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By A.I. Arruda, et al

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Nonetheless, these logics do not question, at least directly, the principle of contradiction, and the possibility of inconsistent but non-trivial theories. l, but the derrogation of the principle of contradiction is "unthinkable" for the intuitionists. On the other hand, although in some manyvalued logics the principle of contradiction is not a thesis, this fact is simply a consequence of the enlargement of the logical values and not of a critical analysis of the principle itself. Consequently, one can argue that in many-valued systems, negation laks most of its power.

We think that this is the most exciting open problem in this area. We would like to think that the present paper is a step in this direction. We 43 44 MIROSLAV BENDA define certain axiom schemas AD, AI' ... which are stronger and stronger, the step from An to An+1 being an addition of the combination if 3 to the axioms in An" The schema AO is equivalent to the induction schema. The schemas are formulated in the language of set theory. It is well-known that PA is (up to relative interpretability) Zermelo's set theory with the axiom that everything is finite replacing the axiom of infinity.

C. A. da Costa and R. , New York, 1-8. pp. L. Aposte1. 1967. Log-ique. , in Logique et Connaissance Scientifique, 197+. t:Uca In. t, in La Formalizazzione della Dialet- (Ed. J. Piaget), Ga11imard, P~ris, tica (Ed. D. Marconi), Rosemberg pp. 357-374. &Se11ier, Torino, to appear. Aristotle. 1955. The Works of Aristotle, vol. 1 (logical works), Oxford University Press. A. I. Arruda. 1964. Considerac;oes sobre os Sistemas Formais NFn (Thesis), Universidade Federal do Parana, Curitiba, Brazil. 1967.

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