Universal proof theory: Semi-analytic rules and Craig interpolation

We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including $U{L}^{-}$, $\mathrm{MTL}$, $R$, Ł_n (for $n⩾3$), G_n (for $n⩾4$), and almost all extensions of $\mathrm{IMTL}$, $Ł$, $\mathrm{BL}$, $R{M}^e$, $\mathrm{IPC}$, $\mathrm{S4}$, and $\mathrm{Grz}$ (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.