That is, if we restrict our attention to rational truth-values then everything we can demonstrate in RealFLPA can already be demonstrated in FLPA; we do not need the additional power of the full real-valued system. So if it is correct that we lose little of interest on the score of vagueness by restricting our attention to rational truthdegrees, then it is also correct that we lose little by using the system FLPA rather than RealFLPA.

13.3 An Alternative Axiomatic System for Tautologies and Validity in FuzzyL Based on BL-Algebras

We will briefly present axiomatic systems for product and Godel¨ fuzzy propositional logics in Sections 13.4 and 13.5. The systems, developed in Hajek´ (1998b), are extensions of his “basic” fuzzy logic axiomatic system. In this section we present the basic system, which we’ll call BLA, and explain how BLA can be expanded to an axiomatic system for Lukasiewicz fuzzy logic.

BLA axiomatizes the basic properties of continuous t-norms and their related residuum operations as captured in BL-algebras. In addition to t-norm and residuum operators & and → we also introduce a special constant formula 0, which always has the value 0. The axiom schemata for BLA are:

BL1. (P → Q) → ((Q → R) → (P → R))

BL2. (P & Q) → P

BL3. (P & Q) → (Q & P)

BL4. (P & (P → Q)) → (Q & (Q → P))

BL5. (P → (Q → R)) → ((P & Q) → R)

BL6. ((P & Q) → R) → (P → (Q → R))

BL7. ((P → Q) → R) → (((Q → P) → R) → R)

BL8. 0 → P

and the single inference rule is, once again, Modus Ponens.

BL1 should come as no surprise, since we have repeatedly seen it in our axiomatic systems. BL2 tells us that we can infer the first conjunct from a t-norm conjunction, and BL3, which formally captures the commutativity of the t-norm operation, will also allow us to infer the second conjunct—we’ll see this in the derivation later. Turning to BL4, we recall that weak conjunction P Q in a fuzzy system is standardly defined as P & (P → Q), so BL4 formally captures the commutativity of weak conjunction. BL5 and BL6 capture the equivalence of the formulas P → (Q → R) and (P & Q) → R. For BL7, note that if the conditional → denotes a residuum operation then for any pair of formulas P and Q, either P → Q or Q → P will have the value 1. So if each of these implies some formula R, then we may conclude that R has the value 1 as well. BL8 says that 0 is the least truth-value.

It may be surprising that BLA has no axioms with a negation operator. But recall that negation is standardly defined in fuzzy logic as ¬P =def P → 0, so that is the

definition that we use here. Here is an example of a derivation of the negated formula ¬0, which is defined to be 0 → 0:

We used Hypothetical Syllogism in this proof; it is derived in BLA exactly as it is derived in FLA. By virtue of this proof, we have the derived axiom schema

BLD1. P → ¬¬P, or P → ((P → 0) → 0)

The converse formula, ¬¬P → P or ((P → 0) → 0) → 0, is not derivable in BLA. This is a good thing, since the converse formula is not a tautology in every variety of fuzzy logic. In particular, it isn’t a tautology in either FuzzyG or FuzzyL.

We said earlier that the system BLA axiomatizes the basic properties of continuous t-norms and their related residuum operations as captured in BL-algebras. We (and the reader, in some exercises) will show that the algebraic counterparts of the axioms of BLA all evaluate to unit in every BL-algebra—they are all BL-tautologies— and that, corresponding to MP, the algebraic rule

If unit ≤ x and unit ≤ x y, then unit ≤ y

is true in every BL-algebra, from which it follows that the BL-algebraic expressions corresponding to theorems of BLA are all BL-tautologies.11

Recall that a BL-algebra is an algebra {L, , ∩, , , unit, zero} that meets the following conditions:

11The proofs later owe largely to Hajek´ (1998b). The converse also holds: every expression that evaluates to unit in every BL-algebra is the algebraic counterpart of a theorem of the axiomatic system BLA (Hajek´ 1998b, pp. 49–54).

For axiom schema BL7, we show that unit ≤ ((x y) z) (((y x) z) z):

BL7:

(((x y) z) (x y)) ((y x) z) ≤ ((x y) z) (x y)

(since x y ≤ x; proof is left as an exercise) so (i) (((x y) z) ((y x) z)) (x y) ≤ ((x y) z) (x y)

(bold meet commutation, association) and (ii) (((x y) z) ((y x) z)) (y x) ≤ ((y x) z) (y x)

(same reason)

thus (iii) ((((x y) z) ((y x) z)) (x y)) ((((x y) z) ((y x) z)) (y x)) ≤

(((x y) z) (x y)) (((y x) z) (y x))

(by (i) and (ii) since if x ≤ z and y ≤ w, then x y ≤ z w— proof is left as an exercise)

In addition,

thus ((x y) ∩ z) ((y x) ∩ z) ≤ z (since x ≤ z and y ≤ z if and only if x y ≤ z—proof is left as an exercise)

and (iv) (((x y) z) (x y)) (((y x) z) (y x)) ≤ z

(bold meet commutation, BL-algebra condition (xi))

Therefore

((((x y) z) ((y x) z)) (x y)) ((((x y) z) ((y x) z)) (y x)) ≤ z (from (iii) and (iv), by ≤ transitivity)

(((x y) z) ((y x) z)) ((x y) (y x)) ≤ z

(since x (y z) = (x y) (x z)—proof is left as an exercise)

(((x y) z) ((y x) z)) unit ≤ z

unit (x y) z) ≤ ((y x) z) z

(bold meet identity, commutation) unit ≤ ((x y) z) (((y x) z) z)

(adjointness)

We leave it as an exercise to show that the algebraic counterparts of axiom schemata BL3, BL4, BL6, and BL8 are all BL-tautologies, and that the rule MP, interpreted algebraically as

If unit ≤ x and unit ≤ x y, then unit ≤ y,

is true in every BL-algebra.

We can extend the axiomatic system BLA with one axiom schema to obtain a system BLLA that is equivalent (in a sense to be explained later) to the axiomatic system FLA for FuzzyL:

BLL9. ¬¬P → P

Note that this explicitly parallels a result presented in Chapter 12: that every BLalgebra with Double Negation is an MV-algebra (recall that the algebraic structures corresponding to FuzzyL are MV-algebras). We have already shown that the converse formula to BLL9 is derivable in the basic system BLA; this additional axiom schema provides full Double Negation.

We can use the definition of bold conjunction:

P & Q =def ¬(P → ¬Q)

to introduce bold conjunction into FLA. The systems FLA and BLLA are then equivalent in the sense that every theorem of FLA is a theorem of BLLA, and vice versa (see Hajek´ 1998b, pp. 65–70). Rather than produce examples of derivations in BLLA, we’ll illustrate the use of BLA in the following axiomatic systems for Godel¨ and product logic.

13.4 An Axiomatic System for Tautologies and Validity in FuzzyG

We obtain an axiomatic system BLGA for FuzzyG by adding the following axiom schema

BLG9. P →G (P &G P)

to BLA (we exclude the Lukasiewicz axiom BLL9), with all of the connectives now subscripted with G to reflect the fact that we are working in FuzzyG. It was left as an exercise in Chapter 12 to show that this axiom, interpreted algebraically, is true in every Godel¨ algebra. (So are the axioms BL1–BL8, since every Godel¨ algebra is a BL-algebra.) BLGA is both sound and strongly complete for FuzzyG (Hajek 1998b, pp. 101–102). The latter means that all entailments in FuzzyG from infinite, as well as finite, sets have corresponding derivations in BLGA.

The strong completeness owes to the fact that Godel¨ fuzzy logic, unlike Lukasiewicz fuzzy logic, is compact (Baaz and Zach 1998). So, for example, consider

Justification:

And here’s the derivation—we omit the subscripted G for readability:

In contrast with FLPA (and BLLA), we have a full Deduction Theorem for BLGA:

Result 11.4 (Deduction Theorem for BLGA): Q is derivable from P in BLGA if and only if P → Q is a theorem.

Proof: Left as an exercise.