Examples like the "this statement is false" thing are just rife with equivocations and other blatant nonsense. Yes, there is such a thing as "neither true nor false", but every example for "both true and false" I've seen is based on lousy thinking and semantic games.

Also, as a JS programmer, the distinction between true/false/neither/ineffable is very familiar (i.e.: true/false/null/undefined, null denoting the absence of a value and undefined denoting the absence of a definition -- though of course in practice the distinction is rarely necessary resulting in a lot of confusion and unnecessary double checks).

It's got nothing to do with "mysticism". It's just arm chair linguistics.