Talk:Monoidal t-norm logic
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[edit] Truth functions?
As a logician, I have (briefly) looked at fuzzy "logic" on a few occasions. It has always seemed bogus to me. Specifically, if one uses the most commonly used assignment of functions to logical operators, there are no tautologies at all. Consequently, one cannot make any logical inferences. However, your article seems (superficially, because I do not understand it yet) to be of a higher quality. But it does not specify what functions (such as maximum or minimum or x+y-1 or whatever) correspond to the logical operators. Could you please specify what the truth functions are? JRSpriggs 03:30, 3 May 2007 (UTC)
- Yes, the article is incomplete yet (that's why I marked it as a stub, even though it is already longer than a typical stub). I'm working on the Semantics section right now (as time allows — unfortunately I'm a bit too busy these weeks) and hope to have a publishable version soon. Meanwhile I can give a quick answer to your question here:
- In the standard (real-valued) semantics, the truth-function of strong conjunction can be any left-continuous t-norm (including the Łukasiewicz t-norm x + y − 1 you mentioned). The truth function of implication is then its residuum. Weak conjunction and weak disjunction are interpreted as the minimum and maximum, respectively, and of course the truth constants 0 (bottom) and 1 (top) denote the truth values 0 and 1. Other connectives (negation, equivalence) are definable from these.
- Regarding your remark on fuzzy logic seeming a bogus — unfortunately it's completely true that a large part of what is called "fuzzy logic" is rather a toolbox of engineering methods than logic, and the rhetorics about its achievements is often exaggerating. Still, some (very small) part of it does make sense from the point of view of formal logic; to distinguish it from the rest, the term mathematical (also symbolic, formal, or deductive) fuzzy logic is often used. Deductive systems of mathematical fuzzy logic are closely related to substructural logics (esp. linear logic), some of them even to intuitionistic logic (e.g., Gŏdel–Dummett logic, which extends intuitionistic logic by the axiom (A → B) ∨ (B → A), happens to be one of important fuzzy logics). I hope to be able to write more WP articles about mathematical fuzzy logic soon and elaborate on what I've just hinted at here. -- LBehounek 20:15, 3 May 2007 (UTC)
