ¬lm 2 iff ¬l11 ∨ l12 , . . . , ¬ln1 ∨ ln2 ; l11 , ¬l12 , . . . , ln 1 , ¬ln 2 ⇒ ¬l11 , l12 , . . . , ¬lm1 , lm2 ; l11 ∧ ¬l12 , . . . , lm 1 ∧ ¬lm 2

iff A f : {1, . . . , n}  {1, 2}Ag : {1, . . . , m }  {1, 2}( ¬ f (1)l1 f (1) , . . . , ¬ f (n)ln f (n) ; l11 , ¬l12 , . . . , ln 1 , ¬ln 2

⇒ l11 , ¬l12 , . . . , lm1 , ¬lm2 ; ¬g(1)l1g(1) , . . . , ¬g(m )lm g(m ) )

iff A f : {1, . . . , n}  {1, 2}Ag : {1, . . . , m }  {1, 2} (incon(σ f ()) or incon(τg ()) or  ∩  = ∅).

Assume that  ⇒  is valid. For any assignment v, there are functions f : {1, . . . , n}  {1, 2} and g : {1, . . . , m }  {1, 2} such that , ¬l12 , . . . , ln 1 , ¬ln 2 v |= ¬ f (1)l1 f (1) , . . . , ¬ f (n)ln f (n) ; l11

implies



, ¬l12 , . . . , lm1 , ¬lm2 ; ¬g(1)l1g(1) , . . . , ¬g(m )lm g(m ) . v |= l11

That is, either (i) El(l, ¬l ∈ σ f ()), or (ii) El(l, ¬l ∈ τg ()), or (iii) σ f () ∩ τg () = ∅, Conversely, assume that there are functions f : {1, . . . , n}  {1, 2} and g : {1, . . . , m }  {1, 2} such that (i’) ¬El(l, ¬l ∈ σ f (), and (ii’) ¬El(l, ¬l ∈ τg ()), and (iii’) σ f () ∩ τg () = ∅. Define an assignment v such that for any variable p, v( p) = 1 iff p ∈ σ f () or ¬ p ∈ τg ().  Then, v is well-defined and v |= σ f (), v |= τg (), which imply v |=  ⇒ .  Proposition 8.5.5 Let ,  be sets of literal statements. Then, |=t  ⇒  if and only if either there is a statement l1  l2 ∈  such that  ⇒   l1  l2 , or there is a statement m 1  m 2 ∈  such that  ⇒   m 1  m 2 . Proof The proposition follows from Propositions 8.1.1 and 8.1.2.

 

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t 8.5.2 Gentzen Deduction System G t Gentzen deduction system G consists of the following axioms and deduction rules:

• Axiom:



(At )

El1  l2 ∈ ( ⇒   l1  l2 ) Em 1  m 2 ∈ ( ⇒   m 1  m 2 ) , ⇒

where ,  are sets of literal statements, and  ⇒   l1  l2 if either   l1  l2 , or   l1  l2 , or ¬ ∪   l1  l2 , or  ∪ ¬  l1  l2 , and ¬δ =∼ δ. • Deduction rules:  ⇒ A  B,  , A  B ⇒  (¬ R ) , A  B ⇒  ⇒ A  B,  , A1  B ⇒   ⇒ C1  D,  (∧ L L ) , A2  B ⇒  (∧ R L )  ⇒ C2  D,  ,   A1 ∧ A2  B ⇒   ⇒ C1 ∧ C2  D,  , A  B1 ⇒   ⇒ C  D1 ,  (∧ L R ) , A  B2 ⇒  (∧ R R )  ⇒ C  D2 ,  , A  B1 ∧ B2 ,   ⇒ C  D1 ∧ D2 ,  (¬ L )

and



 , A1  B ⇒   ⇒ C1  D,  (+∨ L L ) , A1  B ⇒  (+∨ R L )  ⇒ C2  D,  ,   A1 ∨ A2  B ⇒   ⇒ C1 ∨ C2  D,  , A  B1 ⇒   ⇒ C  D1 ,  (+∨ L R ) , A  B2 ⇒  (+∨ R R )  ⇒ C  D2 ,  , A  B1 ∨ B2 ⇒   ⇒ C  D1 ∨ D2 , 

t Definition 8.5.6 A sequent  ⇒  is provable in G , denoted by t  ⇒ , if there is a sequence {1 ⇒ 1 , . . . , n ⇒ n } of sequents such that n ⇒ n =  ⇒ , and for each 1 ≤ i ≤ n, i ⇒ i is an axiom or deduced from the previous t . sequents by one of the deduction rules in G

Theorem 8.5.7 (Soundness theorem) For any sequent  ⇒ , t  ⇒  implies |=t  ⇒ . Proof We prove that each axiom is valid and each deduction rule preserves the validity. (At ) Assume that  and  satisfy the condition in axiom (At ), by Proposition 8.5.5, for any assignment v, v |=  ⇒ . (+∧ L L ) Assume that for any assignment v,

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v(, A1  B) = 1 implies v() = 1; v(, A2  B) = 1 implies v() = 1. For any assignment v, assume that v(, A1 ∧ A2  B) = 1. Then, v(A1 ∧ A2 ) = 1 implies v(B) = 1. If v(A1 ) = 0 or v(A2 ) = 0 then v(A1  B) = 1 or v(A2  B) = 1, and by assumption, v() = 1; if v(A1 ) = 1 and v(A2 ) = 1 then v(B) = 1, i.e., v(A1  B) = 1, v(A2  B) = 1, and by assumption, v() = 1. (+∧1R L ) Assume that for any assignment v, v() = 1 implies v(C1  D, ) = 1. For any assignment v, assume that v() = 1. If v() = 1 then v(C1 ∧ C2  D, ) = 1; otherwise, by assumption, v(C1  D) = 1, and if v(C1 ∧ C2 ) = 0 then v(C1 ∧ C2  D) = 1, and so v(C1 ∧ C2  D, ) = 1; if v(C1 ∧ C2 ) = 1 then v(C1 ) = 1 and by assumption of v(C1  D) = 1, v(D) = 1, i.e., v(C1 ∧ C2  D) = 1, and v(C1 ∧ C2  D, ) = 1. (+∧ L R ) Assume that for any assignment v, either v(, A  B1 ) = 1 implies v() = 1, or v(, A  B2 ) = 1 implies v() = 1, say the former holds. For any assignment v, assume that v(, A  B1 ∧ B2 ) = 1. Then, v(A  B1 ∧ B2 ) = 1. If v(A) = 0 then v(A  B1 ) = 1, and by the assumption, v() = 1; if v(A) = 1 then v(B1 ∧ B2 ) = 1, v(B1 ) = 1, and by the assumption, v() = 1. (+∧ R R ) Assume that for any assignment v, v() = 1 implies v(C  D1 , ) = 1; v() = 1 implies v(C  D2 , ) = 1. For any assignment v, assume that v() = 1. Then, v(C  D1 , ) = 1 and v(C  D2 , ) = 1. If v() = 1 then v(C  D1 ∧ D2 , ) = 1; if v() = 0 then v(C  D1 ) = 1 and v(C  D2 ) = 1. If v(C) = 0 then v(C  D1 ∧ D2 ) = 1, and so v(C  D1 ∧ D2 , ) = 1; otherwise, v(D1 ) = 1, v(D2 ) = 1, i.e., v(C  D1 ∧ D2 ) = 1, and so v(C  D1 ∧ D2 , ) = 1. Similar for other cases.   Theorem 8.5.8 (Completeness theorem) For any sequent  ⇒ , |=t  ⇒  implies t  ⇒ .

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Proof Given a sequent  ⇒ , we construct a tree T as follows: • the root of T is  ⇒ ; • if  ⇒  is a node such that  ,  are sets of literal statements then the node is a leaf; • if  ⇒  is a node of T which is not a leaf then  ⇒  has the direct children nodes containing the following sequents: ⎧ 1 ⇒ A  B, 1 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ 1 , A  B ⇒ 1 ⎪ ⎪ 1 , A1  B ⇒ 1 ⎪ ⎪ ⎪ ⎪  ⎪ 1 , A2  B ⇒ 1 ⎪ ⎨ 1 ⇒ C1  D, 1 ⎪ ⎪  1 ⇒ C2  D, 1 ⎪ ⎪ 1 , A  B1 ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  1 , A  B2 ⇒ 1 ⎪ ⎪ 1 ⇒ C  D1 , 1 ⎪ ⎪ ⎩ 1 ⇒ C  D2 , 1 and

⎧ 1 , A1  B ⇒ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  1 , A2  B ⇒ 1 ⎪ ⎪ 1 ⇒ C1  D,  ⎪ ⎪ ⎨  1 ⇒ C2  D, 1 1 , A  B1 ⇒  ⎪ ⎪ ⎪ ⎪  ⎪ ⎪  1 , A  B2 ⇒ 1 ⎪ ⎪ 1 ⇒ C  D1 , 1 ⎪ ⎪ ⎩ 1 ⇒ C  D2 , 1

if  ⇒  = 1 , A  B ⇒ 1 if  ⇒  = 1 ⇒ A  B, 1 if  ⇒  = 1 , A1 ∧ A2  B ⇒ 1 if  ⇒  = 1 ⇒ C1 ∧ C2  D, 1 if  ⇒  = 1 , A  B1 ∧ B2 ⇒ 1 if  ⇒  = 1 ⇒ C  D1 ∧ D2 , 1

if  ⇒  = 1 , A1 ∨ A2  B ⇒ 1 if  ⇒  = 1 ⇒ C1 ∨ C2  D, 1 if  ⇒  = 1 , A  B1 ∨ B2 ⇒ 1 if  ⇒  = 1 ⇒ C  D1 ∨ D2 , 1

Lemma 8.5.9 If for each branch α ⊆ T, the sequent at the leaf of α is an axiom in t t then T is a proof tree of  ⇒  in G . G Lemma 8.5.10 If there is a branch α ⊆ T such that the leaf of α is not an axiom in t G then there is an assignment v such that v |=  ⇒  for each  ⇒  ∈ α. Proof Let  ⇒  be the leaf of α. By Proposition 8.5.5, there is an assignment v such that v |=  and v |=  . Fix any γ ∈ α, and assume that v |= γ . Case (+∧ L L ). If γ is generated from β ∈ α by (∧ L L ) then there are formulas A1 , A2 , B and β ∈ α such that γ =  , Ai  B ⇒  , β =  , A1 ∧ A2  B ⇒  ,

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245

where i ∈ {1, 2}. By induction assumption, v |=  , A1 ∧ A2  B. Then

v |=  , Ai  B,

and by induction assumption, v |=  . Case (+∧ R L ). If γ is generated from β ∈ α by (∧ R ) then there are formulas C, D1 , D2 such that γ =  ⇒ C  D1 , C  D2 ,  , β =  ⇒ C  D1 ∧ D2 ,  . By induction assumption, v |=  , and v |= C  D1 , C  D2 ,  . Therefore,

v |= C  D1 ∧ D2 ,  .

Similar for other cases.

 

8.6 R-Calculi Given a co-sequent  →  of statements and a sequent (γ , δ) of statements, , γ → , δ is a ⊆-minimal change of  →  by (γ , δ), denoted by |=t  → |(γ , δ) ⇒ , γ → , δ , if (γ , δ ) is minimal such that (i) (γ , δ ) ⊆ (γ , δ) is consistent with  → , and (ii) for any sequent (γ , δ ) ∈ {(γ , δ)} − {(γ , δ )}, , γ → , δ is valid. In this section, we will give an R-calculus Rt , Qt , Pt such that for any sequent  →  and a pair (γ , δ) of statements,  → |(γ , δ) ⇒ , γ → , δ is provable in Rt /Qt , /Pt if and only if , γ → , δ is a R/Q/P-minimal change of  →  by (γ , δ).

8.6.1 R-Calculus Rt R-calculus Rt consists of the following axioms and deduction rules:

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• Axioms:  →   l  l  → |(l  l , δ) ⇒ , l  l → |δ  →   l  l (+AtL0 )  → |(l  l , δ) ⇒  → |δ  →   l  l (−AtL+ )  → |(l  l , δ) ⇒ , l  l → |δ  →   l  l (−AtL0 )  → |(l  l , δ) ⇒  → |δ  →   m  m (+AtR+ )  → |m  m ⇒  → , m  m  →   m  m (+AtR0 )  → |m  m ⇒  →   →   m  m (−AtR+ )  → |m  m ⇒  → , m  m  →   m  m (−AtR0 )  → |m  m ⇒  → 

(+AtL+ )

• Deduction rules:  → |(, A  B; C  D) ⇒  → , A  B|C  D  → |(A  B, C  D) ⇒ , A  B → |C  D  → |(, A  B; C  D) ⇒  → |C  D (¬0L )  → |(A  B, C  D) ⇒  → |C  D  → |(C  D, λ) ⇒ , C  D →  R (¬ +)  → |C  D ⇒  → , C  D  → |(C  D, λ) ⇒  →  (¬0R )  → |C  D ⇒  → 

(¬ L +)

and



 → |(A1  B, C  D) ⇒ , A1  B → |C  D , A1  B → |(A1  B, C  D) ⇒ , A1  B, A2  B → |C  D  → |(A1 ∧ A2  B, C  D) ⇒ , A1 ∧ A2  B → |C  D  → |(A1  B, C  D) ⇒  → |C  D (∧0L L ) , A1  B → |(A2  B, C  D) ⇒ , A1  B → |C  D  → |(A1 ∧ A2  B, C  D) ⇒  → |C  D   → |(A1  B, C  D) ⇒ , A1  B → |C  D LL)  → |(A2  B, C  D) ⇒ , A2  B → |C  D (∨+  → |(A1 ∨ A2  B, C  D) ⇒ , A1 ∨ A2  B → |C  D  → |(A1  B, C  D) ⇒  → |C  D (∨0L L )  → |(A2  B, C  D) ⇒  → |C  D  → |(A1 ∨ A2  B, C  D) ⇒  → |C  D

LL) (∧+

8.6 R-Calculi

and

247



 → |(A  B1 , C  D) ⇒ , A  B1 → |C  D  → |(A  B2 , C  D) ⇒ , A  B2 → |C  D  → |(A  B1 ∧ B2 , C  D) ⇒ , A  B1 ∧ B2 → |C  D  → |(A  B1 , C  D) ⇒  → |C  D (∧0L R )  → |(A  B2 , C  D) ⇒  → |C  D  → |(A  B1 ∧ B2 , C  D) ⇒  → |C  D   → |(A  B1 , C  D) ⇒ , A  B1 → |C  D LR) , A  B1 → |(A  B2 , C  D) ⇒ , A  B1 , A  B2 → |C  D (∨+  → |(A  B 1 ∨ B2 , C  D) ⇒ , A  B1 ∨ B2  → |C  D  → |(A  B1 , C  D) ⇒  → |C  D (∨0L R ) , A  B1 → |(A  B2 , C  D) ⇒ , A  B1 → |C  D  → |(A  B1 ∨ B2 , C  D) ⇒  → |C  D

LR) (∧+

and 

 → |C1  D ⇒  → , C1  D  → |C2  D ⇒  → , C2  D  → |C1 ∧C2  D ⇒  → , C1 ∧ C2  D  → |C1  D ⇒  →  RL (∧0 )  → |C2  D ⇒  →   → |C1 ∧ C2  D ⇒  →    → |C1  D ⇒  → , C1  D RL (∨+ )  → , C1  D|C2  D ⇒  → , C1  D, C2  D  → |C1 ∨C2  D ⇒  → , C1 ∨ C2  D  → |C1  D ⇒  →  RL (∨0 )  → , C1  D|C2  D ⇒  → , C1  D  → |C1 ∨ C2  D ⇒  → , C1 ∨ C2  D

(∧+R L )

and RR) (∧+

RR) (∨+

   → |C  D1 ⇒   → , C  D1   → , C  D1 |C  D2 ⇒   → , C  D1 , C  D2   → |C  D D1 ∧ D2 1 ∧ D2 ⇒   → , C    → |C  D1 ⇒   →  (∧0R R )   → , C  D1 |C  D2 ⇒   → , C  D1 , C  D2   → |C  D1 ∧ D2 ⇒   →     → |C  D1 ⇒   → , C  D1   → |C  D2 ⇒   → , C  D2   → |C  D D2 ⇒   → , C  D1 ∨ D2 1 ∨   → |C  D1 ⇒   →  R R (∨0 )   → |C  D2 ⇒   →    → |C  D1 ∨ D2 ⇒   → 

Definition 8.6.1 A reduction θ =  → |(γ , δ) ⇒ , γ → , θ is provable in Rt , denoted by t θ, if there is a sequence {θ1 , . . . , θm } of reductions such that θm =

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θ, and for each i < m, θi+1 is an axiom or is deduced from the previous reductions by a deduction rule in Rt . Theorem 8.6.2 (Soundness and completeness theorem) For any sequent  →  and a statement pair (γ , δ), t δ iff |= δ, where θ =  → |(γ , δ) ⇒ , γ → , δ .

 

8.6.2 R-Calculus Qt Given a sequent  →  of statements and a sequent (γ , δ) of statements, , γ → , δ is a -minimal change of  →  by (γ , δ), denoted by |=t  → |(γ , δ) ⇒ , γ → , δ , if , γ → , δ is minimal such that (i) (γ , δ )  (γ , δ) is consistent with  → , and (ii) for any (γ , δ ) with (γ , δ ) ≺ (γ , δ )  (γ , δ), , γ → , δ is invalid. R-calculus Qt consists of the following axioms and deduction rules: • Axioms: same as in Rt . • Deduction rules:  → |(, A  B; C  D) ⇒  → , A  B|C  D  → |(A  B, C  D) ⇒ , A  B → |C  D  → |(, A  B; C  D) ⇒  → |C  D (¬0L )  → |(A  B, C  D) ⇒  → |C  D  → |(C  D, λ) ⇒ , C  D →  (¬ R +)  → |C  D ⇒  → , C  D  → |(C  D, λ) ⇒  →  (¬0R )  → |C  D ⇒  → 

(¬ L +)

and



 → |(A1  B, C  D) ⇒ , A 1  B → |C  D , A 1  B → |(A2  B, C  D) ⇒ , A 1  B, A 2  B → |C  D  → |(A1 ∧ A2  B, C  D) ⇒ , A 1 ∧ A 2  B → |C  D ⎧   → |(A1  B, C  D) ⇒ , A 1  B → |C  D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ A1  = λ ⎪ ⎪ ⎨  → |(A1  B, C  D) ⇒  → |C  D ⎣  → |(A1  B, C  D) ⇒ , A  B → |C  D 2 (∨ L L ) ⎪ ⎪ ⎪ ⎪  A2  = λ ⎪ ⎪ ⎪  → |(A1  B, C  D) ⇒  → |C  D ⎪ ⎩  → |(A2  B, C  D) ⇒  → |C  D  → |(A1 ∨ A2  B, C  D) ⇒ , A 1 ∨ A 2  B → |C  D

(∧ L L )

8.6 R-Calculi

⎧ ⎨ A1 ∨ A2 where A 1 ∨ A 2 = A1 ∨ A 2 ⎩ λ

249

if A 1 = λ if A 1 = λ = A 2 and otherwise,

⎧  → |(A  B1 , C  D) ⇒ , A  B1 → |C  D ⎪ ⎪ ⎪ ⎪ B = λ ⎪ ⎪ ⎪⎡ 1 ⎪ ⎨  → |(A  B1 , C  D) ⇒  → |C  D ⎣  → |(A  B2 , C  D) ⇒ , A  B → |C  D L R 2 (∧ ) ⎪ ⎪ ⎪ ⎪  B2 = λ ⎪ ⎪ ⎪  → |(A  B1 , C  D) ⇒  → |C  D ⎪ ⎩  → |(A  B2 , C  D) ⇒  → |C  D  → |(A  B1 ∧ B2 , C  D) ⇒ , A  B1 ∧ B2 → |C  D  → |(A  B1 , C  D) ⇒ , A  B1 → |C  D L R (∨ ) , A  B1 → |(A  B2 , C  D) ⇒ , A  B1 , A  B2 → |C  D  → |(A  B1 ∨ B2 , C  D) ⇒ , A  B1 ∨ B2 → |C  D

⎧ ⎨ B1 ∨ B2 if B1 = λ where B1 ∨ B2 = B1 ∨ B2 if B1 = λ = B2 and ⎩ λ otherwise,

⎧  → |C1  D ⇒  → , C1  D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡C = λ ⎪ ⎪ ⎨  → |C1  D ⇒  →  ⎣  → |C2  D ⇒  → , C2  D (∧ R L ) ⎪ ⎪ ⎪ ⎪  C 2 = λ ⎪ ⎪  → |C1  D ⇒  →  ⎪ ⎪ ⎩  → |C2  D ⇒  →   → |C1 ∧ C2  D ⇒  → , C1 ∧ C2  D  → |C1  D ⇒  → , C1  D R L   → , C  D|C  D ⇒   → , C  D, C  D (∨ ) 2 1 1 2  → |C1 ∨ C2  D ⇒  → , C1 ∨ C2  D ⎧ ⎨ C1 ∨ C2 if C1 = λ where C1 ∨ C2 = C1 ∨ C2 if C1 = λ = C2 and ⎩ λ otherwise,  → |C  D1 ⇒  → , C  D1  → , C  D1 |C  D2 ⇒  → , C  D1 , C  D2  → |C  D1 ∧ D2 ⇒  → , C  D1 ∧ D2 ⎧ ⎪  → |C  D1 ⇒  → , C  D1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ D1 = λ ⎪ ⎪ ⎨  → |C  D1 ⇒  →  ⎣  → |C  D2 ⇒  → , C  D 2 (∨ R R ) ⎪ ⎪ ⎪ ⎪  D 1 = λ ⎪ ⎪  → |C  D1 ⇒  →  ⎪ ⎪ ⎩  → |C  D2 ⇒  →   → |C  D1 ∨ D2 ⇒  → , C  D1 ∨ D2 (∧ R R )

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⎧ ⎨ D1 ∨ D2 if D1 = λ where D1 ∨ D2 = D1 ∨ D2 if D1 = λ = D2 ⎩ λ otherwise. Definition 8.6.3 A reduction θ =  → |(γ , δ) ⇒ , γ → , θ is provable in Qt , denoted by t θ, if there is a sequence {θ1 , . . . , θm } of reductions such that θm = θ, and for each i < m, θi+1 is an axiom or is deduced from the previous reductions by a deduction rule in Qt . Theorem 8.6.4 (Soundness and completeness theorem) For any sequent  →  and a statement pair (γ , δ), t θ iff |=t θ, where θ =  → |(γ , δ) ⇒ , γ → , δ .

 

8.6.3 R-Calculus Pt Given a sequent  →  of statements and a sequent (γ , δ) of statements, , γ → , δ is a  -minimal change of  →  by (γ , δ), denoted by |=t  → |(γ , δ) ⇒ , γ → , δ , if (γ , δ ) is minimal such that (i) (γ , δ )  (γ , δ) is consistent with  → , and (ii) for any pair (γ , δ ) with (γ , δ ) ≺ (γ , δ )  (γ , δ), either , γ → , δ  , γ → , δ and

, γ → , δ  , γ → , δ ,

or , γ → , δ is consistent. R-calculus Pt consists of the following axioms and deduction rules: • Axioms: same as in Rt . • Deduction rules:  → |(, A  B; C  D) ⇒  → , A  B|C  D  → |(A  B, C  D) ⇒ , A  B → |C  D  → |(, A  B; C  D) ⇒  → |C  D (¬0L )  → |(A  B, C  D) ⇒  → |C  D  → |(C  D, λ) ⇒ , C  D →  (¬ R +)  → |(λ, C  D) ⇒  → , C  D  → |(C  D, λ) ⇒  →  (¬0R )  → |(λ, C  D) ⇒  → 

(¬ L +)

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where  → |(, A  B; C  D) = ( → |(λ, A  B))|(λ, C  D), and   → |(A1  B, C  D) ⇒ , A 1  B  → |C  D L L (∧ ) , A 1  B  → |(A2  B, C  D) ⇒ , A 1  B, A 2  B  → |C  D   → |(A1 ∧ A2  B, C  D) ⇒ , A 1 ∧ A 2  B  → |C  D   → |(A1  B, C  D) ⇒ , A 1  B  → |C  D L L (∨ )   → |(A2  B, C  D) ⇒ , A 2  B  → |C  D   → |(A1 ∨ A2  B, C  D) ⇒ , A 1 ∨ A 2  B  → |C  D   → |(A  B1 , C  D) ⇒ , A  B1  → |C  D (∧ L R )   → |(A  B2 , C  D) ⇒ , A  B2  → |C  D   → |(A  B1 ∧ B2 , C  D) ⇒ , A  B1 ∧ B2  → |C  D   → |(A  B1 , C  D) ⇒ , A  B1  → |C  D L R (∨ ) , A  B1  → |(A  B2 , C  D) ⇒ , A  B1 , A  B2  → |C  D   → |(A  B1 ∨ B2 , C  D) ⇒ , A  B1 ∨ B2  → |C  D

and (∧

RL

)

(∨ R L ) (∧ R R ) (∨ R R )

 → |C1  D ⇒  → , C1  D  → |C2  D ⇒  → , C2  D  → |C1 ∧ C2  D ⇒  → , C1 ∧ C2  D  → |C1  D ⇒  → , C1  D  → , C1  D|C2  D ⇒  → , C1  D, C2  D  → |C1 ∨ C2  D ⇒  → , C1 ∨ C2  D  → |C  D1 ⇒  → , C  D1  → , C  D1 |C  D2 ⇒  → , C  D1 , C  D2  → |C  D1 ∧ D2 ⇒  → , C  D1 ∧ D2  → |C  D1 ⇒  → , C  D1  → |C  D2 ⇒  → , C  D2  → |C  D1 ∨ D2 ⇒  → , C  D1 ∨ D2

Definition 8.6.5 A reduction θ =  → |(γ , δ) ⇒ , γ → , θ is provable in Pt , denoted by t θ, if there is a sequence {θ1 , . . . , θm } of reductions such that θm = θ, and for each i < m, θi+1 is an axiom or is deduced from the previous reductions by a deduction rule in Pt . Theorem 8.6.6 (Soundness and completeness theorem) For any sequent  →  and a statement pair (γ , δ), where γ is in disjunctive normal form and δ is in conjunctive normal form, t θ iff |=t θ, where θ =  → |(γ , δ) ⇒ , γ → , δ .

 

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8.7 Conclusions We considered the R-calculi based on three minimal changes and have the following R-calculi for theories:  Tableau proof system Tt , Tf , T t , Tf t f R-calculus R , R  minimal change Rt , Q t , Pt

and for Gentzen deduction system t Gentzen deduction system G minimal change Rt , Qt , Pt .

References Bolc, L., Borowik, P.: Many-Valued Logics (2 Automated Reasoning and Practical Applications). Springer, Berlin (2003) Li, W., Sui, Y.: A sound and complete R-calculi with respect to contraction and minimal change. Frontiers Comput. Sci. 8, 184–191 (2014) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision, International Journal of Intelligence. Science 3, 110–117 (2013) Li, W., Sui, Y., Sun, M.: The sound and complete R-calculus for revising propositional theories. Sci. China: Inf. Sci. 58, 092101:1-092101:12 (2015)