Abstract
Christian List (Noûs, forthcoming, 2018, [24]) has recently proposed a category-theoretic model of a system of levels, applying it to various pertinent metaphysical questions. We modify and extend this framework to correct some minor defects and better adapt it to application in philosophy of science. This includes a richer use of category theoretic ideas and some illustrations using social choice theory.
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- 1.
In Sect. 6.3 we will consider whether this requirement is justified.
- 2.
Of course, there is a sense in which this is true for any application of category-theoretic apparatus, at least insofar as one can represent a category as a set-theoretic structure. But this trivial sense is not the one we have in mind here.
- 3.
In more detail, an inverse limit in a category \(\mathfrak {C}\) can be characterized as the limit of a functor from a partially ordered set, considered as a small category, to \(\mathfrak {C}\), and such limits exist when \(\mathfrak {C}\) has small products and equalizers [9, Theorem 2.8.1]. Posetal categories trivially always have equalizers, but their products are just greatest lower bounds.
- 4.
Thanks to Christian List for this suggestion. Note that the Cartesian product is the product in the category of sets, not in a posetal category—see footnote 3.
- 5.
Proof Suppose for reductio that \([[\phi ]] = \{\omega \}\). Since \(\phi \) is a finite sentence, not every sentence letter can occur in it. So suppose \(P_i\) does not occur in \(\phi \). Then since \(\omega \models \phi \), it must be the case that \(\omega ' \models \phi \), where \(\omega '\) is just like \(\omega \) save that \(P_i \not \in \omega '\) (as the truth value of a sentence in propositional logic is dependent only on the truth values of the sentence letters occurring in it). But then \(\omega ' \in [[\phi ]]\), although \(\omega ' \ne \omega \), so we have a contradiction. \(\square \)
- 6.
- 7.
For discussion of Beth’s theorem, see [17].
- 8.
For a generalisation of Beth’s theorem to many-sorted logics, see [2].
- 9.
- 10.
This is an instance of what Button and Walsh [11] call the “push-through” construction.
- 11.
Namely, that “not every ‘complex’ of ‘lower-grade’ entities will be a higher entity; there is no useful sense in which a slab of marble is a higher entity than the smaller marble parts that make it up” [20, p. 11].
- 12.
For emphasis we have italicized the word “each” in the passage.
- 13.
For a philosophical investigation of the notion of geometric possibility and its role in the relationism-substantivalism debate, see Belot [7].
- 14.
If \(\sigma \) were not total, then \(1_\varOmega \) could not be; if \(\sigma '\) were not surjective, then \(1_\varOmega \) could not be. The same reasoning applies mutatis mutandis to \(\sigma ' \circ \sigma = 1_{\varOmega '}\). Hence, each of \(\sigma \) and \(\sigma '\) is both total and surjective.
- 15.
So on this account, systems of levels of description can be viewed as categories of categories.
- 16.
We use the term “\(\mathbf {L}\)-structure” in its usual model-theoretic sense. So an \(\mathbf {L}\)-structure is an assignment of extensions to the descriptive symbols of \(\mathbf {L}\).
- 17.
For a discussion of the thesis that interpersonal comparisons of utility are meaningless and the inference that such comparisons are therefore impossible, see List [22].
- 18.
“ONC” stands for “ordinal measurability with no interpersonal comparability” and “RFC” is short for “ratio-scale measurability with full interpersonal comparability.” These acronyms are due to List [23].
- 19.
If F is a functor from the category \(\varOmega \) to the category \(\varOmega ^\prime \) and \(\omega _1\) is isomorphic to \(\omega _2\) in \(\varOmega \), then \(F(\omega _1)\) is isomorphic to \(F(\omega _2)\) in \(\varOmega ^\prime \).
- 20.
One might seek to model multiple realizability in this way: for example, perhaps one successful reduction translates “pain” by “firing of C-fibres” whilst another translates “pain” by “firing of D-fibres” (where, let us suppose, human brains have C-fibres and Martian brains have D-fibres). However, it is not clear to us what the prospects for this manoeuvre might be; note that neither translation will map the true higher-level claim “both humans and Martians experience pain” to a true lower-level claim.
- 21.
- 22.
- 23.
This means that their composition FG is naturally isomorphic to the identity functor on C and GF is naturally isomorphic to the identity functor on D.
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Acknowledgements
ND is primarily responsible for Sects. 6.1 and 6.2. SCF is primarily responsible for Sects. 6.3 and 6.5 and for general editing, and secondarily responsible for Sect. 6.1. LH is primarily responsible for Sect. 6.4 and for Proposition 1, and secondarily responsible for Sect. 6.3 and general editing. All authors thank Katie Robertson for many insightful conversations leading to the genesis of this essay, Tomasz Brengos and Christian List for encouraging comments on a previous version, and the audience and organizers of the workshop “New Perspectives on Inter-Theory Reduction” in Salzburg in November, 2017. SCF acknowledges partial support through a Marie Curie International Incoming Fellowship (PIIF-GA-2013-628533).
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Dewar, N., Fletcher, S.C., Hudetz, L. (2019). Extending List’s Levels. In: Kuś, M., Skowron, B. (eds) Category Theory in Physics, Mathematics, and Philosophy. CTPMP 2017. Springer Proceedings in Physics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-030-30896-4_6
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