Two restrictions on possible connectives

If languages could lexicalize arbitrary truth tables as sentential connectives, we should be able to find a great variety of connectives in the world’s languages. However, very few connectives are typologically attested, as has long been known. For example, no known language lexicalizes if-and-only-if, not-and, or McCawley (1972)’s schmor, a connective that returns true exactly when two or more of its arguments are true. In fact, Gazdar (1979) makes the point that the only bona fide non-unary connectives are ∧ and ∨. This typological puzzle calls for an explanation, and indeed several proposals have been suggested in the literature.1 We examine two approaches to restricting the possible connectives. Both follow McCawley (1972), and more specifically Gazdar (1979), in assuming that connectives can only see the set of truth values of their syntactic arguments. As Gazdar notes, this eliminates sensitivity to ordering and repetitions. The first approach, growing out of Gazdar and Pullum (1976) and Gazdar (1979), takes the notion of choice as its starting point: if O is a connective and A its argument, then O(A) ∈ A. For example, if all the arguments are true, A = {1}, and O(A) is 1. We will refer to this as the choice-based approach.2 The second approach, growing out of Keenan and Faltz (1978, 1985), takes ordering as its starting point: the domain of truth values is assumed to be ordered, with 0 < 1, and a connective can only choose the maximum or minimum element within its argument. We will refer to this as the ordering-based approach.3 In the classical domain, choice and ordering seem to predict the same sentential connectives. The perspectives they offer are different, though, offering the hope of divergent predictions if we go beyond the classical domain. A tempting place to look is non-classical semantics, used to implement the Frege-Strawson program for presupposition.4 The challenge here is that there are many trivalent extensions of the classical operators, but only one