LingLang Lunch (12/1/2016): Peter Alrenga (Boston University)

At least and at most: Ignorance and variation in focus

A hallmark feature of the scalar operators at least and at most is their capacity to convey speaker uncertainty: from an utterance of (1), a listener would typically infer that the speaker does not know the exact number of points that LeBron scored.

(1) LeBron scored at least / at most 20 points in last night’s game.

These uncertainty implications tend to disappear in the presence of modals: under their most salient interpretations, neither (2a) nor (2b) need convey any uncertainty regarding what is necessary or required:

(2) a. (In order to win the scoring title), LeBron needs to score at least 45 points
          in tonight’s game.
      b. One person can submit at most one abstract as sole author and one
          abstract as co-author (or two co-authored abstracts).

Rather, the most salient interpretations for these sentences convey variation in what the speaker deems to be sufficient or permissible. Similar variation implications can also be observed in combination with nominal quantifiers:

(3) a. Every player scored at least 10 points in last night’s game.
      b. Individuals can give to as many federal candidates as they want, so long as
          they give at most $2600 to any single candidate in an election cycle.

The question of exactly how at least and at most manage to convey uncertainty and variation in (1)-(3) has attracted considerable scrutiny. Recent work has converged on the view that these implications are implicatures arising from the interaction of the basic semantic properties of at least / at most with general pragmatic mechanisms. A near-universal impulse of these pragmatic approaches is to draw an analogy to disjunction, which gives rise to a similar pattern of uncertainty and variation implications. But capitalizing on this analogy has proven surprisingly difficult. In its most direct form, it amounts to the view that at least and at most form n-ary disjunctions over their associated scalar terms and all higher / lower ones. While such a view correctly characterizes the truth-conditional contribution of at least, it appears to to mischaracterize its pragmatic behavior. And without further amendment, it fails to even adequately capture the truth-conditional contributions of at most.

In the first part of this talk, I argue that a version of the simple view can indeed be maintained for at least, once it is recognized that (i) the scales that at least and at most operate over are fundamentally pragmatic/contextual in nature, and (ii) these scales are never ordered by entailment. While the simple n-ary disjunction view cannot be maintained for at most, I show how its essential insights into at most‘s pragmatic behavior nevertheless can be. In the second part of the talk, I apply the resulting analysis to certain unresolved problems concerning the interactions of these scalar operators with modals and other quantifiers.