From the subformula property to cut-admissibility in propositional sequent calculi

While the subformula property is usually a trivial consequence of cut-admissibility in sequent calculi, it is unclear in which cases the subformula property implies cut-admissibility. In this paper, we identify two wide families of propositional sequent calculi for which this is the case: the (generalized) subformula property is equivalent to cut-admissibility. For this purpose, we employ a semantic criterion for cut-admissibility, which allows us to uniformly handle a wide variety of calculi. Our results shed light on the relation between these two fundamental properties of sequent calculi and can be useful to simplify cut-admissibility proofs in various calculi for non-classical logics, where the subformula property (equivalently, the property known as 'analytic cut-admissibility') is easier to show than cut-admissibility.¹