Many-Valued Logics. A Mathematical and Computational Introduction. Luis M. Augusto

Transcription

1 Many-Valued Logics A Mathematical and Computational Introduction Luis M. Augusto

2 Individual author and College Publications 2017 All rights reserved. ISBN College Publications Scientific Director: Dov Gabbay Managing Director: Jane Spurr Printed by Lightning Source, Milton Keynes, UK All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise without prior permission, in writing, from the publisher.

3 Preface xi 1. Introduction Logics, classical and non-classical, among which the manyvalued Logic and mathematics Logic and computation I. THINGS LOGICAL 9 2. Logical languages Formal languages and logical languages Propositional and first-order languages The language of classical logic Clausal and normal forms Literals and clauses Negation normal form Prenex normal form Skolem normal form Conjunctive and disjunctive normal forms Signed logic and signed clause logic Signed logic Signed clause logic Substitutions and unification for FOL Exercises Logical systems Logical consequence and inference Semantics and model theory Truth-functionality and truth-functional completeness Semantics and deduction Matrix semantics Syntax and proof theory Inference rules and proof systems v

4 Syntax and deduction Adequateness of a deductive system The system of classical logic Exercises Logical decisions Meeting the decision problem and the SAT The Boolean satisfiability problem, or SAT Refutation proof procedures Some historical notes on automated theorem proving Herbrand semantics Proving validity and satisfiability Truth tables Axiom systems Natural deduction The sequent calculus LK The DPLL procedure Refutation I: Analytic tableaux Analytic tableaux as a propositional calculus Analytic tableaux as a predicate calculus FOL tableaux without unification FOL tableaux with unification Refutation II: Resolution The resolution principle for propositional logic The resolution principle for FOL Completeness of the resolution principle Resolution refinements A-ordering Hyper-resolution and semantic resolution Implementation of resolution in Prover9-Mace Exercises II. MANY-VALUED LOGICS Many-valued logics Some historical notes Many-valuedness and interpretation Suszko s Thesis Non-trivial many-valuedness Classical generalizations to the many-valued logics Structural properties of many-valued logics vi

5 5.4. The Lukasiewicz propositional logics Lukasiewicz s 3-valued propositional logic L Tautologousness, contradictoriness, and entailment in L n-valued generalizations of L Finitely many-valued propositional logics Bochvar s 3-valued system Kleene s 3-valued logics Finn s 3-valued logic Logics of nonsense: the 3-valued logics of Halldén, Åqvist, Segerberg, and Piróg-Rzepecka Heyting s 3-valued logic Reichenbach s 3-valued logic Belnap s 4-valued logic The finitely n-valued logics of Post and Gödel Post logics Gödel logics Fuzzy logics Quantification in many-valued logics Quantification in finitely many-valued logics Quantification in fuzzy logics Exercises III. REFUTATION CALCULI FOR MANY-VALUED LOGICS The signed SAT for many-valued logics From the MV-SAT to the signed SAT From many-valued formulae to signed formulae General notions and definitions Transformation rules for many-valued connectives Transformation rules for many-valued quantifiers Transformation rules and preservation of structure Translation to clausal form Exercises Signed tableaux for the MV-SAT Introductory remarks Signed analytic tableaux for classical formulae Surma s algorithm vii

6 7.4. Signed tableaux for finitely many-valued logics Propositional signed tableaux FO signed tableaux Signed tableaux for infinitely many-valued logics Exercises Signed resolution for the MV-SAT Introductory remarks Signed resolution for finitely many-valued logics Signed resolution proof procedures Main rules Refinements of signed resolution The main theorem of signed resolution Signed resolution for infinitely many-valued logics Exercises IV. APPENDIX Mathematical notions Sets Functions, operations, and relations Algebras and algebraic structures Lattices Graphs and trees Bibliography 305 Index 319 viii