“School of Mathematics”
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| Paper IPM / M / 47 |
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| Abstract: | |||||
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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 E1, a
theory axiomatized by T→ ⊥. The intersection
CPC ∩E1 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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