Abstract
David Lewis famously argued against structural universals since they allegedly required what he called a composition “sui generis” that differed from standard mereological composition. In this paper it is shown that, although traditional Boolean mereology does not describe parthood and composition in its full generality, a better and more comprehensive theory is provided by the foundational theory of categories. In this category-theoretical framework a theory of structural universals can be formulated that overcomes the conceptual difficulties that Lewis and his followers regarded as unsurmountable. As a concrete example of structural universals groups are considered in some detail.
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Notes
Indeed, Bell discusses the issue of a general category-theoretical account of mereology, but from a mathematically more advanced point of view. The present paper is content to deal with mathematically much more elementary examples.
Among the parthood lattices displayed above, only that of Z3 and Z6 are Boolean. In general, lattices of subgroups are not even distributive as is shown by the examples K and S3. For a comprehensive treatment of subgroup lattices see Schmidt (1994).
In category theory, a C-part of a C-object X is often called a subobject of X.
Already Lewis pointed out that here might be a place for mereology in a theory of universals (cf. Lewis 1986, 218, Endnote 21). He did not realize, however, that this mereology might not be standard Boolean mereology.
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Mormann, T. Structural Universals as Structural Parts: Toward a General Theory of Parthood and Composition. Axiomathes 20, 209–227 (2010). https://doi.org/10.1007/s10516-010-9105-0
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DOI: https://doi.org/10.1007/s10516-010-9105-0