Subsentential utterances provide an ideal testing ground for issues central to the topic of the syntax–pragmatics interface such as the exact nature of the interaction between the information derivable from the structure and information obtained through pragmatic inference or default (automatic) pragmatic adjustments of meaning. Our case study for the purpose of this chapter is incomplete disjunction of the form ‘p or…’, in particular in interrogative constructions, with the second disjunct missing—unpronounced or, in some cases, even ‘unthought’ as our analysis of the corpus data shows. The chapter offers an attempt at a formal treatment, using the theoretical framework of default semantics (DS), of the compositional representation of such expressions, accounting for the sources of information, the processes involved in the recovery of the intended meaning, as well as allowing some preliminary insight into their interaction. It makes use of the database of incomplete disjunctive interrogatives in English complied out of the Great British component of the International Corpus of English, supplemented with examples taken from the Australian National Corpus and thereby offers theoretically motivated explanations for the pragmatic effects that incomplete disjunctive interrogatives have been observed to occasion in naturally occurring interactions. The category of disjunction is semantically and pragmatically complex: It can communicate, among others, that (i) the addressee is given a choice out of a set of alternatives (ii) the addressee has to think of possible alternatives, (iii) the speaker lacks information to make a stronger claim, and (iv) it would not be correct to make an informatively stronger assertion because two or more states of affairs are (equally) plausible. In Sect. 2, we point out the variety of functions that or can adopt. In Sect. 3, we introduce the theoretical problem with incomplete disjunction, and in Sect. 4, we move to the discussion of the semantics and pragmatics of incomplete utterances, focusing on contextualist pragmatic vis-à-vis syntactic ellipsis accounts. Section 5 offers a DS-theoretic analysis of the sources of information about completions and the associated processes that produce the truth-conditional representation. Section 6 lays out theoretical foundations for the mechanism of establishing relevant alternatives by proposing an extension to alternative semantics. Section 7 follows with the presentation of the merger representations for selected examples from the corpus, pertaining to different categories of ‘or…?’ constructions we previously identified.