A restriction on the distribution of exclusive only

(2) a. Three eggs are sufficient to bake this cake. b. Five guests drank over half the beers between them. c. Bob ran to the store in six minutes. These bound inferences can be explained as routine scalar implicatures by observing that, in each case, the degree predicate λn . φ(n) obtained by abstracting over the numeral is either downward scalar (φ(n) entails φ(n−1)) or upward scalar (φ(n) entails φ(n + 1)). For instance, [λn . Alice read n books] is downward scalar, because if she read three books, then she also read two; thus, higher numerals are more informative than lower numerals, and so we draw upper-bound inferences (Horn 1972). Conversely, [λn . n eggs are sufficient to bake this cake] is upward scalar, because if three eggs are sufficient, then so are four (Beck and Rullmann 1999); thus, lower numerals are more informative than higher numerals, and so we draw lower-bound inferences. As is well known, the exclusive only may attach to the sentences in (1) to turn the upper-bound inference into a semantic entailment, suggesting that only happily combines with downward-scalar numerical sentences to exclude higher-numeral alternatives.