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Truth and Falsehood: An Inquiry into Generalized Logical Values 2011 [Hardback]

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  • Formāts: Hardback, 250 pages, height x width: 235x155 mm, weight: 559 g, XIV, 250 p., 1 Hardback
  • Sērija : Trends in Logic 36
  • Izdošanas datums: 15-Sep-2011
  • Izdevniecība: Springer
  • ISBN-10: 9400709064
  • ISBN-13: 9789400709065
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Truth and Falsehood: An Inquiry into Generalized Logical Values 2011Palielināt
  • Formāts: Hardback, 250 pages, height x width: 235x155 mm, weight: 559 g, XIV, 250 p., 1 Hardback
  • Sērija : Trends in Logic 36
  • Izdošanas datums: 15-Sep-2011
  • Izdevniecība: Springer
  • ISBN-10: 9400709064
  • ISBN-13: 9789400709065
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9789400709065.html
Keywords:
Here is a thoroughly elaborated logical theory of generalized truth-values, presenting the idea of a trilattice of truth values - a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity.

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of (basic) truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice (a lattice of truth values with two ordering relations) constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, a trilattice of truth values - a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi.

Recenzijas

This book is an exceptional contribution to philosophical logic; no one who thinks about truth values should miss it. Taking Truth and Falsehood as objects in Frege's way, the authors serve up a compelling combination of (1) authoritative, encyclopedic, and philosophically sensitive history, (2) a careful and persuasive presentation of their beautiful and super-useful theory of sixteen (not just algebraic but really logical) truth values structured as a trilattice, and (3) a dazzling array of related conceptually motivated formal developments that bring the reader to the forefront of current research. Prof. Nuel D. Belnap Truth and Falsehood, two values. What could be simpler? We can all count to 2. Professors Shramko and Wansing in this book build on earlier work of themselves and others (including Nuel Belnap's and my "four valued logic") to show that 2 truth values is barely enough to get started. They consider 4 and especially 16 element truth values, and do not even stop there. Paraphrasing George Gamow, "Two Four Sixteen Infinity." This book is thoughtful and bold, philosophical and mathematical, and very well-written. Prof. J. Michael Dunn Could something be both true and false, and neither true nor false? 'That way,' claimed Bob Meyer, 'lies madness'. But if this be madness, yet there is method in't, as Shramko and Wansing show, unearthing a rich and beautiful family of logical structures. Prof. Graham Priest

Papildus informācija

"This book is an exceptional contribution to philosophical logic; no one who thinks about truth values should miss it. Taking Truth and Falsehood as objects in Frege's way, the authors serve up a compelling combination of (1) authoritative, encyclopedic, and philosophically sensitive history, (2) a careful and persuasive presentation of their beautiful and super-useful theory of sixteen (not just algebraic but really logical) truth values structured as a trilattice, and (3) a dazzling array of related conceptually motivated formal developments that bring the reader to the forefront of current research." (Prof. Nuel D. Belnap) "Truth and Falsehood, two values. What could be simpler? We can all count to 2. Professors Shramko and Wansing in this book build on earlier work of themselves and others (including Nuel Belnap's and my "four valued logic") to show that 2 truth values is barely enough to get started. They consider 4 and especially 16 element truth values, and do not even stop there. Paraphrasing George Gamow, "Two Four Sixteen Infinity." This book is thoughtful and bold, philosophical and mathematical, and very well-written." (Prof. J. Michael Dunn) "Could something be both true and false, and neither true nor false? 'That way,' claimed Bob Meyer, 'lies madness'. But if this be madness, yet there is method in't, as Shramko and Wansing show, unearthing a rich and beautiful family of logical structures." (Prof. Graham Priest)
1 Truth Values
1(18)
1.1 The Idea of Truth Values
1(2)
1.2 Truth Values and the Functional Analysis of Language
3(2)
1.3 The Categorial Status of Truth and Falsehood
5(2)
1.4 The Ontological Background of Truth Values
7(3)
1.5 Logic as the Science of Logical Values
10(2)
1.6 Logical Structures
12(3)
1.7 Truth Values, Truth Degrees, and Vague Concepts
15(4)
2 Truth Values and the Slingshot Argument
19(22)
2.1 An Argument in Favor of Truth Values
19(1)
2.2 Reconstructing the Slingshot Arguments
20(5)
2.2.1 Church's Slingshot
20(2)
2.2.2 Godel's Slingshot
22(2)
2.2.3 Davidson's Slingshot
24(1)
2.3 The Slingshot Argument and Non-Fregean Logic
25(4)
2.4 Non-Fregean Logic and Definite Descriptions
29(5)
2.5 Non-Fregean Logic and λ-Expressions
34(3)
2.6 Non-Fregean Logic and Indefinite Descriptions
37(2)
2.7 Concluding Remarks
39(2)
3 Generalized Truth Values: From FOUR2 to SIXTEEN3
41(22)
3.1 Truth Values as Structured Entities
41(3)
3.2 Generalized Valuations, Four-valued Logic and Bilattices
44(4)
3.3 Taking Generalization Seriously: From Isolated Computers to Computer Networks
48(3)
3.4 Generalized Truth Values and Multilattices
51(2)
3.5 The Trilattice of 16 Truth Values
53(7)
3.6 Another Example of a Trilattice: Truth Values in Constructive Logics
60(3)
4 Generalized Truth Values: SIXTEEN3 and Beyond
63(30)
4.1 Entailment Relations on SIXTEEN3
63(2)
4.2 First-Degree Systems for SIXTEEN3
65(8)
4.2.1 The Languages lt, lf and Systems FDEtt, FDEff
65(4)
4.2.2 The Language ltf for ≤t and ≤f
69(4)
4.3 First-Degree Everywhere
73(6)
4.4 Hyper-Contradictions and Generalizations of Priest's Logic
79(6)
4.5 An Approach to a Generalization of Kleene's Logic: A Tetralattice
85(4)
4.6 Uncertainty Versus Lack of Information
89(4)
5 Axiom Systems for Trilattice Logics
93(20)
5.1 Truth Value Lattices and the Implication Connective
93(2)
5.2 From First-Degree Proof Systems to Proof Systems with Modus Ponens
95(1)
5.3 Odintsov's Axiomatization of Truth Entailment and Falsity Entailment in SIXTEEN3
96(14)
5.3.1 First-Degree Calculi
99(7)
5.3.2 Systems with Modus Ponens as the Sole Rule of Inference
106(4)
5.4 Discussion
110(3)
6 Sequent Systems for Trilattice Logics
113(30)
6.1 Standard Sequent Systems for Logics Related to SIXTEEN3
114(8)
6.2 Alternative Sequent Calculi
122(5)
6.3 Extensions
127(5)
6.4 Sequent Calculi for Truth Entailment and Falsity Entailment in SIXTEEN3
132(11)
7 Intuitionistic Trilattice Logics
143(28)
7.1 Introduction
143(2)
7.2 Sequent Calculus I16
145(10)
7.3 Kripke Completeness for I16
155(5)
7.4 Tableau Calculus IT16
160(4)
7.5 Kripke Completeness for IT16
164(7)
8 Generalized Truth Values and Many-Valued Logics: Harmonious Many-Valued Logics
171(18)
8.1 Many-Valued Propositional Logics Generalized
171(4)
8.2 Designated and Antidesignated Values
175(3)
8.3 Some Separated Finitely-Valued Logics
178(3)
8.4 A Harmonious Logic Inspired by the Logic of SIXTEEN3
181(1)
8.5 Harmony ad Infinitum
182(1)
8.6 Some Remarks on Generalizing Harmony
183(6)
9 Generalized Truth Values and Many-Valued Logics: Suszko's Thesis
189(26)
9.1 Introduction
189(3)
9.2 The Suszko Reduction
192(6)
9.2.1 Malinowski's Analysis of Suszko's Thesis
195(2)
9.2.2 Tsuji's Analysis of Suszko's Thesis
197(1)
9.3 Logical n-Valuedness as Inferential Many-Valuedness
198(8)
9.3.1 What is a Logical Value?
198(3)
9.3.2 Another Kind of Counterexample
201(1)
9.3.3 Examples of Natural Bi-consequence Logics
202(1)
9.3.4 Logically n-Valued Logics
203(3)
9.4 Another Analysis and a Resume
206(1)
9.5 A Bilattice of Four Entailment Relations
207(8)
9.5.1 Definitions of Four Entailment Relations
208(3)
9.5.2 Orderings on Entailment Relations
211(2)
9.5.3 Concluding Remarks
213(2)
10 Further Developments
215(24)
10.1 First-Order Trilattice Logics
215(2)
10.2 Modal Trilattice Logics
217(3)
10.3 Adverbially Qualified Truth Values
220(2)
10.4 Generalized Truth Values: Alternate Interpretations
222(5)
References
227(12)
Index 239