Lewisian Fixed Points I: Two Incomparable Constructions

Our paper studies what one might call “the reverse mathematics of explicit fixed points”. We discuss two methods of constructing such fixed points for formulas whose principal connective is the intuitionistic Lewis arrow \(\rightsquigarrow \) . Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allow several natural kinds of semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a modal system \(\mathsf {L^{\flat }_a}\) , extending the “gathering” axiom \(4_{\textsf{a}}\) with the standard (“box”) version of the Löb axiom. The second one, inspired by de Jongh and Sambin, seemingly simpler, leads to a modal theory \(\textsf{JS}^\flat \) , which proves harder to axiomatize in an elegant way. Apart from showing that both theories are mutually incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixed points of simple formulas. We also show that both \(\mathsf {L^{\flat }_a}\) and \(\textsf{JS}^\flat \) are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfers to its finite extensions.