As we did for FLPA, we will say that a formula P is a theorem to degree n in FL PA if n is the least upper bound of the values m such that there is a derivation of the graded formula [P, m], and 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 .

The system FL PA is fuzzy sound: every formula that is a theorem to degree n in FL PA is an n-tautology of RFuzzyL , and if a formula P is derivable to degree n 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 the set of formulas. (As earlier, interpretations for RFuzzyL are like those for FuzzyL except that, in addition, each of the special formulas added to denote rational truth-values is assigned the truth-value that it denotes.) FL PA is also fuzzy complete: every formula that is an n-tautology in RFuzzyL is a theorem to degree n in FL PA, and a formula P is derivable to degree n from a graded set of formulas if n is the least upper bound of the values that P can have in RFuzzyL given the graded values of the set of formulas (see Novak´ et al. 1999, pp. 147–150).

As a consequence of the non-recursive axiomatizability of FuzzyL (and RFuzzyL ), soundness and (weak) completeness in the traditional sense must fail for FL PA (and for any other axiomatic system for FuzzyL ). Specifically, it is traditional completeness, not soundness, that is problematic: not every formula of RFuzzyL that has the value 1 on every interpretation has a derivation with graded value 1. But Hajek´ (1998b) has proved that a weaker result does hold: that a formula of RFuzzyL will have a derivation with graded value 1 in FL PA if that formula has the value unit on every linear MV-algebraic interpretation in which the algebra includes the rationals in the unit interval along with every glb and lub that is required by the truth-conditions for quantified formulas. (An algebra is linearly ordered if for any two elements x and y in its domain either x ≤ y or y ≤ x.) Hajek´ also proved that a related weaker result holds for a non-Pavelka axiomatization of FuzzyL based on FLA with additional axioms and rules for the quantifiers. The reason that these are weaker results is that there are tautologies over interpretations based on the unit interval [0. .1] that do not evaluate to unit on every algebraic interpretation in the broader classes, and so these formulas are not guaranteed by the weaker results to have derivations with graded value 1.

15.3 An Axiomatic Derivation System for FuzzyG

Here is an axiomatic system BLG A from Hajek´ (1997) that is both sound and complete (in the traditional sense) for FuzzyG , the only one of our three fuzzy first-order systems for which this is possible. BLG Aincludes the axiom schemata for BLGA from Chapter 13, which we here list with the additional prefix since we are now working with a first-order system:

BLG 1. (P →G Q) →G ((Q →G R) →G (P →G R))

BLG 2. (P &G Q) →G P

along with the following axiom schemata for quantifiers:

BLG 9. ( x)(P →G Q) →G (P →G ( x)Q)

where P is a formula in which x does not occur free

BLG 10. ( x)P →G P(a/x)

where a is any individual constant

BLG 11. ( x)(P →G Q) → (( x)P →G Q)

where Q is a formula in which x does not occur free

BLG 12. P(a/x) →G ( x)P

where a is any individual constant

BLG 13. ( x)(P G Q) →G (( x)P G Q)

where Q is a formula in which x does not occur free

(For the last axiom, recall that weak disjunction is identical to strong disjunction in FuzzyG and hence in FuzzyG .) The rules are

MP. From P and P →G Q, infer Q

and

UG. From P(a/x), infer ( x)P

where x is any individual variable, provided no assumption contains the constant a and that P does not contain the constant a.

Note that BLG 9 and BLG 10 are (ungraded versions of) FL PA’s axiom schemata FL G8 and FL G9 in Section 15.2. We showed there that the next two axioms, BLG 11 and BLG 12, are derivable in FL GA. But here we need explicitly to include the remaining axioms to capture quantified claims because the operations are Godel¨ operations. In Lukasiewicz fuzzy logic we can define the existential quantifier in terms of the universal quantifier and negation—that means we don’t need special axioms for the existential quantifier in addition to those for the universal quantifier. But we cannot similarly define the existential quantifier in Godel¨ fuzzy logic because Godel¨ negation behaves differently from Lukasiewicz negation (this will be further explored in an exercise).

We shall show that the conclusion of the Sorites argument using Godel¨ bold conjunction and the Godel¨ conditional:

Ts1

Es2s1 Es3s2 Es4s3

. . .

Es193s192

( x)( y)((Tx &GEyx) →GTy)

Ts193

is derivable from the premises in BLG A. (Recall that weak and bold conjunction are identical in FuzzyG , so the derivation also establishes derivability when the Principle of Charity uses weak rather than bold conjunction.) First, we’ll derive the rule:

BCI (Bold Conjunction Introduction). From P and Q infer P &G Q

The rule is derived as follows (recall that HS is derivable rule in all of the BL-axiomatic systems):

And here’s the Sorites derivation:

which should look familiar—the reasoning is exactly the same as the reasoning displayed in the Sorites derivations in Section 15.2!

Finally, we note that the set of theorems of BLG A is not decidable, not surprising given the undecidability of classical and three-valued first-order systems.

15.4 Combining Fuzzy First-Order Logical Systems; External Assertion

Our bias in favor of Lukasiewicz fuzzy logic has been clear: it (augmented with an external assertion operation) deals quite well with the issues of vagueness that have concerned us, and we also have a Pavelka-style axiomatic system to examine n-degree-validity, decaying validity, and so forth syntactically. On the other hand, Godel¨ negation, which is not definable in FuzzyL , has some interest, and so do the t-norm of product logic and its residuum. We may decide, in the end, that we would also like to have these additional operations available—just as we decided that external assertion was an important operator to have. Mindful of this, and particularly keeping in mind that FuzzyL isn’t recursively axiomatizable, researchers have produced axiomatic systems that combine these three basic fuzzy systems. For example, Esteva, Godo, and Montagna (2001) present complete axiomatizations for a fuzzy propositional system that includes both Lukasiewicz and product connectives (recall here that external assertion is definable as long as we have Lukasiewicz negation and product/Godel¨ negation), and Hajek´ (1998b) has developed an axiomatic system for fuzzy first-order logic that includes all of the Lukasiewicz, Godel,¨ and product connectives, based on work in Takeuti and Titani (1984).

We can also augment either of the axiomatic systems in this chapter with Matthias Baaz’s external assertion axiom schemata, which we repeat here:

∆1. ∆P ¬∆P

∆2. ∆(P Q) → (∆P ∆Q)

∆3. ∆P → P ∆4. ∆P → ∆∆P

∆5. ∆(P → Q)→ (∆P → ∆Q)

and the rule

EA (External Assertion). From P infer ∆P.

In Chapter 13 we illustrated using these axioms in the Pavelka system FL PA. We did note that the resulting system was not fuzzy complete for FuzzyL augmented with external assertion, and similarly here. However, the system is fuzzy sound, so derivations that we can produce will not lead us astray. Here we will give an example using the graded system FL PA. The axiom schemata will all be graded with the value 1, of course. And here is a graded version of EA:

EA (External Assertion). From [P, 1] infer [∆P, 1].