From axioms to synthetic inference rules via focusing

An important application of focused variants of Gentzen's sequent calculus proof rules is the construction of (possibly) large synthetic inference rules. This paper examines the synthetic inference rules that arise when using theories composed of bipolars, and we do this in both classical and intuitionistic logics. A key step in transforming a formula into synthetic inference rules involves attaching a polarity to atomic formulas and some logical connectives. Since there are different choices in how polarity is assigned, it is possible to produce different synthetic inference rules for the same formula. We show that this flexibility allows for the generalization of different approaches for transforming axioms into sequent rules present in the literature. We finish the paper showing how to apply these results to organize the proof theory of labeled sequent systems for several propositional modal logics.