On Some Interpretations of Classical Logic

Abstract

In distinction from the well-known double-negation embeddings of the classical logic we consider some variants of single-negation embeddings and describe some classes of superintuitionistic first-order predicate logics in which the classical first-order calculus is interpretable in such a way. Also we find the minimal extensions of Heyting's logic in which the classical predicate logic can be embedded by means of these translations. MSC: 03B20, 03B55 Key words: Embeddings of classical logic in superintuit.ionistic logics.The first intuitionistic interpretation of classical logic w'a.5 the famous double-negation translation of classical mathematics into int,uitionistic mathematics defined by A. K. KOLMOGOROV [8]. With the help of V. I. GLIVENKO'S result [5], K. GODEL [6]presented a similar relationship between classical and intuitionistic first-order arithmetic. Another interesting double-negation reduction of the classical system into the intuitionistic one was described by S. KURODA [lo]. We mention also the translations considered by H. B. CURRY [3], K . ONO [14], I<. KAwADA and N . MUTI [7), as well as the results obtained by G . KREISEL [9], G . E. MINC and V. P. OREVKOV [12], D. PRAWITZ and P.-E. MALMNAS [lS], C. CELLUCCI [l], D. M. GABBAY [4] (p.41) and D. LEIVANT [ll].