In [modal logic], Dugundji's theorem proves that no modal logic that is a subsystem of can have a finite, truth-value-like semantics.
Assume that we interpret the formulas of a modal logic as elements of a set , traditionally called a matrix. This requires that we interpret all logical connectives of the logic (e.g. , , , ...) as operations on this matrix . Assume further that a subset is used to represent the values that are "true". For example, we could interpret propositional logic by taking to be , and to be .
Then, Dugundji's theorem proves that no finite-value matrix can characterise any logic that is a subset of .
Dugundji's theorem, by Marco Cognilio.
There is a textbook presentation in Chapter 3 of
Carnielli, Walter, and Claudio Pizzi. 2008. Modalities and Multimodalities. Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-1-4020-8590-1.
@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},
}