On finite-valued propositional logical calculi (Q1913635)

The authors propose a general method for constructing Hilbert-type propositional calculi for all truth-complete \({\mathbf C}\)-extending logics. A logic \(L_n\) is said to be \({\mathbf C}\)-extending iff in \(L_n\) one can functionally express the classical operations \(\wedge\), \(\vee\), \(\supset\) and \(\neg\). The completeness of these calculi is proved, using a construction of the realization for formulas and valuations by means of the \(J\)-operators that were introduced by Rosser and Turquette \((J_\alpha (\beta) \equiv [\text{if } \beta= \alpha: 1\text{ else } 0])\) [see \textit{J. B. Rosser} and \textit{A. R. Turquette}, Many-valued logics (1952; Zbl 0047.01503)].