Definition
A modal general frame is a triple, where 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 Boolean operations of (binary) intersection, union, and complement,
- the operation, defined by .
The purpose of V is to restrict the allowed valuations in the frame: a model based on the Kripke frame is admissible in the general frame F, if
- for every propositional variable p.
The closure conditions on V then ensure that belongs to V for every formula A (not only a variable).
A formula A is valid in F, if for all admissible valuations, and all points . A normal modal logic L is valid in the frame F, if all axioms (or equivalently, all theorems) of L are valid in F. In this case we call F an L-frame.
A Kripke frame may be identified with a general frame in which all valuations are admissible: i.e., where denotes the power set of F.
Read more about this topic: General Frame
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