Fuzzy logic and the Liar paradox

[nomadreid]

I have once again (this time in http://www.economist.com/node/2099851) come across the argument that a fuzzy logic solves the liar paradox by assigning the liar sentence a truth value N, other than T or F, with
[[A]] = N ⇒[[~A]] = N. However, I don't see that this gets around the essential point of the liar: the liar uses a predicate ~T, and the assumption of the existence of a predicate ~T leads to a contradiction, for example quickly with the Diagonal Lemma. So if you could build the liar sentence, then the fuzzy logic would be of use to not make it a paradox, granted, but you can't even build the liar sentence in the first place. What am I missing?

 

[TeethWhitener]

Here's the arxiv preprint of the paper that the Economist article refers to:

http://arxiv.org/pdf/cs/0309046v1.pdf

I haven't read the whole thing in depth, but basically, they look at specific instances of "The truth value of B is b." At the first level, you have a set of propositions B∈S₁, which can be assigned a certain truth value Tr(B) (N.B.--in fuzzy logic, Tr(B)∈[0,1]; cf. Boolean logic, where Tr(B)∈{0,1}), and at the second level you have the set of statements C∈S₂ about the truth value of B, namely C = "Tr(B) = b." [It's important to note that S₂⊆S₁.] Of course, Tr(B) is independent of b, but our intuition says that C is true when Tr(B) is actually equal to b and false when Tr(B) and b differ by exactly 1. So they define Tr(C) = 1-|Tr(B)-b|.

The tricky part comes when you have a self-referential formula, like B = "Tr(B) = b." The authors model the Liar sentence as A = "A is false," or A = "Tr(A) = 0," where A∈S₁. But since A is of the same form as C (above), it's also the case that A∈S₂. So the sentence can be recast as C = "Tr(A) = 0." Since A=C, we also have Tr(A)=Tr(C). Taking the definition of Tr(C), you get:

Tr(C) = 1-|Tr(C)-0| = 1-Tr(C) since Tr(C) ≥ 0
2Tr(C) = 1
Tr(C) = 0.5

Whether you agree with it philosophically or not, it does seem to be a consistent way to treat the problem. But it hinges on two things: how you model the Liar sentence, and how you define the truth value of sentences from the set S₂.

 

[nomadreid]

Many thanks, TeethWhitener, both for the article and your excellent summary! Fascinating article, not only the part that you so splendidly summarized, but the rest of it as well. I am still going through it, but the essential part which you explained is indeed rather elegant: a bit like having nested models reflected down to a single syntactic level. Philosophically? I do not see any philosophical objection problem with breaking a truth value assignment into two parts. More of interest are the open questions which the author leaves "for future research", especially: is the set of solutions the base set for a lattice? Thanks again!