Algorithmic Correspondence, Canonicity and Completeness for Possibility Semantics

Possibility semantics. Possibility semantics for modal logic is a generalization of standard Kripke semantics. In this semantics, a possibility frame has a refinement relation which is a partial order between states, in addition to the accessibility relation for modalities. From an algebraic perspective, full possibility frames are dually equivalent to complete Boolean algebras with complete operators which are not necessarily atomic, while filter-descriptive possibility frames are dually equivalent to Boolean algebras with operators. In recent years, the theoretic study of possibility semantics has received more attention. In [23], Yamamoto investigates the correspondence theory in possibility semantics in a frametheoretic way and prove a Sahlqvist-type correspondence theorem over full possibility frames, which are the possibility semantic counterpart of Kripke frames, using insights from the algebraic understanding of possibility semantics. In [15, Theorem 7.20], it is shown that all inductive formulas are filter-canonical and hence every normal modal logic axiomatized by inductive formulas is sound and complete with respect to its canonical full possibility frame. However, the correspondence result for inductive formulas is still missing, as well as the correspondence result over filter-descriptive possibility frames (see [15, page 103]) and soundness and completeness with respect to the corresponding elementary class of full possibility frames. The present paper aims at giving a closer look at the aforementioned unsolved problems using the algebraic and order-theoretic insights from a current ongoing research project, namely unified correspondence.