Henkin style completeness proofs in theories lacking negation.
John L. Pollock · Notre Dame Journal of Formal Logic · 1971
As they are customarily formulated, Henkin style completeness proofs are not applicable to logical theories lacking negation.The purpose of this note is to show that if such a theory contains disjunction, either as a primitive logical constant or as a defined one, then a slight modification of the ordinary constructions can be used to construct a completeness proof.This procedure yields completeness proofs for a large group of truthfunctionally incomplete propositional calculi.Many of these completeness results are already known, but this procedure yields a much simpler proof than the customary ones, and establishes all of these results simultaneously rather than piecemeal.The procedure can also be used in more complex cases, for example, first-order theories lacking negation.