classical logic through refutation and rejection GABRIELE PULCINI Dipartimento di Studi Letterari, FilosoĄci e di Storia dellŠArte, Università di Roma ŞTor VergataŤ ACHILLE C. VARZI Department of Philosophy, Columbia University Final version published in Melvin Fitting (ed.), Selected Topics from Contemporary Logics, London, College Publications, 2021, pp. 667–692 abstract We ofer a critical overview of two sorts of proof systems that may be said to characterize classical propositional logic indirectly (and non-standardly): refutation systems, which prove sound and complete with respect to classical contradictions, and rejection systems, which prove sound and complete with respect to the larger set of all classical non-tautologies. Systems of the latter sort are especially interesting, as they show that classical propositional logic can be given a paraconsistent characterization. In both cases, we consider Hilbert-style systems as well as Gentzen-style sequent calculi and natural-deduction formalisms. 1 introduction In logic we are generally interested in truth-preserving arguments. Given certain premises that we take to be true, we want to see what follows from them, what else must be true. This was AristotleŠs fundamental insight: [C]ertain things being stated, something other than what is stated follows of necessity from their being so. [Prior Analytics 24b18Ű20; 1: 40] As a result, formal logic tends to focus on the development of proof systems that allow us to do just that, to derive truths (and only truths) from truths. In particular, we want the theorems of our proof systems to be exactly those We are grateful to an anonymous referee for their very detailed and helpful feedback on the penultimate draft of this paper. 1