Incompleteness and finite models

Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 12)

In the last section of Chapter 4, we have met a non-canonical modal system. Up to now, our completeness results relied on canonicity, i.e., on the fact that every canonical system is automatically complete. In KGL, the lack of canonicity makes it problematic to prove completeness via the method applied in the preceding sections. However, it does not establish what we may call F -incompleteness, i.e, that there is no class of frames with respect to which the given system is valid. We will be able to show that KGL is complete with respect to the class of irreflexive and wellcovered frames. But we are going to show, nevertheless, that there are modal systems that are F -incomplete, in the sense that no class of frames characterizes them.

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