Two distinct and apparently "dual" traditions of non-classical logic, three-valued logic and paraconsistent logic, are considered here and a unified presentation of "easy-to-handle" versions of these logics is given, in which full naive set theory, i.e. Frege's comprehension principle + extensionality, is not absurd.

In this paper, we present set theories based upon the paraconsistent logic Pac. We describe two different techniques to construct models of such set theories. The first of these is an adaptation of one used to construct classical models of positive comprehension. The properties of the models obtained in that way give rise to a natural paraconsistent set theory which is presented here. The status of the axiom of choice in that theory is also discussed. The second leads to show (...) that any classical universe of set theory (e.g. a model of ZF) can be extended to a paraconsistent one, via a term model construction using an adapted bisimulation technique. (shrink)

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...) is ramifiable” implies the existence of a proper class of inaccessible cardinals. (shrink)

An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK∞+. We show that in these conditions, we can interprete the Kelley‐Morse theory (KM) in GPK∞+ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo‐Fraenkel set theory (ZF) in GPK∞+. We also see what happens in (...) the original GPK theory. Before doing this, we first need to study the basic properties of the theory. This is done in the first two sections. (shrink)

In this paper, we give an alternative to M. Forti and F. Honsell's construction of hyperuniverses. We use this construction to see precisely on what conditions there can exist a $\kappa$-hyperuniverse of given uniform weight. This condition is expressed in terms of mildly ineffable cardinals.