Cylindrification

Cylindrification is a term used in mathematical logic and model theory, specifically in the field of modal logic. It refers to a specific operation that is applied to a set of worlds in a Kripke model. The operation of cylindrification is used to ensure that certain properties hold across all possible worlds in the model.

Definition

In the context of modal logic, the cylindrification of a set A of worlds in a Kripke model M is defined as the set of all worlds in M that have the same truth value for all propositions as some world in A. Formally, if M is a Kripke model and A is a set of worlds in M, then the cylindrification of A in M, denoted by Cyl(A), is defined as follows:

Cyl(A) = {w in W | there exists a world v in A such that for all propositions p, M, w |= p if and only if M, v |= p}

Applications

Cylindrification is used in the study of modal logic to ensure that certain properties, such as bisimulation and invariance under renaming, hold across all possible worlds in a Kripke model. It is also used in the construction of canonical models in modal logic.

See also

• Modal logic
• Kripke model
• Bisimulation
• Canonical model

References

• Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge University Press.
• Hughes, G. E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. Routledge.