It Adds Up After All: Kant's Philosophy of Arithmetic in Light of the Traditional Logic1


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    Work on this paper was supported by a fellowship at the Stanford Humanities Center, which I gratefully acknowledge. The ideas benefitted from exchanges with Solomon Feferman, Michael Friedman, Gary Hatfield, David Hills, Nadeem Hussain, Beatrice Longuenesse, John MacFarlane, Elijah Millgram, John Perry, Lisa Shabel, Alison Simmons, Daniel Sutherland, Pat Suppes, Ken Taylor, Jennie Uleman, Tom Wasow, Allen Wood, and Richard Zach, as well as from audience suggestions after talks at Berkeley's HPLM working group, the William James discussion group at Stanford, the Stanford Humanities Center, HOPOS 2002, and philosophy department colloquia at New York University, Villanova University, the University of Wisconsin, Milwaukee, and the University of Utah. Finally, I am indebted to very helpful comments from two anonymous reviewers for this journal.


Officially, for Kant, judgments are analytic iff the predicate is “contained in” the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like ‘7 + 5=12.’ Kant is correct. Operation concepts (> 7+5 >) bear two relations to number concepts: >7< and >5< are inputs, >12> is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic.