General frame

In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust completeness of the latter.

A modal general frame is a triple ${\displaystyle \mathbf {F} =\langle F,R,V\rangle }$, where ${\displaystyle \langle F,R\rangle }$ is a Kripke frame (i.e., R is a binary relation on the set F), and V is a set of subsets of F which is closed under the following:

The purpose of V is to restrict the allowed valuations in the frame: a model ${\displaystyle \langle F,R,\Vdash \rangle }$ based on the Kripke frame ${\displaystyle \langle F,R\rangle }$ is admissible in the general frame F, if

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