Idempotent variations on the theme of exclusive disjunction

An exclusive disjunction is true when exactly one of the disjuncts is true. In the case of the familiar binary exclusive disjunction, we have a formula occurring as the first disjunct and a formula occurring as the second disjunct, so, if what we have is two formula-tokens of the same formula-type—one formula occurring twice over, that is—the question arises as to whether, when that formula is true, to count the case as one in which exactly one of the disjuncts is true, counting by type, or as a case in which two disjuncts are true, counting by token. The latter is the standard answer: counting by tokens. James McCawley once suggested that, when the exclusively disjunctive construction in natural language (well, in English at least) is at issue, the construction should be treated as involving a multigrade connective whose semantic treatment is sensitive to the set of disjuncts rather than the corresponding multiset. Without any commitment as to whether there actually is such a construction (in English), and conceding that for obvious pragmatic reasons such ‘repeated disjunct’ cases would be at best highly marginal, we note that for the binary case, this requires a nonstandard answer—count by type rather than by token—to the earlier question, and thus, an idempotent exclusive disjunction connective. Section 2 explores that idea and Section 3, a further idempotent variant for which it is the propositions expressed by the disjuncts, rather than the disjuncts themselves, that get counted once only in the case of repetitions. Sections 1 and 4 respectively set the stage for these investigations and conclude the discussion (after noting an intimate connection between the logic of Section 3 and the modal logic S5). More detailed considerations of points arising from the discussion but otherwise in danger of interrupting the flow are deferred to a ‘Longer Notes’ appendix at the end (Section 5.).