Modal Consequence Relations Extending S4.3: An Application of Projective Unification
Wojciech Dzik, Piotr Wojtylak · Notre Dame Journal of Formal Logic · 2016
We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 (the lattice of quasivarieties of S4.3-algebras) is countable and distributive and it forms a Heyting algebra.