The title “Law of the Excluded Middle” suggests its own shortcoming, but it is a reduction of the actual problem, which is that truth and falsehood fall into vagueness. Previous posts explore this claim and the reification of uncertainty. As such, the “middle” which may be considered a zone where it is not possible to discover a truth value, is bordered by the zones of truth and falsehood. The trouble is these borders are also vague, giving rise to a zone where it is unclear whether it is true, or whether it is not possible to know a truth value. The boundaries of these secondary zones give rise to further zones of vagueness. What is the overall shape of this terrain, what kind of mental movement can traverse this landscape?
One model could be the Cantor set, where the middle of an interval is removed, resulting in new intervals with new middles to be removed, and so on. The “excluded middles” are the spaces of uncertainty, and “points” are knowledge. We can conclude from this model that uncertainty and knowledge permeate each other inextricably. Thus, concepts are a composite of truth and falsehood, a composite that is not a matter of degree. As composites, it is natural for them to decay. Kuhn, in true Ivy League form, repackaged the ancient idea that all concepts naturally arise, develop, and pass away in his theory of the paradigm. There is a circular form to be found in the opposition of Truth and Falsehood- one of growth and decay. The model of the Cantor set however, is probably not accurate nor approximate. It is merely a beautiful idea. If a measurement of degree is found for the Cantor set, there will be new middles of intervals to be removed, and the development and decay of theories continues to cycle.
Another model was proposed by Henri Bergson. [more later]