## Section: Scientific Foundations

### Combining Classical and Intuitionistic Proof Systems

In order to develop an approach to proof certificates that is as comprehensive as possible, one needs to handle theorems and proofs in both classical logic and intuitionistic logic. Yet, building two separate libraries, one for each logic, can be inconvenient and error-prone. An ideal approach would be to design a single proof system in which both classical and intuitionistic proofs can exist together. Such a proof system should allow cut-elimination to take place and should have a sensible semantic framework.

Liang and Miller have recently been working on exactly that problem. In their paper [7] , they showed how to describe a general setting for specifying proofs in intuitionistic and classical logic and to achieve one framework for describing initial-elimination and cut-elimination for such these two logics. That framework allowed for some mixing of classical and intuitionistic features in one logic. A more ambitious merging of these logics was provided in their work on “polarized intuitionistic logic” in which classical and intuitionistic connectives can be used within the same formulas [14] .