Iterating Both and Neither: With Applications to the Paradoxes

A common response to the paradoxes of vagueness and truth is to introduce the truth-values “neither true nor false” or “both true and false” (or both). However, this infamously runs into trouble with higher-order vagueness or the revenge paradox. This, and other considerations, suggest iterating “both” and “neither”: as in “neither true nor neither true nor false.” We present a novel explication of iterating “both” and “neither.” Unlike previous approaches, each iteration will change the logic, and the logic in the limit of iteration is an extension of paraconsistent quantum logic. Surprisingly, we obtain the same limit logic if we use (a) both and neither, (b) only neither, or (c) only neither applied to comparable truth-values. These results promise new and fruitful replies to the paradoxes of vagueness and truth. (The paper allows for modular reading: for example, half of it is an appendix studying involutive lattices to prove the results.)