Full Text (With omissions as indicated)
[ … snip …]. Supplementation then emerges in the familiar guise [ … snip …].
- As we found it, mereology was dominated by a single theory: classical extensional mereology (CEM), present in two logical guises—the Calculus of Individuals and Mereology---each in a number of variants. CEM is algebraically neat: only a complete Boolean algebra is neater. It is also strong: how strong can be seen from § 1.4. CEM is
… (1) tenseless
… (2) non-modal1
… (3) upholds extensionality of parts; and
… (4) upholds the conditioned existence of general sums.
Of these characteristics (1) and (2) are privative, (3) and (4) are positive. Among approaches at variance with CEM, most retain (1)-(3) and drop (4) in favour of some weaker conditional existence principle (see § 2.9).
- In the face of apparent temporal and modal2 variation, two major strategies have been followed. The first ignores modality3 and attempts to retain (3) by recourse to an ontology of four-dimensional objects. This fails because modality4 still distinguishes objects which (3) would identify. A parallel move to five-dimensional objects, with modality5 as the fifth dimension, has not been seriously contemplated, which in view of the conceptual difficulties facing the four-dimensional strategy is perhaps as well. The second strategy, Chisholm's, takes both time and modality6 seriously, but preserves (3) by putting forward an ontology, opposed to common sense, of modally7 and temporally invariable objects. The problems this approach has are to find good positive arguments in its favour and to account for appearances. In my view, the price paid for retaining (3) is too high whichever strategy is followed.
- Nevertheless, if (3) and (4) are not universally acceptable, there are areas where (3) alone or both together may be correctly applied (Chapter 4). Seeing this involves recognizing two categories of particular against which there exists a deep prejudice in philosophy: pluralities and masses. Taking these into account shows that the concepts of mereology have not one but a number of analogous interpretations; what these have in common are the formal properties of these concepts as captured in their algebra.
- The question is then what mereology looks like when none of (1) - (4) is followed. We followed up consecutively the effects of adding time (Part II) and modality8 (Part III). But here let us isolate the result of giving up (3) and (4) in a non-temporal, non-modal9 context. What is the minimum we can require of a relation if it is to be one of proper part to whole? I suggest we need the following four principles10:
We have left the quantifiers off: if we universally quantify, the first principle is derivable from the usual quantifier laws in free logic, but if we take the variables instead to be free (i.e. as parameters), Falsehood is required. Notice which logical concepts are presupposed: identity and existence. The import of the principle Supplementation is clearer if we define
This is then the acceptable Weak Supplementation Principle of § 1.4, adjusted to allow for free logic as a basis.This, I suggest, is the formal skeleton of the meaning of ‘part'. For the temporal version, modify all modifiable predicates by ‘at t' and slip in ‘∀t' after other quantifiers with wide scope, i.e. add ‘always'. For modal11 and modal/temporal12 versions, replace universal closure by necessary universal closure (§7.1). Now we see the point of the principle of Falsehood in the basic version.If this is all there essentially is to the part-relation, why can stronger principles sometimes apply? The answer lies not in the part-relation itself but in the nature of the objects to which it applies. Among certain regions of objects we have extensionality and essentiality of parts; such objects fulfil these principles, but the principles are not constitutive of the part-relation, which is formal, i.e. applies in all regions. So we must distinguish global mereology, for which the four principles above provide the formal properties, and various local mereologies, where these alone do not suffice to capture the mereological properties of the objects in question. The fault of CEM is essentially that of making global what is only local.The net effect of rejecting CEM in full generality is to make mereology more complicated, but also more interesting. That most modern ontology passes mereology by is due to the inadequacy of CEM as a conceptual instrument capable of use in the variety of issues found in ontology, coupled with a historically misinformed supposition that mereology is something for nominalists only. If I am right about the formal nature of mereology, it should be neutral on the issue of nominalism/realism. If mereology can be applied universally (and that has not been shown here, because we have not discussed abstract objects), then it should regain a central position in ontology; along with existence and identity, it should take us to the heart of many ontological issues. The topics covered in Parts II and III are meant to show this: Part II for existence in and through time, for identity, matter, and form, and Part III for essence, dependence, substance, unity, integrity, and form. It is notable how many of the issues in Part III are under-represented in the contemporary literature, although they loom large in traditional ontology, where it was felt to pay to be discriminating about different kinds of parts, as the quotation from Aquinas13 at the beginning of this section shows.The contemporary field ontologist is better equipped than his predecessors because he is familiar with formal systems, a device we owe to Leibniz. The acquisition of this tool does not render the old resources — experience, wit, authority, the lore of language — obsolete, but it shifts the ontologist's role. He now has a theoretically endless supply of formal templates to hold up to the untamed phenomena, and his job now consists in fair part in constructing such formal systems and testing them for their applicability. It is tempting to be led by the attraction of internal properties of the formalism either into taking the world to be tamer than it is, or into a relativistic, pragmatic attitude to ontology which can be seen at its most significant in Quine. Such attraction, for which again Leibniz is responsible, lies behind CEM's two errors of omission and two of commission. For different regions, we need different templates, and it is mainly the templates which must be bent to fit, not the world. In the case of mereology, this fails to descend to utter relativism because the theory has a formal skeleton and a range of analogous fleshings out which provides unity in the diversity.As to the content of the ontological theses I have upheld as emerging from a rejection of CEM, I am aware of a chastening old-fashionedness in having emphasized, among other things
… (1) the variety of meanings of ‘part' and cognate concepts
… (2) their analogous connections
… (3) the centrality of continuants in ontology
… (4) the paradigms of which are natural units, especially organisms
… (5) the distinction of matter and form (structure)
… (6) the importance of composition and constitution
… (7) the distinction of essential from accidental and normal parts
… (8) the distinction between dependent and independent particulars
… (9) the idea of integrity, and its degrees
all of which points, within limits and suitably qualified, back to Aristotle.
Annotated printout filed in "Various - Papers on Identity Boxes: Vol 16 (S1: Sa-Sl)".
Footnote 10: The logical formulae are omitted from this transcript, both immediately below and where indicated by “[ … snip …]” tags. I may restore them in due course if I have time to work out the HTML. The text cannot be understood without them; refer to the book.
Footnote 13: I’ve omitted the (Latin) quotations from Aquinas and Leibniz.
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