Comments to “Paradoxes of fuzzy logic, revisited”

✍ Scribed by E. Trillas; C. Alsina

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 52 KB
• Volume: 26
• Category: Article
• ISSN: 0888-613X
• DOI: 10.1016/s0888-613x(00)00066-9

✦ Synopsis

What follows are concrete comments to [3] concerning the main goal of [2], the study in Fuzzy Set Theory of the law:

(i) Provided the system in [1] contains P 0 such that tP 0 0 (if not, what?), (*) yields MaxtP Y 1 À tP 1 for any assertion P. Hence, from the very beginning the system can only contain assertions whose truth values are in f0Y 1g and obviously, either tP 1 tP 2 or tP 1 1 À tP 2 for any P 1 Y P 2 . With ``numerical'' truth values Elkan's theoretical argument is a triviality that says nothing on Fuzzy Logic. But posed with fuzzy sets, as it was done in Ref.

[11] in [2], the question is not so trivial and deserves to be reconsidered.

The system in [1], although called there formal system'', is one whose form'' (structure, initial laws) is actually hidden. Hence, it is unknown how to make inferences within it, and easy to impose much simpler laws than (*) to obtain only the numerical truth values 0 and 1.

(ii) In [3], Prof. Elkan asserts that his theorem is not about fuzzy sets but on something simpler and of great importance: fuzzy truth values attached to individual logical assertions. In [1,3] he also asserts that his theorem applies to fuzzy set theory, and that provided AxY BxY F F F refer to truth values of