In this paper we construct the five-parameter generating solution of N = 8 BPS regular supergravity black holes as a five-parameter solution of the N = 2 ST U model. Our solution has a simpler form with respect to previous constructions which have already appeared in the literature and moreover, through the embedding [SL(2)] 3 ⊂ SU (3, 3) ⊂ E 7(7) discussed in previous papers, the action of the U -duality group is well defined. This allows us to reproduce via U -duality rotations any other solution, such as those corresponding to R-R black holes whose microscopic description is given by intersecting D-branes.

The paper studies hereditarily complete superintuitionistic deductive systems, that is, the deductive system which logic is an extension of the intuitionistic propositional logic. It is proven that for deductive systems a criterion of hereditary structurality -similar to one that exists for logics -does not exists. Nevertheless, it is proven that many standard superintuitionistic logics (including Int) can be defined by a hereditarily structurally complete deductive system.

Abstract. This work was intended to be an attempt to introduce the meta-language for working with multiple-conclusion inference rules that admit asserted propositions along with the rejected propositions. The presence of rejected propositions, and especially the presence of the rule of reverse substitution, requires certain change the definition of structurality.

Positive logics are $\{ \wedge , \vee , \to \}$ -fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

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Abstract. We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics.