Dugundji's theorem

In modal logic, Dugundji's theorem proves that no modal logic that is a subsystem of \textsf{S5} can have a finite, truth-value-like semantics.

Assume that we interpret the formulas \phi of a modal logic as elements of a set \mathcal{M}, traditionally called a matrix. This requires that we interpret all logical connectives of the logic (e.g. \land, \lor, \Box, ...) as operations on this matrix \mathcal{M}. Assume further that a subset \mathcal{T} \subseteq \mathcal{M} is used to represent the values that are "true". For example, we could interpret propositional logic by taking \mathcal{M} to be \{ 0 , 1\}, and \mathcal{T} to be \{ 1 \}.

Then, Dugundji's theorem proves that no finite-value matrix can characterise any logic that is a subset of \textsf{S5}.

References

• Dugundji's theorem, by Marco Cognilio.

• There is a textbook presentation in Chapter 3 of

@book{carnielli_2008,
	address = {Dordrecht},
	title = {Modalities and {Multimodalities}},
	url = {http://link.springer.com/10.1007/978-1-4020-8590-1},
	publisher = {Springer Netherlands},
	author = {Carnielli, Walter and Pizzi, Claudio},
	year = {2008},
	doi = {10.1007/978-1-4020-8590-1},
}