Towards a Proof System for Admissibility

In [4] a basis for the admissible rules of intuitionistic propositional logic IPC was given. Here we strengthen this result by presenting a system ADM that has the following two properties. \(\phantom{~} A\vdash_{\sf ADM}B\) implies that A admissibly derives B. ADM is complete in the sense that for every formula A there exists a formula \(\phantom{~} A\vdash_{\sf ADM}\Lambda_A\) such that the admissibly derivable consequences of A are the (normal) consequences of \(\phantom{~} \Lambda_A\). This work is related to and partly relies upon research by Ghilardi on projective formulas [2, 3].