- •Contents
- •Preface
- •1 Introduction
- •1.1 Issues of Vagueness
- •1.3 The Problem of the Fringe
- •1.4 Preview of the Rest of the Book
- •1.5 History and Scope of Fuzzy Logic
- •1.6 Tall People
- •1.7 Exercises
- •2 Review of Classical Propositional Logic
- •2.1 The Language of Classical Propositional Logic
- •2.2 Semantics of Classical Propositional Logic
- •2.3 Normal Forms
- •2.4 An Axiomatic Derivation System for Classical Propositional Logic
- •2.5 Functional Completeness
- •2.6 Decidability
- •2.7 Exercises
- •3.2 Semantics of Classical First-Order Logic
- •3.3 An Axiomatic Derivation System for Classical First-Order Logic
- •3.4 Exercises
- •4.1 Numeric Truth-Values for Classical Logic
- •4.2 Boolean Algebras and Classical Logic
- •4.3 More Results about Boolean Algebras
- •4.4 Exercises
- •5.2 Lukasiewicz’s Three-Valued Logic
- •5.3 Bochvar’s Three-Valued Logics
- •5.5 Normal Forms
- •5.7 Lukasiewicz’s System Expanded
- •5.8 Exercises
- •6.3 Exercises
- •7.3 Tautologies, Validity, and “Quasi-”Semantic Concepts
- •7.4 Exercises
- •8.3 Exercises
- •9.3 MV-Algebras
- •9.4 Exercises
- •11 Fuzzy Propositional Logics: Semantics
- •11.1 Fuzzy Sets and Degrees of Truth
- •11.2 Lukasiewicz Fuzzy Propositional Logic
- •11.3 Tautologies, Contradictions, and Entailment in Fuzzy Logic
- •11.4 N-Tautologies, Degree-Entailment, and N-Degree-Entailment
- •11.5 Fuzzy Consequence
- •11.7 T-Norms, T-Conorms, and Implication in Fuzzy Logic
- •11.9 Product Fuzzy Propositional Logic
- •11.10 Fuzzy External Assertion and Negation
- •11.11 Exercises
- •12 Fuzzy Algebras
- •12.2 Residuated Lattices and BL-Algebras
- •12.3 Zero and Unit Projections in Algebraic Structures
- •12.4 Exercises
- •13 Derivation Systems for Fuzzy Propositional Logic
- •13.1 An Axiomatic System for Tautologies and Validity in Fuzzy
- •13.2 A Pavelka-Style Derivation System for Fuzzy
- •13.3 An Alternative Axiomatic System for Tautologies and Validity in FuzzyL Based on BL-Algebras
- •13.7 External Assertion Axioms
- •13.8 Exercises
- •14.1 Fuzzy Interpretations
- •14.2 Lukasiewicz Fuzzy First-Order Logic
- •14.3 Tautologies and Other Semantic Concepts
- •14.4 Lukasiewicz Fuzzy Logic and the Problems of Vagueness
- •14.6 Product Fuzzy First-Order Logic
- •14.8 Exercises
- •15.3 An Axiomatic Derivation System for Fuzzy
- •15.4 Combining Fuzzy First-Order Logical Systems; External Assertion
- •15.5 Exercises
- •16 Extensions of Fuzziness
- •16.2 Fuzzy “Linguistic” Truth-Values
- •16.3 Other Fuzzy Extensions of Fuzzy Logic
- •16.4 Exercises
- •17 Fuzzy Membership Functions
- •17.2 Empirical Construction of Membership Functions
- •17.3 Logical Relevance?
- •17.4 Exercises
- •Bibliography
- •Index
298 |
Derivation Systems for Fuzzy First-Order Logic |
Note that if a formula P has any value less than 1, ∆P will have the value 0. But we don’t bother to include this as part of the rule EA because we can derive any formula with graded value 0.
We’ll show that if Ta has at least the value .5, in FuzzyL augmented with the external assertion operator then ( x)¬Tx is clearly not clearly true (recall the reading of the external assertion operator as clearly):
1 |
[Ta, .5] |
|
Assumption |
2 |
[1/2 → Ta, 1] |
1, TCI |
|
3 |
[¬Ta → ¬1/2, 1] |
2, GCON |
|
4 |
[¬1/2 → 1/2, 1] |
FL P6.1 |
|
5 |
[¬Ta → 1/2, 1] |
3,4 HS |
|
6 |
[( x)¬Tx → ¬Ta, 1] |
FL P9, with ( x)¬Tx / ( x)Px, a / a |
|
7 |
[( x)¬Tx → 1/2, 1] |
5,6 HS |
|
8 |
[ ( x)¬Tx → ( x)¬Tx, 1] |
3, with ( x)¬Tx / P |
|
9 |
[ ( x)¬Tx → 1/2, 1] |
7,8 HS |
|
10 |
[¬1/2 → ¬ ( x)¬Tx, 1] |
9, GCON |
|
11 |
[1/2 → ¬1/2, 1] |
FL P6.2 |
|
12 |
[1/2 → ¬ ( x)¬Tx, 1] |
10,11 HS |
|
13 |
[ ( x)¬Tx → ¬ ( x)¬Tx, 1] |
9,12 HS |
|
14 |
[( ( x)¬Tx → ¬ ( x)¬Tx) → ¬ ( x)¬Tx, 1] |
1, with ( x)¬Tx / P |
|
15 |
[¬ ( x)¬Tx, 1] |
13,14 MP |
|
16 |
[ ¬ ( x)¬Tx, 1] |
15, EA |
15.5 Exercises
SECTION 15.2
1 Construct a derivation that justifies the derived axiom schema
FL PD21. [P(a/x) → ( x)P, 1] in FL PA.
2Present a derivation in FL PA based on the method of reasoning in the corresponding “chain” derivation in Section 13.3 of Chapter 13, that shows that if the Principle of Charity premise in the weak conjunction version of the Sorites argument has at most the value 191/192 and the other premises have at most the value 1, then the conclusion has at most the value 0.
3Present a derivation in FL PA that shows that if the Principle of Charity premise in the bold conjunction version of the Sorites argument has at most the value
191/192 and the other premises have at most the value 1, then the conclusion has at most the value 0.
4Present a derivation that shows that if the premises of the Kleenean version of the Sorites argument all have at most the value .6, then so does the conclusion.
15.5 Exercises |
299 |
5Consider the non-Pavelka axiomatic system FL A that consists of axiom schemata FL P1–FL P4, FL P8 and FL P9, and the rules MP and UG, all with grades removed from the formulas. We know that this system must be incomplete, as explained in Section 15.1. Nevertheless, the Lukasiewicz weak conjunction Sorites argument, the Lukasiewicz strong conjunction Sorites argument, and the Kleene conditional Sorites arguments are all valid in this system. Show this.
SECTION 15.3
6Explain why we cannot define the existential quantifier in Godel¨ fuzzy firstorder logic as we did in classical logic; that is, explain why ( x)P and ¬ ( x)¬GP are not in general equivalent in FuzzyG . You may do this by giving an instance of these formulas and an interpretation on which these instances have different truth-values.
7Construct a derivation of the modified conclusion of the Godel¨ conjunctive version of the Sorites paradox from its premises in BLG A (remember that ¬P is defined as (P → 0) → 0):
Ts1
Es2s1
Es3s2
. . .
Es193s192
( x)( y)¬G((Tx &G Eyx) &G ¬GTy)
¬G¬GTs193
SECTION 15.4
8Construct a derivation of the graded formula [ ( x)Tx → ( x)Tx, 1] in FL PA.
9Construct a derivation of the graded formula [ ( x)Tx → ( x) Tx, 1] in FL PA.
10Construct a derivation that shows that [¬ ( x)Tx, 1] follows from [ ¬ ( x)Tx, 1] in FL PA.
11Construct a derivation that shows that [ Ra, 1] follows from [¬Pa, .5] and [( x) (Px Rx), .5] in FL PA.