Logical paradoxes for many-valued systems

Abstract

As we know, the system of material implication is led into an inconsistency by the Russellian class, defined as λx. Nϵxx. This class, however, does no harm to many other systems, for example, the three-valued system L3 given by J. Łukasiewicz. A natural question is whether or not there exist classes which affect some of these other logical systems. The main result of the present paper is to answer this question affirmatively. At the end of this paper we point out that the interpretation given by J. Łukasiewicz for the system L3 is not satisfactory, and propose a new interpretation.B. Russell deduced the mentioned inconsistency by the aid of the notion of negation. Later on, H. B. Curry pointed out that we could get the same result without the aid of that notion. None of these results affects the system L3 and other similar systems. But these systems may be involved.To show this, we need the following definitions.A function of two variables Cpq will be called an implication when the following “implication rule” is valid:Under this definition we should note that material equivalence Epq, for example, is an implication.Let C be such an implication. Then the symbol “(Cp)iq” is defined recursively byThe class an is defined as λx. (Cϵxx)np, where p is a propositional variable. The rule of absorption of order n, denoted by (An), is: