Note that every formula of the form P → P will still (as in L3) have the value 1, since min (1, 1−V(P) + V(P)) = 1, no matter what the value V(P) is. Every formula of the form P ¬P will have the value 1, since V(P ¬P) = min (1, V(P) + 1 −V(P)) = min (1,1). Similarly, every formula of the form ¬(P & ¬P) will always have the value 1. On the other hand, a formula of the form P ¬P may have a value as low as .5 (but no lower)—it has that value when V(P) = .5—and the same holds for formulas of the form ¬(P ¬P). Thus the Laws of Excluded Middle and Noncontradiction hold in FuzzyL when expressed using the bold connectives but fail when expressed using the weak connectives.

We remind the reader of the rationale for the truth-conditions of theLukasiewicz conditional. By the definition in clause 4, whenever V(P) is less than or equal to V(Q), V(P → Q) = 1. This is obviously as it should be if Q is “truer” than P. Whenever V(P) is greater than V(Q), V(P → Q) = 1 – V(P) + V(Q), which is 1 – (V(P) – V(Q)). Thus the value of the conditional is 1 minus the value representing how much “truer” P is than Q. As Graham Priest explains it: “If A is more true than B, then there is something faulty about the conditional: its truth value must be less than 1. How much less? The amount that the truth value falls in going from A to B. In particular, if it falls all the way from 1 to 0, then the value of A → B is 0 (Priest 2001,

p.215).

11.3Tautologies, Contradictions, and Entailment in Fuzzy Logic

All of the semantic results for L3 in Chapter 5 except those concerning expressive power (which we’ll address in Section 11.6) carry over to FuzzyL when we consider weak conjunction and disjunction to be the counterparts of the classical conjunction and disjunction operators. In this section we consider tautologousness, contradictoriness, and entailment.

First, the system FuzzyL is normal, so we have

Result 11.1: Every FuzzyL tautology is a classical tautology, and every FuzzyL contradiction is a classical contradiction,

where a tautology is defined to be a formula that always has the value 1 and a contradiction is defined to be a formula that always has the value 0. If we define entailment to hold between a set of formulas and a formula P whenever P has the value 1 on any fuzzy truth-value assignment on which all the members of have the value 1, normality also guarantees that

Result 11.2: Every FuzzyL entailment is a classical entailment.

As a consequence of 11.2, every argument that is valid in FuzzyL is also classically valid.

The converses do not hold:

Result 11.3: Not every classical tautology is a FuzzyL tautology, and not every classical contradiction is a FuzzyL contradiction.

Result 11.4: Not every classical entailment is a FuzzyL entailment.

The classical tautology P ¬P isn’t a FuzzyL tautology, and the classical contradiction P ¬P isn’t a FuzzyL contradiction. The classically valid argument

¬ (P ↔ Q)

(P ↔ R) (Q ↔ R)

fails to be valid in FuzzyL. When P has the value 1, Q has the value 0, and R has the value .5, the premise has the value 1 but the conclusion has the value .5.

Because bold conjunction and disjunction are also normal connectives, Results 11.1 and 11.2 hold as well when the bold connectives are substituted for the classical connectives. Moreover, results 11.3 and 11.4 also hold when we substitute bold connectives for the classical ones. Both P → (P & P) and (P P) → P fail to be FuzzyL tautologies although P → (P P) and (P P) → P are classical tautologies. When P has the value .5, for example, so do both P → (P & P) and (P P) → P. The argument

P P

P

is not valid in FuzzyL—when P has the value .5, the premise of the argument has the value 1 while the conclusion has the value .5—but the argument

P P

P

is classically valid.

Because every truth-value assignment for L3 is also a truth-value assignment for FuzzyL, we have the following additional result:

Result 11.5: Every FuzzyL tautology is an L3 tautology, and every FuzzyL contradiction is an L3 contradiction.

That is, a formula that has the value 1 on every FuzzyL assignment must thereby be true on every L3 assignment, and similarly for formulas that have the value 0 on every FuzzyL assignment.7 The converse, however, does not hold:

Result 11.6: Not all L3 tautologies are FuzzyL tautologies, and not all L3 contradictions are FuzzyL contradictions.

7Note that we cannot replace a bold connective of FuzzyL with a weak connective of L3 and expect the result to obtain. For example, P ¬P is a tautology of FuzzyL but P ¬P isn’t a tautology of L3. The result does hold whenever we use exactly the same connectives—weak or strong—in both FuzzyL and L3.

182 Fuzzy Propositional Logics: Semantics

An example L3 tautology is the formula (P → ¬P) → P) → P (this can be expressed as (P → ¬P) P using disjunction: either P implies ¬P, or P). When V(P) = .7 in FuzzyL, for example, V((P → ¬P) → P) → P) = .7 as well (as the reader can easily verify); so the formula isn’t a FuzzyL tautology. The difference in the formula’s behavior in the two systems shouldn’t be surprising: the only case where (P → ¬P) isn’t true in L3, that is, where P is truer than ¬P, occurs when P has the value T, making the second disjunct of (P → ¬P) P true. But in FuzzyL, obviously, P will be truer than ¬P whenever P has a value greater than .5 – so the nontruth of (P → ¬P) doesn’t require P to be true. Comparison of L3 and FuzzyL entailment is left as an exercise.

We found that the set of tautologies of classical propositional logic as well as the sets of tautologies in each of our three-valued propositional systems are all decidable, because in each case we can construct a truth-table to decide the status of a formula. We can’t generalize this decision method to fuzzy propositional logics because it is impossible to produce a truth-table for all fuzzy truth-value assignments. Consider the simple case of the negated atomic formula ¬P: a truth-table for ¬P in a fuzzy system would have to list the infinitely many values from the unit interval that can be assigned to P! So the question arises; Are there other decision procedures for the sets of tautologies of fuzzy propositional systems? The answer, interestingly, is yes for FuzzyL (and for each of the systems presented in this chap- ter)—and that means that the set of theorems of the basic axiomatic system for FuzzyL will also be decidable.

Several decision procedures for FuzzyL tautologousness are known; we’ll describe a procedure from Aguzzoli and Ciabattoni (2000).8 First, we define n-valued Lukasiewicz propositional logics for each integer n ≥ 3: Ln is the propositional logic whose truth-values are the n members of the set {0, 1 / (n − 1), . . ., (n − 2) / (n − 1), 1} and whose truth-conditions are the clauses 1–7 for L3 (stated at the beginning of Section 11.2). Note that when n = 3 the truth-value set is the one we used for L3 in Chapter 9: {0, 1/2, 1}.

Second, we define #O(P) to be the number of occurrences of atomic formulas in P. Thus, #O(B) and #O(¬B) are both 1, while both #O(B C) and #O(B B) are 2. Aguzzoli and Ciabattoni proved that a formula P containing only negation and bold disjunction as connectives is a tautology of FuzzyL if and only if P is a tautology of Ln where n = 2#O(P) + 1. Thus, for example, the formula B ¬B is a tautology of FuzzyL if and only if it is a tautology of L5, while the formula ¬B (B ¬C) is a tautology of FuzzyL if and only if it is a tautology of L9.

Now, the set of tautologies in any finite-valued Lukasiewicz propositional logic is decidable by using truth-tables. Moreover, every FuzzyL formula can be mechanically converted to an equivalent FuzzyL formula containing only negation and bold disjunction (proof is left as an exercise), so we can decide whether a formula P of

8Actually, Aguzzoli and Ciabattoni only prove the if part; the only if part was established in Ackermann (1967, pp. 60–63). Another decision procedure for FuzzyL is presented in Gottwald (2001, Section 9.1.4).

concept in terms of the greatest lower bound of a formula’s possible truth-values, rather than the minimum such truth-value, because not all sets of fuzzy truth-values have minimum members. Tautologies per se as we have defined them are then 1- tautologies. By our definition, we can also have 0-tautologies, perhaps somewhat oddly called; these are formulas that can only have the value 0 (or values arbitrarily close to 0).10

The formula P ¬P is a .5-tautology of FuzzyL. We claimed earlier that this formula has the value .5 when P has the value .5, and that it has no lower value. This is because V(P ¬P) = max (V(P), 1 – V(P)). If V(P) = .5, then max (V(P), 1 – V(P)) = .5. If V(P) =.5, then either V(P) or 1 – V(P) is greater than .5, so max (V(P), 1 – V(P)) > .5. In fact, we can use normal forms to prove that every classical tautology involving only the connectives ¬, , and is at least a .5-tautology in FuzzyL (using the weak connectives for conjunction and disjunction).11

In our proof we will also use the concept of an n-contradiction, for which we need to define the concept of a least upper bound: a number n in [0. .1] is the least upper bound of a set R of real numbers in [0. .1] (denoted as lub(R))12 if and only if

(a)n is greater than or equal to each member of R (thus n is an upper bound), and

(b)there is no m in [0. .1] such than m is also greater than or equal to each member of R but m is less than n (n is the least upper bound). We define a formula of FuzzyL to be an n-contradiction if n is the least upper bound of the set of truth-values that the formula can have.

The following normal form equivalences hold in FuzzyL:

(The reader will be asked to verify these in an exercise.) Given these equivalences, it follows that every FuzzyL formula containing only negation, weak conjunction, and weak disjunction is equivalent to a formula in conjunctive normal form and to one in disjunctive normal form, where the normal forms are defined as before using the weak connectives:

A literal is a phrase.

If P and Q are phrases, so is (P Q).

Every phrase is in disjunctive normal form.

10Novak,´ Perfilieva, and Mockoˇˇr (1999) also acknowledge this oddity, commenting that it is a consequence of the fuzzy “principle of equal importance of [all] truth-values” (p. 102).

11This was first proved by Lee and Chang (1971). In particular, Lee and Chang are responsible for Results 11.7–11.11.

12The least upper bound of a set is also called the supremum of the set, or sup for short.

If P and Q are in disjunctive normal form, so is (P Q).

A literal is a clause.

If P and Q are clauses, so is (P Q).

Every clause is in conjunctive normal form.

If P and Q are in conjunctive normal form, so is (P Q).

We now prove

Result 11.7: A clause of FuzzyL is a .5-tautology if and only if it contains a complementary pair of literals.

Proof: If a clause C is a .5-tautology, then it must contain a complementary pair of literals, for if it didn’t then every literal in C could be assigned a value less than .5 since the literals would have independent truth-conditions, and so C itself would have a value less than .5. Conversely, if C contains a complementary pair of literals then on any truth-value assignment at least one of the literals in the pair will have the value .5 or greater, by the truth-condition for negations. Thus the disjunction C itself will have at least the value .5. Moreover, if all of the literals in C have the value .5 on an assignment, then so will C; so .5 is the greatest lower bound of the values that C can have.

Result 11.8: A phrase of FuzzyL is a .5-contradiction if and only if it contains a complementary pair of literals.

Proof: Left as an exercise.

As a consequence of these two results we have:

Result 11.9: A formula P of FuzzyL that is in conjunctive normal form is a .5- tautology if and only if each clause in P contains a complementary pair of literals.

Proof: The value of a (weak) conjunction in FuzzyL is the minimum of the values of the conjuncts, so the greatest lower bound of the values the conjunction can have is the greatest lower bound of the values of its conjuncts. Thus P is a .5- tautology if and only if all of the conjuncts (clauses) are .5-tautologies (a clause cannot be greater than a .5-tautology because it will have the value .5 whenever all of its literals do), and by Result 11.7 all of these clauses will be .5-tautologies if and only if each of them contains a complementary pair of literals.

Result 11.10: A formula P of FuzzyL that is in disjunctive normal form is a .5- contradiction if and only if each phrase in P contains a complementary pair of literals.

Proof: Left as an exercise.

In Chapter 2 we proved analogues of Results 11.9 and 11.10 for classical propositional logic, except that in the classical case the results state that P is, respectively,

a tautology or a contradiction. Because every formula in classical logic is equivalent to one in either conjunctive or disjunctive normal form, and every formula of FuzzyL that contains only negation, weak conjunction, and/or weak disjunction is equivalent to a formula in either conjunctive or disjunctive normal form, it follows that

Result 11.11: A formula of FuzzyL that contains only negation, weak conjunction, and weak disjunction as connectives is a .5-tautology if and only if it is a classical tautology, and is a .5-contradiction if and only if it is a classical contradiction.

Given Result 5.15 of Chapter 5, it also follows that

Result 11.12: A formula of FuzzyL that contains only negation, weak conjunction, and weak disjunction as connectives is a .5-tautology if and only if it is a quasi-tautology in L3 and is a .5-contradiction if and only if it is a quasicontradiction in L3.

Classical tautologies that contain the conditional or the biconditional can fail to be at least .5-tautologies in FuzzyL. One example is the formula ¬(P (P → ¬P)).13 The value of the formula is 1 – min (V(P), 2 – (2 ·V(P))), and this is (for example) .4 when V(P) is .6 or .7. It is left to the reader to prove that this formula is specifically a 1/3-tautology. The classically tautologous formulas ¬(A → ¬A) ¬(¬A → A) and ¬(A ↔ ¬A), formulas that can have the value F in L3 as we showed in Chapter 5, have the value 0 in FuzzyL when V(A) = .5. So we have

Result 11.13: Not every formula that is a tautology in classical logic is at least a .5-tautology in FuzzyL; indeed some classical tautologies are 0-tautologies in FuzzyL.

Analogous counterexamples can be found when we substitute the bold connectives for classical conjunction or disjunction; the reader will be asked to explore these in an exercise.

We revise the three-valued definition of degree-entailment so that it applies now to FuzzyL: We will say that a set of formulas degree-entails a formula P if on every fuzzy truth-value assignment the value of P is greater than or equal to the greatest lower bound of the values of members of on that assignment. (In the case where is a finite set, this can be restated as: on every fuzzy truth-value assignment, P is at least as true as the least true member of —because a finite set has a finite number of values on a given assignment, and hence there is a minimum member of that set of values.) An argument is degree-valid in FuzzyL if the set of its premises degreeentails its conclusion. Obviously, every argument that is degree-valid in FuzzyL is

13This example is from Kenton F. Machina (1976). Machina attributes the example to Lawrence Eggan.

also valid simpliciter in FuzzyL; but the converse does not hold. For example, the argument

P

¬(P → ¬P)

is valid simpliciter in FuzzyL (and therefore also in classical logic), but it is not degree-valid in FuzzyL since the conclusion has the value 0 when the single premise has the value .5. More strikingly, the argument

P

P → Q

Q

also fails degree-validity (although it is valid simpliciter). If V(P) = .5 and V(Q) = .4, for example, then V(P → Q) = .9 and so the least value of the premises is .5 but the conclusion has the value .4.

On the other hand, of course, some valid arguments in FuzzyL are also degreevalid, for example, the argument

P

P Q

This is because the value of P Q is the maximum of the values of P and Q and so cannot be less than the value of P. Here then are the relevant results:

Result 11.14: Every FuzzyL degree-entailment is a classical entailment.

Result 11.15: Not every classical entailment is a FuzzyL degree-entailment.

Degree-validity plays an important role in dealing with arguments with vague premises that may be less than completely true. In such a case (as with the Sorites, which we will discuss in Chapter 14), we may well be interested in whether the conclusion is at least as true as the least true premise.

The concept of degree-validity captures whether the conclusion of an argument preserves the smallest degree of truth of its premises. We can go even further and ask, If an argument such as

P

P → Q

Q

is not degree-valid, then to what extent does it approximate degree-validity? We shall measure this as a function of how much truth can be lost going from the premises to the conclusion. When the difference between the greatest lower bound of the truth-values of members of a set of formulas on a fuzzy truth-value assignment V and the truth-value of a formula P on that assignment is greater than 0, we will call this difference the downward distance between and P on V. If the

difference is less than or equal to 0, we will say that the downward distance between

and P on V is 0. For example, the downward distance between the set of formulas {P, P → Q} and the formula Q on a fuzzy assignment V such that V(P) = .9 and V(Q) = .2 is .3 − .2 or .1, since the greatest lower bound of values of formulas in the set is .3, the value of P → Q. The maximum downward distance between and P is the least upper bound of the downward distances between and P on any truth-value assignment. Finally, we will say that a set of formulas n-degree- entails P if 1 – n is the maximum downward distance between and P, and that an argument is n-degree-valid if the set of its premises n-degree-entails its conclusion. So, for example, if the maximum downward distance between and P is .3, then

.7-degree-entails P, but if the maximum downward distance is .7, then only

.3-degree-entails P. We note that 1-degree-validity and 1-degree-entailment coincide with degree-validity and degree-entailment simpliciter, since by definition the former cases have a maximum downward distance of 0.

To determine the n-degree-validity of the argument

P

P → Q

Q

in FuzzyL, we must find the maximum downward distance between the set of formulas {P, P → Q} and the formula Q. We first note that when the least truth-value of one of the formulas P and P → Q is 1—that is, when both formulas are true, the downward distance to the value of Q is 0 since Q must have the value 1 in this case. More generally, the downward distance must be 0 whenever V(P) ≤ V(Q), trivially. What is the downward distance when V(P) > V(Q)? Well, in this case V(P → Q) is 1 – V(P) + V(Q), so V(Q) = V(P → Q) – 1 + V(P). Consequently, when V(P) > V(Q) the gap between the value of the least true of the formulas {P, P → Q} and that of Q is min (V(P), V(P → Q)) – (V(P → Q) + V(P) – 1) or min (1 – V(P), 1 – V(P → Q)). We now consider three cases:

a.If V(P) = .5, then 1 – V(P) = .5, V(P → Q) ≥ 5, and 1 – V(P → Q)) ≤ .5. In this case, then, min (1 – V(P), 1 – V(P → Q)) may be as great as .5 (as will happen when V(Q) = 0).

b.If V(P) > .5 then 1 – V(P) < .5, and so min (1 – V(P), 1 – V(P → Q)) < .5.

c.If V(P) < .5 then V(P → Q) > .5, so 1 – V(P → Q) < .5 and min (1 – V(P), 1 – V(P → Q)) < .5.

Thus the greatest downward distance when V(P) > V(Q) is .5. It follows that the maximum downward distance between the set of formulas {P, P → Q} and the formula Q in any case is .5 and that the argument

P

P → Q

Q

is therefore (1 − .5)-degree-valid, or .5-degree-valid.

Interestingly, we have

Result 11.16: Every n-degree-entailment in FuzzyL with n > 0 is also a classical entailment (whether we use the weak or the bold connectives as the counterparts to the classical ones).

Proof: If a set n-degree-entails a formula P with n > 0 then by definition the downward distance between the (greater lower bound of) values of members of and P on any fuzzy assignment V is less than 1. Because FuzzyL is normal, it follows that any classical valuation that makes all of the members of true must make P true as well—because if P were false the downward distance in this case would not be less than 1.

On the other hand, we have

Result 11.17: Some classical entailments are only n-degree-entailments in FuzzyL for very small values of n, including 0.

Proof: An example of a classically valid argument that is 0-degree-valid in FuzzyL is

B

¬ (A → ¬A) ¬(¬A → A)

The argument is classically valid because the conclusion is a classical tautology. However, as we saw earlier, the conclusion will have the value in 0 in FuzzyL when V(A) = .5. Since V(B) can be 1 when V(A) = .5, the maximum downward movement from the truth-value of the single premise in FuzzyL to the truth-value of the conclusion is 1 and the argument is thus 0-degree-valid.

Another important (for our purposes) example illustrating Result 11.17 is the classically valid argument

A

A → B

B → C

C → D

D → E

E → F

F → G

G → H

H → I

I → J

J

This argument is .1-degree-valid in FuzzyL. When V(A) = .9, V(B) = .8, V(C) = .7, . . ., V(I) = .1 and V(J) = 0, all of the premises have value .9 but the conclusion has the value 0. So this argument is at most .1-degree-valid. It is left as an exercise to prove that it is exactly .1-degree-valid. If we add another conditional premise J → K and