On the completeness of some first-order extensions of C

We show the completeness of several Hilbert-style systems resulting from extending the propositional connexive logics C and C3 by the set of Nelsonian quantifiers, both in the varying domain and in the constant domain setting. In doing so, we focus on countable signatures and proceed by variations of the Henkin construction. We compare our work on the first-order extensions of C3 with the results of an earlier paper by Omori and Wansing and answer several open questions naturally arising in this respect. In addition, we consider possible extensions of C and C3 with a non-Nelsonian universal quantifier preserving a specific rapport between the interpretation of conditionals and the interpretation of the universal quantification which is visible in both intuitionistic logic and Nelson’s logic but is lost if one adds the Nelsonian quantifiers on top of the propositional basis provided by C and C3. We briefly explore the completeness of systems resulting from adding this non-Nelsonian quantifier either together with the Nelsonian ones or separately to the two propositional connexive logics.