A Note on the Cardinalities of Sets of Scalar Alternatives

Formal theories of scalar implicature appeal crucially to a set of alternatives. These are the alternative statements that a speaker could have made but chose not to in pragmatic accounts, and the alternative statements that figure in the computation of exhaustivity operators in grammatical approaches. I show that the three sufficiently explicit theories of alternatives in the literature generate sets of alternatives that grow at least exponentially as a function of the input, and that these theories generate very large sets even for relatively small inputs. For pragmatic accounts of scalar implicature, I argue these results are hard or impossible to square with what we know independently about manipulating alternatives from the psychology of human reasoning. I propose that they pose a weaker but more general challenge for grammatical approaches, since alternatives as required by exhaustivity operators occur elsewhere in grammar, for example as part of the semantics of operators like “only” and “even.”