 Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold.Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation formula_100 In this way, the left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives (see the section "Standard semantics" below) determine the truth values of complex propositional formulae in [0, 1].Formulae that always evaluate to 1 are then called "tautologies" with respect to the given left-continuous t-norm formula_101 or "formula_102tautologies."