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Home , Logical consequence, Mereology

Simons, P., 1987. Part I is a standard The of structure is of central im- for extensional . portance to mereology (for the following Parts II and III treat further topics and see also Koslicki 2008, ch. IX). Histori- question some assumptions of - cal contributions had this clearly in view; ality. Has an extensive bibliography. for example, as is brought out in Harte 2002, in numerous dialogues grap-

ples with the question of how a whole and further readings which has many parts can nevertheless be a single unified and ultimately Cartwright, R., 1975, “Scattered Ob- endorses a structure-based response to jects”, as reprinted in his Philosophical this question (see Plato). Wholes, ac- Essays, 1987, Cambridge, MIT Press: cording to Plato’s mature views (as de- 171-86. veloped primarily in the Sophist, Par- Lewis, D., 1991, Parts of Classes, Ox- menides, Philebus and Timaeus), have a ford: Blackwell. dichotomous , consisting of both material as well as structural compo- Sanford, D. H., 2003, “Fusion Confu- nents; it is the job of structure to unify sion”, Analysis 63: 1-4. and organize the plurality of material Sanford, D. H., 2011, “Can a sum change parts that are present in a unified whole. its parts?”, Analysis 71: 235-9. A similar conception is taken up and worked out further by who Simons, P., 1987, Parts: A Study in On- famously believed that ordinary material tology, Oxford: Clarendon Press. objects, such as houses, are compounds Tarski, A., 1929, “Foundations of the of (viz., the bricks, wood, etc.) of Solids” as reprinted in his and form (viz., the arrangement exhibited , , , by the material components for the pur- Oxford: Oxford University Press, 1959: pose of providing shelter). 24-29. In , due to the development in the Van Cleve, J., 2008, “The Moon and early 20th century of a often re- Sixpence: A Defence of Mereological ferred to as ‘standard mereology’, based Universalism”, in Contemporary De- on the work of Stanislaw Leśniewski and bates in , edited by Sider, T. (see also Tarski et. al., Oxford: Blackwell: 321-40. 1937, 1956; Leonard and Goodman 1940), the notion of structure has been van Inwagen, P., 1990, Material , largely absent from more recent mereo- Ithaca: Cornell U. P. logical frameworks. (A notable excep- van Inwagen, P., 2006, “Can Mereologi- tion, however, is the Third Logical In- cal Sums Change their Parts?”, The vestigation of Husserl 1900-1.) Because Journal of Philosophy 103: 614-30. the founders of standard mereology were primarily interested in providing a nomi- Wiggins, D., 1980, Sameness and Sub- nalistically acceptable alternative to stance, Cambridge, Harvard U. P. theory, according to standard mereology David H. Sanford wholes (also known as ‘mereological sums’, ‘fusions’ or ‘aggregates’) are conceived of as completely unstructured entities. On analogy with the of in , the Structure and of a mereological sum is determined exclusively on the basis of Koslicki 2008; Simons 1987). To these the existence and identity of its parts; the theorists, it seems quite clear that the arrangement or configuration of these material objects we encounter in ordinary parts is immaterial to the existence and life and scientific practice cannot have identity of the sum they compose. In fact, the conditions of identity and individua- because standard mereology does not tion that are attributed to mereological recognize a distinction analogous to that sums by standard mereology: for, unlike between and membership, mereo- mereological sums, not only are these logical sums are, if anything, even more objects quite obviously capable of sur- unstructured than sets, since all the enti- viving changes with respect to their ties recognized by standard mereology parts, while mereological sums (like sets) are of the same ontological type, viz., so- have their parts essentially; but, in con- called ‘individuals’. Finally, as a result trast to the completely unstructured na- of its endorsement of the now controver- ture of mereological sums, the existence sial of Unrestricted Composi- and identity of these objects is also evi- tion (according to which any plurality of dently tied to the arrangement or config- objects itself composes a further object, uration of their parts. For example, as is viz., their mereological sum), standard pointed out in Fine 1999, a ham sand- mereology is committed to a plenitude of wich does not in fact come into existence potentially gerry-mandered objects, such until a slice of ham is placed between as David Lewis’ notorious ‘trout-turkey’, two slices of bread; and the ham sand- an object composed of, say, the (still wich does not remain in the existence attached) upper half of a trout and the unless the parts in question continue to (still attached) lower half of a turkey (see exhibit this arrangement. Given the ap- Lewis 1986). parent clash between the conditions of identity and individuation of material Because standard mereology has been objects, as we ordinarily conceive of and perhaps still is the most well-worked them, and those of mereological sums in out and widely accepted conception of the standard sense, there seems to be parthood and composition in recent his- plenty of room, then, for the develop- tory, it was that, insofar as ordi- ment of alternative structure-based mer- nary material objects are wholes (i.e., eologies. composite objects made up of parts), they must therefore be conceptualized as One obstacle that has stood in the way of mereological sums in the standard sense. the pursuit of such alternative systems is This seemingly consensus that the notion of structure, given its among contemporary metaphysicians, traditional affiliation with Platonic forms however, is now beginning to be called or Aristotelian , in the of into question by the arrival of some dis- many contemporary metaphysicians and senting voices, who have turned their mereologists inherits much of the philo- attention to the development of alterna- sophical baggage that is associated with tive non-standard mereological frame- its historical precursors. Aristotle already works, and in particular to the re- intro- criticized Platonic forms for so far duction of the notion of structure into the removed from the sensible particulars analysis of parthood and composition, whose characteristics they were sup- especially as it aims to capture the mere- posed to explain that they became, in his ological characteristics of ordinary mate- view, causally inert and explanatorily rial objects (see for example Fine 1982, useless. Plato’s invocation of the partici- 1994, 1999; Harte 2002; Johnston 2002; pation , which was meant to con- nect Platonic forms to sensible particu- or nodes for other objects to occupy; in lars, did not improve the situation, in order to be admissible occupants of these Aristotle’s since he found this positions, the objects in question must relation to be utterly unexplained and satisfy two different sorts of constraints: mysterious. In reaction to the Platonic (i) constraints concerning the type of model, Aristotle made an effort to con- object which may occupy the position in nect his own explanatory and causal question; and (ii) constraints concerning much more intimately to the the configuration or arrangement which matter/form compounds whose behavior must be exhibited by the occupants of the and characteristics they were supposed to positions made available by the structure. make comprehensible. However, in the Secondly, a particularly noteworthy course of doing so, Aristotle’s own con- characteristic of structures or structural ception of form or became asso- features across different domains is that ciated with philosophically loaded no- the numerical identity of the particular tions such as his actuality/potentiality objects occupying the positions made distinction and the accompanying Ho- available within a structure inevitably monymy Principle (according to which tends to be immaterial to the question of an ‘axe’ that cannot cut, for example, is whether the structure or structural feature an ‘axe’ in alone), which in turn in question is implemented; as long as made Aristotelian forms or essences the occupants in question satisfy the two acceptable only to philosophers who constraints just mentioned, they are con- share his general teleological outlook. sidered indistinguishable and hence in- When we look more closely at the vari- terchangeable from the point of view of ous disciplines in which the notion of the structure. Thus, the notion of a struc- structure obviously plays a central and ture or structural feature should be significant role, however, we realize that thought of as going along with a distinc- Aristotle’s notion of structure as form tion between what is considered to be need not be conceived of as the causally variable and what is considered to be and explanatorily inert metaphysical invariable within a given domain or invention ridiculed by Descartes and ; variability, in this connection, others. Rather, in such disciplines as amounts to the interchangeability of mathematics, logic, chemistry, linguistics objects in the domain relative to certain and music, for example, we find that the admissible transformations which leave notion of structure is alive and well, the structural features at issue un- whatever exactly its metaphysical status changed. turns out to be. Although the notion of Finally, in each case, the discipline in structure, as it is applied in each case, is question is interested in particular in tailored to the particular concerns of each capturing, usually by means of a system such discipline, we can nevertheless of laws, , and the like, the charac- recognize certain general characteristics teristics and behavior of those features that go along with any such domain- that are taken as invariable, i.e., the specific conception of structure. (The structural features within the domain in general characteristics I am about to question. The particular nature of those single out will be illustrated shortly by elements that occupy the positions made means of examples from particular disci- available by a given structure, i.e., ele- plines.) ments which are considered to be varia- First, structures in general are entities ble within the domain at issue, on the which make available ‘slots’, positions other hand, tends not to lie within the purview of the significant generalizations and q in this schema is merely formulated by the theory in question, to mark places that may be occupied by since these elements in any case are any non- logical expression of the right taken as interchangeable as far as the grammatical category (viz., in this case, a structure is concerned, provided that the sentence). Thus, as far as the of type and configuration constraints im- the argument schema in question is con- posed by the structure remain satisfied. I cerned, the of the non- now turn to the illustration of these gen- logical vocabulary may vary, while that eral principles governing the notion of of the logical vocabulary stays fixed. The structure by means of examples taken -rules of a particular logical from particular disciplines. Mathematical system aim in particular to describe the Structure. Structures within mathematics role played by the logical vocabulary in are defined as ordered n-tuples consisting generating valid argument . of a set of objects (the or do- Chemical Structure. The chemical struc- main of ) along with ‘a list of ture of a compound is determined on the mathematical operations and basis of (i) the types of constituents of and their required properties, commonly which it consists, viz., its formula; and given as axioms, and often so formulated (ii) the spatial (i.e., geometrical or topo- as to be properties shared by a number of logical) configuration exhibited by these possibly quite different specific mathe- constituents. In the 18th and 19th centu- matical objects’ (Mac Lane 1996, 174). ry, it was discovered, in connection with Widely studied examples of mathemati- the phenomenon of ‘isomers’ or ‘chiral’ cal structures include for example (‘handed’) molecules, that chemical groups, metric spaces, topological spac- substances which are composed of same es, rings, fields, orders and lattices. constituents, i.e., have the same chemical Mathematical structures can be com- formula, can nevertheless exhibit dramat- pared and contrasted by means of various ically different behavior under certain relations, such as embedding, homomor- circumstances, if these constituents are phism, isomorphism, and the like. As any arranged differently. (Cases in point are two isomorphic structures satisfy the for example silver cyanate and silver same axioms and are thus indistinguisha- fulminate as well as racemic and tartaric ble from the point of view of the theory acid.) This discovery led to a three- in question, structures are often said to dimensional conception of molecular be describable only ‘up to isomorphism’. shape, which is still to this day widely Logical Structure. A logically valid ar- employed across many of the natural gument is one that is not only necessarily sciences to explain the processes under- - preserving, but is so in virtue of its gone by organic and inorganic com- or structure: to illustrate, pounds. while the first requirement is satisfied in Linguistic Structure. Linguistic structure the argument, ‘Roses are red; therefore, bears a remarkable similarity to chemical roses are colored’, the second is not. The structure. For example, the syntactic notion of logical form makes sense only structure of a linguistic compound is relative to a particular of logical similarly determined on the basis of (i) vocabulary: for example, because of the the types of constituents of which it con- assigned to the logical constant, sists (e.g., noun-phrases, verb phrases, ‘and’, any instance of the modifiers, and the like) as well as (ii) the ⟔ ⁊ p and q; therefore q is valid within hierarchical arrangements exhibited by classical sentential logic. The role of p these constituents; the latter is typically represented by means of a spatial (i.e., ‘laws’, e.g., the laws of tonality, which geometrical or topological) vocabulary, constrain how smaller musical units consisting of such notions familiar for (e.g., tones) may be organized into larger example from the tree-diagrams used musical wholes (e.g., chords, patterns, within the Chomskyan tradition as ‘being motifs, melodies, and the like) relative to to the left of’, ‘being higher up than’, the principles of composition that govern ‘being connected via a continuous a particular musical tradition. The sorts downward path to’ and so on These two of arrangements into which individual aspects of syntactic structure help ex- tones enter are again characterized by plain why linguistic compounds which means of a quasi-three-dimensional vo- on the surface look very similar (e.g., cabulary invoking space and , ‘John is reluctant to leave’ versus ‘John e.g., ‘high’, ‘low’, ‘fast’, ‘slow’, etc. is likely to leave’) may nevertheless The study of structure, as this is exhibit very different behavior under relevant in particular to the development certain transformations (e.g., ‘*It is re- of non- standard systems of mereology, luctant that John leaves” versus “It is confronts several important metaphysical likely that John leaves’). The numerical questions which at this point remain identity of the lexical items filling the relatively underexplored especially in the various positions within a syntactic struc- context of the contemporary literature on ture is again immaterial from the point of parthood and composition. (1) Ontologi- view of the structure, as long as the syn- cal Category. To what ontological cate- tactically relevant features mentioned in gory do structures belong? Are they (i) and (ii) remain unchanged; thus, inso- objects, properties, relations, or some- far as two lexical items belong to the thing else entirely? (2) Grounding Prob- same syntactic category and fit into the lem. How is the modal or essential pro- same hierarchical arrangements, they are file of a structured whole connected to indistinguishable from the point of view the structure that is present within it? of the and are hence interchange- That is, what sorts of contributions does able without affecting the grammaticality the presence of a structure within an of the resulting construction. object make to the nature of that struc- Musical Structure. Musical structure, tured whole? (3) Mereological Con- unlike the other examples considered straints. What sorts of mereological con- thus far, of course concerns a perceived straints do structures impose on the or phenomenal order, a kind of ordering wholes they organize? To what extent or organization which comes about when and in what way do they dictate the mer- sound waves interact with creatures like eological make-up of a structured whole? us who are equipped with the sort of (4) Individual vs. Species Forms. What cognitive apparatus required to hear sorts of structural features are shared by sound as music. The of hear- the members of a single kind or species? ing sound as music sets up in such a To what extent should structures be hearer certain expectations as to how the thought of as incorporating haecceitistic tones he hears are going to be organized features that are peculiar to individual with respect to the principles of pitch, members of a kind? (5) Structural rhythm, melody and harmony. Relative Change. To what extent can structured to certain musical traditions, e.g., the wholes change with respect to their Western tradition of ‘tonal music’, it is structural features? Through what sorts even possible to speak (though somewhat of structural changes can they persist? metaphorically no doubt) of a system of The of these questions would contribute much to the advancement of Husserl, E., 1900-1, Logische Unter- alternative structure-based systems of suchungen, 1st edn., Halle, Germany: M. mereology vis-a-vis standard mereology. Niemeyer Verlag. Johnston, M., 2002, “Parts and Princi- ples: Axioms in Mereology”, Phil- See also > osophical Topics 30 (1), 129-166.

Koslicki, K., 2008, The Structure of Bibliographical remarks Objects, Oxford: Oxford University Press. Fine, K., 1999. A neo- Aristotelian theo- ry of the nature of material objects and Le Poidevin, R., 2000, “Space and the their parts. Chiral Molecule”, in N. Bhushan and S. Rosenfeld, eds. Of Minds and Molecules: Harte, V., 2002. A historical study of New Philosophical Perspectives on different conceptions of parthood and Chemistry, New York: Oxford Universi- composition considered in the works of ty Press. Plato. Leonard, H., and Goodman, N., 1940, Johnston, M., 2002. A neo-Aristotelian “The Calculus of Individuals and Its approach to mereology. Uses”, Journal of Symbolic Logic 5, 45- Leonard, H., and Goodman, N., 1940. 55. The classic text introducing standard Lésniewski, S., 1916, “Podstawy ogólnej mereology to the English- speaking teoryi mnogoœci I” [Foundations of a world. General Theory of Manifolds], Prace Rescher, N., and Oppenheim, P., 1955. A Polskiego Ko³a Naukowe w Moskwie, Gestalt-theoretic exploration of the con- Sekcya matematyczno-przyrodnicza, 2, cepts of part, whole and structure. Moscow. Tarski, A., 1966. An analysis of what it Lésniewski, S., 1927-30, “O Podstawach means to be a logical notion in terms of Matematyki” [On the Foundations of invariance under a sufficiently wide Mathematics], Przeglad Filozoficzny, 30 conception of transformations. (1927), 164-206; 31 (1928), 261-291; 32 (1929), 60-101; 33 (1930),75-105, 142-

170. References and further readings Lewis, D., 1986, On the Plurality of Fine, K., 1982, “Acts, Events and Worlds, Oxford: Blackwell. Things”, Language and , Pro- ceedings of the 6th International Witt- Mac Lane, S., 1996, “Structure in Math- genstein Symposium, Wien: Hölder- ematics”, Philosophia Mathematica 4 Pichler-Tempsky, 97-105. (3), 174-183. Fine, K., 1994, “Compounds and Aggre- Nozick, R., 2001, Invariances: The gates”, Nous 28 (2), 137-158. Structure of the Objective World, Cam- bridge, MA: Harvard University Press. Fine, K., 1999, “Things and Their Parts”, Midwest Studies in Philosophy 23, 61-74. Rescher, N., and Oppenheim, P., 1955, “Logical Analysis of Gestalt ”, Harte, V., 2002, Plato on Parts and The British Journal for the Philosophy of Wholes: The Metaphysics of Structure, Science 6 (22), 89-106. Oxford: Clarendon Press. Scruton, R., 1997, The Aesthetics of known than, for example, Languages of Music, Oxford: Clarendon Press. Art (1986). It is, in fact, a heavily revised version of Goodman's Ph.D. thesis, A Sider, T., 2001, Four-Dimensionalism: Study of Qualities (Goodman 1941, short An Ontology of Persistence and , SQ). SA presents ‘constructional’ sys- Oxford: Clarendon Press. tem that, just like the constitution system Simons, P., 1987, Parts: A Study in On- in 's Der logische Aufbau tology, Oxford: Oxford University Press. der Welt (1928), shows how from a basis of primitive objects and a basic relation Tarski, A., 1936, “On the Concept of between those objects all other objects ”, in J.H. Woodger, can be obtained by alone. In transl. and ed. with an introduction by SA (and already in SQ) Goodman ap- John Corcoran, Logic, Semantics, Meta- plies a mereological system, the Calculus mathematics, 2nd Edition, Indianapolis: of Individuals, which he developed joint- Hackett, 1983, 409-420 (article was first ly with Henry Leonard (first published in published in Polish and German in Goodman and Leonard 1940). The use of 1936). mereology allows him to avoid certain Tarski, A., 1937, “Appendix E”, in J.H. technical problems that Carnap’s system Woodger, ed., The Axiomatic Method in encounters. Biology, Cambridge: Cambridge Univer- In the Aufbau, Carnap investigates the sity Press, 161-172. example of a world built up from primi- Tarski, A., 1956, “Foundations of the tive of the totality of ex- Geometry of Solids”, i: A. Tarski, Logic, periences of a (the so-called Semantics and Metamathematics, transl. ‘elementary ’ or just ‘erlebs’) by J.H. Woodger, Oxford: Clarendon and thus faces the problem of abstrac- Press, 24-29. tion: how can qualities, properties and their objects in the world be abstracted Tarski, A., 1966, “What Are Logical from our phenomenal experiences. Notions?”, History and Philosophy of Erlebs, which are time slices of the to- Logic 7 (1986), 143- 154. tality of our experiences, can be part- Tranöy, K. E., 1959, Wholes and Struc- similar with each other in a variety of tures: An Attempt at a Philosophical ways. Perhaps two slices are similar with Analysis, Interdisciplinary Studies from respect to what is in our visual field at the Scandinavian Summer University, the time in question, or they are similar Copenhagen: Ejnar Munksgaard. with respect to what we hear or smell. However, since the time-slices are primi- Woolley, R.G., 1978, “Must a Molecule tives in the system, we cannot yet even Have a Shape?”, Journal of the American talk about these respects or ways in Chemical Society 100, 1073-1078. which the slices should be similar in Kathrin Koslicki order to be considered experiences of the same feature (for example, the same color). Carnap’s ingenious is to group exactly those erlebs together that are mutually part-similar, thereby group- Structure of Appearance, Goodman’s ing exactly those that (pre-theoretically The Structure of Appearance (1951, speaking) share a . short SA) is perhaps 's Carnap tries to show that by using this main work, although it is less widely method of ‘quasi-analysis’ all the struc-