Talk:Monoidal t-norm logic

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)

Motivation and jargon

Very pleased to see a § Motivation section before launching into the technicalities, as it's often difficult for non-specialists to get a handle on why we'd bother exploring such arcane and abstract mathematical structures. Yet, unfortunately, beyond the early link to T-norm, much of this text duplicates that in T-norm fuzzy logics § Motivation. This is not very helpful! As an encyclopaedia, Wikipedia tries to explain technical subjects with a minimum of jargon, but a maximum of precision within the constraints of the necessary paraphrases. But here, even the lead paragraph is so full of technical jargon that it would scare off all but the most determined enquirer. Frankly, I'm not at all sure how to make this material more accessible to intelligent enquiry. Yet the writing needs improving, without sacrificing accuracy.

Does anybody know of a more accessible (popular or undergraduate) exposition of any of this particular material on fuzzy logic? Or know enough to help translate the ideas into less formal ones? Or have some hints on increasing the readability of this text – say, for example, using diagrams? yoyo (talk) 14:59, 6 July 2018 (UTC)

It occurs to me that what should be the key point in the motivation is that the residuation property reflects the equivalence in deductive logic between $A,B\vdash C$ and $A\vdash B\to C$ (a proof of either can be transformed into a proof of the other, by adding an extra modus ponens or implication introduction step). The left-continuity of the t-norm is only needed to have equivalence between $a*b\leqslant c$ and $a\leqslant b\Rightarrow c$. 130.243.68.38 (talk) 12:57, 25 November 2019 (UTC)

I (same editor as previous remark, though at a different IP address) have today finished a rewrite along those lines, and removed the "too technical" cleanup template from the Motivation section. 95.199.149.236 (talk) 08:42, 18 December 2019 (UTC)

Motivation still missing: the other and

Still missing from the motivation section is any insight into why we want an idempotent ‘and’ $\wedge$ in addition to the t-norm ‘and’ $\otimes$. 95.199.149.236 (talk) 08:42, 18 December 2019 (UTC)

Meaning of monoidal in topic

The article should eventually explain what the attribute 'monoidal' in the title signifies. Presumably not just that $*$ is the operation in a monoid, since any t-norm is already by the definition of that stated to be the operation in a (commutative) monoid. Is it perhaps rather related to the way the attribute figures in monoidal category (which besides associativity of $\otimes$ also places requirements on its interaction with the $\mathrm {Hom}$ functor $\longrightarrow$)? 130.243.68.38 (talk) 13:11, 25 November 2019 (UTC)