There are simple, purely syntactic axiomatic proof systems for both the logical truths and the logical falsehoods of propositional logic. However, to date no such system has been developed for the logical contingencies, that is, formulas that are both satisfiable and falsifiable. This paper formalizes the purely syntactic axiomatic proof systems for the logical contingencies and proves its soundness as well as completeness.
"A Sound and Complete Proof Theory for Propositional Logical Contingencies." Notre Dame J. Formal Logic 48 (4) 521 - 530, 2007. https://doi.org/10.1305/ndjfl/1193667709