Talk:T-norm

Citation needed for origin of name

Personally, I'd mark "probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces." with a "citation needed" because I have not yet read this despite having looked in multiple books about Fuzzy logic. To be precise: There is a general agreement that t means triangular but the explanation that it is called triangular norm because it generalized the triangle equality is not something I was readily able to find. Maybe someone can provide a reference for this claim? — Preceding unsigned comment added by 141.3.74.172 (talk) 08:54, 13 June 2017 (UTC)

See Triangular Norms by Klement, Mesiar and Pap - In about this book section when you search for it (or search for Menger 1942). "The history of triangular norms started with the paper "Statistical metrics" [Menger 1942]. The main idea of Karl Menger was to construct metric spaces where probability distributions rather than numbers are used in order to de­ scribe the distance between two elements of the space in question. Triangular norms (t-norms for short) naturally came into the picture in the course of the generalization of the classical triangle inequality to this more general set­ting." Slimeythej (talk) 09:16, 17 January 2025 (UTC)

or of course the Menger text, available on J-stor. Slimeythej (talk) 09:18, 17 January 2025 (UTC)

The title does not represent the containts

I think t-norm and t-conorm should be in different articles or the title of this article should be different and include other fuzzy logic operations as negation. Your thoughts? --trylks 21:43, 15 September 2007 (UTC)

I have put t-conorms in the same article as t-norms since the former are just duals of the latter, and I did not plan to write much more on t-conorms myself; so it seemed to me that a separate article would at present just repeat (or dualize) the contents of the article on t-norms. However, if anybody thinks that t-conorms (or residua) deserve a separate article and that there is enough material for one, I am not going to oppose. I suppose that in the long run, many other fuzzy-logic operators would deserve their own articles as well. I wrote only on t-norms (and their derivative operators) since they are needed for t-norm based fuzzy logics like MTL, which were my primary aim. Any expansion of articles related to t-norms or other fuzzy-logic operators is of course very much welcome. -- LBehounek 18:44, 27 September 2007 (UTC)

Should add "probabilistic sum"

$\sum _{\text{prob}}(x,y)=x+y-x\cdot y$

--Divof (talk) 12:49, 3 January 2021 (UTC)

Continuity of real functions in two variables

Currently, the article says: "Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]², [...]" I think it should say "one variable" instead of "each variable" because if a function f is continuous in both variables then f is continuous. Daniel Hernández (talk) 17:37, 3 April 2023 (UTC)

No, what is written is correct and your contrary assertion is wrong. The function $f(x,y)={\begin{cases}0&x=y=0\\{\frac {xy}{x^{2}+y^{2}}}&{\text{o.w.}}\end{cases}}$ is a standard example. --JBL (talk) 17:51, 3 April 2023 (UTC)