This paper is divided into an empirical part and a theoretical part. The empirical part rediscovers a peculiar creature which was found by [Cooper, 1983] and then plunged back into the abyss of oblivion — indexical presuppositions, a special kind of "presuppositions" which can only, according to Cooper, be satisfied in the actual world. Cooper claimed that presuppositions of the gender features of free anaphoric pronouns induce such non-projecting "presuppositions". I enlarge the empirical scope of the discovery by showing that, first, indexical presuppositions are induced by gender features of all anaphoric pronouns, not just the free ones; and second, that their satisfaction requirements are more complicated than simple "satisfaction in the actual world only." On the theoretical side, I sketch three possible analyses of indexical presuppositions. The first one relies on direct copying of features from the antecedent, but meets certain serious problems. More work is needed in order to figure out if that theory can actually be worked out. The second theory employs a very complex constraint on choosing the name for the world variable of the pronoun, crucially using both syntax of the sentence and the model against which it is interpreted. The third one, in a sense the least conventional of all three, moves all the work to semantics, and analyzes indexical presuppositions as conditions on a rich structure of worlds, somewhat doubling the structure of embedding in the formula, which replaces the standard single possible world evaluation parameter.
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