Scalar implicatures are cancellable inferences that appear to arise from considering certain alternative sentences. When first described by Paul Grice (1967), scalar implicatures were taken to be the result of processes of pragmatic inference. In a nutshell: starting from what the speaker literally said, the hearer will consider what stronger statements the speaker chose not to utter, and conclude that the speaker believes all such statements to be false, as long as their falsehood is compatible with what she literally said. More recent approaches reject the pragmatic dimension of the phenomenon, proposing unpronounced operators that derive scalar implicatures as a matter of literal meaning rather than the result of pragmatic inferences (Chierchia et al., 2012).
In this note, I show that there is one technical component, universally employed in formal theories of scalar implicature, that has very puzzling consequences under the assumption that these theories ought to be psychologically tenable. The issue can be summarized as follows. Every modern theory of scalar implicature makes crucial use of a set of scalar alternatives, sentences that are in a precise sense related to the sentence uttered by a speaker. These are the alternatives that a hearer will take into consideration when thinking of what the speaker could have said but chose not to. They are the alternatives that unpronounced operators in non-pragmatic approaches must access in order to deliver truth conditions for sentences containing them. I show that the cardinalities of these sets increase at very fast rates, and moreover that, even for sentences with a relatively small number of coordinated clauses, the cardinalities of alternative sets are very large numbers. If the theories of alternatives considered here are making claims about psychological processes, then these claims are very difficult to square with what we know about alternatives in grammar and in reasoning. On the other hand, if these theories are to be taken as mathematical idealizations or theories of pragmatic competence, then it becomes necessary to investigate what psychologically tenable heuristics might implement this competence. It does not follow from the facts I report here that modern formal approaches to scalar implicature are doomed. Instead, I point out a collection of puzzling and previously unnoticed facts that require answers.