Conditionals in constructive logics
Grigory Olkhovikov
We consider the problems arising in figuring out the right counterparts to the basic conditional logic \(\mathsf{CK}\) when the propositional basis of the logic is no longer assumed to be classical. We argue that, as long as the new underlying logic is constructive, this problem shows essential resemblance to the problem of figuring out the right intuitionistic counterparts to the well-known classical modal logics as addressed, e.g. in [3], where the famous set of six requirements was put forward.
Among these requirements, the last and the most important one demands an explanation of the semantics of conditionals/modalities in terms of the first-order version of the underlying non-classical logic, and we fundamentally agree with A. Simpson’s intepretation of this explanation as the faithfulness of the embedding into the first-order version of the underlying logic provided for the candidate conditional/modal logic by the standard translation borrowed from the classical case.
However, both the choice of the underlying non-classical logic and the peculiar features of the conditional logic may pose additional challenges.
We illustrate these challenges by the examples \(\mathsf{IntCK}\) and \(\mathsf{N4CK}\), the two recently proposed analogues of \(\mathsf{CK}\) (see [1] and [2]) based on the intuitionistic propositional logic and on the paraconsistent variant of Nelson’s logic of strong negation, respectively.
Bibliography
- G. Olkhovikov. An intuitionistically complete system of basic intuitionistic conditional logic. Submitted, preprint available at (2023).
- G. Olkhovikov. A basic system of paraconsistent Nelsonian logic of conditionals. Submitted, preprint available at (2023).
- A. Simpson. The Proof Theory and Semantics of Intuitionistic modal Logic, PhD Thesis, University of Edinburgh (1994).