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Many-valued Logics : Many-Valued Logics 1 Theoretical Foundations Volume 1 - Leonard Bolc

Many-valued Logics

Many-Valued Logics 1 Theoretical Foundations Volume 1

Hardcover Published: 3rd December 1992
ISBN: 9783540559269
Number Of Pages: 288

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Many-valued logics were developed as an attempt to handlephilosophical doubts about the "law of excluded middle" inclassical logic. The first many-valued formal systems weredeveloped by J. Lukasiewicz in Poland and E.Post in theU.S.A. in the 1920s, and since then the field has expandeddramatically as the applicability of the systems to otherphilosophical and semantic problems was recognized.Intuitionisticlogic, for example, arose from deep problemsin the foundations of mathematics. Fuzzy logics,approximation logics, and probability logics all addressquestions that classical logic alone cannot answer. Allthese interpretations of many-valued calculi motivatespecific formal systems thatallow detailed mathematicaltreatment.In this volume, the authors are concerned with finite-valuedlogics, and especially with three-valued logical calculi.Matrix constructions, axiomatizations of propositional andpredicate calculi, syntax, semantic structures, andmethodology are discussed. Separate chapters deal withintuitionistic logic, fuzzy logics, approximation logics,and probability logics. These systems all find applicationin practice, in automatic inference processes, which havebeen decisive for the intensive development of these logics.This volume acquaints the reader with theoreticalfundamentals of many-valued logics. It is intended to be thefirst of a two-volume work. The second volume will deal withpractical applications and methods of automated reasoningusing many-valued logics.

Preliminariesp. 1
Set Operationsp. 1
Relationsp. 2
Partial Functions and Functionsp. 4
Indexed Families of Sets and Generalized Set Operationsp. 5
Natural Numbers, Countable Setsp. 5
Equivalence Relations, Congruencesp. 6
Orderingsp. 7
Treesp. 9
Inductive Definitionsp. 10
Abstract Algebrasp. 12
Logical Matricesp. 20
Many-Valued Propositional Calculip. 23
Remarks on Historyp. 23
The Definition of a Propositional Calculusp. 25
Many-Valued Calculi of Lukasiewiczp. 27
Finitely Valued Calculi of Lukasiewiczp. 30
The Formalized Language of Propositional Calculip. 30
Algebraic Characterization of the n-valued Calculi of Lukasiewiczp. 32
Latticesp. 32
Quasi-Boolean Algebras and Heyting Algebrap. 33
Proper Lukasiewicz Algebrasp. 37
The Lukasiewicz Implicationp. 39
Stone Filters in Proper n-valued Lukasiewicz Algebrasp. 41
The Axiom System for the n-valued Propositional Calculus of Lukasiewiczp. 42
Many-Valued Calculi of Postp. 46
Bibliographical Remarksp. 46
Post Algebrasp. 46
Post Algebra Filtersp. 49
The Axiom System for the n-valued Post Calculusp. 51
Many-Valued Post Calculi with Several Designated Truth Valuesp. 54
Definability of Functors in the n-valued Post Logicp. 57
Survey of Three-Valued Propositional Calculip. 63
The Three-Valued Calculus of Lukasiewicz (L[subscript 3])p. 63
The Three-Valued Calculus of Bochvarp. 65
The Three-Valued Calculus of Finnp. 66
The Three-Valued Calculus of Halldenp. 68
The Three-Va]ued Calculus of Aqvistp. 69
The Three-Valued Calculi of Segerbergp. 70
The Three-Valued Calculus of Pirog-Rzepeckap. 71
The Three-Valued Calculus of Heytingp. 73
The Three-Valued Calculus of Kleenep. 74
The Three-Valued Calculus of Reichenbachp. 75
The Three-Valued Calculus of Slupeckip. 76
The Three-Valued Calculus of Sobocinskip. 77
Some n-valued Propositional Calculi: A Selectionp. 79
The Many-Valued Calculus of Slupeckip. 79
The Many-Valued Calculus of Sobocinskip. 82
The Many-Valued Calculi of Godelp. 84
The Many-Valued Calculus Cnrp. 85
Intuitionistic Propositional Calculusp. 95
The Intuitionistic Propositional Logic in an Axiomatic Settingp. 95
The Natural-Deduction Method for the Intuitionistic Propositional Logicp. 98
Characterization of the Intuitionistic Propositional Logic in Terms of the Consequence Operator Cn[subscript I]p. 100
Algebraic Characterization of the Intuitionistic Propositional Logicp. 101
Kripke's Semantics for the Intuitionistic Propositional Calculusp. 102
First-Order Predicate Calculus for Many-Valued Logicsp. 105
The Language of the First-Order Predicate Calculusp. 105
Free Variables and Bound Variablesp. 107
The Rule of Substitution for Individual Variablesp. 108
Fundamental Semantic Notionsp. 109
The Many-Valued First-Order Predicate Calculus of Postp. 113
The Method of Finitely Generated Trees in n-valued Logical Calculip. 123
Introductory Remarksp. 123
Finitely Generated Trees for n-valued Propositional Calculip. 123
The Existence of Models for the Propositional Calculusp. 130
Finitely Generated Trees for n-valued First-Order Predicate Calculip. 133
Finitely Generated Trees for n-valued Quantifiersp. 137
Fuzzy Propositional Calculip. 143
Introductory Remarksp. 143
Fuzzy Setsp. 143
Syntactic Introductionp. 144
Semantic Basis for Fuzzy Propositional Logicsp. 154
Remarks on the Incompleteness of Fuzzy Propositional Calculip. 171
First-Order Predicate Calculus for Fuzzy Logicsp. 192
Introductory Remarksp. 192
Generalized Residual Latticesp. 192
The Language of the Fuzzy First-Order Predicate Calculusp. 195
Semantic Consequence Operationp. 199
Syntax of the Fuzzy First-Order Predicate Calculusp. 202
Syntactic Consequence Operationp. 203
An Axiom System for the Fuzzy First-Order Predicate Calculusp. 204
Fuzzy First-Order Theoriesp. 206
Approximation Logicsp. 209
Introductionp. 209
Rough Setsp. 209
Rough Logics with a Chain of Indistinguishability Relationsp. 212
Basic Conceptsp. 212
Approximate Logical Systemsp. 214
Approximation Theoriesp. 219
Approximation Logics with Partially Ordered Sets of Indiscernibility Relationsp. 221
Plain Semi-Post Algebrasp. 221
Approximation Logic of Type Tp. 225
Approximation Logics of Type T with Many Indiscernibility Relationsp. 228
Probability Logicsp. 231
Introductionp. 231
Lukasiewicz' Idea of Logical Probabilityp. 232
An Algebraic Description of Probability Logicp. 233
Syntaxp. 233
Semanticsp. 234
Constructionsp. 237
Probabilistic Consequencep. 239
Axiomatic Approach to Probability Logicp. 243
Syntaxp. 243
Probability and Probabilistic Consequencep. 245
Completeness of Probability Logicp. 247
Applicationsp. 252
Unreasonable Inferencep. 253
Referencesp. 255
Index of Symbolsp. 285
Author Indexp. 287
Subject Indexp. 289
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540559269
ISBN-10: 3540559264
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 288
Published: 3rd December 1992
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.9
Weight (kg): 1.34