Hume's big brother: Counting concepts and the bad company objection

Synthese 170 (3):349 - 369 (2009)
A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising.
Keywords Frege  Neo-logicism  Abstraction  Arithmetic  Higher-order logic  Bad company objection
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DOI 10.2307/40271356
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