 Non-bivalent logics of semantic presupposition, dating back to Łukasiewicz (1930) and Kleene (1952), generally posit (at least) two not-operators, the distinction arising lexically rather than (as for Russell) scopally; see the entry on  many-valued logic  and  Section 2  below. The ordinary, presupposition-preserving internal or choice negation is the only one countenanced by Frege and Strawson; on this reading, Santa is not white, like Santa is white, is neither true nor false, given that Santa does not exist. The presupposition-cancelling or exclusion negation always determines a classical value. With exclusion negation, Santa is not white (or perhaps more plausibly It is not the case that Santa is white) is true even if there is no Santa. Thus there is no excluded middle; any affirmation and its corresponding exclusion negation are contradictories rather than contraries (see the entry on  presupposition  for elaboration and further details).