We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E l , a theory axiomatized by T -+ 1. The intersection C P C n El is axiomatizable by the Principle of the Excluded Middle A V -A. If B is a formula such that (T -+ B) -+ Bis not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula (T + B) -+ B is derivable exactly when B is provably equivalent to a formula of the form ((T .-t A) + A ) .-t (T + A ) .Mathematics Subject Classification: 03B20, 03B55, 03C90.

No abstract

No abstract

In "Intuitionistic validity in T-normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory T, that of all T-normal Kripke structures H (T ) for which he gave an r.e. axiomatization.In the language of arithmetic Iop and Lop denote PA − plus Open Induction or Open LNP, iop and lop are their intuitionistic deductive closures. We show H (Iop ) = lop is recursively axiomatizable and lop i c iop, while i∀ 1 lop.If iT proves PEM atomic but not totality of a classically provably total Diophantine function of T, then H (T ) ⊆ iT and so iT ∈ range(H ). A result due to Wehmeier then implies i 1 ∈ range(H ). We prove Iop is not ∀ 2 -conservative over i∀ 1 . If Iop ⊆ T ⊆ I∀ 1 , then iT is not closed under MR open or Friedman's translation, so iT ∈ range (H ). Both Iop and I∀ 1 are closed under the negative translation.

We study the linear Lindenbaum algebra of Basic Propositional Calculus, called linear basic algebra. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)