A TEST FOR COMPLETENESS WITH RESPECT TO IMPLICIT REDUCIBILITY IN THE CHAIN SUPER-INTUTIONISTIC LOGICS

Abstract

We examine chain logicsC2, C3, . . . ,which are intermediary betweenclassical and intuitionistic logics. They are also the logics of pseudo-Boolean algebrasof type< Em,&,∨,⊃,¬>, whereEmis the chain 0< τ1< τ2<···< τm−2<1 (m= 2,3, . . .).The formulaFis called to be implicitly expressible in logicLbythe system Σ of formulas if the relationL⊢(F∼q)∼((G1∼H1) &. . .& (Gk∼Hk))is true, whereqdo not appear inF, and formulasGiandHi, fori= 1, . . . , k, areexplicitly expressible inLvia Σ. The formulaFis said to be implicitly reducible inlogicLto formulas of Σ if there exists a finite sequence of formulasG1, G2, . . . , GlwhereGlcoincides withFand forj= 1, . . . , lthe formulaGjis implicitly expressibleinLby Σ∪{G1, . . . , Gj−1}. The system Σ is called complete relative to implicitreducibility in logicLif any formula is implicitly reducible inLto Σ.The paper contains the criterion for recognition of completeness with respect to im-plicit reducibility in the logicCm, for anym= 2,3, . . .. The criterion is based on 13closed pre-complete classes of formulas.