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Abstract

Part/whole is said in many ways: the leg is part of the table, the subset is part of the set, rectangularity is part of squareness, and so on. Do the various flavors of part/whole have anything in common? They may be partial orders, but so are lots of non-mereological relations. I propose an “upward difference transmission” principle: x is part of y if and only if x cannot change in specified respects while y stays the same in those respects.

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Fig. 1
Fig. 2

Notes

  1. Lewis (1991). See also Yablo (1992).

  2. Residuated partial orders, we might call them (a residuated lattice is a partially ordered set with operations \(\wedge\) and \(\leftarrow\) related roughly as intersection and relative complementation. The existence of difference-makers corresponds to closure under the second operation).

  3. I am not taking a stand on this either way. The thought was only that we have must have some other way of thinking of part/whole, that lets us maintain our focus on it while questioning (e.g) transitivity.

  4. West with the Night, Beryl Markham.

  5. Lewis (1988a, b).

  6. See below; m is connected to \(\mathbf{m}^\prime\) iff they have a non-trivial part in common.

  7. Lewis (1988b). (Fig. 1).

  8. m and n are disconnected, then, insofar as there is no non-trivial p whose cells are subdivided both by the cells of m and by those of n. This will be the case if disjunctions of ways for things to be m-wise can never agree with disjunctions of ways for things to be n-wise, unless both are the set of all worlds whatsoever. Yablo (2014) shows how orthogonality and disconnectedness can come apart. See also Humberstone (2000).

  9. I assume that each n-cell intersects at least one m-cell.

  10. Noted already by Ramsey in his review of the Tractatus.

  11. Pronounced similarly.

  12. The example of Goats eat cans is due Benj Hellie.

  13. Related definitions are given in Gemes (1997) and Fine (2013).

  14. If E is The world will end in fire or ice (Frost), then worlds where it ends in fire should differ where E’s subject matter is concerned from worlds where it ends in ice.

  15. Also S’s ways of being false, but we ignore this until the very end of the paper.

  16. The two are equivalent if subject matters are equivalence or similarity relations.

  17. \(\dot{\vee }\) is exclusive disjunction.

  18. The definition as written makes p \(\vee\) q part of p & q. This can be blocked by stipulating that B’s minimal countermodels \(\overline{\beta }\), too, should be in each case included in some minimal countermodel \(\overline{\alpha }\) of A. The definition then becomes: B \(\le\) A iff

    1. (i)

      \(\forall \alpha\,\exists \beta\) such that \(\beta \subseteq \alpha\) (A implies B),

    2. (ii)

      \(\forall \beta\,\exists \alpha\) such that \(\beta \subseteq \alpha\) (A’s subject matter includes B’s), and

    3. (iii)

      \(\forall \overline{\beta }\,\exists \overline{\alpha }\) such that \(\overline{\beta }\subseteq \overline{\alpha }\) (A’s subject anti-matter includes B’s).

    The reason p \(\vee\) q is not part of p & q is that the former’s only minimal countermodel assigns falsity to both p and q, whereas the latter’s minimal countermodels assign falsity only to one or the other; an assignment to both is not included in an assignment to either.

  19. Assume for these purposes that sets can survive changes of membership. Alternatively we could speak, perhaps, of one set being replaced by another.

  20. G is how F is possessed by a in w iff a is F in w by being G there.

  21. Which I tried to work up into an analysis in Yablo (1999).

  22. I don’t say that this simple two-way distinction is all we’d ever need, or that the distinction is clear, or that “intensive/extensive” is the right way to think of it. The richness of this terrain is brought out by Kratzer: “Paula didn’t paint apples and bananas apart from painting a still life. Painting apples and painting bananas was part of her painting a still life, like my arms and legs are part of me” (Kratzer 1989, 609). The fruit-painting was part of painting the still life in what sense? The analogy with arms and legs suggests extensive parthood; the rest of the time she was painting a pheasant and wildflowers. Even if the still life consisted entirely of apples and bananas, though, the fruit-painting would still not an “extra” thing she does. Consider a simpler case: Paula paints a red tomato by painting a Scarlet Runner. Now its the thinner, less specific, event that plays the part role, which suggests we’re talking about intensive part. See below for Kratzer’s notion of lumping.

  23. Allis and Koetsier (1991, 1995).

  24. This alternative has its own problems, Schaffer showed in email correspondence. But they are not the problems we’re worried about now.

  25. I admit it took me a while to appreciate the force of this worry. My first thought was that one just had to be more careful about the time: y is part of x over a certain interval of time only if intrinsic changes in y over that interval are passed up to x. No such luck, however.

  26. Nor could they have been forbidden, without the whole of nature outside of y having to stop in its tracks.

  27. Lewis (2000).

  28. See the “Dialogue with a Lunatic” in Kratzer (1989). “Lunatic: What did you do yesterday evening? Paula: The only thing I did yesterday evening was paint this still life…Lunatic: This is not true. You also painted these apples and you also painted these bananas. Hence painting this still life was not the only thing you did yesterday evening.” \(\psi\) is not an “extra” thing done, for Kratzer, if \(\varphi\) “lumps” \(\psi\), that is to say, every actual \(\varphi\)-supporting situation is a situation in which \(\psi\). holds. A disjunction, for instance, lumps its true disjunct. (To avoid overdetermination worries, as when a disjunction has two true disjuncts, we should say that \(\varphi\) lumps \(\psi\) if there exists a \(\varphi\)-supporting situation each of whose \(\varphi\)-supporting sub-situations also supports \(\psi\).)

  29. Thanks to Louis DeRosset for urging this line of response.

  30. Conjecture: Part and way are duals in the case of content parts. An initial hypothesis along these lines: C is a way for A to be true iff \(\lnot\) C \(\le\) \(\lnot\) A, and C is part of all stronger Ds such that \(\lnot\) D \(\le\) \(\lnot\) A.

  31. Assume for these purposes that an unripe tomato is green.

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Yablo, S. Parts and differences. Philos Stud 173, 141–157 (2016). https://doi.org/10.1007/s11098-014-0433-6

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