Axiom schemes for <i>m</i>-valued propositions calculi

In an m -valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m -valued propositional calculi, and it should be observed at the outset that two m -valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.