Numbers and Cardinalities: What's Really Wrong with the Easy Argument for Numbers?

Abstract: This paper investigates a certain puzzling argument concerning number expressions and their meanings, the Easy Argument for Numbers. After finding faults with previous views, I offer a new take on what’s ultimately wrong with the Argument: it equivocates. I develop a semantics for number expressions which relates various of their uses, including those relevant to the Easy Argument, via type-shifting. By marrying Romero’s (Linguist Philos 28(6):687–737, 2005) analysis of specificational clauses with Scontras’ (The semantics of measurement, 2014) semantics for Degree Nouns, I show how to extend Landman’s (Indefinites and the type of sets, Blackwell, Oxford, 2004) Adjectival Theory to numerical specificational clauses. The resulting semantics can explain various contrasts observed by Moltmann (Philos Stud 162:499–536, 2013a), but only if Scontras’ contention that degrees and numbers are sortally distinct is correct. At the same time, the Easy Argument can establish its intended conclusion only if numbers and degrees are mistakenly assumed to be identical.

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