- •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
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Alternative Semantics for Three-Valued Logic |
9.4 Exercises
SECTION 9.1
1Develop numeric truth-condition clauses for Bochvar’s internal conditional and biconditional.
2Develop numeric truth-condition clauses for Bochvar’s external connectives.
3Develop numeric truth-value clauses for Bochvar’s internal existential quantifier.
4Develop numeric truth-value clauses for Bochvar’s external quantifiers.
SECTION 9.2
5Prove that L3 weak (or KS3) conjunction and disjunction satisfy distributed lattice meet and join conditions for the domain {1,1/2, 0}, with 1 and 0 serving respectively as unit and zero elements.
6Do the L3 bold conjunction and disjunction operations produce a distributed lattice structure over {1, 1/2, 0}? Defend your answer.
7Show that the second part of the orthocomplement condition vii, unit = zero, follows from the first part of condition vii (zero = unit) and condition viii, x = x.
8Prove that LKL is a DeMorgan algebra.
9Prove that every Boolean algebra is a DeMorgan algebra, where the Boolean complement serves as the DeMorgan algebra’s orthocomplement.
10Prove Result 9.1.
11Prove that BIA meets conditions v and vi specifying distributivity and unit and zero elements:
v.x (y ∩ z) = (x y) ∩ (x z), and x ∩ (y z) = (x ∩ y) (x ∩ z)
vi.x zero = x, and x ∩ unit = x.
12Prove that BEA is a distributed lattice.
13Is Bochvar’s external negation an orthocomplement for the algebra BEA? Explain. Is it a complementation operation (such that x x = unit and x ∩ x = zero)? Explain.
14There does not exist a result analogous to Result 9.1 that relates BI3’s truthtables and three-valued distributive dual systems of semi-lattices with unit and zero elements and an orthocomplement because every Kleene algebra satisfies the conditions of these structures and we know that three-valued Kleene algebras produce the KS3 truth-tables. Here’s your task: find one or more conditions that can be added to i–iii, v, vi, vii, and viii that will force a threevalued algebra to produce the truth-tables for BI3 disjunction, conjunction, and negation.
SECTION 9.3
15Prove that the equation V(P & Q) = max (0, V(P) + V(Q) – 1) correctly defines the truth-conditions for Lukasiewicz bold conjunction.
9.4 Exercises |
173 |
16Complete the proofs that the bold disjunction and bold conjunction operationsL and L are both commutative and associative.
17Prove that conditions iii–ix for MV-algebras hold for L3MV.
18Derive lattice join commutation, the dual of lattice meet commutation, from the conditions defining MV-algebras.
19Derive the second complementation condition for MV-algebras using the other conditions defining MV-algebras.
20Show that the second DeMorgan Law for MV-algebras follows from the first DeMorgan Law and conditions i–v and vii–ix for MV-algebras.
21Show that the duality of zero and unit, and lattice meet commutation also hold for all Boolean algebras.
22Prove that in an MV-algebra, x ∩ y = x if and only if x y = unit, where
x∩ y =def x (x y).
23Prove that
a.the MV-algebra lattice meet operation ∩ defined as x ∩ y =def x (x y) is idempotent
b.the MV-algebra lattice join operation defined as x y =def (x ∩ y ) is commutative, associative, and idempotent
c.these two operations satisfy the lattice absorption conditions x (x ∩ y) = x and x ∩ (x y) = x.
24Prove Result 9.2.
10 The Principle of Charity Reconsidered and a New Problem of the Fringe
It’s time to face two problems that we sidestepped while exploring three-valued logical systems for vagueness.
Although the Sorites argument is valid in all of the systems we’ve presented, we claimed that the paradox can nevertheless be dissolved in three-valued logic because the Principle of Charity premise is not true on any reasonable interpretation. The first problem concerns the exact nature of the principle’s nontruth. Our sample interpretations rendered the premise false in Bochvar’s external system, which didn’t sound right because its negation—which states that 1/8 does make a difference—must then be true. However, the situation looked more promising in the other three systems, where the Principle of Charity and its negation were neither true nor false. But now let us recall that the Principle of Charity is so called by virtue of the colloquial reading, One-eighth of an inch doesn’t make a difference. Put that way, the Principle of Charity seems true, or close to it, doesn’t it? If you shrink a tall person by 1/8 , surely that person will still be tall. (If you disagree, change the shrinking to 1/100 —we’ll still get the paradox, but surely 1/100 doesn’t make a difference.) Three-valued accounts can avoid the paradox by claiming that the Principle of Charity is either false or neither true nor false, but that leaves another puzzle: why does the principle seem to be true?
We’ll see that in fuzzy logical systems we can do better: we’ll be able to say that the Principle of Charity, although not exactly true, is very close to true, because we’ll allow sentences to have one of infinitely many truth-values—ranging from true at one end to false at the other, and reflecting various degrees of truth in between. The alternative, which three-valued theorists can embrace, is to supplement the logical solution to the paradox with an explanation of why the apparent truth of the Principle of Charity is in fact illusory.1 So we might say that fuzzy logic takes the apparent truth at face value.
The second and more serious problem, which we mentioned in footnote 2 to Chapter 7 and which we will call the New Problem of the Fringe, is that our threevalued interpretations for vague predicates assume clear cutoff points between the extension of a predicate and its fringe and between the fringe and the counterextension. This can’t be right. Where is the cutoff between the extension of tall and
1 |
Kit Fine (1975), for example, takes this route. |
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The Principle of Charity and a New Problem of the Fringe |
175 |
the fringe of tall, for example? Is it 5 11 , as we assumed in our interpretations in Chapter 7? Or is it 5 10 , or 5 9 , or perhaps 6 ? And even if we can agree on where to make the cutoff, its mere existence shows that an eighth of an inch can make a difference, contrary to the Principle of Charity. That is, you can go from being tall to being neither tall nor not tall by shrinking 1/8 , and that seems plain wrong.
Indeed, Bertrand Russell recognized that the existence of a fringe seems to require “higher-order” fringes between a vague predicate’s extension and fringe and also between the predicate’s fringe and counterextension—to replace the objectionable cutoff points.2 But now if we countenance these higher-order fringes, we may find that we need yet higher-order fringes to set them off—lest we posit an exact cutoff between, say, the extension of a predicate and the higher-order fringe that separates it from the predicate’s fringe. We thus find ourselves going from three values to five values to nine values to . . . an infinite number of truth-values. And that is exactly what we have in fuzzy logic, which is a type of infinite-valued logic.
So let’s get fuzzy!
2 |
Bertrand Russell (1923, p. 87). The term fringe is from Black (1937), not from Russell. |
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