I understand that classical logic preserves truth, is bivalent, and contains non-constructive proofs. In contrast, intuitionistic logic preserves justification/verification for a proof, is not bivalent, and contains only constructive proofs.
Is there an example of a logic that preserves justification/verification, but is neither constructive nor bivalent? Is there something about a logic that preserves proof that requires it to be constructive?
There is a logic that is neither bivalent nor constructive: Lukasiewiz 3-valued logic. The chief reason it is not more widely employed appears to be that in the usual and original form, it is not truth-preserving. Neither modus ponens nor the transitive law of the conditional hold, which make deduction difficult, to say the least.
What is not generally recognized is that they should not. Lukasiewicz logic allows doubtful conditionals. If modus ponens held, it would be possible to combine a doubtful premise and a doubtful conditional to yield a false conclusion. Likewise, if the transitive law held, it would be possible to chain two doubful conditionals to start from a true premise and reach a false conclusion.
It is also not generally recognized that it is possible to easily define a strict conditional that is truth preserving. I note that this is not the same as the strict conditional employed in the Lewis systems of modal logic.