Embracing the technicalities: Expressive completeness and revenge

The Revenge Problem threatens every approach to the semantic paradoxes that proceeds by introducing nonclassical semantic values. Given any such collection Δ of additional semantic values, one can construct a Revenge sentence: This sentence is either false or has a value in Δ.The Embracing Revenge view, developed independently by Roy T. Cook and Phlippe Schlenker, addresses this problem by suggesting that the class of nonclassical semantic values is indefinitely extensible, with each successive Revenge sentence introducing a new 'pathological' semantic value into the discourse. The view is explicitly motivated in terms of the idea that every notion that seems to be expressible (e.g., has a value in Δ, for any definite collection of semantic values Δ) should, if at all possible, be expressible. Extant work on the Embracing Revenge view has failed to live up to this promise, since the formal languages developed within such work are expressively impoverished. We rectify this here by developing a much richer formal language, and semantics for that language, and we then prove an extremely powerful expressive completeness result for the system in question.