The fuzzy logic discussed in this reply to Entemann [1] is due to Zadeh, in which singular propositions such as Pa take on truth values in the real-valued interval [0.0..1.0], with the truth function t defined as follows: t(A ∧ B) = min(t(A),t(B)), t(A ∨ B) = max(t(A),t(B)), and t(¬A) = 1 - t(A).

Entemann makes three claims that contradict Pelletier’s prior work, and Pelletier replies to them one by one in sections 3, 4, and 5.

The first claim is that fuzzy logic does not generate results that contradict classical logic. Pelletier replies by pointing out that A ∨ ¬A is a theorem in the latter but not in the former, while A ∧ ¬A is a contradiction in the latter but not in the former. Entemann means, however, and says, that provided one uses only the fuzzy values 0.0 and 1.0, fuzzy logic reduces to classical logic. This, of course, is also true, if trivially so. Our interlocutors are talking past each other here.

The second claim is that fuzzy logic can have a proof theory. Technically, it cannot, as Pelletier shows, but with the limiting conditions Entemann specifies, which perhaps trivialize his proof theory by excluding “true” or “full” fuzzy logic, Entemann does provide what might be termed (mathematicians will not be amused) a fuzzy proof theory.

The third claim is that fuzzy logic is proof-theoretic complete. Because fuzzy logic does not have the compactness property that classical logic has, namely that a conclusion that follows from an infinite number of premises must also follow from some finite subset of those premises, a simple argument akin to diagonalization demonstrates argument-incompleteness (proofs are necessarily finite).

In the final section, Pelletier considers a restriction on fuzzy logic, the elimination of the fuzzy value “indeterminate” or .5, that would make Entemann’s claims true, but concludes correctly that this is to sap fuzzy logic of just what makes it special. He also considers an extension of fuzzy logic to include his own parametric operators, which provide a method within the logic (not just in the metalanguage) of getting at the specific fuzzy truth values, and making the sort of claim, not available within Zadeh’s fuzzy logic, that one fuzzy full (.8) glass of water is fuller than another fuzzy full (.6) glass of water.

Pelletier also observes that none of the applications of fuzzy logic use the nondenumerable range [0.0..1.0]; rather, from an artificial intelligence (AI) point of view, all that is needed are finite, multi-valued logics. This, too, is correct, but because in such applications one wants to keep one’s options open, it is impossible to say which finite values one may need in advance. So, if the full power of fuzzy logic is never used, perhaps one should speak instead of the real-valued interval [0.0..1.0] as being a schema from which infinitely many [finite] multi-valued logics are drawn.

Is fuzzy logic a Logic (with a well-defined deduction operator (&tstile;)), as Entemann claims, or is it no such thing, as Pelletier proves? Entemann need only argue that “is a Logic” (L) is a fuzzy predicate that, when predicated of fuzzy logic (f), namely Lf, results in a fuzzy true statement. Indeed, Entemann’s key problem is that he uses what one might call “logician’s reserved terms” for things that logicians do not intend by them, even though he is competent enough.