- •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
11Fuzzy Propositional Logics: Semantics
11.1Fuzzy Sets and Degrees of Truth
Chapter 10 noted two problems that crop up for three-valued approaches to vagueness. The first problem is that the Principle of Charity seems to be, if not completely
true, then at least very close to true: 1/ doesn’t make a significant difference where
8
tallness is concerned. Three-valued logic has no obvious way to capture “very close to true.”1 The other problem is that the three sets used to interpret predicates— the extension, counterextension, and fringe—require clear cutoff points. Heights of 5 11 and greater might be clearly in the extension of tall, heights of 5 3 or less might be clearly in the counterextension of tall, and 5 7 might be clearly in the fringe—but can we classify all heights in this way? If so, there is a sharp cutoff point between being tall (the extension of tall) and being neither tall nor not tall (the fringe of tall), and one between being neither tall nor not tall and being not tall (the counterextension of tall), something like:
Height
4 7 |
|
5 3 |
5 11 |
|
6 7 |
||
} |
|
} |
} |
|
} |
||
|
←------- |
-------not tall |
--------→←neither tall nor not tall→ ← |
tall |
-------- |
→ |
|
Or may be the cutoff points are at 5 2 and 5 10 , or . . .? But wherever we draw them, it doesn’t seem true to the facts, to wit: there are no sharp cutoff points between an extension, fringe, and counterextension for our ordinary concept tall (or for any other vague concepts).2
Rather, there are infinitely many degrees of tallness, which we may indicate with values between 0 and 1 inclusive. Gina Biggerly, at 6 7 , is tall to degree 1 (i.e., clearly tall), while Tina Littleton, at 4 7 , is tall to degree 0 (clearly not tall). Mary Middleford, at 5 7 , is perhaps tall to degree .5—smack in the middle between being tall and being not tall; Anne, at 5 8 , is perhaps tall to degree .6: somewhat closer to
1Unless we use N to stand for very close to true—but then we will have no way also to capture very close to false, and so forth.
2The claim here is independent of intersubjective agreement or relativity of concepts. Just consider your own idea of tallness for some class of people: you’ll find that there are not sharp cutoff points for the application of that concept; any attempt to fix such points seems arbitrary.
176
11.1 Fuzzy Sets and Degrees of Truth |
177 |
tall than to not tall. Crystal, at 5 2 , is perhaps tall to degree .1—not as clearly not tall as Tina, but almost there.
Deciding exactly how to assign degrees of tallness is interesting, and we’ll return to this issue in Chapter 17. But our logic doesn’t depend on any specific way of assigning these degrees, so for now we’ll introduce one way this might be done in order to explore the logic that unfolds. Let’s consider only the heights between 4 7 and 6 7 inclusive. We first express the heights on our previous scale as heights in excess of 4 7 :
Height
4 7 |
5 3 |
5 11 |
6 7 |
||
} |
} |
} |
} |
||
|
|
|
|
|
|
Height in excess of 4 7 |
|
|
|
|
|
0 |
8 |
16 |
24 |
||
} |
} |
} |
} |
||
That is, 4 7 is 0 inch greater than 4 7 , while 6 7 is 24 inches greater than 4 7 . This gives us a scale ranging from 0 to 24. But we want a scale ranging from 0 to 1, so we’ll divide these values by 24 to arrive at the degree of tallness for each height:
Height
4 7 |
5 3 |
5 11 |
6 7 |
||
} |
} |
} |
} |
||
|
|
|
|
|
|
Height in excess of 4 7 |
|
|
|
||
0 |
8 |
16 |
24 |
||
} |
} |
} |
} |
||
|
|
|
|
|
|
Degree of tallness |
|
|
|
|
|
0 (= 0/24) |
0.333 . . . (= 8/24) |
0.666 . . . (= 16/24) |
1 (= 24/24) |
||
} |
} |
} |
} |
||
|
|
|
|
|
|
Thus 4 7 is tall to degree 0, 5 3 is tall to degree .333 . . . , 5 11 is tall to degree
.666 . . . , and 6 7 is tall to degree 1.
We’ve just described what is known as a fuzzy set: a fuzzy set of heights between 4 7 and 6 7 . A non-fuzzy (crisp)3 set is a collection of entities, such that each entity either is or isn’t a member of the set. But with fuzzy sets we don’t have entities either being or not being a member of the set; rather, we have entities being members of the set to some degree. More technically, a fuzzy set is defined by a function that assigns to each entity in its domain a value between 0 and 1 inclusive, representing the entity’s degree of membership in the set. 4 7 is a member of our fuzzy set of heights to degree 0; 5 3 is a member to degree .333 . . . ; and so on.
We can in fact describe a crisp set in fuzzy terms: it is a set for which the degree of membership of any entity is either 1 (the entity is in the set) or 0 (the entity is not in the set). Our three-valued interpretations for vague predicates in earlier chapters may also be characterized as fuzzy sets: the entities in the extension of the predicate
3 |
Crisp is the fuzzy community’s term for classical sets, crisp contrasting with fuzzy. |
|
178 |
Fuzzy Propositional Logics: Semantics |
are members of the fuzzy set corresponding to the predicate to degree 1; the entities in the counterextension are members to degree 0; and the entities in the fringe are members to degree .5. So what we have done in this text is to move from two distinct degrees of set membership (classical logic) to three degrees of membership (threevalued logic) to an infinite number of degrees of membership in the sets that are used to interpret predicates.
Just as we used the extension, counterextension, and fringe assigned to predicates to determine the truth-value of simple subject-predicate sentences, we may now use fuzzy sets assigned to predicates to determine truth-values. Our truthvalues will be values between 0 and 1 inclusive, and in the case of simple sentences will correspond directly to degrees of membership. If Anne’s height is tall to degree 1, for example, we will assign the value 1 to the sentence Anne is tall. If her height is tall to degree .2, then we will assign the value .2 to the sentence Anne is tall, and so on. We call these values degrees of truth. A logical system in which sentences may have any of an infinite number of degrees of truth (e.g., values between 0 and 1) is an infinite-valued logical system. When the bases for assigning the degrees of truth are fuzzy sets, we call the system a fuzzy logic.4
The move to fuzzy logic does more than just settle—or eliminate—the problem of exact boundaries for the fringe of a vague predicate. With a variety of degrees of truth we can also address the other major problem that hounds three-valued accounts of vagueness: we can accommodate the intuition that the Principle of Charity is very close to true by assigning it a high degree of truth.5
11.2 Lukasiewicz Fuzzy Propositional Logic
We’ll use fuzzy sets explicitly in fuzzy first-order logics in Chapter 14, but first we present propositional systems, just as we did for three-valued logic, so that we can examine some of the logical principles in a simpler setting.
To specify a full fuzzy propositional logic, we begin with an assignment V of fuzzy truth-values, between 0 and 1 inclusive, to the atomic formulas of the language. We call the set of real numbers between 0 and 1 inclusive the unit interval, and we use the notation [0. .1] to designate the unit interval. So we may say that for each atomic formula P, V(P) is a member of the unit interval, or in more concise notation, V(P) [0. .1]. We can then use the same numeric clauses that we presented for
4There is disagreement in the literature about whether this is enough to call a logic fuzzy (as opposed to an infinite-valued logic), or whether the logic also needs to include fuzzy semantic and syntactic concepts like n-degree-validity that will be introduced later in this chapter. Our logics will be fuzzy in any case since we are introducing these latter concepts.
5Graham Priest (2001) notes that many Sorites paradoxes involve discrete steps (like the tallness
version in which we subtract 1 each time) rather than continuous ones (such as a version that
/
8
said that any reduction of height of 1 or less can’t take us from a tall person to one who’s not
/
8
tall) and that in these cases a finite-valued logic would suffice for a solution to the paradox. But he adds (and we agree) that “the continuum-valued semantics [the semantics that includes all real numbers between 0 and 1 as truth-values] is more general, and can be applied to all [S]orites paradoxes, giving, what is clearly desirable, a uniform account” (p. 214).
11.2 Lukasiewicz |
Fuzzy Propositional Logic |
179 |
Lukasiewicz three-valued logic to obtain the Lukasiewicz fuzzy system FuzzyL:6
1.V(¬P) = 1 – V(P)
2.V(P Q) = min (V(P), V(Q))
3.V(P Q) = max (V(P), V(Q))
4.V(P → Q) = min (1, 1 – V(P) + V(Q))
5.V(P ↔ Q) = min (1, 1 – V(P) + V(Q), 1 – V(Q) + V(P))
6.V(P & Q) = max (0, V(P) + V(Q) – 1)
7.V(P Q) = min (1, V(P) + V(Q))
Some of these connectives can be defined using others, as we did in L3. For example, in Section 9.3 of Chapter 9 we noted that P Q is definable as P & (¬P Q) in L3, and V(P & (¬P Q)) = max(0, V(P)+min(1, 1−V(P)+ V(Q))−1)= max(0, min(V(P)+ 1−1, V(P)+1−V(P)+V(Q)−1)) = max(0, min(V(P), V(Q)) = min(V(P), V(Q)).
Assuming that V(P) = 1, V(Q) = .75, and V(R) = .5, here are the values of various compound formulas:
Formula |
Value |
¬P |
0 |
¬Q |
.25 |
¬R |
.5 |
Q P |
.75 |
Q R |
.5 |
P ¬P |
0 |
Q ¬Q |
.25 |
Q R |
.75 |
P ¬P |
1 |
R ¬R |
.5 |
P → Q |
.75 |
P → R |
.5 |
Q → R |
.75 |
Q → Q |
1 |
Q → ¬Q |
.5 |
P & Q |
.75 |
Q & R |
.25 |
Q ¬Q |
0 |
P R |
1 |
R ¬R |
1 |
6Apropos of footnote 4, Lukasiewicz developed an infinite-valued logic but his work predated by a good 40 years the introductions of fuzzy sets and of fuzzy semantic concepts. Lukasiewicz’s
work on infinite-valued logic is discussed in Lukasiewicz and Tarski (1930). We also note that Lukasiewicz assumed a range of truth-values consisting of the rational numbers, rather than all of the real numbers, in the unit interval. However, it turns out that the set of tautologies obtained when the truth-values are restricted to the rationals is identical to the set of tautologies when all of the real numbers in the unit interval are used as truth-values. See Gottwald (2001, pp. 191–192), for a concise proof of this equivalence.