A strong completeness theorem for $3$-valued logic.
H. Goldberg, Hugues Leblanc, George E. Weaver · Notre Dame Journal of Formal Logic · 1974
We establish here that Wajsberg's axiomatization of SC 3 , the 3-valued sentential calculus, is strongly complete, Theorem 1, p. 329, and by rebound weakly complete, Theorem 2, p. 329.Theorem 2 is a familiar result, obtained by Wajsberg himself in [5], and Theorem 1 can be recovered from results in [3], But because of its simplicity and directness our proof of Theorem 1 may be worth reporting.1 The primitive signs of SC 3 are '~', '=», '(', ')', and a denumerable infinity of sentence letters, say '/>', ( q 9 , 'r' r , 'p", ( q", 'r", etc.The wffs of SC 3 are those sentence letters, plus all formulas of the sort ~A9 where A is a wff, plus all those of the sort (A ^ B), where A and B are wffs.The length l(P) of a sentence letter P is 1; the length l(~A) of a negation ~A is l(A) + 1; and the length l((A => B)) of a conditional {A D B) is l(A) + l(B) + 1.We abbreviate the wff ( ~(p p) 9 as