2.2.1 Vagueness and Indeterminate Boundaries—Are Boundaries always Crisp?

When thinking of physical boundaries (as contrasted with e.g. political, legal or linguistic boundaries) we usually think of clear-cut lines separating one entity from its environment. In case of three-dimensional entities the boundaries are surfaces and thus entities of two dimensionality—they are not three-dimensional bodies themselves. And in case of surfaces the physical boundaries are lines and not regions. One could conclude that, regardless of whether boundaries are fiat or bona fide, they are necessarily crisp in this respect that they are always of a dimensionality one less than their hosts' dimensionality.

When dealing with real entities and their actual boundaries, however, a supposedly different picture emerges. In some cases, like for instance the boundaries of many geographical objects such as deserts, dunes, or the Caribbean Sea, it seems that a given entity cannot be delineated by crisp boundaries—we have troubles to identify a single surface that demarcates a desert, dune, or the Caribbean Sea. Instead, they are delineated by border zones, i.e. by boundary-like regions, which are indeterminate to some degree [6]–[10], [17]. Consequently, we would have to distinguish between crisp (i.e. sharp) and indeterminate (i.e. fuzzy, vague) boundaries [18], [19], with crisp boundaries always possessing a dimensionality one lower than their host. Thus, one could ask whether an indeterminate boundary, on the other hand, always shares the dimensionality with its host.

An indeterminate boundary could be interpreted as the region in which an ontologically crisp boundary must be located, but we simply cannot narrow down its actual location. In this case, the indeterminacy would represent a conceptual issue that is owed to linguistic or epistemological problems instead of ontological ones (cf. [14]). Thus, we must distinguish an ontological from an epistemic-conceptual interpretation of the indeterminacy of some physical boundaries. The epistemic-conceptual interpretation of the indeterminacy problem, which is favored by various authors (e.g. [7], [8], [10], [17], [20], [21]), shifts the indeterminacy problem to a supposed vagueness of the respective concepts, instead of looking for an ontological reason (e.g. quantum mechanics). They argue that indeterminate boundaries can be defined in principle, but they cannot be determined precisely. Interestingly, the examples that are commonly used in this context usually refer to fiat entities. Therefore, one could also argue that the vagueness of the respective boundaries is owed to their fiat nature and thus to the fiatness of the entities involved. It is thus the vagueness of the corresponding concepts themselves, resulting from their mind-dependent conceptual nature, that is responsible for the indeterminacy (e.g. [8], [17]). Examples are for instance entities that are heterogeneously composed, with two clearly distinguishable poles that are not sharply delimited from each other by a crisp boundary. Instead, they are bound by a region in which they merge seamlessly, as it is often the case with different and overlapping chemical gradients in biological objects (e.g. the polarity of a blastula during embryogenesis, with the animal pole and the vegetale pole).

The indeterminacy of time dependent boundaries, like for instance coastlines or river banks, is another example for epistemic-conceptually indeterminate boundaries. Coastlines are shifting borders, and since they are crisp and bona fide at any given moment in time, their indeterminacy can be attributed to the problem of the time dependence of their actual location [9]. Their indeterminacy does not pose fundamental problems to ontology design, since the vagueness is just an epiphenomenon of the dynamics of reality.

Thus, as long as indeterminate boundaries can be restricted to fiat entities or attributed to the dynamics of reality, they do not pose fundamental problems to ontology design, as they nicely match with the basic categorial distinction of bona fide and fiat boundaries and the accompanied basic categorial distinction of bona fide objects and fiat entities respectively.

But are bona fide boundaries really always crisp at a given moment in time? The distinction of what counts as physical discontinuity or qualitative heterogeneity on the one hand and physical continuity and qualitative homogeneity on the other hand, is not crisp. There are cases of ontological indeterminacy—bona fide boundaries that are less bona fide than others, so to speak.

It cannot be denied that discontinuities come in various degrees of abruptness. The cutting edge of a sharp knife comes more abruptly than the edge of the white cliffs of Dover. From a purely perceptual point of view, the edges of the letters that appear on your screen while typing an email are crisper than the color changes in a rainbow. Obviously, discontinuity is a matter of scale and therefore of granularity. With increasing resolution spatial boundaries of physical entities become increasingly fuzzy. This results in what has been called the Problem of the Many [22], [23], which Lewis characterizes as follows:

“Think of a cloud — just one cloud, and around it a clear blue sky. Seen from the ground, the cloud may seem to have a sharp boundary. Not so. The cloud is a swarm of water droplets. At the outskirts of the cloud, the density of the droplets falls off. Eventually they are so few and far between that we may hesitate to say that the outlying droplets are still part of the cloud at all; perhaps we might better say only that they are near the cloud. But the transition is gradual. Many surfaces are equally good candidates to be the boundary of the cloud. Therefore many aggregates of droplets (…) are equally good candidates to be the cloud. Since they have equal claim, how can we say that the cloud is one of these aggregates rather than another? But if all of them count as clouds, then we have many clouds rather than one. And if none of them count, each one being ruled out because of the competition from the others, then we have no cloud. How is it, then, that we have just one cloud? And yet we do.” [23]

Whereas one could argue that the Problem of the Many represents a purely linguistic and epistemological problem of referencing and the use of language rather than an ontological problem, it nonetheless results from the vagueness or fuzzyness of boundaries. The relevance of this problem becomes apparent when considering that at the subatomic level, any physical entity resolves into a swarm of subatomic particles, of which no clearly determinable boundary can be specified—the location and shape of the outer surface of a material entity involves some degree of arbitrariness at these fine levels of granularity [9], [14], [17]. Thus, at least at very fine levels of granularity, the idea of abrupt physical discontinuities seems to be questionable [8].

At coarser levels of granularity, however, the dichotomy between fiat and bona fide boundaries and their respective physical entities can be maintained and seems to be a reasonable distinction [8], [9], [17]. From which granularity level onwards this dichotomy applies depends on the entity and cannot be determined universally. In other words, when increasing the resolution some entities turn fuzzy in coarser levels of granularity than others.

Whether the indeterminacy at finer levels of granularity and the granularity-dependence of the bona fideness of boundaries as such can be solely ascribed to conceptual and epistemological issues, or whether it is evidence for an underlying ontological vagueness, represents an open question.

2.2.2 Continua in Biological Objects and the Distinction of Fiat and Bona Fide Boundaries—Is this Distinction really Absolute?

According to the above discussed spatio-structural notion of boundaries, the question whether a given boundary is a bona fide boundary or a fiat boundary only depends on the question of what counts as physical discontinuity or qualitative heterogeneity on the one hand and what as physical continuity and qualitative homogeneity on the other hand. It is the question of where to draw the line between these two conditions. This decision immediately concerns the distinction of objects and fiat object parts.

According to its definition in BFO, an object is a material entity that is maximally self-connected and self-contained [11]. This definition draws on a principle of connectedness that allows only ‘true’ or ‘false’ as possible values, while at the same time functioning as a principle of unity for the entity to be delineated (e.g. [24]). This is insofar problematic, as connectedness comes in degrees, which at its turn results from an underlying continuity problem [12], [17] that frequently impedes consistently distinguishing bona fide from fiat boundaries.

When considering physical connectedness, one has to deal with various types of connectedness, ranging from gravitational forces to all kinds of electro-chemical bonds (covalent bonds, ionic bonds, metallic bonds, hydrogen bonds, etc.) and physical connections and junctions (screws, bolts, staples, nails, etc.). Part of the problem is also the fact that what counts as maximally self-connected is granularity-dependent and deals with the problem of how aggregations of bona fide objects of finer levels of granularity constitute a single bona fide object of a coarser level of granularity—how does an aggregate of individual cells, with each cell having its own bona fide boundary, constitute a single multicellular organism at a coarser level of granularity that also possesses its own boundary (cf. [25], [26])?

Continuity: Although continuity problems affect all kinds of material entities, they are especially serious in biology. Biological objects are the product of evolution and exhibit a high degree of variability that constitutes a complex network of relations of similarities and differences between all objects of the enlivened nature. Accordingly, biologists have to deal with a continuum of forms and functions that spans a complex morphological property space, in which usually no two objects occupy exactly the same place [12]. This alone poses considerable conceptual problems [27], and many of the concepts used for referring to specific types of entities within this continuum are delineated by fiat (cf. fiat concepts; [8]). But this problem of delineating different types of entities is not exclusively of conceptual nature: if, due to the high degree of variability, no two cells are identical, any aggregation of cells exhibits qualitative heterogeneity between any two neighboring cells. When distinguishing biological objects above the cellular level one is thus confronted with the question where to draw the line between relevant and irrelevant heterogeneity, since heterogeneity is always present but not always relevant. As a consequence, at supracellular levels of granularity the distinction of bona fide and fiat concepts is not crisp anymore and involves some fuzziness that cannot be explained by referring to conceptual problems alone. Instead, this fuzziness is the result of an underlying ontological continuum.

Connectedness: Another problem in biology is the fact that, except for whole organisms, all anatomical objects of the cellular and supracellular levels of granularity are connected to neighboring anatomical objects via conduits, tunnels, vessels, ducts, nerve cords, intercellular spaces, pores, channels, and junctions (cf. [12], [28]). These connections are products of evolution, are functionally necessary, and are characteristic to complex systems of interacting subsystems, such as multicellular organisms and their parts. As a consequence, anatomical objects possess regions within their otherwise bona fide outer boundary that are fiat [12]. If the dichotomy of fiat and bona fide boundaries is absolute, as it has been claimed [9], [10], the respective entities would have to be treated as fiat entities. Consequently, no bona fide biological objects would exist at levels of granularity coarser than the molecular level, since even organelles exhibit such connections.

Some authors argue, however, that in many cases the fraction of fiat boundaries is very small compared to the total outer boundary of anatomical objects and, therefore, can be ignored in these cases (e.g. [28]). The question, then, is what are the criteria on which to decide whether a given portion of fiat boundaries can be ignored? How much fiatness can be tolerated? And how could this fit with the claimed categorial and thus absolute distinction of fiat and bona fide boundary?

Granularity-dependence: Apparently, regarding continuity and connectedness we are, again, dealing with a granularity-dependence of the bona fideness of boundaries. Unlike the general granularity-dependence of physical bona fide boundaries discussed above, however, in which all physical boundaries lose their overall bona fideness at very fine levels of granularity and become fuzzy, the specific granularity-dependence of biological objects is gradual. For example the liver of a cow: The liver is surrounded by a compact layer of extracellular matrix and the peritoneum, which at its turn surrounds most organs of the trunk. The liver is thus located in a large basal swell of the peritoneum, which almost completely surrounds it. As a consequence, one can easily demarcate the liver from the rest of the cow at first glance by using traditional preparation techniques and inspection by eye. After closer inspection, however, one will realize that the liver is connected to the rest of the body by various blood vessels, bile ducts and nerve cords. Moreover, when using a light microscope, one will realize after preparation that the number of vessels, bile canaliculi and nerve cords is much higher than expected. However, on the light microscopic level of resolution one would still think that the combination of cell membranes of the outer most liver cells provides the liver a continuous bona fide boundary that is only interrupted by the lumina and axons of the afore noticed vessels, capillaries and nerve cords. After increasing resolution by using electron microscopy, however, one will see that even the cell membranes themselves do not provide bona fide boundaries for their cells, because each membrane is interrupted by gap junctions, which connect the cytoplasm of the liver cells with each other and with the surrounding tissue. As a consequence, the total outer boundary of each liver cell is interrupted by a small fraction of fiat boundaries and the fraction of fiat boundaries of the total outer boundary of the cow liver is larger than expected by light microscopy. Therefore, at some level of granularity, the bona fide total boundary of a supramolecular biological object will turn into a bona fide boundary interrupted by very few portions of fiat boundaries. When further increasing the resolution, the proportion of fiat to bona fide boundary often increases as well. This is independent of and qualitatively different from the general granularity-dependence of bona fide boundaries discussed further above, which only comes into effect at very fine levels of granularity and results in a switch from a bona fide total boundary to a fiat total boundary.

The specific granularity-dependence poses fundamental problems to ontology design in the biomedical domain. Since the connections (i.e. conduits, tunnels, vessels, ducts, nerve cords, intercellular spaces, pores, channels, and junctions) play a fundamental causal role in those biological systems to which they belong, they are of genuine interest to the biomedical domain and cannot be ignored in ontology design. Whereas for most domains the general granularity-dependence of the bona fideness of boundaries might be unproblematic, since it is usually restricted to levels of granularity that are outside their scope and focus, this is not the case for the specific granularity-dependence of biological objects, which is not restricted to the finer levels of granularity. As a consequence, at least the specific granularity-dependence of the bona fideness of physical boundaries and therefore also the granularity-dependence of the distinction of objects and fiat object parts must be accounted for in ontology design in the biomedical domain. For an approach how this can be achieved in principle via using different representations for the same type of entity for different levels of granularity see Vogt et al. [26].

2.2.3 Fiat Boundaries and Bona Fide Landmarks—Spatio-Structural Fiatness comes in Degrees.

The discussion above demonstrates that any given bona fide boundary can be bona fide at some coarser level of granularity and fiat at finer ones. It seems as if fiatness, or bona fideness respectively, comes in degrees. This impression is reinforced when considering that the possibility to reliably specify and re-locate any given fiat boundary requires some bona fide landmarks and coordinates (i.e. bona fide parts of the same or lower dimensionality that are used to locate a fiat boundary, as for instance the juncture of two blood vessels as a bona fide landmark for locating a fiat boundary between the two parts of the vessel, the part before and the part after the juncture) or other pragmatic or even scientifically justified criteria [7], [8], [29]. In other words, fiat entities are to varying degrees supervenient on bona fide objects on finer levels of granularity [7], or some other unambiguously identifiable landmarks. From this follows that, because their specification and re-location involves real properties of the underlying factual materials, fiat entities usually owe their existence not exclusively to human fiat [10]. These real properties also constrain the range of locations of fiat boundaries that are relevant in the scientific discourse.

In its pure and strictest meaning, fiatness implies mind-dependence, and thus a fiat boundary is a boundary that is determined by human fiat, lacking any natural indication. Fiat boundaries in this sense, however, would be inapplicable in any practical context. Instead, fiat boundaries of interest usually rest on:

1. threshold values as fiat landmarks within a continuous heterogeneous field, the values being based on some legal or otherwise specified convention and agreement, as for instance isobars, the International Date Line, or meters over mean sea level;
2. bona fide mathematical and topological landmarks, for example the center of mass of a material entity, the upper and lower hemisphere of a rotating sphere, or the saddle point of a curve (cf. [9], [10]);
3. bona fide landmarks within a homogeneous field that are based on natural units, as for instance the classification of chemical elements based on their characteristic number of units of mass;
4. spatio-structural bona fide landmarks, as for instance the demarcation of an apple hanging from a tree, which is supervenient on the branching point of the stalk from the branch it is connected to, or the fiat boundaries of your right upper arm, which are supervenient on your armpit and elbow that relate to the position and function of your humerus and its associated muscles, all of which are bona fide objects themselves;
5. the identification of causal subsystems, i.e. spatio-structural parts that actively participate in causal processes, which are characteristic to the subsystem and that play a causal role within the system as a whole, as for instance the apple on a tree as a unit of reproduction, your thumb or your right upper arm as units of locomotion, the mesodermal germ layer as a unit of embryogenesis, or the active center of an enzyme as a physiological (biochemical) unit.

These examples of different types of fiat boundaries demonstrate the broad range of degrees of fiatness that can be involved when dealing with biological entities, ranging from (a), full-blown fiatness that is exclusively based on convention, to (e), for which we argue that it is actually a bona fideness, since it is exclusively based on real properties of a causally dynamic reality. To the latter case belong all those entities that are delimited on grounds of their causal properties and dispositions—functional units, as for instance your right upper arm. Whereas one can argue that the boundaries of such functional units are to some degree fuzzy and indeterminate, this indeterminacy is granularity-dependent like all other bona fide boundaries, and one could argue that they can be assigned to conceptual rather than ontological reasons.

A biological object can be looked upon from very different perspectives (e.g. spatio-structural, developmental, physiological, evolutionary), with each perspective putting a different focus on the real properties of the object. As a consequence, when partitioning an object, the resulting partition will differ from perspective to perspective [30]. Whether a given partition will yield fiat parts or bona fide objects will not only depend on the real properties of the entity and the level of granularity of focus, but also on the perspective applied and thus, to a certain degree, also on the interests of the person conducting the partition.

Unfortunately, many aspects of the ontological theory of fiatness have been developed in the context of geographical use cases and applications (e.g. [6]–[10], [17]). Anatomical structures and biological objects in general have been touched upon only briefly and await closer examination in terms of consequences for the criteria of distinguishing fiat from bona fide boundaries and their status as a categorial ontological distinction (for exceptions see [12], [28], [31]). Since biological entities actively participate in many different types of causal processes as causal agents in a dynamic reality, we will take a closer look at the different types of dispositions and historical relations that can be used for partitioning and demarcating biological material entities. Biological processes range from evolution to individual development and embryogenesis, from physiological processes within an organism to all kinds of ecological and social interaction. We think that the discussion about what criteria must be used for distinguishing fiat from bona fide boundaries, and fiat from bona fide entities, respectively, must not only consider intrinsic spatio-structural properties but also the causal roles and historical relations of material entities and the ontological nature of the respective systems in which they constitute subsystems.

2.3.1 Dispositions Independent of Morphogenesis.

Three different basic types of functional units that refer to dispositions that do not involve morphogenesis can be distinguished for biological material entities.

1. Locomotory dispositions & functional units of locomotion: Mobile organisms usually possess various parts that are bearers of the disposition to move or to be moved relative to the position of the organism as a whole. Your right upper arm, for instance, is spatio-structurally bounded by a portion of bona fide boundary, i.e. the surface of your skin, but also by a portion of fiat boundary, i.e. the demarcation from your right forearm and your trunk. According to Smith's [6]–[10] notion of fiatness, your right upper arm would therefore be a fiat body part. Moreover, following this notion of fiatness, its demarcation would exclusively rest on grounds of mental and linguistic activities—your right upper arm would represent an artificially delimited part of your body. The recognition of your right upper arm as a functional unit of locomotion, however, is not exclusively the product of mental and linguistic activities, and its delimitation from the rest of your body is not arbitrary and does not merely rest on acts of human decision. When leaving aside BFO's strictly spatio-structural framework and, instead, employing a framework of functional systems that focuses on the locomotory musculoskeletal system, your upper arm is a bona fide entity—a genuine natural (i.e. mind-independent) unit of locomotion that is delimited by its locomotory dispositions. Your right upper arm is a functional unit or element of locomotion that can move or be moved independent from the rest of your body. Granted, purely spatio-structurally, your upper arm is delimited from the rest of your body involving fiat boundaries. However, these fiat boundaries rest on bona fide landmarks, as for instance the proximal and distal limits of your right humerus, which is a bona fide object, and are thus also spatio-structurally not completely fiat in a strict sense. In fact, they are very close to bona fideness. This bona fideness is additionally affirmed by the locomotory function of the respective entity.
   In the same way one can argue that your right forearm, your hand and each of your fingers are bona fide functional units of locomotion, although they are fiat entities from a strictly spatio-structural point of view. The important point here is that their delimitation is not the product of mental or linguistic activities, but reflects reality, only from a locomotory-functional perspective instead of a purely spatio-structural perspective. This can result in spatio-structurally fuzzy delimited bona fide entities, as the example of your right upper arm indicates.
2. Physiological dispositions & functional units of physiology: Organisms usually possess various parts that are bearers of the disposition to actively participate in physiological processes within the organism, which are more or less vital for sustaining the integrity of the organism as a whole, keeping it operating and alive. These parts physiologically interact with other parts of the same organism or with ingested biotic and abiotic substances. Spatio-structurally, many of these parts are fiat entities, as for instance the human heart, which is delimited from its connecting veins and arteries by fiat boundaries. Physiologically, however, due to the functional role the heart has as a pumping organ maintaining the blood flow, it is a bona fide functional unit. The active center of an enzyme is another example of a bona fide functional unit of physiology that is spatio-structurally a fiat entity. Its spatio-structural demarcation involves fiat boundaries, which, however, rest upon bona fide landmarks (e.g. transmembrane domain, extracellular domain, transition between alpha helix and pleated sheet, coils, folds, indentations, grooves). Functional units of physiology exist independent of human mental or linguistic activities in the same way as functional units of locomotion.
3. Physiological dispositions & functional units of ecology: Organisms possess various parts that are bearers of the disposition to actively participate in causal processes involving parts of other organisms or material entities from their respective abiotic environment. In other words, every organism possesses structures with which it interacts with its biotic and abiotic environment and which are vital for sustaining the integrity of the organism as a whole. Because the respective functions of these structures involve a larger surrounding system, including the biotic and abiotic environment, they are ecologically relevant. Spatio-structurally, many of these interacting parts are fiat entities, as for instance the eye pits of gastropod species of Patella, or the pinhole eye of Haliotis species, the abalones, which represent specific concave regions of the epidermis, in which the epithelial cells are differentiated to photoreceptor cells but remain continuously connected to the rest of the epidermis. Although these eyes are spatio-structurally fiat entities, due to the functional role they take in for the organism as a whole in interacting with its environment, they are ecologically bona fide functional units. Like the other types of functional units discussed above, functional units of ecology exist independent of human mental or linguistic activities.

2.3.2 Dispositions Involving Morphogenesis.

Two different basic types of functional units that refer to dispositions that involve morphogenesis can be distinguished for biological material entities.

1. Morphogenetic dispositions & functional units of development: Biological objects originate, transform, mutate, merge, and differentiate. They are in a constant flux. The processes of genetically and environmentally induced changes of the spatio-structural composition and qualities of an organism are generally referred to as its development. All parts of an organism bear the disposition to develop. Each part has its own genetically determined developmental sequence of changes, called its morphogenesis. Often, the development of one part is causally dependent on the development of another part or both developments are controlled by the same genetic control mechanism. As a consequence, their development is causally linked and coordinated with one another—they cannot develop independently from each other. In other words, they form a functional unit of development.
   For instance, the distinction of the three germ layers mesoderm, endoderm and ectoderm of undifferentiated tissue in early embryogenesis is the distinction of three functional units of development that can be characterized by the different morphogenetic dispositions they bear. Usually, the three germ layers cannot be differentiated on purely spatio-structural grounds. Despite their spatio-structural fiatness, the layers are nevertheless bona fide functional units of development, because they exist independent of human mental or linguistic activities. Other examples are the 4 d cell in spiralian metazoans, which gives rise to the entire entomesoderm of these animals, the spermatogonia that gives rise to sperm cells, and the apical meristem in higher plants that gives rise to stem and leaves and their derivates.
2. Morphogenetic dispositions & functional units of reproduction and propagation: Organisms have the disposition to reproduce, either sexually or asexually. During the course of evolution, various structures (and behavioral strategies) have evolved that bear the function to facilitate the reproduction of the organism and the propagation of its offspring. These structures form functional units of reproduction and propagation. Typical examples for functional units of reproduction are sexual organs, as for instance flowers, and anatomical structures that attract the attention of potential partners for mating, as for instance the colorful plumage of males in some bird species, which are usually spatio-structurally demarcated from the rest of the organism by fiat boundaries. This holds for many functional units of propagation as well, as for instance fruits, the parachutes of Taraxacum species, and the hooks of the seed of Arctium species. Despite their spatio-structural fiatness, these units nevertheless exist independent of human mental or linguistic activities and thus represent bona fide functional units of reproduction and propagation.

2.3.3 Structural Anatomy versus Functional Anatomy.

The examples above demonstrate that entities can be demarcated not only exclusively on grounds of their intrinsic spatio-structural properties, but also on grounds of a combination of their spatio-structural properties and their dispositions. Whereas the former is important when delineating entities of structural anatomy, the latter is important for delineating entities of functional anatomy (cf. [31]). The boundaries of entities of structural anatomy are determined on spatio-structural grounds, the boundaries of entities of functional anatomy are determined in such a way that the entity delineated is a function bearer—a unit that, as a whole, bears the disposition to perform a certain function [31]. Entities of functional anatomy, however, often involve spatio-structurally fiat boundaries that are only bona fide from a functional point of view. Some functional units, as for instance metanephridial systems, even lack physical connectedness and are spatio-structurally discontinuous, because they are composed of a spatially separated group of material entities, i.e. the filtration site formed by podocytes at the blood vessels and the nephridial duct draining the coelomic space that store the filtrate. Instead of physical connectedness, they exhibit functional connectedness.

The identification of entities of functional anatomy often rests on the use of spatio-structural bona fide landmarks that are subparts of the entity, and the localization of the entity's boundary usually involves some degree of fuzziness. This fuzziness and indeterminacy, however, cannot be used as a categorial argument against their possible bona fide ontological nature, because the respective indeterminacy is in the same way granularity-dependent as the indeterminacy of spatio-structural bona fide boundaries at finer levels of granularity (see above).

From the fact that boundaries are not only granularity-dependent but also perspective-dependent, we conclude that for any biological organism one can always distinguish spatio-structural partitions from spatio-functional partitions. Moreover, as the examples from above indicate, several different spatio-functional partitions can be differentiated (e.g. locomotory, physiological, ecological, developmental, reproductive). What is seen as fiat and what as bona fide depends on the perspective: a given entity can be fiat from a spatio-structural point of view, but bona fide from a functional point of view. As a consequence, one has to distinguish a taxonomy of structural anatomy from a taxonomy of functional anatomy (see also [31]), or its more specific taxonomies of various types of functional anatomies.

Restricting the criteria for delimiting fiat from bona fide boundaries to intrinsic spatio-structural properties and ignoring all dispositional properties of an entity reflects the position of the proponents of form of the traditional opposition of form versus function (see the famous controversy between Etienne Geoffroy St. Hilaire and Georges Cuvier [32]–[34]). Hitherto, most ontology authors focused on form rather than function when discussing the ontological nature of boundaries and basic categories of material entity—they propagate, so to speak, Etienne Geoffroy St. Hilaire's position. We want to bring Georges Cuvier's position into the discussion.

2.4.1 Structural Integrity and Stability of an Entity over Time.

Three different basic types of historical units can be distinguished that refer to a common causal history that is responsible for maintaining the structural integrity and stability of the respective biological material entities over time. (Please note that we ignore all questions regarding temporal boundaries of the time of existence of material entities. Their discussion goes beyond the scope of this paper.)

1. Developmental relations & historical units of development: Whereas a functional unit of development is a material entity that is delimited by bearing a specific developmental disposition, i.e. a promise to behave in a certain way in future developmental processes, a historical unit of development, in contrast, is a material entity that is delimited by the fact that all its parts share the same developmental history. Thus, the developmental processes have already occurred and certain particular structures shared the same developmental history. A historical unit of development is composed of those particular parts of a particular individual organism that have proven in the past that they have developed in concert, as a unit of development, thereby maintaining the spatio-structural integrity of the entity as a whole. In other words, the entity, with all its parts, is delimited on grounds of the fact that it has acted as a functional unit of development in the past. Its identity is thus based on a common morphogenetic history and not on morphogenetic dispositions.
   One can, indeed, argue that the defining properties of a historical unit of development, i.e. their shared developmental origin, supervene on the morphogenetic dispositions and thus on the intrinsic functional properties of its corresponding functional unit of development, since the former is an effect caused by the latter. Epistemologically, however, when it comes to identifying morphogenetic dispositions and with them functional units of development, one must identify the respective historical units of development that serve as empirical evidence for the existence of morphogenetic dispositions in the first place. Anyhow, just like functional units of development, historical units of development exist independently of human mental or linguistic activities, although in many cases they possess spatio-structurally fiat boundaries.
2. Heredity relations & historical units of heredity: Whereas historical units of development relate to the integrity of particular parts of a particular organism during its individual development, population biologists also talk about other units that show integrity over time, thereby crossing the spatio-temporal boundaries of individual organisms (e.g. species as individuals, see [35]). Many bio-species or bio-populations, for instance, maintain a high degree of structural integrity or homogeneity of their member organisms over time. And so do genes or morphological traits that exhibit considerable stability within a population over a certain period of time. These structures are historical units of heredity that result from reproductive mechanisms that guarantee a high degree of similarity between the original morphological structure and its copies. And, again, also historical units of heredity exist independently of human mental or linguistic activities, although they in many cases possess spatio-structurally fiat boundaries.
3. Heredity relations & historical units of evolution: If we take a look at historical units of heredity at a coarser time-resolution, additional mechanisms must be in effect in order to still maintain structural integrity or stability. It requires stabilizing selection pressures for a given inheritable structure to maintain its spatio-structural integrity through an evolutionary period of time. The respective structures form historical units of evolution. For instance the “members” of a phylogenetic character state (i.e. a set of particular inheritable traits that are structurally identical throughout representatives of various species due to homology) are good examples for historical units of evolution (see also modularity [36]–[39]). Historical units of evolution also exist independently of human mental or linguistic activities, although they possess in many cases spatio-structural fiat boundaries.

2.4.2 Lineages—Constituent Historical Relations of Entities distributed in Time and Space.

Whereas the examples discussed above concern material entities that maintain their structural integrity and stability during some period of the lifespan of an organism or even across the spatio-temporal boundaries of single individuals, biological material entities can also historically relate to one another independently of any shared structural stability. When common historical origin is the only defining criterion, the respective sets of structures usually form spatio-temporally scattered groups of material entities whose delineation as a single material entity requires a principle of connectedness (i.e. principle of unity; see also [24]) other than the principles of physical connectedness, like electro-chemical bonds or physical junctions. The example of the functional connectedness of the scattered parts of a metanephridial system mentioned above already demonstrated that it is in principle possible to apply different perspectives for delineating scattered entities, i.e. groups of spatially separated material entities (for groups see [25]). Common historical origin may serve as another principle of connectedness that allows the delineation of groups of spatio-temporally scattered material entities. (Again, please note that we ignore all questions regarding temporal boundaries.)

1. Relations of common developmental origin & developmental lineages: When a cell divides into two daughter cells during a cell fission event, the two daughter cells share the same developmental origin. Although they may migrate to locations separated in space and may proliferate further into separate tissues (e.g. 4 d cell derivates), they nevertheless constitute, together with their parent cell from which they originated and which does not exist anymore, a developmental lineage. Developmental lineages, although spatio-structurally and spatio-temporally delimited by fiat boundaries, nevertheless exist independently of human mental or linguistic activities and are in this sense bona fide in nature.
2. Relations of common kinship & genealogical lineages: Whereas developmental lineages are restricted to the spatio-temporal boundaries of a single organism, the respective structures can be inherited to offspring and thereby constitute genealogical lineages that cross this boundary. In biology we are talking about gene lineages, lineages of morphological traits or even the kinship relations (i.e. parent-child relations) of families of individual organisms. They represent groups of entities that relate to one another across the spatio-temporal boundaries of a particular individual organism, based on common kinship. The phenomenon of infertile castes within some social insects, like for instance the worker caste of ants, demonstrates that these kinship relations can also have a biological impact (i.e. explaining the existence of infertile workers with otherwise zero fitness) and are thus not merely a construct of human mental or linguistic activities. Therefore, also genealogical lineages, although spatio-structurally and spatio-temporally delimited by fiat boundaries, nevertheless can be bona fide in nature.
3. Relations of common ancestry/descent & evolutionary lineages: If we take genealogical lineages to a coarser time scale and allow them to also cross the spatio-temporal boundaries of bio-populations and species, we are looking at evolutionary lineages. When we are talking about homologues, gene families, apomorphic characters and monophyletic groups of species, we are talking about evolutionary lineages. Evolutionary lineages are spatio-structurally and spatio-temporally delimited by fiat boundaries, but nevertheless exist independently of human mental or linguistic activities and are in this sense bona fide in nature.

2.4.3 Structural Anatomy versus Historical/Evolutionary Anatomy.

The examples given above demonstrate that some material entities, be they scattered or not, can be demarcated on grounds of a combination of their historical relations and their intrinsic spatio-structural properties. Their delineation is relevant for historical/evolutionary anatomy. The boundaries of historical/evolutionary entities of anatomy are determined in a way that the entity delineated is a whole, either spatio-temporally scattered or connected: every historical/evolutionary entity of anatomy is composed of parts which share the same historical/evolutionary origin, with no entity not belonging to it sharing the same historical origin.

Just like with entities of functional anatomy, historical/evolutionary entities of anatomy often involve spatio-structurally fiat boundaries that are only bona fide from a historical/evolutionary point of view. Their specification usually involves the use of spatio-structural bona fide landmarks of subparts of the entity, and their demarcation often involves some degree of fuzziness as well. While the determination of the boundaries of historical entities is often conducted by observation with the aid of specific instruments (e.g. video tracing, 4D microscopy), the actual determination of the boundary of evolutionary entities is usually the result of an extensive comparison of the spatio-structural properties of various structures and necessarily remains hypothetical.

3.1.1 Granular Partitions and Basic Categories of Boundaries.

So far, we have shown that the distinction between fiat and bona fide boundaries is not as straightforward as it is usually assumed. Instead, we are dealing with a variety of different foundational categories of boundaries that are independent of the distinction between fiat and bona fide itself: Before one can decide whether a given boundary is fiat or bona fide, one first has to specify from which perspective this distinction is being made. In other words, the ontological status of a given boundary is perspective-dependent.

In a series of papers, Smith and coauthors (e.g. [15], [16], [40], [41] have introduced a formal theory of granular partitions and discussed its consequences and implications for various problems of reference and truth, including the abovementioned Problem of the Many. Smith and Brogaard argue that whenever we use an expression to refer to some real entity, this brings about “a partition of reality into two domains: the foreground domain, within which the object of reference is located, and the background domain, which comprehends all entities left in the dark” [16] (see also [15], [41]). As a consequence, every partition has its granularity built in and is an artifact of perception, judgment or classification [41]. Judgments, at their turn, “come along with partitions of reality of various sorts, whose type, granularity and scope depend upon the contexts in which our judgments are made” [16].

Therefore, according to this theory of granular partitions, every partition is judgment-dependent and context-dependent (instead of the term ‘context’ we use the term ‘perspective’, which serves a similar function—we prefer ‘perspective’, because it refers to Keet's formal theory of granularity [42] and we already used it in [26], [30]). A context, at its turn, is of a certain granularity, and it is understood to be “a portion of reality associated with a given conversation or perceptual report and embracing also the beliefs and interests and background knowledge of the participants, their mental set, patterns of language use, ambient standards of precision, and so forth” [16]. A context is thus “a matter of what is paid attention to by participant speakers and hearers on given occasions” [16]. Since granular partitions are context-dependent, a granular partition is “a device for focusing upon what is salient and also for masking what is not salient” [16]. As a consequence, a granular partition that puts a particular entity into its foreground will not necessarily recognize all the entity's parts and subparts: when a math teacher counts the number of students in her class, she will use a partition that does not recognize the students on the level of their cellular and subcellular composition. Accordingly, when she has to grade the students' performance in her class, she will use a partition that only judges their performance in math and not in English.

The theory of granular partitions can also manage vague boundaries: “vagueness de dicto is captured at the partition level via multiple ways of projecting crisply” [15] (‘projection’ here refers to the relation from judgment to reality, i.e. the relation from a partition cell to its corresponding portion of reality [40]). A granular partition is considered to be crisp if it projects onto reality in a single and unique manner, whereas it is considered to be vague if it involves a multitude of projections onto reality, thereby interpreting vagueness as a semantic property of names and predicates [15]. Thus, the theory of granular partitions introduced by Smith and coauthors provides an adequate formal framework for dealing with the problem of vague and indeterminate boundaries.

Moreover, in combination with Keet's formal theory of granularity [42], the theory of granular partitions also provides the formal framework for dealing with problems of the granularity-dependence and perspective-dependence of boundaries. Thereby one should note that every partition by itself is entirely fiat in nature [16]. However, as Smith and Brogaard argue, some partitions “are coordinated with bona fide demarcations on the side of objects in reality and some of them merely with fiat demarcations which we ourselves have introduced into reality in our various dealings with nature” [16]. Smith and Brogaard thus distinguish partitions that track bona fide boundaries in reality from partitions that track boundaries that only exist as a result of acts of human fiat. Unfortunately, Smith and coauthors do not provide any criteria for distinguishing fiat and bona fide boundaries—they merely discuss the possibilities of fiat and bona fide partitions.

The hitherto prevailing notion of boundaries restricted itself to the spatio-structural perspective that does not account for the role of time and thus processes in reality and how they affect the ontology of boundaries and the ontology of basic types of material entities. In other words, the strictly spatio-structural perspective is ignorant of causality. Above we have shown that there are other types of entities besides purely spatio-structurally defined ones, all of which are epistemologically relevant and empirically accessible. The main difference between them and spatio-structurally defined entities is that only spatio-structural entities are exclusively demarcated on grounds of their intrinsic structural properties, their qualities. In contrast, the other entities are defined either (i) in terms of dispositions and therefore involve types of processes that are repeatable in principle (i.e. based on a notion of universal causality), or (ii) they are defined in terms of historical relations and therefore involve a particular sequence of processes that has taken place in the past and the effects this sequence had on a collection of particular spatio-structural entities (i.e. based on a notion of particular causality). The resulting general distinction of foundational types of boundaries and of material entities thereby reflects the very basic distinction of scientific disciplines or aspects of empirical sciences, which can be classified into the following three basic categorial perspectives (i.e. context categories):

1. Spatio-structural perspective: the view on what is given now, at a particular point in time, i.e. what is intrinsically inherent in material entities; descriptive and inventory-oriented; restricted to passive observation; represents reality in a way that is analog to the painting of a still life (cf. Etienne Geoffroy St. Hilaire's structuralism position of priority of form).
2. Predictive perspective: the view on what can happen in the future, dealing with dispositional/functional aspects of reality and thus with potentiality; predictive and systems-oriented; involves experimentation and the active interference and manipulation of a human-independent reality by an investigator; represents reality as a dynamic system and describes material entities with a focus on their potential future interactions, i.e. models an entity's causal space of possible (inter-)action (cf. Georges Cuvier's functionalism position of priority of function).
3. Retrodictive (diachronic) perspective: the view on what has happened in the past, retrodictive and history-oriented; involves the observation and description of particular processes or the reconstruction of past processes by comparing present distribution patterns of spatio-structural properties and their bearers; represents reality as a dynamic system and describes material entities with a focus on their historical interactions and common origins (Karl Ernst von Baer's position of embryology; Charles Darwin's position of evolution).

Above, we have argued that one can meaningfully distinguish fiat from bona fide boundaries and fiat from bona fide entities within all three perspectives. In all three perspectives fiatness implies mind-dependent (i.e. purely epistemological/conceptual) delimitation, whereas bona fideness implies mind-independent (i.e. natural, ontological) delimitation. As a consequence, some entities are spatio-structurally demarcated by fiat, although functionally they are bona fide. One can argue that these spatio-structurally fiat boundaries are nevertheless mind-independent and, as a consequence, bona fide, but only from a non-spatio-structural point of view. Therefore, one must distinguish between mind-dependent and mind-independent spatio-structurally fiat boundaries.

Our analysis implies that if one does not restrict the distinction of fiat and bona fide boundaries to a purely spatio-structural framework, but, instead, focuses on the less restrictive and thus more general framework of mind-dependence and mind-independence, several different categories of fiat and bona fide boundaries must be distinguished. Our examples demonstrate that we should not talk about fiat and bona fide boundaries without specifying the perspective we use, because many biological objects can be partitioned into bona fide components in various different ways. Consequently, the distinction of fiat and bona fide material entities necessarily depends on the perspective as well (for a discussion of granularity perspectives cf. [42], [30]).

Any perspective that itself is mind-independent provides a principle of identity (cf. [24]) that depends on the specific aspects of reality that are taken into account (i.e. intrinsic spatio-structural properties versus functional dispositions versus historical integrity or historically caused distribution patterns) and has its own two categories of boundaries, fiat and bona fide respectively. Therefore, one has to distinguish spatio-structural from functional and historical boundaries, and thus also spatio-structural from functional and historical types of material entities. This results in a taxonomy of spatio-structural entities alongside a taxonomy of functional entities and a taxonomy of historical entities. Moreover, in case one further differentiates these foundational perspectives into several more specific perspectives that are still ontologically independent from each other (i.e. perspectives must not supervene on each other), it results in even more taxonomies. This is the case for the different types of functional entities, i.e. locomotory, physiological, ecological, developmental, and reproductive entities, with each corresponding perspective resulting in its own specific taxonomy of material entities.

It should be noted that distinguishing types of boundaries in terms of fiatness versus bona fideness and at the same time in terms of different mind-independent perspectives results in the recognition of distinctions that are ontologically independent from each other and thus truly categorial. Muscles of multicellular animals, for instance, usually can be clearly delineated and distinguished from each other on grounds of their spatio-structurally bona fide boundaries. This is not necessarily the case when employing a locomotory perspective, since sometimes several muscles are innervated by the same nerve and thus cannot be moved independently from each other. As a consequence, at least from a locomotory perspective, not every individual muscle does constitute a bona fide functional unit of locomotion. Instead, individual muscles exist that are functionally fiat entities of locomotion. Thus, some spatio-functionally fiat entities are spatio-structurally bona fide objects, and vice versa, which demonstrates the ontological independence of the respective categories and their underlying perspectives.