On Fuzzy Logic III. Semantical completeness of some many‐valued propositional calculi

Now we come to speak of the first non-trivial question which, in a way, puts to the test all the definitions presented so far, viz. the question of balance of syntax against semantics. Every choice of an enriched complete residuated lattice & = ((L, 8, +), S ) of a certain type ( A T : A + N,, Ex: A + N*), plus a set P of propositional variables, poses the problem of axiomatizability of the 8-valued propositional calculus over P. According to [3], M. 25, the question actually goes like this: (Q) Do there exist an L-fuzzy set of logical axioms A : F(P, L, A ) --f L and a set W of L-valued rules of inference in F(P, L, A ) such that for any L-fuzzy theory X: F(P, L, A ) + L over P and any formula e) E F(P, L, A ) the degree ( g y ( p , 8 ) X ) e), to which q~ follows from X in the L-semantical system ( F ( P , L , A ) , Y ( P , a)), equals exactly the degree ( q A , @ X ) q ~ , to which q~ is provable from X in the L-syntactical system ( F ( P , L, A ) , A, W ) ? l ) In this paper we investigate the case when the underlying lattice of B is a chain. an abstract set of formulas ([3]);