 For a general t-norm there remains the possibility that formula_75 has a jump discontinuity at formula_76, in which case formula_77 could come out strictly larger than formula_28 even though formula_79 is defined as the least upper bound of formula_80s satisfying formula_81; to prevent that and have the construction work as expected, we require that the t-norm formula_7 is left-continuous.The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic.More algebraically, we say that an operation formula_46 is a residuum of a t-norm formula_7 if for all formula_26, formula_28, and formula_87 it satisfies
This equivalence of numerical comparisons mirrors the equivalence of entailments
that exists because any proof of formula_92 from the premise formula_40 can be converted into a proof of formula_94 from the premise formula_5 by doing an extra implication introduction step, and conversely any proof of formula_94 from the premise formula_5 can be converted into a proof of formula_92 from the premise formula_40 by doing an extra implication elimination step.