An Automated Derivation of Two Alternate Axiomatic Bases for Łukasiewicz's Sentential Calculus

The optimization of computing systems incorporating Boolean-circuit-based computing equipment must be expressed at some level in Boolean behaviors and operations. Boolean behaviors and operations are part of a larger family of logics -the logic of sentences, also known as the "sentential calculus". Two logics are implicationally equivalent if the axioms and inference rules of each imply the axioms of the other. Characterizing the inferential equivalences of various formulations of the sentential calculi is thus foundational to the optimization of Boolean-oriented computing systems. Using an automated deduction system, I show that one of the most austere formulations of the sentential calculus, Łukasiewicz's CN, has at least two alternate axiomatic bases; the bases appears to be novel. The proofs further demonstrate a natural proving order that both informs and constrains optimization strategies.