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
3 - Review of Classical First-Order Logic
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
The language of classical first-order logic
First-order logic (sometimes called predicate logic) includes all of the connectives of propositional logic. Unlike propositional logic, however, first-order logic analyzes simple sentences into terms and predicates. We use uppercase roman letters as predicates, lowercase roman letters a through t as (individual) constants, and lowercase roman letters u through z as (individual) variables. Predicates, constants, and variables may be augmented with subscripts if necessary, thus guaranteeing an infinite supply of each.
Constants function like names in English, and variables function like pronouns. Together constants and variables count as terms. Predicates have arities, where an arity is the number of terms to which a predicate applies. In English, for example, the arity of the predicate runs in John runs is 1 – it combines with a single term, John in this case – while the arity of the predicate loves in John loves Sue is 2 – it combines with two terms. Atomic formulas are formed by writing predicates in initial position followed by an appropriate number of terms (determined by the predicate's arity). John runs and John loves Sue might thus be symbolized as Rj and Ljs.
There are two standard quantifiers in first-order logic, the universal and the existential quantifiers. We'll use ∀ as the universal quantifier symbol and ∃ as the existential quantifier symbol.
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- An Introduction to Many-Valued and Fuzzy LogicSemantics, Algebras, and Derivation Systems, pp. 39 - 56Publisher: Cambridge University PressPrint publication year: 2008