Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Review of Classical Propositional Logic
- 3 Review of Classical First-Order Logic
- 4 Alternative Semantics for Truth-Values and Truth-Functions: Numeric Truth-Values and Abstract Algebras
- 5 Three-Valued Propositional Logics: Semantics
- 6 Derivation Systems for Three-Valued Propositional Logic
- 7 Three-Valued First-Order Logics: Semantics
- 8 Derivation Systems for Three-Valued First-Order Logic
- 9 Alternative Semantics for Three-Valued Logic
- 10 The Principle of Charity Reconsidered and a New Problem of the Fringe
- 11 Fuzzy Propositional Logics: Semantics
- 12 Fuzzy Algebras
- 13 Derivation Systems for fuzzy Propositional Logic
- 14 Fuzzy First-Order Logics: Semantics
- 15 Derivation Systems for Fuzzy First-Order Logic
- 16 Extensions of Fuzziness
- 17 Fuzzy Membership Functions
- Appendix: Basics of Countability and Uncountability
- Bibliography
- Index
5 - Three-Valued Propositional Logics: Semantics
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Review of Classical Propositional Logic
- 3 Review of Classical First-Order Logic
- 4 Alternative Semantics for Truth-Values and Truth-Functions: Numeric Truth-Values and Abstract Algebras
- 5 Three-Valued Propositional Logics: Semantics
- 6 Derivation Systems for Three-Valued Propositional Logic
- 7 Three-Valued First-Order Logics: Semantics
- 8 Derivation Systems for Three-Valued First-Order Logic
- 9 Alternative Semantics for Three-Valued Logic
- 10 The Principle of Charity Reconsidered and a New Problem of the Fringe
- 11 Fuzzy Propositional Logics: Semantics
- 12 Fuzzy Algebras
- 13 Derivation Systems for fuzzy Propositional Logic
- 14 Fuzzy First-Order Logics: Semantics
- 15 Derivation Systems for Fuzzy First-Order Logic
- 16 Extensions of Fuzziness
- 17 Fuzzy Membership Functions
- Appendix: Basics of Countability and Uncountability
- Bibliography
- Index
Summary
Kleene's “strong” three-valued logic
We began Chapter 1 by noting that sentences concerning borderline cases of vague predicates pose counterexamples to the Principle of Bivalence. For example, the sentence Mary Middleford is tall appears to be neither true nor false. We begin our exploration of logics for vagueness by dropping the Principle of Bivalence and allowing sentences to be either true (T), false (F), or neither true nor false (N – if you like, you may also say that N is neutral). This gives rise to three-valued (trivalent) systems of logic. We use the same language as classical propositional logic. Truth-value assignments can now assign N (as well as T or F) to atomic formulas, and we'll use this value to signal the application of a vague predicate to a borderline case.
How are the truth-functions for the standard propositional connectives defined over the three values? There are several plausible choices, and the set of truth-functions we choose will define a specific system of three-valued logic. In this chapter we present four well-known systems of three-valued logic. Many others have been developed, but these four systems are sufficient to explore the flavor of three-valued logics and how they might be used to tackle problems associated with vagueness.
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- An Introduction to Many-Valued and Fuzzy LogicSemantics, Algebras, and Derivation Systems, pp. 71 - 99Publisher: Cambridge University PressPrint publication year: 2008