Parts: A Study in Ontology
Simons (Peter)
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Cover Blurb

  1. The relationship of part to whole is one of the most fundamental there is, yet until now there has been no full-length study of this concept. This book shows that mereology, the formal theory of part and whole, is essential to ontology.
  2. Peter Simons surveys and criticizes previous theories, especially the standard extensional view, and proposes a more adequate account which encompasses both temporal and modal1 considerations in detail. This has far-reaching consequences for our understanding of such classical philosophical concepts as identity, individual, class, substance and accident, matter, form, essence, dependence, and integral whole.
  3. It also enables the author to offer new solutions to long-standing problems surrounding these concepts, such as the Ship of Theseus Problem2 and the issue of mereological essentialism.
  4. The author shows by his use of formal techniques that classical philosophical problems are amendable to rigorous treatment, and the book represents a synthesis of issues and methods from the analytical tradition and from the older continental tradition of Franz Brentano and the early Edmund Husserl.


Clarendon Press, Oxford, paperback edition, 2003 reprint

"Simons (Peter) - Parts: A Study in Ontology - Introduction"

Source: Simons - Parts: A Study in Ontology, 1987, Introduction

Introduction (Full Text)
  1. This book has two major aims. The first is to give a connected account of the various kinds of mereology, or formal theory of part, whole, and related concepts, which exist, widely scattered, in the literature. This aim is fulfilled mostly in Part I. The second and more important aim is to expose the philosophical defects of most of this tradition, and to suggest why, where, and how it should be put right.
  2. The standardly accepted formal theory of part-whole is classical extensional mereology, which is known in two logical guises, the Calculus of Individuals of Leonard and Goodman, and the Mereology of Lesniewski. Despite the discrepancies between the underlying logics of these two approaches, there is a precise sense in which both say the same things about parts and wholes. Classical extensional mereology (CEM) is subject to two major criticisms. The first is that, either out of conviction or for reasons of algebraic neatness, it asserts the existence of certain individuals, mereological sums, for whose existence in general we have no evidence outside the theory itself. The second and more fundamental criticism is that the theory is not applicable to most of the objects around us, and is accordingly of little use as a formal reconstruction of the concepts of part and whole which we actually employ.
  3. There are two reasons for this. The first is that the underlying logic of CEM does not have the resources to deal with temporal and modal1 notions in connection with mereology, such as temporary part, temporal part, essential part, or essential permanent part. This is not an internal criticism of CEM, since one could envisage suitably extending it to cope with temporal and modal2 concepts. However there is an internal reason why CEM is not suitable for such extension, and this concerns mereological extensionality. This is the thesis that objects with the same parts are identical (by analogy with the extensionality of sets, whereby sets with the same members are identical). If mereological extensionality is accepted, then two prima facie facts need to be accounted for. The first is that certain things, like human beings, have different parts at different times: they are mereologically variable or in flux. An object with different parts at different times cannot be identical with the sum of its parts at any time, for then it would be different from itself. The second problem is that some objects (again, like human beings) might have had some parts other than those they in fact have, and yet still have been the same objects. In other words, they are not modally3 rigid in their parts. If we accept mereological extensionality in a modally4 strengthened form, to the effect that objects with the same parts must be identical, then no object could have had parts other than those it actually has, a thesis called mereological essentialism, and associated with Chisholm.
  4. In the face of these two problems, a number of strategies may be adopted to preserve extensionality, some of them extreme. One may try revising the logic of identity, or denying that objects have undetached parts. The modal5 problem may be ignored by refusing to take modality6 seriously. If it is taken seriously, then it seems that mereological essentialism is the best option. One then faces the problem of explaining why it appears that ordinary objects are not modally7 mereologically rigid. Chisholm accounts for appearances by construing such objects as logical constructions out of objects for which mereological essentialism indeed holds.
  5. Since it is more difficult to ignore time than modality8, the problem of flux has been faced more often. The analogous solution to mereological essentialism is to deny that objects do vary their parts: they are mereologically constant. Chisholm holds that objects properly so called are indeed mereologically constant, and the appearance of variability among objects is to be explained by construing these, once again, as logical constructions. I argue that the price Chisholm pays is in each case too high to warrant upholding mereological extensionality.
  6. A second and more popular solution to the flux problem is to propose replacing the things (continuants) of our usual ontology by processes, which have temporal parts. I argue that the difficulties involved in such a revision have been greatly underestimated, and that in any case the move fails to save mereological extensionality because such four-dimensional objects fall prey to the modal9 argument.
  7. There are nevertheless places where extensional mereology is appropriate, in particular among events and among non-singular objects, that is, classes and masses: in the last two cases the full classical theory can apply. It emerges that `part', like other formal concepts, is not univocal, but has analogous meanings according to whether we talk of individuals, classes, or masses.
  8. It remains to be considered how mereology looks when extensionality is rejected. Part II considers the mereology of continuants, which may be in flux. The rejection of extensionality has as a consequence that more than one object may have exactly the same parts at the same time, and hence occupy the same position. Another somewhat controversial thesis defended is that an object may exist intermittently in special circumstances. These views are applied together to give a novel solution of the Ship of Theseus10 problem.
  9. Consideration of the conditions under which distinct things may be in the same place at the same time leads to a discussion of the nature of composition, constitution, and matter in their mereological ramifications. With the rejection of extensionality, it becomes possible to distinguish different concepts of proper-or-improper-part which enrich our conceptual palette and allow disputes to be resolved as turning on equivocation.
  10. Modal11 mereology has received almost no attention because the logical opinions of mereologists and modal logicians12 have usually been fundamentally opposed. Part III brings modality13 and mereology together as they are found in the work of Husserl at the beginning of the century and later in that of Chisholm. Mereological essentialism is rejected as a general doctrine, though again there are regions in which it is appropriate, essentially those where extensionality applies. For most continuants some parts are essential and others are not.
  11. Husserl used modal14 mereology as a tool in developing various concepts of ontological dependence of objects on other objects, a study which quickly leads into some of the central topics of ontology, concerning substance and conditional and unconditional existence. The modal15 approach developed here is used to re-examine traditional problems in this light, and to reassess Husserl's achievement in this field.
  12. The arguments have so far turned mainly on extensionality and what happens if it is rejected. In the last chapter I return to mereological sums, and ask what it is they seem to lack that other paradigmatic objects have. The leading idea here is that of a family of objects which is maximally connected under some relation; an object composed of such a family is integrated under the relation; such integrity — provided the relation involved is not merely formal — is what arbitrary sums and incomplete fragments lack. Among the kinds of relation constitutive of such integrity we consider forms of ontological and functional dependence, and give an account, based on work by Grelling and Oppenheim, of the characteristic structure or Gestalt of such integral wholes.
  13. A word should be said about the use of symbolic formulae and formal systems. Since most of the existing work on mereology uses such means of expression, their use is unavoidable if we are to survey such theories and compare their relative strengths. As for the rest of the book, since the aim is to discover the most suitable mereological concepts and the formal principles governing them, though it is in principle possible to do without symbols, in practice we need them for their brevity and clarity, and to enable us to test the consequences of formulae using the established means of formal logic. Nevertheless, I want to stress that this book is about ontology, not logic. The level of logical sophistication required to understand the text is not high: a journeyman's acquaintance with first-order predicate logic with identity will suffice. Even the modal16 sections require no more, provided one is prepared to indulge the fiction of possible worlds. The unfamiliar language of Lesniewski is introduced in such a way that a reader without foreknowledge may follow subsequent discussion; but in general I stay closer to predicate-logical means of expression. I have tried to make the notation as unfussy as possible; a survey of the notation used generally, in particular the conventions which allow parentheses to be almost entirely dispensed with, can be found in §2.2. Apart from this, the symbolically dense Chapter 2 is not a prerequisite for understanding the main argument. It is a survey exhibiting the riches of the predominant extensional tradition, and gathers material which is otherwise widely scattered.
  14. Finally, I should mention that the book does not deal with two areas where mereology overlaps with other important philosophical issues. The first is vagueness. Apart from some remarks in Part II, where mention of vagueness is unavoidable, this subject remains within the haven of bivalence. The second issue is whether there can be a mereology of abstract objects (if there are any). The discussion is confined to the concrete, except for a few remarks in §4.10. The reason for avoiding these two areas is that to discuss them with the required thoroughness would involve bringing in a good deal of material which is not in itself mereological, and which is also not relevant to the main argument, which is that extensional mereology is inadequate even for its primary intended sphere of application – concrete individuals.

COMMENT: Annotated printout filed in "Various - Papers on Identity Boxes: Vol 16 (S1: Sa-Sl)".

"Simons (Peter) - Concepts and Principles of Extensional Mereology"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 1

  1. Basic Concepts;
  2. An Example;
  3. The Lesniewskian Approach;
  4. Mereological Principles;
  5. Classical Extensional Mereology;
  6. The Question of Atomism;
  7. Summary

"Simons (Peter) - Survey of Extensional Mereology"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 2

  1. Introduction;
  2. Presentation of Formal Systems;
  3. Calculi of Individuals Without Sets;
  4. Calculi of Individuals With Sets;
  5. A Free Calculus of Individuals;
  6. Lesniewski's Mereology;
  7. Alternatives and Developments in Lesniewskian Mereology;
  8. Affinities;
  9. Non-Classical Extensional Mereologies;
  10. The Boundary with Topology;
  11. Appendix: The Notational Jungle

"Simons (Peter) - Problems"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 3

  1. Historical Remarks;
  2. Criticisms;
  3. The Flux Argument;
  4. The Fourth Dimension;
  5. Applications of Extensional Mereology

"Simons (Peter) - Occurrents, Classes, and Masses"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 4

  1. Occurrents;
  2. Activities and Performances;
  3. Plural Reference;
  4. Pluralities: Groups and Classes;
  5. Analogies Between Ontology and Mereology;
  6. Mass Reference and Masses;
  7. Outline of a Comprehensive Theory;
  8. Extending the Analogy?;
  9. Parts of Groups;
  10. Further Possible Applications of Extensional Mereology

"Simons (Peter) - Temporary Parts and Intermittent Existence"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 5

  1. Continuants;
  2. Temporary Parts;
  3. Chisholm's Entia Successiva;
  4. Intermittent Existence1 and Part-Replacement;
  5. Solution of the Difficulty;
  6. Corroboration Found in an English Idiom;
  7. Further Cases of Intermittence, and Conclusion

"Simons (Peter) - Superposition, Composition and Coincidence"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 6

  1. Superposition and Coincidence;
  2. Mixtures;
  3. Can Things of a Kind Be Superposed?
  4. Composition;
  5. Constitution;
  6. The Problem of Matter;
  7. Coincidence Again: A Resolution

"Simons (Peter) - Essential Parts"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 7

  1. Modality1 and Essence;
  2. Some Theses of Modal2 Mereology;
  3. Essential Parts for Continuants;
  4. Mereological Essentialism for Continuants;
  5. Essential Parts for Occurrents;
  6. Normal Parts;
  7. Appendix: Quantified S5

"Simons (Peter) - Ontological Dependence"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 8

  1. Introduction;
  2. Dependence in General;
  3. Ontological Dependence;
  4. Independence, and More Dependence;
  5. Some Related Concepts: Substance, Accident, Disturbance;
  6. Non-modal1 Theories of Foundation and Dependence;
  7. Unconditional Existence

"Simons (Peter) - Integral Wholes"

Source: Simons - Parts: A Study in Ontology, 1987, Chapter 9

  1. Sums and Complexes;
  2. Integrity in General;
  3. Mereo-topological Integrity;
  4. Ontological Integrity;
  5. Functional Integrity;
  6. Temporal Integrity and Persistence;
  7. Structure and Gestalt

"Simons (Peter) - Parts: A Study in Ontology - Concluding Remarks"

Source: Simons - Parts: A Study in Ontology, 1987, Concluding Remarks

Full Text (With omissions as indicated)
  • 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:
    → Falsehood
    → Asymmetry
    → Transitivity
    → Supplementation
    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
    [ … snip …]. Supplementation then emerges in the familiar guise [ … snip …].
    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.

COMMENT: Annotated printout filed in "Various - Papers on Identity Boxes: Vol 16 (S1: Sa-Sl)".

In-Page Footnotes ("Simons (Peter) - Parts: A Study in Ontology - Concluding Remarks")

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 in these places 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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