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.

Definition

A modal general frame is a binary relation that is closed under the following:

• the Boolean operations of (binary) intersection, union, and complement,

• .

They are thus a special case of fields of sets with additional structure is admissible, if

1. for every propositional variable.

for every (not only a variable).

is valid. A normal modal logic, if all axioms (or equivalently, all theorems-frame.

denotes the power set.

Types of frames

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

• differentiated,

• tight,

• compact with the finite intersection property has a non-empty intersection,

• atomic contains all singletons,

• refined, if it is differentiated and tight,

• descriptive, if it is refined and compact.

Kripke frames are refined and atomic. However, infinite Kripke frames are never compact. Every finite differentiated or atomic frame is a Kripke frame.

Descriptive frames are the most important class of frames because of the duality theory (see below). Refined frames are useful as a common generalization of descriptive and Kripke frames.

Operations and morphisms on frames

induces is defined as

The fundamental truth-preserving operations of generated subframes, p-morphic images is a generated subframe), and

A p-morphism (or bounded morphism, and satisfies the additional constraint

1. .

The disjoint union is the disjoint union, and

The refinement defined as follows. We consider the equivalence relation

. Then we put

Completeness

is complete with respect to a single descriptive is complete with respect to its canonical model, and the general frame induced by the canonical model (called the canonical frame) is descriptive.

Jónsson–Tarski duality

General frames bear close connection to modal algebras is closed under Boolean operations, therefore it is a subalgebra of the power set Boolean algebra. It also carries an additional unary operation is a modal algebra, which is called the dual algebra.

In the opposite direction, it is possible to construct the dual frame has a Stone space is the set of all ultrafilters consists of the clopen is defined by

.

is descriptive if and only if.

become a pair of contravariant functors between the category of general frames, and the category of modal algebras. 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.

Intuitionistic frames

The frame semantics for intuitionistic and intermediate logics can be developed in parallel to the semantics for modal logics. An intuitionistic general frame is a partial order is a set of upper subsetss (cones that contains the empty set, and is closed under

• intersection and union,

• .

is called

• tight,

• compact with the finite intersection property has a non-empty intersection.

Tight intuitionistic frames are automatically differentiated, hence refined.

is the Heyting algebra is the set of all prime filters is inclusion of the form

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.

See also

• Neighborhood semantics

References

• Alexander Chagrov and Michael Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.

• Patrick Blackburn, Maarten de Rijke, and Yde Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.