General frame

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.

that is closed under the following: They are thus a special case of fields of sets with additional structure.

is to restrict the allowed valuations in the frame: a model

may be identified with a general frame in which all valuations are admissible: i.e.,

In full generality, general frames are hardly more than a fancy name for Kripke models; in particular, the correspondence of modal axioms to properties on the accessibility relation is lost.

This can be remedied by imposing additional conditions on the set of admissible valuations.

is called Kripke frames are refined and atomic.

However, infinite Kripke frames are never compact.

is defined as The fundamental truth-preserving operations of generated subframes, p-morphic images, and disjoint unions of Kripke frames have analogues on general frames.

is a generated subframe of the Kripke frame

, and satisfies the additional constraint The disjoint union of an indexed set of frames

Then we put Unlike Kripke frames, every normal modal logic

is complete with respect to a class of general frames.

is complete with respect to a class of Kripke models

is closed under substitution, the general frame induced by

is complete with respect to a single descriptive frame.

General frames bear close connection to modal algebras.

It also carries an additional unary operation,

In the opposite direction, it is possible to construct the dual frame

has a Stone space, whose underlying set

A frame and its dual validate the same formulas; hence the general frame semantics and algebraic semantics are in a sense equivalent.

is descriptive if and only if it is isomorphic to its double dual

These functors provide a duality (called Jónsson–Tarski duality after Bjarni Jónsson and Alfred Tarski) between the categories of descriptive frames, and modal algebras.

This is a special case of a more general duality between complex algebras and fields of sets on relational structures.

The frame semantics for intuitionistic and intermediate logics can be developed in parallel to the semantics for modal logics.

is a set of upper subsets (cones) of

that contains the empty set, and is closed under Validity and other concepts are then introduced similarly to modal frames, with a few changes necessary to accommodate for the weaker closure properties of the set of admissible valuations.

is called Tight intuitionistic frames are automatically differentiated, hence refined.

are a pair of contravariant functors, which make the category of Heyting algebras dually equivalent to the category of descriptive intuitionistic frames.

It is possible to construct intuitionistic general frames from transitive reflexive modal frames and vice versa, see modal companion.