nLab
Kripke frame

Frames in Modal Logics

Warning

(The term frame is used in a different sense in modal logic than in geometric logic where it stands for an abstraction of the algebraic structure corresponding to the lattice of open sets of a topological space; see frame.)

Frames in Monomodal Logics

We will start with the simplest case, namely a frame for the basic modal language, ω(1)\mathcal{L}_\omega(1), (so with one unary modal operator, denoted \Diamond).

Definition

A frame for ω(1)\mathcal{L}_\omega(1) is a pair 𝔉=(W,R)\mathfrak{F} = (W,R) with WW a non-empty set and RR a binary relation on WW.

(This is also called a basic Kripke frame, after the philosopher and logician Saul Kripke.)

The terminology often used refers to WW as the set of possible worlds. Its elements are sometimes called worlds, sometimes states, sometimes points, depending on the context and the whim of the writer. The relation RR is called the accessibility relation so Rwv R w v says ‘vv is accessible from ww’.

Frames in Multimodal Logics

(N.B. Here we are still restricting to multimodal logics in which the modal operators are unary. The generalisation to allowing more general nn-ary modalities will be considered later.)

The generalisation is not difficult. In ω(1)\mathcal{L}_\omega(1), the single unary modality \Diamond is modeled by one binary relation. In ω(n)\mathcal{L}_\omega(n), there are nn-unary modalities so the frames have nn-binary relations. Explicitly we have

Definition

A frame for ω(n)\mathcal{L}_\omega(n) is a (n+1)(n+1)-tuple 𝔉=(W,{R i} {i=1,,n})\mathfrak{F} = (W,\{R_i\}_{\{i=1,\ldots, n\}}) with WW a non-empty set and for each i=1,,ni=1,\ldots, n, R iR_i a binary relation on WW.

(More to go here … frames with unary and more general relations.)

Models in Modal Logics

To give the standard (geometric) semantics of modal logics one needs the models and these will be discussed in the entry geometric models for modal logics and the companion algebraic models for modal logics.

References

Generally this entry is based on

  • P. Blackburn, M. de Rijke and Y. Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,

(any mistakes or errors of interpretation are due to ….!)

Revised on January 16, 2014 03:46:41 by David Corfield (84.93.103.209)