Evaluation Trees for Proposition Algebra - The Case for Free and Repetition-Proof Valuation Congruence

Proposition algebra is based on Hoare’s conditional connective, which is a ternary connective comparable to if-then-else and used in the setting of propositional logic. Conditional statements are provided with a simple semantics that is based on evaluation trees and that characterizes so-called free valuation congruence: two conditional statements are free valuation congruent if, and only if, they have equal evaluation trees. Free valuation congruence is axiomatized by the four basic equational axioms of proposition algebra that define the conditional connective. A valuation congruence that is axiomatized in proposition algebra and that identifies more conditional statements than free valuation congruence is repetition-proof valuation congruence, which we characterize by a simple transformation on evaluation trees.