Like FuzzyL, FuzzyG has a decision procedure for tautologies based on truthtables for finite-valued logics, in this case finite-valued Godel¨ logics. We define n- valued Godel¨ propositional logics for each integer n ≥ 3: Gn 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 1–5 for FuzzyG. For any propositional formula P of FuzzyG, let #A(P) be the number of distinct atomic formulas in P—here we only count one occurrence of each atomic subformula. Thus, #A(B), #A(¬B), and

#A(B B) are all 1, while #A(B C) is 2. It turns out that P is a tautology of FuzzyG if and only if P is a tautology of Gn, where n = #A(P) + 2.23 This gives us a decision procedure for FuzzyG: to determine whether P is a tautology of FuzzyG just examine the truth-table for P in the appropriate finite-valued Godel¨ system.

11.9 Product Fuzzy Propositional Logic

The truth-conditions for formulas formed with the major connectives of product fuzzy logic,24 or FuzzyP, are

1.V(¬PP) = 1 if V(P) = 0

0otherwise

2.V(P &P Q) = V(P) · V(Q)

3.V(P P Q) = (V(P) + V(Q)) – (V(P) · V(Q))

4.V(P →P Q) = 1 if V(P) ≤ V(Q)

V(Q) / V(P) otherwise

5.V(P ↔P Q): left as an exercise

6.V(P P Q) = min (V(P), V(Q))

7.V(P P Q) = max (V(P), V(Q))

The negation here, defining ¬PP as P →P 0, is identical to FuzzyG’s negation, and so the Law of Double Negation also fails for FuzzyP. We’ve already discussed (in Section 11.7) the rationale behind the algebraic product and algebraic sum that are used to define product bold conjunction and disjunction, and the way the associated residuum operation for the product conditional is derived. The reader will be asked in the exercises to prove that the truth-conditions 6 and 7 for FuzzyP weak conjunction and disjunction follow from the definitions

P P Q =def P &P (P →P Q)

and

P P Q =def ((P →P Q) →P Q) P ((Q →P P) →P P).

23See Hajek´ (1998b, p. 157), for a proof.

24Goguen (1968–1969) developed product fuzzy logic and studied its application to Sorites paradoxes. As we noted in Chapter 1, Goguen’s article was the beginning of formal fuzzy logic.

equivalence), and both double negations (which are identical) are true when P has any value other than 0, but Godel/product¨ negation is not fuzzy Bochvarian external negation. Singly-negated P has the value 0 in the Godel¨ and product systems when P has a value strictly between 1 and 0, for example.

The fuzzy versions of Bochvar’s external assertion and negation have the following truth-conditions:

V( P) = 1 if V(P) = 1 0 otherwise V( P) = 1 if V(P) =1 0 otherwise

Here we have used the symbol ∆ for external assertion because this is the standard fuzzy symbol, introduced by Matthias Baaz in (1996) (although Baaz did not present the operation as a generalization of Bochvar’s three-valued external assertion as we are doing here; he called it 1-projection and the fuzzy community has subsequently called it the Baaz delta operation). We have used the symbol for external negation to make it clearly distinguishable from the standard negations for our fuzzy systems.

In Chapter 5 we read Bochvar’s external negation formula aP as: P is true. But because we now have degrees of truth, we’ll follow Gottwald and Hajek´ (2005) and read the fuzzy formula ∆P as P is absolutely true. Similarly, we’ll read the fuzzy formula P as P is not absolutely true. Note that these two connectives are interdefinable in FuzzyL using either ∆P =def ¬L P or P =def ¬L∆P. Moreover, they are interdefinable in the same way using FuzzyG/FuzzyP negation in place of ¬L. So we’ll focus on just one of the connectives, fuzzy external assertion.

We have the important result that

Result 11.20: Fuzzy external assertion is not definable in FuzzyL, FuzzyG, or FuzzyP.

Proof: In Section 11.6 we explained that Bochvar’s external connectives aren’t definable in FuzzyL because the external operations are not continuous.

However, both FuzzyG and FuzzyP include noncontinuous operations, so the same reasoning can’t establish the negative result in these latter cases. Rather, we’ll use the following facts, each of which can be readily verified, to prove that external assertion isn’t definable in either of these two systems:

a.¬GP and ¬PP have the value 0 if and only if P does not have the value 0.

b.P &G Q and P &P Q have the value 0 if and only if at least one of P and Q has the value 0.

c.P G Q and P P Q have the value 0 if and only if both P and Q have the value 0.

d.P →G Q and P →P Q have the value 0 if and only if P does not have the value 0 while Q does have the value 0.

Because the weak conjunction and disjunction in FuzzyG and FuzzyP are definable using the other connectives, we have considered in (a)–(d) all of the operations in terms of which other operations of these systems are definable.

Now, for fuzzy external negation to be definable in a system it must be possible to find a formula formed from an atomic formula P and zero or more connectives that has the value 1 when P has the value 1 and that has the value 0 when P has any value other than 1. In particular, focusing on an arbitrary value between 1 and 0, say .5, the formula must have the value 1 when P has the value 1 and must have the value 0 when P has the value .5. We can show that there is no such formula either FuzzyG or FuzzyP, because every formula Q formed from the single atomic formula P and zero or more connectives has the

collapsing property: Q has the value 0 when P has the value .5 if and only if Q has the value 0 when P has the value 1.

Every atomic formula P obviously has the collapsing property. Moreover, it follows from facts (a)–(d) earlier that any formula Q built up from P and one of more of the connectives has the collapsing property if its immediate subformulas have that property. For example, if Q has the form ¬R then from fact (a) we have: ¬R has the value 0 when P has the value .5 if and only if R does not have the value 0 when P has the value .5. If R has the collapsing property, then R does not have the value 0 when P has the value .5 if and only if R does not have the value 0 when P has the value 1. And it follows again from fact (a) that R does not have the value 0 when P has the value 1 if and only if ¬R has the value 0 when P has the value 1. Thus in this case Q has the collapsing property as well. Similar reasoning using facts (b)–(d) establishes that for any other form that Q may have, Q will have the collapsing property if each of its immediate components does. It follows, then, that no formula of FuzzyG or FuzzyL can express fuzzy external assertion.

So if we want fuzzy external assertion (and thus fuzzy external negation) in any of our three fuzzy systems, we explicitly have to add it. Fortunately systems with this additional connective have been developed in recent years, as we shall see when we turn to algebras and derivation systems for fuzzy logic. We do, however, note an interesting positive expressibility result: fuzzy external assertion (and consequently fuzzy external negation) can be defined in terms of FuzzyG/FuzzyP negation ¬G and FuzzyL negation ¬L: ∆P =def ¬G¬LP. This will be verified in an exercise.

Adding fuzzy external assertion to FuzzyL, the formula ¬ P ¬ ¬P captures the sense of negation needed for a true rendition of Black’s formula P is not true and P is not not true where P is a vague assertion. This formula is true in FuzzyL (augmented with the new connective) when P has any value other than 0 or 1, and it is false otherwise. Because of the fact that external assertion always produces 1 or 0 as its value, the formula ¬ P ¬ ¬P is equivalent to ¬ P & ¬ ¬P in FuzzyL.