Toward a Logic of Cumulative Quantification

ABSTRACT. This paper studies entailments between sentences like "three boys kissed five girls" involving two or more numerically quantified noun phrases that are interpreted as expressing cumulative quantification in the sense of Scha (1984). A precise characterization of when one such sentence entails another such sentence that differs from it only in the numerals is crucially important to evaluate claims about scalar implicatures arising from the use of those sentences, as pointed out by Shimada (2014). This problem turns out to be non-trivial and surprisingly difficult. We give a characterization of these entailments for the case of sentences with two noun phrases, together with a complete axiomatization consisting of two pairs of simple inference rules. We also give some valid inference rules for sentences with three noun phrases.