To see that the only if part holds in FLA as well, assume that there is a derivation of Q from P:

2R1

3R2

. . .

k+1 Rk k+2 Q

We’ll show how to construct a new derivation, without assumptions, in which each of the formulas S from the previous derivation appears in a conditional P → (P → (P → ( . . . → (P → S) . . . ) with some finite number of antecedent P’s— for then we can conclude that in particular this holds true for the last formula, Q. For simplicity, we assume that the given derivation does not use any derived axioms or rules. We begin our new derivation (which may use derived axioms and rules) with

For each of the formulas Ri that is an axiom, we include the following lines in the new derivation:

If a formula Ri was derived using Modus Ponens, then there were earlier formulas Re and Re → Ri in the given derivation from which Ri was derived. In our new derivation we will already have earlier formulas P → (P → (P → ( . . . → (P → Re) . . . ) with a finite number m of antecedent P’s and P → (P → (P → ( . . . → (P → (Re → Ri)) . . . ) with a finite number n of antecedent P’s. We use TRAN n times to derive Re → (P → (P → ( . . . → (P → (P → Ri)) . . . ) from the latter formula, and then we use GHS to derive P → (P → (P → ( . . . → (P → Ri) . . . ), with m+n antecedent P’s, from the former formula and this new one (both TRAN and GHS are shown to be derivable in FLA in Exercise 1).

13.2 A Pavelka-Style Derivation System for FuzzyL

We will now add atomic formulas that stand for truth-values to the language of FuzzyL.5 When we did this for L3, we included a name for each of the three truthvalues. However, in the case of FuzzyL we have an infinite number of truth-values.

5The resulting system FLPA is formulated as in Novak,´ Perfilieva, and Mockoˇˇr (1999). They use simplifications of Pavelka’s formulation developed by Hajek´ (1995a, 1995b).

Recall that our language already has infinitely many atomic formulas, since we can subscript our sentence letters with any integer. So adding another infinite supply of atomic formulas that name truth-values seems unproblematic. However, there is a complication: there are uncountably many truth-values in the unit interval of truthvalues [0. .1].6 This means that in order to name every truth-value in the unit interval we would need to add uncountably many atomic formulas to the language. Why do we care about this? Well, the major issue is that many standard and important metatheoretic proof techniques for logical systems assume that the language of those systems is countable.

We will therefore introduce atomic formulas to name only a countably infinite subset of values in [0. .1]. The standard subset that is used for this purpose is the set Rat of rational numbers in the unit interval, numbers that can be expressed as fractions with integral numerators and denominators such as 1/2, 2/5, and 599/6432. 0 and 1 are included here, since 0 is 0/n for any integer n, and 1 is 1/1 (or n/n for any positive, nonzero n).7 We will call the system that results from adding to the language of FuzzyL an atomic formula that denotes each value in Rat, RFuzzyL. Our atomic formulas will be boldface fractional expressions, such as, 1/2 and 599/6432, except that we will denote 0 and 1 with the boldface numerals 0 and 1. Note that there are infinitely many fractions that generate each rational number, for example, 1/2 = 2/4 = 3/6 = 4/8 = 5/10 = . . . . We will always use the expression with the smallest numerator, so for example 1/2 will be the unique atomic formula that we use to denote 1/2.

The graded axiomatic system FLPA contains Lukasiewicz’s axioms FL1–FL4, all with the grade 1 and renamed to reflect the fact that we are now working in a Pavelka-style system:

FLP1. [P → (Q → P), 1]

FLP2. [(P → Q) → ((Q → R) → (P → R)), 1]

FLP3. [(¬P → ¬Q) → (Q → P), 1]

FLP4. [((P → Q) → Q) → ((Q → P) → P), 1]

In addition, there are infinitely many (countably infinitely many) axioms, graded with 1, relating the truth-values under the conditional and negation operations:

FLP5.1. Includes every graded formula [(m → n) → p, 1] where m, n, and p are atomic formulas denoting rational truth-values m, n, and p in the unit interval such that p = min (1, 1−m + n)

FLP5.2. Includes every graded formula [p → (m → n), 1] where m, n, and p are as in FLP5.1

6See the Appendix for a definition of uncountability and a proof that there are uncountably many truth-values in the unit interval.

7See the Appendix for a proof that the set Rat is countable.

FLP6.1. Includes every graded formula [¬m → p, 1] where m and p are atomic formulas denoting rational truth-values m and p such that p = 1−m

FLP6.2. Includes every graded formula [p → ¬m, 1] where m and p are as in FLP6.1.

Some of these new axioms are

[(1 → 1) → 1, 1] [1 → (1 → 1), 1]

[(1/2 → 3/4) → 1, 1] [1 → (1/2 → 3/4), 1] [(1/2 → 1/3) → 5/6, 1] [5/6 → (1/2 → 1/3), 1] [¬1 → 0, 1]

[0 → ¬1, 1]

[¬1/2 → 1/2, 1] [1/2 → ¬1/2, 1] [3/5 → ¬2/5, 1] [¬2/5 → 3/5, 1]

We also add the special axiom schema

FLP7. Includes [m, m] for any rational value m in the unit interval, where m is the atomic formula that denotes the value m

and the truth-value constant introduction rule:

TCI. From [P, m] infer [m → P, 1],

where m is the atomic formula that denotes the value m.

The graded Modus Ponens rule for FLPA is

MP. From [P, m] and [P → Q, n], infer [Q, p]

where p is the result of applying theLukasiewicz bold conjunction operation to m and n, i.e., p = max (0, m + n−1)

To understand the graded value p inferred for Q, recall the Adjointness Condition from Chapter 11 for a t-norm tn and its corresponding residuation operation :

m tn n ≤ p if and only if m ≤ n p, for all m, n, p [0. .1]

This condition defines the value produced by the residuation operation. The inequalities in the Adjointness Condition can be restated thus:

the value p is at least m tn n if and only if the value n p is at least m

or:

the value p is at least the value m tn n, where n p is at least m

When the t-norm and residuation operations are Lukasiewicz’s bold conjunction and conditional, this condition entails:

the value p of Q is at least the value that results from applying the bold conjunction operation to the least value m that P → Q might have and the least value n that P might have

or:

if the value of P is at least n and the value of P → Q is at least m, then the value p of Q is at least max (0, m + n−1).

And this is the formula used in the graded Modus Ponens rule.

Before proceeding to derivations, we pause to address an important issue raised by our specification of the axioms FLP5.1, FLP5.2, FLP6.1, and FLP6.2 relating truthvalues under the operations. We said in FLP5.1, for example: where m, n, and p are atomic formulas denoting rational truth-values m, n, and p in the unit interval such that p = min (1, 1−m + n). The issue is, Given any pair of rational truth-values m and n from the unit interval, will the value p defined as (1, 1−m + n) also be a rational value from the unit interval? The answer is yes, and proof is left as exercise. Similarly, we would like to be certain that if m is a rational value in the unit interval then so is 1−m. Proof of this latter claim, which is also true, is left as an exercise.

The following derivation produces a graded formula asserting that the Law of Excluded Middle using bold disjunction has at least the value 1 in RFuzzyL. We reexpress A ¬A as ¬A → ¬A, since we define P Q as ¬P → Q.

Note that we have used an axiom shown to be derivable in FLA and given it the graded value 1. All axioms derivable in FLA may be used as derived axioms in FLPA with graded value 1, since they are derivable in FLPA from axioms, all of which have graded value 1—and the rules of FLPA always preserve the value 1. For a similar reason, all rules derivable in FLA may be used in FLPA with all of the grades set to 1 (we may want to tune the grades more finely, but in that case we need to produce a derivation justifying the fine-tuning, as we will do shortly).

Recall from Chapter 6 that a formula of the form n → P, where n is an atomic formula denoting the truth-value n, means the truth-value of P is at least n, and a formula of the form P → n means: the truth-value of P is at most n. The following derivation, with all formulas graded with 1, establishes the theoremhood (in FLPA) of a formula claiming that if Q has at least the value 9/10 then P → Q also has at least the value 9/10:

Justification:

but we produce a longer derivation here in order to draw some more general conclusions:

Now, this derivation shows only that J has at least the value 0. The conclusion that J has at least the value 0 is in fact consistent with a stronger claim: that J has at least the value .1, and even a much stronger claim: that J has at least the value 1! Now, because of the soundness of FLPA (more on this later), we know that there is no derivation from the graded premises to these stronger conclusions since we know from Chapter 11 that the argument is .1-degree-valid. But keep in mind the general point, which we may restate as: the claim that J has at least the value 0 does not

entail that it has at most the value 0. If we want to support the latter claim in FLPA, we will need a derivation with that claim as its conclusion. We will produce such a derivation in a moment.

But first, note that if we had ended the derivation at line 18 we would have a nontrivial conclusion about I (in contrast with the trivial conclusion about J on line 19: trivial because every formula has at least the value 0). The derivation ending at line 18 shows that if each of A, A → B, B → C, . . . , H → I has at least the value .9, then I must have at least the value .1. So even though our longer derivation establishes a trivial conclusion, the “subconclusions” reached along the way and used in subsequent reasoning in the derivation are not themselves trivial. This, by the way, meshes with what we noted about Modus Ponens in Chapter 11: that longer chains of Modus Ponens reasoning have lower n-degree-validities than do shorter ones.

We will now produce a derivation that shows that when A and each of the conditional premises have at most the value .9, the conclusion J has (at most) the value 0. We introduce two derived rules to capture repeated inference patterns in our derivation. The first is

CV (Consequent Value). From [P →m, 1], [Q →m, 1] and [(P → Q) → n, 1], infer [Q → ¬(n → ¬m),1]

The derived rule does not require that m and n be atomic formulas denoting rational values—in fact we can use any formula in their place—but we shall only be using this rule when atomic formulas denoting rational values are used in place of m and n, hence the notation and the name. Given, for example, [A → 9/10, 1], [B → 9/10, 1], and [(A → B) → 9/10, 1], the rule allows us to derive [B → ¬(9/10→ ¬9/10), 1]. Once we do this, we shall use a second derived rule to simplify the conditional containing the names of truth-values to a single truth-value so that from [B → ¬(9/10→ ¬9/10), 1] we will be able to derive [B → 9/10, 1]. Here’s the derivation justifying CV:

Note that we used several derived rules in this derivation, rules that have been shown in Section 13.1 and Exercise 1 to be derivable in FLA. In each case the derived rule has been applied to formulas with grade 1 and the derived formula also has grade 1, which we pointed out earlier is always allowed since the axioms used in the rule derivations in FLA have grade 1 in FLPA and the rule MP preserves grade 1 in FLPA.

The second derived rule, for deriving a simplified value, is

VS (Value Summary). From [m → ¬n, 1], [p → (q →m), 1] and [¬r →p, 1], infer [¬(q → ¬n) →r, 1]

This will allow, for example, the move from [B → ¬( 9/10 → ¬9/10), 1] to [B → 9/10, 1]. It is left as an exercise to show that that this rule is derivable in FLPA. As in the previous case, any atomic formulas can be used in place of m, n, p, q, and r—but since we’ll be using this rule when these are all formulas denoting rational values, we choose the name and notation appropriate for this intended use.

Here, then, is the derivation showing that when A and each of the conditional premises in the earlier argument have at most the value .9, the conclusion J has at most the value 0.8 The basic strategy is that once we have placed an upper bound n on the antecedent of one of the conditional premises, then we then derive an upper bound of n−1/10 on the consequent of that premise, and hence on the antecedent of the next conditional premise in the chain—we can derive this upper bound because the greatest value that m can be when the formula (n → m) → 9/10 is true is 1/10 less than n. (The reader will be asked to explain this in an exercise.) Having an upper bound of 9/10 for A on line 1, we can then derive an upper bound for B that is 1/10 less by first deriving an upper bound of 9/10 for B (lines 11–12), and then an upper bound of 8/10 (lines 13–18), and so on. . . .

8I am grateful to Petr Hajek´ (personal correspondence dated March 1, 2005) for helping me to produce this derivation. It was he who suggested the derivation steps involved in the CV rule. Any inelegance is mine, not his, since he showed me these steps using the bold connectives of FuzzyL rather than just the two connectives—negation and implication—that I have chosen to include in FLPA.

where we have continued after line 54 by deriving, for each of the atomic formulas F through J, the formulas placing respective upper bounds of 4/10, 3/10, 2/10, 1/10, and 0 on their values. For each atomic formula, we first derive the upper bound we did for the previous atomic formula, using the earlier formulas with the obvious substitutions; then we add instances of FLP6.2, FLP5.2, and FLP6.1 that will give an upper bound that is 1/10 less than the final upper bound for the previous letter. Again, it should be obvious which instances of FLP6.2, FLP5.2, and FLP6.1 are used in each case. This derivation, along with the derivation establishing that J has at least the value 0 when A and each of the conditional premises all have at least the value 9/10, allows us to conclude that J has exactly the value 0 when A and the conditional premises all have exactly the value 9/10.

The definitions of theoremhood and derivability in the graded Pavelka system FLPA are more complicated than in FLA, for reasons similar to those for the Pavelka system for L3 in Chapter 6. We say that a formula P is a theorem to degree n in FLPA if n is the least upper bound of the values m for which there is a derivation (without assumptions) of the graded formula [P, m]. As in the three-valued system, there may be two different values j and k such that there are derivations of the graded formula [P, j] and of the same formula graded as [P, k]. In this case we want to say that P is a theorem to at least the greater of the two values (particularly when we recall that for any formula P, including FuzzyL tautologies, there is always a derivation of [P, 0]!). But unlike in the three-valued case, in FLPA we may have a single formula for which there are derivations with an infinite number of different grades, and there may be no value m that is the largest. For this reason, we define the degree n to which P is a theorem as the least upper bound of all of those grades.

In a similar vein, we say that a formula P is derivable to degree n from a set of graded formulas if n is the least upper bound of the degrees m such that [P, m] is derivable from the graded formulas in . It follows from these definitions that in general we cannot directly establish the degree of theoremhood or derivability of a formula with a single derivation, since the single derivation does not imply that there are no other derivations with higher graded values for that formula. Again, this is most obvious when we recall that any formula can be derived with the graded value 0: it doesn’t follow that 0 is the highest graded value derivable for that formula. All that a single derivation can show is that a formula is a theorem, or derivable from a graded set of formulas, to at least the degree established by the derivation. There is one exception: if we can derive a formula with graded value 1, then we may conclude that it is a theorem, or derivable, to degree 1—since no higher degree is possible.

Special case aside, the fuzzy soundness of FLPAmay sometimes allow us to reason more generally about degrees of theoremhood and derivability. FLPA is fuzzy sound in the following sense: every formula that is a theorem to degree n in FLPA is a tautology to degree n in RFuzzyL, and if a formula P is derivable to degree n in FLPA from a graded set of formulas then n is the greatest lower bound of the values

that P can have in RFuzzyL given the graded values of formulas in .9 In the latter case, P is a member to degree n of the fuzzy consequence of the fuzzy set in which each formula of is a member to the degree indicated by its graded value and all other formulas are members to degree 0. (We note that the truth-value assignments for RFuzzyL are like those for FuzzyL except that, in addition, each of the special formulas added to denote rational truth-values is always assigned the truth-value that it denotes. This isn’t a surprise, but we need to state it explicitly!)

So, for example, we have produced a derivation showing that A ¬A is derivable in FLPA (without assumptions) to at least degree .5. Given fuzzy soundness we can conclude something stronger: because this formula is a .5-tautology fuzzy soundness tells us that there is no derivation of this formula with a higher graded value; A ¬A is a theorem to degree .5 in FLPA. Similarly, we have produced a derivation showing that J is derivable to at least the degree 0 from the graded formulas [A, .9], [A → B, .9], [B → C, .9], [C → D, .9], [D → E, .9], [E → F, .9], [F → G, .9], [G → H, .9], [H → I, .9], and [I → J, .9]. From the fuzzy soundness of the system we can draw this stronger conclusion: Because the set {A, A → B, B → C, C → D, D → E, E → F, F → G, G → H, H → I, I → J} .1-degree-entails J, there is no derivation of J from these graded formulas in which J has a graded value higher than 0.

The converse of fuzzy soundness holds as well: FLPA is fuzzy complete for RFuzzyL. FLPA is fuzzy complete in the sense that every formula that is a tautology to degree n in RFuzzyL is a theorem to degree n in FLPA, and a formula P is derivable to degree n from a graded set of formulas if n is the greatest lower bound of the values that P can have in RFuzzyL given the graded values of the formulas in .

FLPA is also sound in the traditional sense: if there is a derivation (without assumptions) of the graded formula [P, 1] then P is a tautology of RFuzzyL, and if a formula P is derivable in FLPA with graded value 1 from a set of formulas all of which have the graded value 1, then the (ungraded) set of formulas in entails the formula P in RFuzzyL. And the system is weakly complete in the traditional sense: for every tautology P of RFuzzyL there is a derivation of P (without assumptions) with graded value 1 in FLPA. Strong completeness fails for the same reason that it failed for our non-Pavelka axiomatic system FLA for FuzzyL: semantic entailment is not compact in FuzzyL. On the other hand, traditional completeness does hold in FLPA for entailments from finite sets: if a finite set of formulas entails the formula P in RFuzzyL, then there is a derivation in FLPA of [P,1], from the set consisting of each of the formulas in with graded value 1.

There is one possibly confusing relationship between fuzzy completeness and traditional completeness for FLPA. In the case of fuzzy completeness, we have strong fuzzy completeness: completeness holds for entailments from infinite sets as well as entailments from finite sets. Why doesn’t it follow that we have strong

9See Hajek´ (1998b, pp. 80–83), for a proof of these claims and of the claims about fuzzy completeness, traditional soundness, and traditional weak completeness later.

traditional completeness as well? Consider our earlier counterexample to compactness: the set consisting of the infinite sequence of formulas (¬P → P) → Q,

(¬P → (¬P → P)) → Q, (¬P → (¬P → (¬P → P))) → Q, (¬P → (¬P → (¬P → (¬P → P)))) → Q, . . . semantically entails the formula Q, but Q is not entailed by any finite subset of . Strong fuzzy completeness tells us that the least upper bound of the graded values with which Q is derivable is 1 when the formulas in the set are themselves graded with the value 1. Then how can this semantic entailment be a counterexample to strong traditional completeness for FLPA, as we just claimed?

The answer is that there is no derivation of Q in FLPA with exactly the graded value 1 from a finite subset of formulas from with the graded value 1, but there are derivations with values that approach as close to 1 as you can without actually getting there, just as there are n-entailments of Q from finite subsets of in RFuzzyL with n as close to 1 as you can be without actually getting there (this will be examined in the exercises). The least upper bound of a set of values—which is featured in the definition of fuzzy completeness—need not be a member of that set, and in this case it is not. So FLPA can be strongly fuzzy complete but only weakly traditionally complete, since strong traditional completeness would require a derivation of Q with exactly the value 1.

As a consequence of the strong fuzzy completeness of FLPA we have fuzzy compactness. Recall that a fuzzy set of the formulas of a language is a fuzzy set to which every formula of the language belongs to some degree (including possibly degree 0). We will say that a fuzzy set of formulas is a finite base of a fuzzy set of formulasif a finite subset of the formulas that are members of to a degree greater than 0 are members of to the same degree (as in ), and all other formulas of the language are members of to degree 0. Here is the result:

Result 13.4: FuzzyL and RFuzzyL are fuzzy compact: for every formula P and fuzzy set of formulas , P is a member to degree n of the fuzzy consequence ofif and only if P is a member to degree n of the fuzzy consequence of at least one finite base of .

Degree-entailment and n-degree-entailment are not standardly studied in connection with fuzzy Pavelka derivation systems, where fuzzy consequence is the predominant semantic relation. Unlike entailment simpliciter and fuzzy consequence, these semantic concepts talk about all possible combinations of values that the members of a set of formulas might have. In the case of entailment simpliciter, we are only interested in what happens when all of the members of have the value 1. For fuzzy consequence, we are only interested in one combination of values for the formulas in . But for degree-validity and n-degree-validity we are interested in the relationship between the values that the members of might have, no matter what those values might be, and the value that the entailed formula has.

However, just as we can define degree-entailment in terms of fuzzy conse- quence—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

244 Derivation Systems for Fuzzy Propositional Logic

which the greatest lower bound of the values of members of is n and all other formulas of FuzzyL have the value 0—we can define a syntactic counterpart to degree-entailment as: P is degree-derivable from a set of formulas if for every n in [0. .1], [P, n] is derivable from each set * in which the formulas of have all been graded with values whose greatest lower bound is n. In the context of three-valued logic, we were able to produce derivations establishing degree-validity because we only had a finite number of combinations of values to consider. But in general, no finite number of derivations can establish a claim about all possible combinations of values for some set of formulas in RFuzzyL (an exception to this general rule is the case where we can establish that P is a theorem to degree 1, for here it doesn’t matter what graded values may be associated with members of ).

Before closing this section we would like to address the question, In RFuzzyL and the axiomatic system FLPA we’ve included constants to name only the rational truth-values in the unit interval so that our supply of atomic formulas will be countable, but do we lose anything important by failing to include constants denoting the nonrational members of [0. .1]? Well, we won’t be able to prove things like if P has some specific irrational value i and Q has a larger irrational value j, then P → Q has the value 1. Is this a significant loss? Is it important to be able to state explicitly relationships between irrational degrees of membership in fuzzy sets? As a first answer, we note that in our analyses of vagueness all of our examples have used rational truth-values, and it is not clear that issues of vagueness require even finer distinctions involving irrational truth-values (in fact, some have claimed that only finitely many truth-values are required for issues of vagueness).10

Second, Petr Hajek,´ Jeff Paris, and John Shepherdson (2000) have proved an important relevant theorem. Let RealFuzzyL be the logical system FuzzyL with the addition of atomic formulas naming each value, irrational and rational alike, in the unit interval [0. .1]. So RealFuzzyL is an extension of RfuzzyL in the sense that the formulas of the latter are a subset of those of the former. Let RealFLPA be the Pavelka-style axiomatic system that is like FLPA except that it also includes axioms analogous to FLP5.1, FLP5.2, FLP6.1, FLP6.2, and FLP for all of the real values in [0.

.1], not just the rational ones. RealFLPA is thus an extension of FLPA in the sense that every derivation in FLPA is also a derivation in RealFLPA. Hajek´ and colleagues’ theorem states that RealFLPA is a conservative extension of FLPA:

Conservative Extension Theorem for FLPA: If a formula P of RfuzzyL is a theorem to degree n of RealFLPA then it is also a theorem to degree n of FLPA, and if a formula P of RfuzzyL is derivable to degree n from a graded set of RfuzzyL formulas in RealFLPA then P is also derivable to degree n from that graded set of formulas in FLPA.

10E.g., Morgan and Pelletier (1977). They suggest that this is a reason for sticking to finitely manyvalued logics. We won’t go that far, since fuzzy logics are interesting in their own right.