let K be the conclusion of the new argument, we will have an argument with an even smaller n-degree-validity. Assigning 10/11 to A, 9/11 to B, 8/11 to C, . . ., 1/11 to J, and 0 to K, all of the conditional premises will have the value 10/11, as will A, and so the downward distance from the premises to the conclusion in this case is 10/11 giving the new argument at most 1/11-degree-validity.

More generally, if we increase the number of atomic formulas to m, add corresponding conditional formulas, and set the consequent of the last conditional as the argument’s conclusion, the resulting argument will be at most 1/m-degree-valid. We can assign (m − 1)/m to A, (m − 2)/m to B, (m − 3)/m to C, . . . , and 0 to the conclusion in which case the conditional premises will all have the value (m − 1)/m. So the downward distance to the conclusion will also be (m − 1)/m, making the argument at most 1/m-degree-valid. There is no upper limit on the number of premises an argument can have, so as we add more atomic formulas the n-degree-validity of longer arguments of this form will approach (but never reach) 0. Yet every such argument is classically valid. The astute reader will have noticed the affinity between this pattern of argument and the Sorites arguments—and hence may surmise that when we turn to the first-order version of FuzzyL, we will find that Sorites arguments have very low n-degree-validity.

11.5 Fuzzy Consequence

It is customary in fuzzy logic to analyze arguments semantically in yet another way, using the concept of fuzzy consequence. First, some preliminaries: A fuzzy truth-value assignment defines a fuzzy set of the formulas of FuzzyL, where each formula’s degree of membership in the set is its degree of truth on that truth-value assignment. Let us call every fuzzy truth-value assignment that makes each formula in a fuzzy set at least as true as its degree of membership in a consonant truthvalue assignment for the set . So, for example, for the fuzzy set that contains the formula A to degree 1, the formula A B to degree .3, and all other formulas to degree 0, the consonant truth-value assignments include all (and only) truth-value assignments that assign 1 to A, any value ≥ .3 to B, and any combination of values to the other atomic formulas.14

The fuzzy consequence of a fuzzy set of formulas is defined to be another fuzzy set, the fuzzy set in which the degree of membership for a formula P is the greatest lower bound of the truth-values that P can have on any consonant truthvalue assignment for . We may summarize this loosely by saying that a formula P is a member to degree n in the fuzzy consequence of a fuzzy set of formulas if the

14Note that some fuzzy sets have no consonant truth-value assignments. For example, any fuzzy set in which the FuzzyL formula ¬(P → P) has a degree of membership greater than 0 does not have any consonant truth-value assignments since there is no fuzzy truth-value assignment that will give this formula a value greater than 0. Similarly, there are no consonant truth-value assignments for any fuzzy set that contains P with a degree of .5 and ¬P with a degree of .6.

value of P is guaranteed to be at least n given the truth-degrees of the members of. We’ll use FC( ) to denote the fuzzy consequence of .

As an example, consider the fuzzy set of formulas in which P is a member to degree 1, P → Q is a member to degree .9, and all other formulas of the language are members to degree 0. The consonant truth-value assignments for are those fuzzy assignments that assign P the value 1 and that assign Q a value between .9 and 1 inclusive. FC( ) is the fuzzy set that assigns to each formula of FuzzyL the greatest lower bound of its truth-values on the consonant truth-value assignments for . A formula R that does not contain P or Q will be a member of FC( ) to degree n if it is an n-tautology. This is because the consonant truth-value assignments forare free to assign any combination of values to atomic formulas other than P and Q, and so they will between them assign all the possible values that R can have on any truth-value assignment. So, for example, the formula S is a member of FC( ) to degree 0, and S ¬S is a member to degree .5. P and P → Q are members of FC( ) to the degree that they are members of , namely, 1 and .9. Q is a member of FC( ) to degree .9 because it has at least that value on any consonant truth-value assignment for . P Q is also a member of FC( ) to degree .9, while P ¬Q is a member to degree 0. The latter is because some consonant truth-value assignments for assign the value 1 to Q; these assignments give ¬Q (and hence P ¬Q) the value 0.

A special case of fuzzy consequence occurs when the fuzzy set has no consonant truth-value assignments. In this case FC( ) is the fuzzy set that contains every formula to degree 1. This is because the set of values assigned to any formula by the consonant truth-value assignments is the empty set Ø, and 1 is the greatest lower bound (in the unit interval) of all the values in the empty set: it is trivially true that every member of Ø is at least as true as 1, since there are no such members.

Another, more interesting, special case of fuzzy consequence arises when a fuzzy set of formulas of FuzzyL is crisp, that is, when every formula of FuzzyL is a member of to either degree 1 or degree 0. The fuzzy consequence of a crisp set is a fuzzy set in which the formulas that are entailed simpliciter in FuzzyL by the (degree 1) members of have the value 1, and all other formulas have a degree less than 1. So we may define entailment simpliciter in FuzzyL in terms of fuzzy consequence in FuzzyL: a set of formulas entails the formula P in FuzzyL if P is a member to degree 1 of the fuzzy consequence of the crisp set in which the formulas of are members to degree 1 and all other formulas are members to degree 0. More generally, degreeentailment (of which entailment simpliciter is a special case) is definable in terms of fuzzy consequence: a set of formulas degree-entails the formula P if for every n in [0. .1], P is a member to at least degree n of the fuzzy consequence of each fuzzy set in which the greatest lower bound of the values of members of is n and all other formulas of FuzzyL have the value 0. N-degree-entailment is also definable in terms of fuzzy consequence—this is left as an exercise for the reader.

Some theoreticians feature the concept of fuzzy consequence as the centerpiece of fuzzy logic. But others, as we noted in footnote 4 to Section 11.1, merely require an infinite number of truth-values. Our graded axiomatic systems for fuzzy logic will

capture the concept of fuzzy consequence, although we will also have ungraded axiomatic systems that simply capture the concept of validity simpliciter. In this book we take an inclusive view of what fuzzy logic is, since in our investigations of vagueness it is illuminating to study not only fuzzy consequence, but all of the infinite-valued concepts that we have introduced in this section.

11.6 Fuzzy Generalizations of KS3, BI3, and BE3; the Expressive

Power of FuzzyL

In Chapter 5 we saw that the connectives of KS3 are definable in L3. The definitions of the connectives of KS3 for numerical values generalize to an infinite number of truth-values, and all of these generalizations are definable in FuzzyL. FuzzyKS negation, conjunction, and disjunction coincide with the FuzzyL (weak) versions of those connectives because the numerical definitions of the connectives coincide in the three-valued case. The clauses for the FuzzyK conditional and biconditional are

V(P →K Q) = max (1−V(P), V(Q))

V(P ↔K Q) = min (max (1−V(P), V(Q)), max (1−V(Q), V(P)))

and these are the truth-conditions we get when we define these connectives in FuzzyL just as we did in the three-valued case:

P →K Q =def ¬P Q

P ↔K Q =def ¬(P ¬Q) ¬(¬P Q)15

Unlike FuzzyL conditionals, not all conditionals of the form P →K P are 1-tautologies; their value can drop as low as .5 when P has the value .5 (but no lower; every conditional of the form P →K P is at least a .5-tautology).

Nicholas Rescher (1969) proposed an infinite generalization of BI3 in which the truth-clause for negation is identical to Lukasiewicz’s infinite-valued negation and the truth-clauses for the binary connectives are based on the rule that a formula will have the value .5 if it contains any components with truth-values other than the classical values 1 and 0. So, for example, the clause for conjunction is

V(P BI Q) = 1/2 if 0 < V(P) < 1 or 0 < V(Q) < 1 min (V(P), V(Q)) otherwise

None of the binary connectives for the resulting system, FuzzyBI, are definable in FuzzyL. This is because the truth-functions for these FuzzyBI connectives are not continuous.

15In these definitions as well as everywhere else in this and subsequent chapters we are using the convention, introduced in Chapter 6, that connectives without subscripts are the Lukasiewicz connectives. In this chapter, of course, they are the Lukasiewicz connectives for fuzzy logic.

A continuous unary function f (that is, a continuous function of one argument) over the unit interval meets the intuitively stated criterion that as m approaches n, for any value n in the unit interval, f(m) approaches f(n). A precise definition tells us that small changes in m result in small changes in f(m): a unary function f over the unit interval is continuous if for any ε > 0 there is a δ > 0 (with ε and δ members of the unit interval) such that whenever }m1 – m2}< δ, }f(m1) – f(m2)}< ε.

The negation function defined as f(m) = 1 – m is continuous: as m approaches (gets closer to) 1, for example, f(m) approaches 0, and the same is true for any value n that m may be approaching. But the binary connectives of FuzzyBI are not continuous. A binary function f over the unit interval is continuous if for any ε > 0 there is a δ > 0 (with ε and δ members of the unit interval) such that whenever }m1 – m3}< δ and }m2 – m4}< δ, }f(m1, m2) – f(m3, m4)}< ε. Intuitively, the binary FuzzyBI connectives fail to be continuous because as the truth-values of their arguments approach 1 from anywhere above 0, the functions’ values stay fixed at .5 and then noncontinuously jump to either 1 or 0 once the arguments have both reached 1. For example, if the truth-values of P and Q are .1 and .2, then V(P BI Q) = .5, and if we increase the values of P and Q the value of the conjunction remains at .5 until the values of P and Q both reach 1, in which case the value of the conjunction noncontinuously jumps to 1. More technically in this case, consider the value .5 for ε. Continuity would require that there is some δ > 0 such that whenever the values of P and of Q change by less than δ, the value of P BI Q changes by less than .5. But there is a jump in the value of P BI Q from .5 to 1 when we take any values of P and Q that are arbitrarily close to 1 and change them to 1, so there can be no such value.

In 1951 Robert McNaughton proved that a function must at least be continuous to be definable in FuzzyL, so it follows that the binary FuzzyBI functions are therefore not definable in FuzzyL. In fact, McNaughton precisely characterized the functions definable in FuzzyL:

Result 11.18 (McNaughton’s Theorem). For any n ≥ 1, the n-ary truth-functions definable in FuzzyL are exactly the continuous n-ary functions f for which “there are a finite number of distinct polynomials λ1, . . ., λµ, each λj = bj + m1jx1 +· · ·+ mnjxn, where all the b’s and m’s are integers, such that for every (x1, . . ., xn), 0 ≤ xi ≤ 1, 1 ≤ i ≤ n, there is a j, 1 ≤ j ≤ µ, such that f(x1, . . ., xn) = λj(x1, . . ., xn).” (McNaughton [1951], p. 3)

See McNaughton’s article for a discussion of this result along his proof, which is beyond the scope of this text.

In the external system BE3 the value .5 is treated in the same way as 0; that suggests that Bochvar’s external connectives should treat all values other than 1 and 0 the same way 0 is treated. So, for example, (external) negation in FuzzyBE maps 1 to 0 and all other values to 1. This function, like the infinite-valued binary functions for the internal connectives, is noncontinuous: as m approaches 1 the negation function remains fixed at 1, but at 1 the function noncontinuously drops