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Basic Propositional Calculus I
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AbstractWe 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 V ∨ ⌝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).
Title: Basic Propositional Calculus I
Description:
AbstractWe 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 V ∨ ⌝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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