Parts and partitions

Theoria

Volumn 66, Issue 1, 2000, Pages 41-82

  Humberstone, Lloyd ^a  

^a MONASH UNIVERSITY   (Australia)

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EID: 33845944452     PISSN: 00405825     EISSN: 17552567     Source Type: Journal    
DOI: 10.1111/j.1755-2567.2000.tb01144.x     Document Type: Review

References (41)

1. On p.3 of Eleanor Rosch et al., Basic Objects in Natural Categories (Working Paper No. 43, Language Behavior Research Laboratory, University of California, Berkeley, 1975), one reads that "the aim of the present research was to show that the world does, in a sense, contain 'intrinsically separate things'." The authors oppose a view which they say (p.2) "would be reasonable only if the world were entirely unstructured". The notion of structure employed turns out to be psychological (defined in terms of stimulus discrimination), explaining the parenthetical 'in a sense'. The research reported also bears more on the question (as philosophers might put it) of the type a given token is most readily classified as falling under, rather than - what is more to the point here - how a part of reality comes to attention as a token about whose type-status such a question might arise
2. See the third (new) paragraph on p. 62 of David Lewis, On the Plurality of Worlds (Blackwell, Oxford 1986), for an illustration of this point. The intrinsic/extrinsic distinction amongst properties in play in our discussion is the same as that explained by Lewis; although the property of having parts related in such-and-such ways is intrinsic on this understanding, it would not be appropriate to regard it as non-relational, at least as picked out under the (schematic) description just given, with its conspicuous appeal to how the parts are related
3. A more detailed exposition of this and related points may be found in my 'Intrinsic/Extrinsic' (Synthese 108 (1996), 205-267)
4. Possible Worlds and Subject Matter', pp. 124-137 in Sture Allén (ed.), Possible Worlds in Humanities, Arts and Sciences, de Gruyter, Berlin 1989. (The copyright date is 1988; the conference whose proceedings are here collected was held in 1986.)
5. Perry draws attention to the distinction with which we shall be concerned here, between parts of the world and aspects of the world, as a distinction between 'chunks' and aspects (using 'part' to subsume both), at the base of p. 101 of his paper 'From Worlds to Situations', Journal of Philosophical Logic 15 (1986), 83-107, the general moral of which is supposed to be that the apparatus of possible worlds is too undiscriminating to draw many important distinctions
6. Statements Partly About Observation', Philosophical Papers 17, 1-31
7. and 'Relevant Implication', Theoria 54, 161-174
8. The final sentence of note 20 of the former paper applies no less to Perry's argument (Theoria op. cit., p. 129) as to the earlier ruminations of Nelson Goodman which provoked it
9. Statements Partly About Observation', p. 12; the inset quotation appearing next is from the same page. Note for younger readers: with the reference to Styrofoam, Lewis is making a gently humorous allusion to an influential 1970 paper on mass terms (by Terence Parsons) which featured memorable examples along the lines of "All blue Styrofoam is styrofoam"
10. See note 5 (p. 13) of Peter Simons, Parts: A Study in Ontology, Clarendon Press, Oxford 1987
11. Though he does not think this gives a good model of the part-whole relation amongst material objects, Simons (in the opening paragraph of Section 3.5, p. 127, Peter Simons, Parts: A Study in Ontology, op. cit.) accepts what he calls extensional mereology for regions of space, meaning that the part-whole relation here has the structure of a boolean algebra with its zero element removed. (Of course, we have taken the further liberty, in the hope that the passage from material objects to spatial regions makes it more palatable, of leaving the zero element in place.)
12. See for example Theorem 4.6 on p. 15 of S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York 1981
13. th Bell number, whose values for k = 8,9, as given in the text, I have taken from p. 26 of Martin Gardner's 'Mathematical Games' column on this topic in Scientific American 238 (Issue 5, May 1978, 24-30
14. A discussion of this distinction, and of the interrelations between the two notions of consequence, may be found in my papers 'Some Structural and Logical Aspects of the Notion of Supervenience', Logique et Analyse 35 (1992), 101-137
15. and 'Functional Dependencies, Supervenience, and Consequence Relations', Journal of Logic, Language and Information 2 (1993), 309-336; because the uninterpreted language of sentential logic is there to the fore, the role played by possible worlds in the and of the present text is played instead by suitable valuations (truth-value assignments)
16. A simpler example along with some other pertinent information may be found in Appendix C of my paper 'A Study in Philosophical Taxonomy', Philosophical Studies 83 (1996), 121-169
17. An excellent introduction to the topic is provided by R. A. Jaeger, 'Action and Subtraction', Philosophical Review 82 (1973), 320-329
18. the title of which is inspired by a well known remark of Wittgenstein's in the Philosophical Investigations
19. and the reply by J. L. Hudson, 'Logical Subtraction', Analysis 35 (1975),130-135
20. Additional published discussions include R. A. Jaeger, 'Logical Subtraction and the Analysis of Action', Analysis 36 (1976), 141-146
21. and J. Hornsby, 'Arm Raising and Arm Rising', Philosophy 55 (1980), 73-84. I have beard the suggestion from several people (including Peter Lavers and David Makinson) that an approach to logical subtraction analogous to the treatment of contraction in the theory of belief revision may show promise, though I would prefer something more 'objective' than this (not relying on anything like relative entrenchment of beliefs, for example). My own first venture into this area was a paper 'Reciprocal Connectives' delivered in 1973 to the weekly philosophy staff/graduate seminar at the University of York, written without knowledge of any of the above publications and motivated by a concern to fill in what seemed to be an expressive gap in the logical vocabulary of (e.g.) English. A later version ('Towards a Non-Truth-Functional Account of Logical Subtraction') was presented at the annual conference of the Australasian Association of Philosophy in Wollongong, 1979 and a still later version ('Logical Subtraction: Problems and Prospects') to a University of London philosophy colloquium (University College, London) in 1981. These papers were not published, and the current incarnation of this material is in a section on logical subtraction in the 'And' chapter of a book in preparation (working title: The Connectives), on some of which the present discussion is based
22. F.C. Jackson, Perception, Cambridge University Press, Cambridge 1977; the following quotation is from pp. 4-5
23. The notoriety (of potential unwanted sensitivity to choice of primitives) comes from such discussions as Nelson Goodman's 'grue' example, though more directly bearing on the way I have put the point here, the most important reference is perhaps David Miller, 'Popper's Qualitative Theory of Verisimilitude', British Journal for the Philosophy of Science 25 (1974), 166-177
24. This is essentially the example used in the Arizonan/Minnesotan discussion in Section 3 of Miller British Journal for the Philosophy of Science op. cit.; the intertranslatability point relies on the equivalence in classical logic of (A ↔ B) ↔ B with A. (By contrast with the earlier point about the non-conservativeness of an extension with the unrestricted Subtraction Rule, the present point is highly sensitive to the choice of classical logic.)
25. As Miller remarked in a later paper ('Verisimilitude Redeflated', British Journal for the Philosophy of Science 27 (1976), 363-381)
26. the first person to make such a point was Max Black, in whose A Companion to Wittgenstein's Tractatus (Cambridge University Press, Cambridge 1964) it appears on p.46, though the example there is slightly more complicated than Miller's. We describe Miller's example in Section 5 below
27. For let B be such a formula. Then B is variable-disjoint from p ∨ q and so also from p, even though B, p ∨ q ⊢ p ; the inconsistency of B then follows by what D. J. Shoesmith and T. J. Smiley (Multiple-Conclusion Logic, Cambridge University Press, Cambridge 1978) call the Cancellation Property (mid-page 272): 'Cancellation' here refers to simply deleting a formula from a consequence statement, rather than the 'canceling-out' we have in mind in the Cancellation Rule above
28. While we are addressing such terminological matters, it is worth pointing out that the 'Canceling Out' fallacy described in P. T. Geach, Reference and Generality (Cornell University Press, Ithaca 1968) does indeed have as a special case the invoking of an unrestricted version of our Cancellation Rule - whence the need for the restrictions, in fact
29. 2, as explained in Section 3; thus the latter is complete logical independence
30. This usage, reminiscent of Popper, of the term 'content' is found for instance in Lewis, 'Statements Partly About Observation'; in the unpublished work mentioned in note 20, I used this idea of content to examine the suggestion that we explain A - B for arbitrary A, B, by stipulating that the content of A - B is to be the content of A 'minus' (in the sense of relative complementation or set-theoretic difference) the content of B: as the reader may check, this makes the proposition expressed by A - B the same as that expressed by B ⊃ A, confirming that it is this is the correspondence that Hudson has in mind. (Of course, in view of what was said at the start of Section 3 above, I think any such truth-functional account is a non-starter amongst candidate explications of logical subtraction. A more thorough discussion of this matter would need to consider the role of implications other than that of classical propositional logic, as having converses worth considering as candidates for logical subtraction. In this connection, it is worth recalling that the theory of BCK-algebras originated with the observation that arithmetical subtraction and set-theoretic difference share many formal similarities with the converse of implication in some weaker logics
31. See the survey article by K. Iséki and S. Tanaka, 'An Introduction to the Theory of BCK-Algebras', Mathematica Japonica 23 (1978), 1-26.)
32. This point is structurally analogous to Max Black's point (Mathematica Japonica op. cit., p.47) about the constitution of Wittgenstein's atomic facts out of objects. Note that there has been no resort to facts - whether of the eccentrically Tractarian variety or of a more familiar sort - in the present account; my impression is that unless something like the mereological emphasis of our discussion is injected into the discussion of facts, issues of independence and orthogonality will remain unresolved and we might as well have remained at the linguistic level
33. 2 example of $3)
34. Thus we should in the present context be especially careful to avoid such tak as Peter Unger's, of 'parts of the whole truth about everything' which are claimed (inter alia) to be the proper objects of knowledge in his paper, 'Truth', pp. 257-291 in M. K. Munitz and P. K. Unger (eds.), Semantics and Philosophy, New York University Press, New York 1974
35. (This material also appears in Chapter VII of Unger, Ignorance, Clarendon Press, Oxford 1975.)
36. R. Stalnaker, 'Anti-Essentialism', pp. 343-355 in P. A. French, T. E. Uehling and H. K. Wettstein (eds.) Midwest Studies in Philosophy, Vol. IV: Studies in Metaphysics, University of Minnesota Press, Minneapolis 1979
37. D. H. Sanford, 'Independent Predicates', American Philosophical Quarterly 18 (1981), 171-174
38. P. Gärdenfors, 'Frameworks for Properties: Possible Worlds vs. Conceptual Spaces', pp. 383-407 in L. Haaparanta, M. Kusch, and LNiiniluoto (eds.), Language, Knowledge and Intentionality: Perspectives on the Philosophy of Jaakko Hintikka (Acta Philosophica Fennica 49) 1990
39. § 6.6 of Graham Oddie, Likeness to Truth, Reidel, Dordrecht 1986
40. From p. 86 of 'Moral Beliefs', pp. 83-100 in P. Foot (ed.), Theories of Ethics, Oxford University Press, London 1967
41. I am indebted to audiences in seminars at the Australian National University and Monash University for several suggestions and corrections which have been incorporated into the present version. I am also deeply grateful to David Lewis for his enthusiastic interest in the material on logical subtraction, as it has undergone successive modifications over the years (see note 20). Lewis himself considers what seems to be a closely related issue in his paper 'A Problem about Permission', pp. 163-175 in E. Saarinen et al. (eds.), Essays in Honour of Jaakko Hintikka, Reidel, Dordrecht 1979