Abstract
AbstractA variety of intuitionistic versions of modal logic $$ K $$ have been proposed in the literature. An apparent misconception is that all these logics coincide on their $$\Box $$-only (or $$\Diamond $$-free) fragment, suggesting some robustness of ‘$$\Box $$-only intuitionistic modal logic’. However in this work we show that this is not true, by consideration of negative translations from classical modal logic: Fischer Servi’s $$ IK $$ proves strictly more $$\Diamond $$-free theorems than Fitch’s $$ CK $$, and indeed $$i K $$, the minimal $$\Box $$-normal intuitionistic modal logic.On the other hand we show that the smallest extension of $$i K $$ by a normal $$\Diamond $$ is in fact conservative over $$i K $$ (over $$\Diamond $$-free formulas). To this end, we develop a novel proof calculus based on nested sequents for intuitionistic propositional logic due to Fitting. Along the way we establish a number of new metalogical results.
Publisher
Springer Nature Switzerland
Reference36 articles.
1. Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence);M Acclavio,2021
2. Arisaka, R., Das, A., Straßburger, L.: On nested sequents for constructive modal logic. Log. Methods Comput. Sci. (2015)
3. Bellin, G., De Paiva, V., Ritter, E.: Extended curry-howard correspondence for a basic constructive modal logic. In: Proceedings of Methods for Modalities, vol. 2 (2001)
4. Blackburn, P., van Benthem, J.F., Wolter, F.: Handbook of Modal Logic. Elsevier, Amsterdam (2006)
5. Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic, vol. 53. Cambridge University Press, Cambridge (2001)