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Paper IPM / M / 47  


Abstract:  
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_{1}, a
theory axiomatized by T→ ⊥. The intersection
CPC ∩E_{1} is axiomatizable by the Principle of the Excluded
Middle A∨¬A. If B is a formula such that
(T→ B)→ B is 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→A)→ A) → (T→ A).
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