We emphasize the relativity of wholes and parts in whole/part perception, and suggest that consideration must be given to what the terms "whole" and "part" mean, and how they relate in a particular context. A formal analysis of the part/whole relationship by Rescher & Oppenheim, (1955) is shown to have a unifying and clarifying role in many controversial issues including illusions, emergence, local/global precedence, holistic/analytic processing, schema/feature theories and "smart mechanisms". The logic of direct extraction of holistic properties is questioned, and attention drawn to vagueness of reference to wholes and parts which can refer to phenomenal units, physiological structures or theoretical units of perceptual analysis.
1. A recurrent theme in the psychological literature on visual and auditory perception and pattern recognition has been the separation and opposition of whole and part processing. Issues arising from this basic opposition can be traced back at least as far as Aristotle (Barnes, 1984) through medieval mereology (Henry, 1991), John Stuart Mill's "mental chemistry" (Mill, 1874), Wundt's "creative synthesis" (Wundt, 1904), to Gestalt psychology (Wertheimer, 1925/1967). Contemporary investigations have been carried out under many headings: templates versus features; distinctive features versus prototype learning; local versus global precedence; part versus whole perception; holistic versus analytic processing.
2. We emphasize consideration of the relativity and exact meanings of the terms "whole" and "part" and how wholes and parts are related in particular contexts. As an aid, we informally outline a formal analysis of the whole/part relationship by Rescher & Oppenheim (1955), and demonstrate its potential for disambiguating the terms "whole" and "part" and for clarifying the relationships in which wholes and their parts may stand - see also Smith (1988) for comment on this formalism. Additionally, we consider the potential of the formalism as a unifying principle and possible basis of a common nomenclature for many current controversial issues in perception and pattern recognition.
3. The formalism does not oppose any particular theoretical orientation, but highlights the conceptual issues that have been neglected in much of the theoretical and experimental work on whole/part processing. Implicitly, the formalism directs attention to detailed specification of mechanisms that may underlie our perception of wholes and their parts, and leads ultimately to doubts about the possibility of mechanisms that can extract global attributes from spatial, temporal and spatiotemporal patterns without prior, elaborate analysis of their local elements.
4. We argue that in some psychological contexts, theorists and experimenters have failed to specify whether the wholes and parts they are dealing with are: 1) neurophysiological structures; 2) hypothesized objective units of input for perceptual systems (or computer simulations of those systems) such as "lines", "edges", "curves" and "vertices" or waveform descriptors such as "frequencies", "durations" and "amplitudes"; or 3) phenomenal units of analysis and attributes reported by participants in perceptual tasks.
5. There are similarities across perceptual domains (Julesz & Hirsch, 1972; Pick, 1965; Stevens & Latimer, 1991) and to increase the generality of our arguments, we propose that visual, auditory and tactile perception can be described as spatial, temporal or spatiotemporal. Vision, while predominantly spatial, can be conceived of as having a temporal dimension in the serial allocation of attention. Audition, while essentially temporal, possesses a spatial dimension (Bregman, 1990). The perception of tactile displays has obvious spatial and temporal dimensions. Finally, we demonstrate the efficacy of the formalism in theory construction and experimental investigations by reference to theories and experiments that have arisen from, been devised within and executed according to, our conceptual analysis of the part/whole relationship.
6. In experimental studies of whole/part processing, a typical strategy has been to nominate some pattern class for study - geometric forms, faces, dot patterns, alphanumeric characters, words, sentences etc. Having provided ostensive definitions of "whole", "part", "local" and "global", experimenters proceed with manipulations of these as independent variables and with observations of the effects of their manipulations on dependent measures such as recognition time, accuracy and magnitude of illusion. Conclusions are then drawn on the type and order of processing - local, global, local before global, global before local, etc. Such a strategy involves at least two difficulties. First, the choice of stimulus analysis and what will constitute a part and what will be regarded as a whole are arbitrary; any pattern or pattern class can bear an indefinite number of descriptions (Taylor, 1964). While in some cases investigators provide a rationale for their choice of analysis in terms of evidence (Gibson, 1969) or some operational measure of perceptual grouping such as selective attention (Pomerantz & Garner, 1973) "goodness" and "badness" as determined by gestalt laws (Bower, & Glass 1976), natural and unnatural parsing, (Backer Cave & Kosslyn, 1993), the experimenter's intuition remains a major factor in the choice of analysis.
7. The second difficulty is that adopted notions of whole and part rarely proceed or derive from any general theory of wholes, parts and their relationships. Indeed, a survey of the literature quickly reveals great divergence in the assumptions underlying the use of terms such as "whole", "part", "local", and "global" not to speak of the bewildering nomenclature used to describe the units of analysis: "templates", "prototypes", "canonical forms", "features" , "formons", "geons", "blocks", "wedges", "cylinders and cones", "textons", "local and global properties", "raw and refined images", "components", "schemata and subschemata", "attributes", "dimensions", "blobs", "perceptual primitives", "features", "aspects", "spatial frequency components", "edges" and "bars". Although some theorists have tried to impose order where none has existed, (Garner, 1978; Leeuwenberg, 1971; Minsky, 1963; Schyns, Goldstone & Thibaut, 1997; Uttal, 1988), systematic attempts at comparison and reconciliation of the terms and assumptions in current usage are rare, and problems remain in much of the literature. Often one cannot be sure whether a theorist is using a term to refer to some memory representation of the stimulus, some extracted unit(s) to be compared with the memory representation, or, indeed, the stimulus pattern itself.
8. The lack of general theory and common nomenclature for wholes and parts makes it difficult to appreciate the implied antitheses of, say, template and feature theory, local and global processing. For example, Gibson (1969) cites evidence (Hubel & Weisel, 1965) in support of her use of "discontinuity" as a "feature". Lockhead (1972), who argues for an "holistic" model, cites the same evidence in support of his notion of "blob processing". If these theories are accepted as exemplifying the putative local/global opposition, then perhaps, in some cases, the opposition rests on nothing more than a difference in the theorists' definitions of "whole" and "part", and, if this is so, then it is difficult to see how any empirical evidence could ever be brought to bear on the issue. Tanaka & Farah (1993) in a study on whether faces are recognised on the basis of their individual features or more holistically, suggest that, "One factor that has contributed to the difficulty of resolving this issue is the lack of clear, generally accepted definitions of the concepts 'holistic' and 'feature'. Without clear definitions of what these terms mean, it is difficult to operationalise them in experimental tests." Arnheim (1986), Hochberg (1986), Perkins (1986), and Voneche (1986) have also alluded to the central difficulties in definition of wholes and parts.
9. It is to these two difficulties - the need for explicit, psychologically valid stimulus analysis, and a general theory and terminology for the whole/part relationship - that this paper is directed. Attempts to dispel some of the ambiguity inherent in terminology have been made by Garner (1978) and Hoffman, (1966), but one analysis which attempts to characterize the whole/part relationship formally, and which may provide a basis for the necessary general theory and terminology of wholes and parts is that of Rescher & Oppenheim (1955), hereafter R&O.
10. R&O have provided a conceptual framework for the precise explication of the gestalt concepts of "whole", "gestalt quality" and "system" as they are employed in various branches of empirical science. They summarize the intuitive requirements or conditions of adequacy which underlie talk of wholes in three conditions: "i) the whole must possess some attribute in virtue of its status as a whole, an attribute peculiar to it and characteristic of it as a whole; ii) the parts of the whole must stand in some special and characteristic relation of dependence with one another; they must satisfy some special condition in virtue of their status as parts of a whole; and iii) the whole must possess some kind of structure in virtue of which certain specifically structural characteristics pertain to it." (p. 90).
11. To specify some concept of part, it is necessary at the very outset, to state the conditions under which an object "x" is to be considered as a part of a whole of object "y". The specification of a particular part-whole relation thus determines for a given object, or whole, just which objects are its parts. Only when the context makes clear which specific part relation is intended may we even speak of parts and wholes and, until definitions of "whole" and "part" are proffered, discussion of these terms cannot proceed. As an example, consider that uppercase Character "A" presented on a computer screen may be designated as a whole whose parts are the small black and white segments represented by screen pixels.
12. R&O point out that, for any object, there can exist an indefinite number of decompositions. For example, while Character "A" may be decomposed into its pixel segments, it still has other decompositions that do not maintain the integrity of these originally designated parts. A physiological psychologist may choose to adopt a very fine-grained decomposition of Character "A" more in keeping with the high resolution afforded by retinal receptors or a more coarse decomposition in keeping with the receptive fields of cortical cells. A cognitive psychologist investigating attention may wish to decompose Character "A" into several regions attracting high levels of eye fixation. In each of these cases, the new decomposition does not maintain the integrity of the original parts (pixel segments of Character A). Thus, depending on context and intention, wholes and parts may assume an indefinite number of definitions and decompositions, and as a result, without specification of context and precise definition, ambiguity, confusion and pseudo controversies are spawned by the relativity of the terms.
13. R&O introduce attributes which can be either shared or unshared by wholes and their parts. Relative to a given decomposition of a given whole, any attribute which is possessed by none of the elements of the decomposition of the whole is said to be "unshared". For example, when an auditory whole, decomposed into a series of "falling" Shepard tones, has the (illusory) attribute of "rising" (Risset, 1989), then this is an instance of a whole not sharing the attribute "rising" with its parts. In vision, a color wheel comprised of the spectral colors looks white when spun (Mill, 1829/1967); the whole wheel has the phenomenal attribute "white" which is not a phenomenal attribute of any of the components.
14. Attributes of a whole may also be "shared" by their parts. For example, a chordal sequence (CEG, EGC, GCE) with the attribute "tonic C major" can share this attribute with each of its component chords and, indeed, in the visual domain, lack of control of this factor can lead to confounding in experimental design. In the construction of compound stimuli consisting of large (global) letters composed of small (local) letters, Crassini (1991) has demonstrated that it may not be enough simply to make the large letters consistent or inconsistent with the small letters. Instead, it is necessary to delineate precisely and control the degree to which attributes of the whole (large letter) such as "curvature" are shared or unshared by the parts (small letters). For example, a large Letter "O" constructed out of small Letter "C"s shares the attribute of "curvature" with its parts, whereas a large Letter "O" constructed from small Letter "H"s does not, and whether or not large letter stimuli share or do not share attributes of their component small letters can affect recognition time and errors in studies using such stimuli.
15. An attribute of a whole may be said to be an "underivable" attribute, relative to a designated decomposition and to a set of attributes of the elements of that decomposition. Such an attribute is underivable if it is not a logical consequence of the characterisation of the elements of the decomposition and their attributes. The weight of a pile of stones is an underivable attribute, because the weight is not a direct logical consequence of the individual weights, but calculation requires the extra physical law that weight is additive. The holistic attribute "vertical symmetry" of Character "A" is underivable when its decomposition and attributes do not specify the spatial relationships of the parts of "A".
16. An attribute may be said to be a "derivable" attribute if, relative to a particular decomposition and a set of attributes, it is a logical consequence of the decomposition and its characterisation. Derivability of the various visual symmetries is assured in many connectionist models of symmetry detection where the input units are set to register not just the parts of the patterns, but also attributes of the parts such as their relative locations within an input array (Latimer, Joung & Stevens, 1994; Sejnowski, Kienker & Hinton, 1986). Indeed, most computer models of spatial and spatiotemporal pattern recognition achieve their success through the use of decompositions and sets of attributes that permit derivation of global attributes, (Fukushima, 1988; Marr & Nishihara, 1978; Uhr & Vossler, 1963). When short musical compositions are decomposed as series of pitches with attributes of duration and volume across time, global attributes such as smooth and jagged "melodic contour" become derivable. Derivability and underivability of propositions about a particular attribute of a whole can, of course, only be determined if the propositions describing the relevant decomposition and its attributes are set out formally - see Kyburg (1968) for examples of possible formal presentation of propositions in psychological theory.
17. In addition to decomposition and characterisation, derivability may also depend on the assumptions of an appropriate, clearly specified theory. The attribute "curvature" becomes underivable in this sense when it is somehow extracted directly by curvature detectors in the visual system and is not derived from the products of local analysis (Riggs, 1973; 1974). An alternative theory-dependent, derivable conception holds that large curved lines may be registered by cells whose optimum stimulus is a straight line. Thus, an arc may be derived by way of the outputs of short line detectors located so that they are tangents to the larger arc (Blakemore, 1974; Crassini & Over, 1975), and indeed, the weight of evidence supports this theory.
18. The above definitions have many implications for the construction of theories and the conduct of experiments in perception and pattern recognition. At a general level, the analysis highlights the relativity of the notions of whole, part, local and global, and this is not, of course, how these terms have been treated in the psychological literature. The Gestaltists believed that wholes had primacy and were necessarily underivable and greater than the sums of their parts. The R&O analysis, so far, demonstrates that for any whole or class of wholes, there is an indefinite number of decompositions and sets of attributes of those decompositions. There is also an indefinite number of possible background theories. Rather than wholes being underivable in any absolute Gestaltist sense, the question of whether or not the attributes of wholes are shared/unshared or derivable/underivable, depends entirely upon the choice of decomposition, set of attributes and background theory. This conception of wholes, parts and their inter-relationships undercuts much of the supposed controversy in contemporary theory and experimentation in whole/part perception. Several examples serve to illustrate this point.
19. Derivability/underivability of attributes and shared/unshared attributes lie at the very heart of the doctrine of emergence. Emergent attributes are attributes of a whole that are not attributes of its parts (unshared attributes), and are said to emerge or somehow become evident only when the parts are assembled. However, the derivability/underivability of such holistic attributes is again relative to chosen decomposition, attributes and background theory. In James Mill's time, a spinning color wheel's appearance of white would have been unpredictable and emergent, but the assumptions of the Young-Helmholtz trichromatic theory of color perception render the phenomenon explainable, predictable and non-emergent. An alternative theory-dependent conception of this emergent phenomenon involves hue cancellation; the complementary hues of the colour wheel cancel leaving the ever present white, and contemporary physiology provides the neural underpinnings of this explanation with its spectrally-opponent chromatic cells and non-opponent luminosity cells. In the auditory domain, we take Bregman (1990, p. 139) to be arguing that, given a clear, detailed and precise description of auditory streams in terms of, say, "pitch", "amplitude" and "duration", it would be possible to derive so-called emergent attributes such as "melodic form". For less clearly defined within-stream components, "melodic form" etc., may remain underivable and emergent.
20. The major general implication of the R&O formalism is its bearing on the issue of just what perceptual analysis or analyses the human system applies at different levels in its recognition patterns. The formalism can serve as an heuristic in the search for acceptable models of perception and pattern recognition, by directing questions such as: For a given experimental task and set of stimuli, what perceptual decomposition (if any) is employed by participants? What are the likely attributes of the decomposed stimulus parts? Are global attributes shared/unshared or derivable/underivable from the decomposition and the attributes of the stimulus parts? What background theories are required to underpin hypotheses about the derivability or underivability of global attributes of the stimulus patterns? Further questions need to be asked about the ontological status of any wholes and parts under investigation. Are they neurophysiological structures? Are they hypothesized objective units of analysis such as "lines", "curves" or "edges" etc.? Are they phenomenal units of analysis reported by experimental participants during perceptual tasks?
21. Rather than militating against any particular theoretical orientation or stimulus analysis, the R&O formalism makes a general appeal for precise definitions of the terms "whole" and "part" as well as implicitly directing attention towards possible mechanisms for derivation of holistic attributes. In essence, what sort of machine does the human system have to be in order to derive holistic attributes from prior local analysis or, by contrast, to extract holistic attributes directly?
22. The holistic versus analytic opposition has a long history in psychology: are patterns recognized by way of processes that extract holistic attributes, local attributes or some combination of these? The majority of the many studies on this issue have been purely empirical, and investigators have been content to elicit experimental evidence for one form of processing or the other. It is quite clear from the descriptions of analytic theories (Fukushima, 1988; Marr & Nishihara, 1978; Uhr & Vossler, 1963) what the decompositions of the patterns to be recognized are, and that the attributes of their parts are those of "orientation" and "location". While holistic attributes such as "symmetry" would be derivable in the models for these analytic theories, such attributes are apparently unnecessary and play no part in the successful recognition of the patterns under consideration. In Lockhead's (1972) theory of "blob processing", no decomposition in early processing is proposed nor is decomposition necessary; the initial unit of processing is the whole pattern - see Melara, Marks & Potts (1993) for criticism of blobs. Others continue to maintain that early processing is holistic, and that the analysis of stimuli into dimensions is a secondary, cognitive, effortful process (Kemler Nelson, 1993). The implications of the R&O formalism for holistic and analytic processing are fairly clear. There is a need to spell out objectively and in detail adopted decompositions, their attributes, and any background theoretical assumptions. Only then will it be possible at a conceptual level to determine whether or not holistic attributes are shared/unshared or objectively derivable/underivable from the products of local analysis and whether the processes of each theory can be tested for sufficiency by embodiment in some working system. Logically prior conceptual issues of what will constitute a whole, its parts and their status require attention before any experimental investigation. After such analysis, it may be that the proponents of primary holistic processing can demonstrate the existence of truly underived and underivable holistic properties of stimulus patterns and specify explicit mechanisms for their extraction, but as yet, they have done neither.
23. Similar problems arising from inattention to conceptual issues surface in the local versus global precedence debate. In this opposition, it is proposed by one class of theorists that global or holistic attributes are extracted prior to the extraction of local attributes such as particular "lines", "curves", "angles" etc., in local regions of the patterns. Navon (1977; 1981; 1991), using as stimuli large alphabetic characters constructed from smaller alphabetic characters, argued that the attributes of the higher level unit are processed first, followed by analysis of the attributes of lower level units. Others have argued that local features are the primary objects of extraction (Thompson & Massaro, 1989). For recent comprehensive reviews of this debate, see Kimchi (1992), Treisman (1986) and the thoughtful commentary by Uttal (1988). The intention here is not to deny the findings of global-to-local interference and faster responses to global structure - both of which occur under particular experimental conditions. Rather, the intention is to demonstrate that the R&O formalism can provide a basis for questioning explanation of those findings in terms of a theory that proposes extraction of global attributes prior to analysis of local elements and their attributes. The question is: what mechanism could recognize patterns in this way? Leaving aside the definitional issues of what will constitute the decomposition, attributes and background theory, it is difficult to conceive of any device or computer program that is capable of extracting holistic attributes like "symmetry" directly from patterns. In all cases, it is first necessary to compute a rich description of local elements and their relationships to each other prior to the computation of global attributes. (Latimer et al., 1994; Sejnowski et al., 1986).
24. In defence of primary holistic processing and global precedence, it may be argued that there are programs, procedures and machines that can demonstrate sufficiency of holistic theory by extracting global attributes directly from patterns: the gestalt methods (Zusne, 1970); autocorrelation methods (Uttal, 1975) which use optical methods that are claimed to be totally continuous; holistic schema and prototype theories (Evans, Arnoult, Hoffman & Zinser, 1968); devices for figure-ground segregation (Marr & Nishihara, 1978; Tunstall, 1975); so-called "smart mechanisms" for direct extraction of global properties (Pomerantz & Kubovy, 1981; Runeson, 1977); global stereopsis (Julesz, 1971). However, in each and every case it can be demonstrated that: 1) the holistic properties or units are not extracted directly but are derived from the products of prior analysis of parts; 2) there has been insufficient attention to the relativity and precise definitions of the notions of whole and part.
25. R&O consider the dependence of certain characteristics of one part of a whole upon those of other parts, that is, dependent attributes. Consider a configuration consisting of a set of magnetic needles of equal strength inserted in pieces of cork and floated with all like poles upwards in a basin of water. When a magnet of unlike pole is suspended above the small magnets, they arrange and re-arrange themselves in a pattern of one or more concentric circles depending on the number of magnets. Various dependencies exist in this type of configuration: for example, the distance of one cork from the nearest adjacent cork depends on the magnetic strengths and the number of other magnets. A configuration may then be characterized by specifying the various dependence relations in which its constituents, or parts, must stand. The Muller-Lyer illusion is a good psychophysical example of a configuration with dependent attributes: lines of equal length appear unequal when additional lines (fins) are drawn in appropriate positions with respect to the original lines. The magnitude of the illusion can be plotted as a function of fin angle and size. Dependence may also exist in a configuration of colored areas where perception of the color of a given area in the configuration depends upon the colors of all the constituent areas by virtue of the laws of biophysics and the physiology of color perception.
26. Illusory contours and illusions of misalignment such as the Poggendorff and Morinaga effects serve to illustrate the implications of dependence and derivability. These illusions may be generated by configurations of parts which, by virtue of their attributes of location, orientation, length and color etc., may form dependence systems. An enduring issue concerning spatial and spatiotemporal illusions is very close to that of local versus global precedence discussed earlier. It concerns whether the apparent misalignments and illusory contours are holistic attributes derived by early visual processes which act upon the local elements or parts of a configuration (bottom-up or data-driven processes) or whether they are holistic attributes which, when extracted directly from a configuration, act upon and distort perception of the local elements (top-down or conceptually-driven processes).
27. Again, the implications of the R&O formalism for this controversy are clear. In any case, where there is talk of wholes and parts, it is necessary to consider what exactly constitute the decomposition, attributes and possible theories relevant to each illusory configuration, and, in terms of these, to specify some mechanism which could respond to the configuration in an illusory manner. These conditions are partially met by a theory which proposes that illusory contours are the result of normal neurophysiological processes that act upon the local lines, intersections and vertices of configurations, (Peterhans & Von der Heydt, 1991). Recording from alert monkeys, the investigators found that many cells in area V2 respond to anomalous-contour figures as if the contours were given by edges or lines. These cells fire when an anomalous contour passes over their response field. Recent psychophysical evidence also indicates that relatively sharper illusory contours can be produced by the inclusion of boundary markers such as line-ends, points and, presumably, other small local elements (Day & Jory, 1980). Thus, there is a possibility of explanation of illusory contours in terms of a dependence system with derivable holistic attributes. It may be that the visual system derives illusory contours from stimulus configurations by way of a bottom-up, data driven system that works on local elements and their inter-relationships, and explanatory mechanisms for such systems already exist. On the other hand, it is not obvious how a top-down, underivable conception of illusory contours would work, let alone how the blueprints for a machine that could perceive them directly would be provided.
28. R&O consider the third item in the list of conditions for wholes, viz. that the whole must possess some kind of structure in virtue of which certain specifically structural characteristics pertain to it. This catches the important conception of a whole as a "structured" organisation of its elements. A structured whole involves three things: its parts; a domain of positions which the parts occupy (this need not be spatial or temporal but can be any kind of topological structure); an assignment specifying which part occupies each of the positions of the domain. A particular performance of a musical composition is an example of a whole in this sense: the time interval of the performance is the domain of positions; the various tones and their attributes of pitch, volume and duration, are the temporally distributed parts; the musical score is the assignment of the parts to the positions in the domain. See Stevens & Latimer (1992; 1994) for illustrations of structure in this sense in the characterisation of short musical compositions.
29. Often concern is not for individual parts, but only the type of part. When parts are treated this way, a structured whole may be viewed as a "complex". A complex is characterized by the following three features: a set of topologically structured attributes; a topologically structured space constituting a domain of positions: an assignment of exactly one attribute to each position". As such, the notion of complex provides a basis for determining isomorphism between and invariance across patterns.
30. The above considerations underlie R&O's final definition, that of "complexial features". "Given some subgroup Z of the group of all transpositions of a complex, we designate as a complexial feature (relative to Z) of this complex any attribute which it shares with all complexes differing from it only by Z-transformations, i.e. any attribute invariant under these transpositions." (p. l04). An example of such a complex would be a grouping of black dots on a white background. The phenomenal grouping of the ensemble is a complexial feature relative to the group Z of certain changes of distance (the contractions and dilations) which leave invariant this attribute of grouping.
31. The concepts of complex and complexial features could also be used to guide theorizing in spatial, temporal and spatiotemporal perception. Feature location could be held constant while the effects of orientation and size transformations of features on recognition are recorded. In the same manner, the extent to which complexial features of patterns are derivable or not derivable from transformations of individual features may be investigated. The aim of this sort of research would be the establishment of the levels at, and the order in which the human system extracts and/or derives such attributes as orientation, location, size, contour, frequency, amplitude and envelope of patterns.
32. If holistic and analytic theories of feature extraction are to be taken seriously, then there must be some way in which stimulus sets can be decomposed non-arbitrarily in terms of psychologically valid or functional units and their attributes. The present analysis draws attention to the need for clearly stated and precise stimulus decompositions and attributes that are psychologically valid. Psychologically sound decompositions allow a more extensive investigation of further important attributes of the parts, for example, their discriminating or differentiating capacity, the weight assigned to them in perception and recognition and whether or not such attributes are derivable or underivable. Space does not allow description of empirical work, but illustrations of applied R&O principles are provided in studies of the recognition of uppercase characters, words, geometric forms and short musical compositions (Latimer, 1978; 1988; 1990; 1992; Latimer et al., 1994; Latimer & Stevens, 1993a; 1993b; Stevens & Latimer, 1991; 1992; 1994).
33. We have tried to demonstrate that the logical analysis of gestalt concepts by Rescher and Oppenheim (1955) can have a unifying, clarifying and disambiguating influence in the exceedingly diverse area of whole/part perception and pattern recognition. The formalism does not oppose any particular theoretical orientation in part/whole processing, but emphasizes the relativity of the terms "whole" and "part". We have argued that in the veritable babel of terms, concepts and confusing nomenclature, the R&O formalism may provide the basis for a clear, simple and precise terminology for discussion of wholes, parts and their relationships.
34. R&O argue that wholes should meet three conditions: A whole should possess some attribute in virtue of its status as a whole; the parts of the whole should stand in some relation of dependence on one another; a whole should have structure and structural characteristics.
35. Considering the first condition, wholes can be anything from the smallest microscopic objects to the universe, so it is essential that, in each research context, investigators specify in detail what wholes are being considered, what decomposition of the wholes (if any) is being employed, what attributes of the decomposed parts are relevant, and which are shared and/or unshared by the whole and its parts. If holistic attributes form part of the investigation, questions regarding their derivability or non-derivability with or without background theory and assumptions may be asked.
36. The second condition of wholes raises questions regarding how the parts of a whole and their attributes may stand in relations of dependence, one with another. Parts, by virtue of possessing certain attributes, may exert influence on other parts, and, in consequence, form dependence systems. Wholes may also possess dependent attributes which, given a particular decomposition and attributes, may be derivable or underivable from the parts.
37. The third condition suggests that wholes have structure and structural characteristics. For example, a whole may have its parts and their attributes assigned to a domain of positions and be considered as a complex with derivable and underivable complexial attributes. Such structure and structural characteristics provide an important basis for investigations of invariance across classes of wholes; under certain transpositions and groups of transpositions, classes or families of wholes may retain attributes peculiar to their class.
38. One function of the R&O formalism and its implications may be to direct more psychological interest to mechanism. However, before mechanisms for part/whole processing can be specified, it is necessary to know, in each situation, the exact meanings and referents of the terms "whole" and "part". Only then may it be asked what sort of machine can: derive holistic attributes; respond to dependence systems and derive dependent attributes; impose structure, derive structural attributes and compute transpositions which preserve invariants? On the other hand, what must the machine be like that can extract holistic attributes and structure directly from patterns without recourse to analysis of local elements? It has been suggested here that the specifications for such a machine would be very difficult. What the Gestaltists and their heirs, who uphold direct detection of holistic attributes, may need are unconditionally, underivable holistic attributes which remain so under any decomposition into parts and their attributes.
39. Unquestionably, the problems of wholes, parts and their perception will eventually be solved by proper empirical investigation, but such investigation may not proceed as planned without prior consideration of the conceptual issues raised here. Without knowledge of what exactly is meant by wholes and parts in each case, we shall, as Voneche (1986) suggests, "...navigate in a sea of imprecision and vagueness: (p. 903)."
This research received support from University of Sydney and Australian Research Commission Grants, together with an Australian Commonwealth Postgraduate Research Scholarship and an Australian Research Commission Postdoctoral Fellowship awarded to the second author. During the period of working through the ideas in the paper, we derived great benefit from discussions with Professor Phil Sutcliffe. We were also helped in discussions with Margaret Charles, Joel Michell, George Oliphant, Agnes Petocz and Peter Wenderoth. We thank Professor Ross Day for drawing our attention to the possible existence of dependence systems and dependent attributes in visual configurations with subjective contours and illusions of misalignment such as the Poggendorff and Morinaga. We also greatly appreciate his many comments on a draft of our manuscript. Finally, we thank our anonymous referees for their insightful and most helpful comments.
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