In fuzzy logics, a binary truth function serving to generalize the classical notion of conjunction. In semantics in which the set of truth values is the real interval $\left[0,1\right]$ (or a subset of this interval), a triangular norm is a binary function ∗ mapping pairs of values of $\left[0,1\right]$ to $\left[0,1\right]$ such that for all $m,n,m^{\prime },n^{\prime }\in \left[0,1\right]$, the following conditions are satisfied:

1. 1 $m\ast n=n\ast m$

2. 2 $\left(m\ast n\right)\ast m^{\prime }=m\ast \left(n\ast m^{\prime }\right)$

3. 3 If $m\le m^{\prime }$ and $n\le n^{\prime }$, then $m\ast n\le m^{\prime }\ast n^{\prime }$

4. 4 $m\ast 1=m$

An associated notion is that of a residuum of a t-norm (corresponding to the residuum of meet), i.e., a binary function $\to$ on the real interval $\left[0,1\right]$ fulfilling the criteria:

• • $m\le m^{\prime }\to n$ if and only if $m\ast m^{\prime }\le n$

When a particular t-norm has a residuum, the residuum is frequently taken to define a conditional connective. Hence, in t-norm fuzzy logics, the t-norm not only determines a notion of conjunction, but provides a foundation for the treatment of the conditional as well.