Formalisations of Many‐Valued Propositional Calculi with Variable Functors

Let C and N be the implication and negation functors respectively of LUKASIEWICZ (see [ 7 ] ) and let T be the tertium functor of SLUPECKI (for m = 3 see [25] , for 3 i m < N~ see [ 2 2 ] ) . Let 6, = { F , , . . ., Fk} (k 2 2 ) be a set of n-argument functors (?L 2 2 ) and let G, , . . ., G, be a (possibly empty) list of the functors N and T with no repetitions. We consider here the m-(2 5 m < N ~ ) and valued propositional calculi in which the primitive symbols are propositional variables, the constant functors G I , . . . , G, (T is not relevant in the No-valued case, of course) and variable functors taking values from the set €,. For examples of such propositional calculi which satisfy definability conditions to be given in $ 1 we give complete formalisations (see $9 2 , 3). The methods we describe were developed in the special case of variable functors taking values from the set (C, C’} (see [4], [ 5 ] ) and having N as an additional primitive constant functor (see [6]). The only designated truth value is 1.