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Paper   IPM / M / 47
School of Mathematics
  Title:   Basic propositional calculus I
1.  W. Ruitenburg
2.  M. Ardeshir
  Status:   Published
  Journal: Math. Logic Quart.
  Vol.:  44
  Year:  1998
  Pages:   317-343
  Supported by:  IPM
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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