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This document is the continuation of the systematic and thematic exposition of Aristotelian metaphysics as a theory of natures, or the ontology of the individual thing.
Sequel to the VIA PRAEDICATIONIS, the Predicables.
We have discussed the so-called 'essential parts' which are features of given beings that express what that given being specifically is. Such a part cannot be removed without destroying the specific identity of that being. The mentioned being can be an accident, but here we are exclusively dealing with essential parts of complete beings, such as an organism or a crystal, or, if not, then we are dealing with the carrier-only of such a complete being. And these parts come in three types.
The mereotopological path ultimately leading towards a characterization of essential parts proceeds with some undefined basic concepts and definitions.
Among the basic undefined concepts are the notions of "an individual", "is necessarily such that", ( proper and improper ) "part" ). While an improper part, or just part, can also mean the whole, this is excluded by the notion of proper part.
X, Y and Z are variables standing in for proper names of individuals and non-individuals.
The item to be defined will be connected to the definition by the sign " = ", which means : " if and only if ". The Definiendum ( = item to be defined ) is placed between brackets (...), so also the Definiens ( = the definition).
In all of the following we will apply the mereotopological definitions almost exclusively to crystals, because they are, in contrast to organisms, let alone man, much more clear with respect to what they precisely are (for applications to other things see the mentioned document in First Part of Website).
Mereotopology is about real concrete things. Roughly every entity which is not a scattered aggregate is a thing in the mereotopological sense. Such a thing is called a Mereo-totality.
Because it is possible that one (and the same) dynamical system generates a scattered entity, a scattered aggregate (that is, an aggregate of which the parts lie away from each other, that is, they are not contiguous [let alone continuous] ), the dynamical systems approach cannot distinguish between scattered and non-scattered aggregates (which is evidently important for any substance-accident metaphysics). So in such cases a mereotopological treatment is necessary. Generally, substance-accident metaphysics cannot fulfil its task without consulting mereotopology, that is, without bringing mereotopology into its domain.
In Mereotopology there are two categories of Mereo-totalities (concrete real things):
Lets now proceed :
( X is disjoint from Y ) = ( X and Y have no parts in common )
Entities can never be totally disjoint because ultimately they all come together in the concept of Being.
(X is discrete from Y ) = ( X and Y are individuals which have no individual parts in common )
This Salt crystal and that Salt crystal are both individuals, and although they do have an essential part in common, namely their chemical composition, NaCl, they do not have individual parts in common, so they are discrete from each other.
In order to understand the relation between a Substance and its Accidents we must work out the notion of inherence. To capture at least part of what is involved in that notion we need the notion of specific dependence (SMITH, B., 1997) :
( X is specifically dependent on Y ) = (  X is discrete from Y, and  X is necessarily such that it cannot exist unless Y exists. )
In this Definition it is not yet decided whether this dependency goes both ways (two-sided [ = mutual] dependency), or only one way (one-sided dependency).
However, if the dependency is mutual in the same respect, then X and Y cannot be discrete from one another, and, consequently, the dependency does not satisfy  of Definition 3, as can be seen in the following examples :
A crystal of a given species, say A, cannot be an entity that is discrete from its Space Group PLUS Chemical Composition, which (composed entity) can be written as
"(S + C)[A]". This is, because (S + C)[A] is the total content of the definition of species A.
(S + C)[A] is a content (a form) that pervades the whole crystal, that is, it is everywhere present in the crystal. So a crystal of the species A, as representing a content, has (all its) parts in common with (S + C)[A], and thus are not discrete from one another.  of Definition 3 is, accordingly, violated.
If, on the other hand, the dependency is a mere g e n e r i c one, it cannot be mutual :
An Analcime crystal is generically dependent on the content 'symmetry according to the Space Group Ia3d '. There exist other crystal species with the same Space Group. The extension (range of reference) of the concept expressing this content (form) is larger than that of the species Analcime. The Ia3d symmetry is like a genus with respect to 'Analcime crystal'.
However, we cannot say that the content 'symmetry according to the Space Group Ia3d ' is, for it to actually exist (for it to be materialized), generically dependent on an Analcime crystal. On the contrary, it is not at all dependent on the existence of Analcime crystals (Only ' this Ia3d symmetry ' [while pointing to an Analcime crystal] is dependent on ' this Analcime crystal ' ), because it already exists when there exist Garnet crystals (which have a different chemical composition, and do exist, but crystallize in the same Space Group Ia3d ).
And because the Ia3d symmetry pervades the whole Analcime crystal, they (the crystal and its symmetry) cannot be discrete from one another. So generic dependency does not rule out coincidence (commoness) of parts, and, consequently, does not automatically comply with  of definition 3.
As has been said, Definition 3 does not decide whether the specific dependence is two-sided or one-sided. The next Definition is about one-sided specific dependence, and there we will give examples from the world of crystals.
So, with respect to Definition 3, we still must give an example of the 'two-sided '-possiblity left open by this Definition.
It is, however, clear that a specific dependency, involving X and Y, that goes both ways (in the same respect) -- and where X (or Y) is an ontologically independent entity in the sense that it is not an accident (in order to find an example at all -- implies that X and Y cannot be discrete from one another and thus violates  of Definition 3.
If, on the other hand, the dependency is generic, then it is not mutual.
A possible example (given by SMITH) of two entities that are mutually specifically dependent is the north and south poles of a magnet. They are mutually, and in the same respect, dependent on each other : The north pole cannot exist without the south pole, and vice versa. Despite of that they are discrete from one another.
This example (which has its analogue in electrically charged crystals) concerns a condition (state) of some Mereo-totality (a thing, or a genuine Substance), namely its being magnetized. But it does not involve this Mereo-totality. It concerns the dependency of entities that reside in the accidental domain, entities that must be carried by something. Nevertheless it correctly exemplifies Definition 3 while at the same time not exemplifying Definition 4 (one-sided specific dependence).
So far so good. But in aiming at an understanding of the inherence of an accident in a Mereo-totality we are much more interested in finding an example of, to begin with, mutual specific dependence of two discrete entities of which one of them is a Mereo-totality. And such an example we have not yet found.
A special, and important case (of Definition 3) is : one-sided specific dependency, and this is in fact inherence :
Definition 4 (Inherence)
( X is one-sidedly specifically dependent on Y ) = (  X is discrete from Y, and  X is necessarily such that it cannot exist unless Y exists, but not vice versa )
We will see that inherence not exclusively implies specific dependence, but also generic dependence.
A snow crystal is sectored-plated (See Figure above ).
We assume (which assumption is probably correct) that sectored plates can develop on snow crystals but never on any other crystal species.
Here X = sectored-plated. Y = snow crystal. A given (but whatever) snow crystal is discrete from (its being) 'sectored-plated ', because (1) they are both individual, and (2) their content does not overlap : we cannot say that a snow crystal as such is partly sectored-plateness (evident from the empirical fact that not all snow crystals have developed sectored plates). We can also not say that sectored-platedness is partly a snow crystal, because sectored-plateness as such is just a certain geometrical structure, it is not ice, although it is realized only in ice. And, lastly (3), 'sectored-plated ' is just a condition (or, equivalently, a state) of the given crystal, that is, a condition in which that crystal finds itself. But such a condition is replaceable (by another condition, such as 'fern-like branched ' (See HERE for such a crystal)). And because it is replaceable it must be discrete from that (species of) substrate that supports the alternative conditions.
Of course, the term 'sectored-plated ' (and the general term 'condition', or 'state' ) does refer to this substrate, and thus partly coincides with it. But this reference is only implicit. Explicitly the term refers only to the form 'sectored-plateness ' and this form has no parts that coincide with the substrate (which in the present case is a historical individual of a snow crystal [that is, the carrier-only of the snow crystal] ).
So 'snow crystal ' and 'sectored-plated ' are discrete from one another (but only) insofar as the explicit content of the terms is considered.
And with all this it is clear that  of Definition 4 is satisfied.
'Sectored-plated ' is for its actual (material) existence, ontologically dependent on 'snow crystal ' (as its ontological carrier), but 'snow crystal ' does not depend on actually being sectored-plated for its existence (while it does depend on possessing the ability to form sectored plates under certain external conditions). And that means that the dependency is not mutual. So  of Definition 4 is also satisfied.
So 'sectored-plated ' is specifically dependent on 'snow crystal ' (as such this snow crystal represents an Ice crystal), because it can (ex hypothesi) only be materialized in the species ' Ice crystal '.
On the other hand, Ice crystals can adopt (under certain external conditions) a hexagonal columnar shape (and will adopt other hexagonal shapes under other conditions), but ' hexagonally columnarly shaped ' is, for it to exist, not specifically dependent on the existence of Ice crystals, because many other crystals can adopt such a shape. So ' hexagonally columnarly shaped ' is not at all dependent on the existence of Ice crystals. There are enough other substrates existing that are appropriate to receive the determination ' hexagonally columnarly shaped '. But for its very existence it is dependent on being carried by one or another appropriate substrate. And because the extension (range of reference) of the notion 'appropriate substrate' is larger than that of the notion ' hexagonal columnar shape' , the latter depends on the former as if this (former) were a genus. So this dependency, that is, the dependency of ' hexagonally columnarly shaped ' on an appropriate substrate, is a generic one. And because ' hexagonally columnarly shaped ' is just a condition (state) of a given crystal, they (that is, the crystal and its shape) are discrete with respect to one another. So  of Definition 4 is satisfied. Further, the dependence is not mutual, because an Ice crystal does (in order to be an Ice crystal) not depend on it actually being hexagonally columnarly shaped. It can be just as well hexagonally tabloidly shaped (while it does depend on its having the ability to adopt such a shape). So  of Definition 4 is also satisfied. So Definition 4 turns out, in addition to specific, to allow for generic (one-sided) dependency as well.
There are many other accidents like this one that are not dependent on some specific substrate, like for instance the determination 'white'. But all of them must be carried by one or another appropriate substrate in order to exist.
So, generally, the dependency of an accident on a substrate can be specific (as we saw with 'sectored-plated ' ), or it can be generic (as we saw with ' hexagonally columnarly shaped ' ).
We have said that the content 'sectored-plated ' does not (at least it is not explicitly so expressed by the [concrete] term) overlap with the content 'snow crystal '. And because 'sectored-plated ' is just a possible condition (state) of a snow crystal ( 'snow crystals sometimes happen to be sectored-plated ' ), they (crystal and condition) must be discrete from one another, again suggesting that they qualitatively lie outside each other, implying that 'sectored-plated ' is formally independent of 'snow crystal '.
But later we stated that 'sectored-plated ' is dependent on 'snow crystal '. How can that be?
Well, we could say that 'sectored-plated ' is only dependent on 'snow crystal ' for its very existence (what that might be), because 'pure' sectored-platedness needs a carrier, as is indeed indicated by the concrete term 'sectored-plated ', which connotes a carrier or substrate.
But in the present case the dependency is not about mere existence, because here the dependency is specific : Although the condition 'sectored-plated ' is replaceable by another condition, (ex hypothesi) only (particular) ice crystals (and not crystals of other species) can be sectored-plated at all. So here the dependency involves, in addition to possible existence, some (qualitative) content, that is, in addition to the condition of having to be carried by something, some formal dependency turns out also to be involved. And this means that there must be some qualitative overlap between 'sectored-plated ' and 'snow crystal ', which in turn would mean that they are not discrete, and consequently violate  of Definition 4 .
Even in cases where the one-sided dependency of X on Y is only generic, the substrate cannot be just a carrier. It must be an appropriate substrate with respect to a given accident ( = replaceable determination).
All this suggests that clause  of Definition 4 must be removed or changed (this was already evident when discussing Definition 3).
These Definitions were given (Def. 3 explicitly, Def. 4 implicitly) by SMITH, B., in "On Substances, Accidents and Universals : In Defence of a Constituent Ontology", Philosophical Papers, 26 (1997), 105-127. Let us see how he conceives and exemplifies them.
SMITH ' example that should illustrate Definition 4 is :
My headache is specifically dependent on me (as also on my head).
But because Aristotelian metaphysics is a study of natures, that is, of repeatable contents, 'my headache' is not of particular interest, but ' headache ' is, because ' headache ' is ' headache-as-such', and indeed is a repeatable content. This content is per accidens with respect to 'my headache' , because it is per accidens with respect to 'my'. So we want to investigate the ontological and existential dependencies of this content ' headache ' (serving as an example). Does it need a carrier (substrate)? If so, does it need a specific carrier, that is, a carrier that is qualitatively appropriate for carrying ' headache ', while no other species of carrier can? Or is its dependency on a substrate just generic, or is it not at all involving qualitative content?
A headache is a temporary state of its carrier. It can fade away, making room for some other state. Therefore headache and carrier must be entities discrete from one another. So  of Definition 4 is satisfied. And because the body (or head) is not dependent on ' headache ', while ' headache ' does depend, for its existence at least, on the body (or head),  of definition 4 is also satisfied. So ' headache ' is one-sidedly specifically dependent on a [living] body or a head (namely that of humans or also that of all animals having genuine brains).
With all this we see that we have to do with replaceable conditions. Therefore we must -- in characterizing inherence -- stick to clause  of Definition 4 , that is to say, we must concede the discreteness of X and Y with respect to each other.
However, we have seen that, despite of this discreteness, there must exist some qualitative overlap between the carrier and its accident. This means that we must somehow qualify clause , or clause .
The substrate Y of X must, in a way, qualitatively a n t i c i p a t e possible accidents, which possibly are going to determine it, among which is accident X. In the case of X = 'sectored-plated ' and Y = 'Ice crystal ' (snow crystal) this qualitative anticipation is rather intensive, that is, detailed (because 'sectored-plateness' can [ex hypothesi], when it exists, only exist in the species 'Ice crystal ' ( NOTE 85a-1 )), while it is not so intensive in the case of X = ' hexagonally columnarly shaped ' and Y = 'Ice crystal ' (snow crystal).
So inherence is only possible when there is a partly overlap of the content of X with that of Y. Only then the carrier and the accident 'lock' into each other, only then the accident can be properly carried by the substrate.
When, on the other hand, the overlap is (qualitatively) total, as in the case of 'crystal of species A and (S + C)[A] ', the dependency is mutual in the same respect, and thus we do not have a case of true inherence anymore.
So for inherence to be possible there must be a partial overlap between the content of X and that of Y. And this we could express by X and Y being only 'partially discrete' from one another. Such a 'splitting up' of an accident (resulting in two parts : one that already (also) belongs to the substrate, and one exclusively belonging to the accident) could be based on some pre-existent partition of the accident into two mutually inseparable parts.
Thus it is clear that we must restate clause  of Definition 4 (thereby avoiding the problematic notion 'partially discrete' ) :
There is a proper part of X which is discrete from Y, while another part is not, and there is a proper part of Y that is discrete from X.
The next Figure illustates this diagrammatically :
Clause  then, guarantees the necessary asymmetry between carrier and accident.
( X is one-sidedly specifically or generically dependent on Y ) = (  There is a proper part of X which is discrete from Y, while another part is not, and there is a proper part of Y that is discrete from X, and  X is necessarily such that it cannot exist unless Y exists, but not vice versa )
( X inheres in Y ) = ( X is one-sidedly specifically or generically dependent on Y )
This matches finely with our earlier finding that the carrier-only is equivalent to the historical individual of a given species, because such an individual contains, albeit implicitly and indistinctly, all possible accidents.
That part of the content of a given accident that resides already in the carrier-only, at the same time seems to point to a certain dependency of this carrier-only on that accident. But because the carrier-only anticipates many possible accidents, it is not specifically dependent on that particular accident, provided that, in explicitly showing this independency, this accident is replaced by another accident. So the carrier-only, Y, is not dependent on X, and this complies perfectly well with clause  of Definition 4 and 4a .
An accident is dependent on its substrate. As to the reverse of it, a given substrate only implicitly refers to a range of possible accidents (replaceable determinations), because the notion 'substrate' presupposes entities to be carried. Only in this way a substrate depends on accidents, but this dependency can nevertheless be very specific, as we saw in the case of sectored-plated Ice crystals, where the substrate ( ' Ice crystal ' ) is already a very specific substrate (as such demanded by the possible determination 'sectored-plated ' ). Let's dig a little deeper into these matters.
Earlier we conceded that, in a way, a substrate depends on its possible accidents by anticipating them. But is this a true dependence relation? Couldn't we say, with repect to our examples : "A substrate does not at all depend on ' hexagonally columnarly shaped ' or 'sectored-plated ' for that matter, even when it is an appropriate substrate (i.e. a substrate that seems to be geared to receive these particular determinations as soon as the respective external conditions are right). It could just per accidens be appropriate for carrying the form ' hexagonal columnar shape' or the form 'sectored-plateness' when external conditions permit ( The substrate just happens to be so appropriate) " . However, for the time being, we will interpret the anticipation of becoming determined by possible accidents as a genuine dependency of the substrate on the accidents. But of course, a mere anticipation by the substrate of possible accidents, is not the same as already possessing those accidents, so the dependency is very implicit.
In Part XXIX Sequel-33 of Fourth Part of Website we discussed the generation of sectored-plates in snow crystals. We will reproduce a Figure from that Part, depicting the beginning of the development of a sectored plate at a branch of the snow crystal in a two-dimensional projection along the crystal's c-axis. The net, consisting of equilateral triangles, symbolizes the hexagonal ice-crystal lattice :
Figure above : Around the pre-existent shovel (red) the first stage (blue) of a sectored plate has been completed. At the left the structure has not developed facets because of diffusion problems (this is because on both sides of the structure the same such structures develop nearby). The growth center of the sectored plate is indicated by a white spot (earlier indicated with green). If we draw lines from this center perpendicular onto the four new facets we obtain the rosette of actual relative-growth vectors of this growth center. The three ridges of the shovel have been carried along with the growth of the structure. In the Figure the beginning of the curvature of the two lateral ridges, that is, their angles becoming odd, is shown. It is shown rather crudely because we have so strongly magnified the lattice. The change of direction of the course of these ridges indicates a change of the rosette of actual relative-growth vectors of the growth center of the sectored plate. Often the first stage is marked by a border, either in the form of a weak lining or in the form of a more or less pronounced ridge.
The above Figure shows that the crystal lattice is such that sectored plates can be formed when external conditions are right. The precise and complete contribution of the Ice crystal (in sofar as it is an Ice crystal) to it actually being an appropriate substrate for sectored-plateness can, of course, not be read off from this Figure.
The dependency of an accident on a substrate (which is either specific or generic) and the dependency of a substrate on an accident represent quite different dependencies : The accident is a condition, while the substrate is that of which the accident is a condition.
Later we will call the dependency of a condition (accident) on its substrate "downward dependency", while we will call the dependency of the substrate on a condition (accident) "upward dependency", to express the asymmetry.
Let's summarize our findings :
Where X is ontologically dependent on Y, we have established the following (where " ==> " means : " implies " ) :
In some respects the converse of specific dependence is the relation separability :
( X and Y are mutually separable ) = (  X is not necesarily such that any part of Y exists, and  Y is not necessarily such that any part of X exists. )
While 'sectored-plated' cannot exist without a snow crystal (as its ontological carrier), a sectored plate can. That is, when such a plate is detached from the snow crystal, it can keep on existing, and also the remaining snow crystal torso can keep on existing.
( X is a one-sidedly separable part of Y ) = (  X is a proper part of Y,  some part of Y discrete from X is specifically dependent on X,  X is not specifically dependent on any part of Y discrete from X. )
SMITH gives the following example : X is a human being, and Y is the sum of X together with some one of X 's thought. But here we stick to examples from the world of crystals.
X = a snow crystal. P = prismatic habit. X + P = Y = prismatic snow crystal.
A snow crystal can exist separate from a prismatic habit, because it can have a tabloid habit instead. In fact when a prismatic snow crystal is, while growing, transformed into a tabloid crystal (as a result of changed external growth conditions), the snow crystal is separated from 'prismatic habit'.
Let'see where this leads us.
Let us denote "some part of Y discrete from X, that is specifically dependent on X " as P.
Then in this definition,  implies that P cannot be separated from X, and  implies that X can exist, separated from P.
An individual snow crystal X can exist separated from a habit P (it can have habit T [tabloid] instead), but that habit P cannot exist separately from that snow crystal, or from any material object whatsoever. But in this example "separated" should not mean the result of an (actually executed) division, in this case a division of X + P into X and P, because X and P are not parts of X + P. The snow crystal X cannot be separated from its habit P, neither can P be separated from the snow crystal X. Or, in other words, while X has habit P, it cannot be separated from P. When X has habit P, P is not a part of Y, but a condition, or state, in which X finds itself : P is an Accident of X. And because P is just a condition (Accident) of X, P does not involve any boundary dependence (See below) in relation to X, and therefore X is not an actual or potential part (and consequently not a one-sidedly separable part) of Y. So it seems to me that Definition 6 (given as such by SMITH, 1997) contains an inconsistency (which also is the case when we use SMITH's example of a human being X, who is thinking a thought T, and where X + T = Y : it is demonstrated in the document in First Part of Website), as a result of associating part with specific dependence, which in this context implies inherence. Indeed, on the second page (as SMITH ' article is presented on the Internet) SMITH states :
"Accidents and substances will then be linked not as parts to wholes, but through the special relation of inherence." And on p. 5 he says :
"Note that the relation of boundary dependence [See Definition 10] does not hold between an accident and its substantial carrier." ( The human being X is the carrier of his thought T ). This means that there is no boundary between X and T. And, correspondingly, the snow crystal is the carrier of its prismatic habit. There is no boundary them.
So the phenomenon of inherence, for example the inherence of the thought T in a human being X, or the inherence of the habit P in a snow crystal, is much more subtle than something like " X, that is a one-sidedly separable part of Y, meaning that X is separable from P, but P is not separable from X ". So for inherence we must stick to Definition 4.
( X and Y form a partition of Z ) = (  X and Y are parts of Z,  X and Y are discrete from each other,  no part of Z is discrete from both X and Y. )
The system of cleavage planes (X) and what lies between them (Y) form a partition of a Calcite crystal (Z).
Cleavage planes are definite planes (that is, having definite directions) in the crystal where the chemical bonds are relatively weak, that is, weaker than other bonds in the same crystal. The boundaries of such a partition are common boundaries.
But also when parts of the crystal have come apart as a result of cleavage along some of these cleavage planes, but are macrocopically still contiguous, we have a partition. In this case the parts of the partition have boundaries of their own (so these boundaries are not common boundaries).
( X is atomic ) = (  X is an individual,  X does not involve inherence,  there is no partition of X into mutually separable parts. )
In the context of mereotopology "individual" can mean either "an individual" (an individuum), or just "individual" (something is just individual, like the 'redness' of some individual thing is individual, without being an individual).
Here, in the definition,  should mean : X does not satisfy Definition 4.
Something being 'atomic' can refer either to a carrier-only (substrate of accidents) or to an accident. If that to which is being referred is not specifically dependent on any other entity the reference is to the carrier-only (See next Definition).
When referring to a carrier-only we can, for example, state the following : "An ice crystal (X) (which is individual and thus satisfying  of Definition 8) is called atomic, when we explicitly refer to the carrier of its accidents, and only implicitly to the complete set of replaceable determinations (accidents) of it ". That is to say : Explicitly, the accidents are left out (of consideration), and thus no inherence is explicitly involved. We do this by considering the historical individual instead of the here-and-now individual, because in the historical individual (which is the individual considered during the whole time span of its existence) all replaceable accidents are not (fully) determined, not ultimately specified, and thus, in effect, absent. What is left is the carrier-only, that is, the substrate of those accidents. So  of Defintition 8 is then satisfied.
Replaceable determinations of an ice crystal are for instance : habit, actual shape, being branched, being sectored-plated.
In a dynamical context, and thus involving process and time, X (if not referring to an accident) is the historical individual.
Indeed, if we consider a given ice crystal (or any other given crystal for that matter), not as a here-and-now individual (which it is at any point in time), but as a historical individual (which we also do with people), none of its replaceable accidents (presupposed as types) is fully determined, which is equivalent to them being absent. In this way the historical individual (ice crystal) is the carrier-(of accidents)-only.
We here have defined atomic, and not yet atom in the mereotopological sense (which will be defined in Definition 15). Of course an atom is atomic. But something atomic does not need be an atom, it can be a mere part or region of an atom.
Whether  of Definition 8 is satisfied by a crystal so conceived, we can say the following :
At least two possible partitions can occur in the world of crystals :
When X in Definition 8 refers to something that is specifically dependent on some other entity, while satisfying this definition, X is an accident.
So an accident is also atomic.
The sectored-plateness of this ice crystal is individual (  of Defintion 8 is satisfied).
'Sectored-plateness' explicitly leaves out 'snow crystal' (while 'sectored-plated' explicitly refers to 'sector-plateness', but only implicitly refers (connotes) 'snow crystal' ), and thus 'sector-plateness' as well as 'sector-plated' satisfies  of Definition 8.
'Sectored-plateness' does not have a partition of it into mutually separable parts (because 'sectored-plateness' is a formal content, and a formal content is totally homogeneous). So  of Defintion 8 is satisfied.
( X is substantial ) = (  X is atomic and  X is not specifically dependent on any other entity. )
"Specifically dependent" always, and only, refers to the formal content of something, not to efficient causality.
Something atomic (Definition 8 ) can either be substantial or accidental. Definition 9 defines something being substantial. Commenting on Definition 8 we already dealt with something which is substantial and at the same time not a mere part of a substantial atom but the whole substantial atom, and thus the (whole) carrier-only. But atomic can also mean a proper part or region of an atom, and so something substantial can be just a part of a carrier-only, to which we now turn our attention, while applying the definition to a part of a branched snow crystal-as-historical-individual (thus considered over that part of its life-time in which it was branched [where its branches can vary as to their lengths and shapes, carrying, or not carrying sectored plates, etc., and where all other replaceable accidents are not determined] ).
If, then, we consider an arm of a branched snow crystal, still attached to the crystal, this arm satisfies being substantial, because it satisfies being atomic and is not specifically dependent on any other entity. It does not need any ontological carrier. It is atomic because it is not an aggregate in the mereotopological sense, because any attemted partition (any cut through this arm) involves common boundaries. Such a boundary is an individual part, and this individual part is a common part and so the parts resulting from the attempted partition are not discrete, and consequently they are not mutually separable (because, again, they have a common boundary, and such a boundary is dependent for its existence on both parts). So  of Definition 8 is satisfied. And because the arm is individual, and (if abstractly considered, i.e. considered as the carrier-(of Accidents)-only) does not involve inherence ( NOTE 85a ),  and  of Definition 8 are also satisfied. And because the arm satisfies  of Definition 9, this still-attached arm of the snow crystal is substantial, i.e. it is a part (because it is just an arm) of a Mereo-totality (and moreover of a genuine Substance) (taken as a historical individual), and not (a part of) an accident (i.e. it is not accidental). It is in short a substantial part. It is a part of the carrier-only.
But on actual separation of the arm from the snow crystal's body the arm is not a part anymore and will have a complete boundary of its own. It has become mereotopologically independent, it is a Mereo-totality, i.e. a Substance (in its own right) in the mereotopological sense. But when we supplement this mereotopological treatment with considerations from the dynamical systems approach, then the detached arm is only a Mereo-totality, it is not a genuine Totality, and equivalently, not a genuine Substance within that supplemented approach.
The next Definitions (Definitions 10, 10a and 11) are very important. They are about genuine boundaries of, or within, things.
( X is boundary dependent on Y ) = (  X is a proper individual part of Y, and  X is necessarily such that either Y exists or there exists some part of Y properly including X, and  each individual part of X satisfies . )
For example : X is the surface of a crystal. Also every facet of it can stand for X.
Y is the crystal itself.
With "surface" we do not here mean the skin or rind of some object, but the surface in a geometric sense. This surface is boundary dependent. It is a genuine mereotopological relation.
So the surface of a crystal, and also everyone of its facets (if present) is boundary dependent on the crystal.
Let us analyse the definition.
" X is a proper individual part of Y ", as stated in  of the definition, means that X is not equal to Y : apart from X there is still more of Y.
"properly including X " means that X is contained in something without thereby exhausting that something. So here (in  ) there is a part of Y that is partly at the same time X, partly not.
 and  of Definition 10 are satisfied by X finding itself in the following situation :
Here X is indeed a proper individual part of Y, so  is satisfied.
Y extends beyond X, where we can call it : " Y / X " which means : " Y insofar as it is not X ".
Y / X is discrete from X.
Because X is a part of Y it cannot exist without Y, or at least not without some part of Y that includes X. That is, at least that part of Y that coincides with X is needed for X to exist. But in order to satisfy , X must be dependent either on the whole of Y, or X must be dependent on just a part of it, but this part of Y is then (when we want to satisfy  ) not only that part of Y that precisely coincides with X, but in addition to that (coinciding part) some part, no matter how small, of Y discrete from X (that is, some part of Y / X ). X cannot be just :
We can see that the situation of X in our first drawing satisfies  of Defintion 10 (and it has already satisfied  ).
Now, in order to check , we look to a (proper) part of X :
Here we have a (proper) part of X (yellow). For this part, let us call it Xp , we can say :
Xp , as it is depicted here, that is, existing together with the whole of Y, is necessarily such that either the whole of Y exists, or at least that part of Y must exist (for Xp to exist) that coincides with Xp PLUS some part of Y discrete from Xp. This latter part is contained in Y / Xp :
As one can see, Xp satisfies  : Xp is connected with at least a part of Y that is discrete from it. Therefore X satisfies .
So X, as it is given in our drawing, satisfies Definition 10.
But, although we have now illustrated the definition and demonstrated its actual content, it is clear that the intended meaning of Definition 10 ( X is boundary dependent on Y ) is quite different.
The meaning (that is, the intended meaning) should be that X, as a proper part of Y, is at the extremes of Y, and also all parts of X.
So although I shall not definitively assert that Definition 10, as originally given by SMITH, misses its intention, it is not a very clear definition of the intended state of affairs. Therefore I propose to replace Definition 10 by another Definition that correctly and, above all, clearly, and still in a topological way, expresses the intended meaning :
( X is boundary dependent on Y ) = (  X is a proper individual part of Y, and  X is such that every neighborhood, no matter how small, of any given part of X does contain a part of Y discrete from X.)
In the following diagram the situation of X clearly satisfies Definition 10a :
Here X is indicated by the blue rim at the periphery of Y (red).
In fact the width of this rim must be infinitely small, symbolized by the next diagram :
Here X is the utter extreme part of Y.
The next diagram shows that every possible neighborhood, no matter how small, of any given part of X (as X is given in the diagram) always includes some part of Y that is not X :
So  of definition 10a is satisfied. And because also here X is a proper individual part of Y,  is satisfied too. So X, as depicted in the above diagram , completely satisfies Definition 10a, and is thus boundary dependent.
The surface, or a facet, of a crystal exactly matches this definition, that is, the surface, and also a facet (if present) of a crystal is boundary dependent on that crystal.
When X is a point it can be the extreme of a line, as can be demonstrated by a one-dimensional vanishing neighborhood.
When X is a line or curve it can be the extreme of a plane, as can be demonstrated by a two-dimensional vanishing neighborhood (as is in fact the case in our last two diagrams).
When X is a plane (curved or straight) it can be the extreme of a solid, as can be demonstrated by a three-dimensional vanishing neighborhood.
Indeed, SMITH characterizes the intended meaning of the definition ( X is boundary dependent on Y ) as follows :
Roughly, a boundary of dimension n can never exist alone but exists always only as part of some extended neighborhood of higher dimension. There are no points, lines or surfaces in the universe [that is, in material reality] which are not the boundaries of three-dimensional material things [See next Definition].
( X is a boundary ) = ( X is boundary dependent on some individual )
Because the surface of a crystal, or (of) any of its facets (if present), is boundary dependent on the crystal (that is, satisfies Definition 10a ), it is a boundary of that crystal.
So a genuine boundary of some entity X is not discrete from X, because it has individual parts in common with X (See Definition 2 ).
And consequently a shell, rind, or rim around something (else) is not a genuine boundary of that something. The next Figure illustrates this.
Figure above : The object Y has a rind. This rind has a proper part P (yellow) that is discrete from Y. Therefore this rind is not a genuine boundary of Y.
But the rind is separated from the outside world by a genuine boundary. And this boundary is an entity which indeed is boundary dependent on some individual entity, namely the rind, or, said differently : there is a part of the rind that is boundary dependent on the rind.
Another genuine boundary separates the rind from Y. This boundary is an entity that is boundary dependent on the rind and on Y.
The above Figure shows two types of genuine boundaries, viz., involving one-sided and two-sided boundary dependency. Their difference relates to contiguity (discontinuity) and continuity, and involves non-common and common boundaries. One-sided boundary dependency is about a non-common boundary, while two-sided boundary dependency is about a common(ly possessed) boundary.
( X is a Mereo-totality ) = (  X is substantial,  X has a boundary,  there is no Y that is boundary dependent on X and on some individual that has parts discrete from X. )
The Defintion in fact says that a part of a Mereo-totality is itself not a Mereo-totality.
 excludes substantial parts, i.e. it excludes the possibility that a substantial part is itself a Mereo-totality (and by implication a Substance).
If there were some Y that is boundary dependent on X and at the same time boundary dependent on some individual that has parts discrete from X, then X would be connected with something (by means of Y) that itself extends spatially beyond X, and so X would be just a substantial part of a larger whole comprising X and more. See next Figure.
A zoned crystal is a (single) Substance (and therefore a Mereo-totality). The zones of this zoned crystal are not themselves Substances ( The chemical composition of such a crystal is defined such that the possibility of substitution [of explicitly specified atomic specis] is accounted for.
Parts enclosed by the system of cleavage planes of a given crystal are not themselves substances.
For clarity, we can give Definition 12 in its fully analysed form :
( X is a Mereo-totality ) =
X is atomic Definition 8 ) :
SMITH argues this with the example of two billiard balls which are in contact with each other, as depicted in the next Figure :
Let us now assess the status of one billiard ball, say, X. The boundary, Y, between the two billiard balls, that is, between X and Z, is not boundary dependent on both X and Z, i.e. it is not boundary dependent on X + Z, because this boundary still exists when we destroy one of the balls. To separate X + Z into X and Z we do not have to break the balls away from each other. So X does satisfy  of Definition 12. Moreover X is substantial ( NOTE 85b ), and X has a boundary, so X satisfies Definition 12. This implies that X is a Mereo-totality, also when X is in contact with another ball.
( X is atomic ) = (  X is an individual,  X does not involve inherence,  there is no partition of X into mutually separable parts. )
The complex of the two billiard balls, i.e. the complex X + Z, is clearly partitioned into X and Z. Let us recall the definition of partition, and substituting some of its symbols, to make it easy to read with respect to our example :
( X and Z form a partition of X + Z ) = (  X and Z are parts of X + Z,  X and Z are discrete from each other,  no part of X + Z is discrete from both X and Z. )
Applied to our example :
X and Z are indeed parts of X + Z ( is satisfied), X and Z are indeed discrete from each other, they have no individual parts in common ( is satisfied), no part of X + Z is indeed discrete from both X and Z ( is satisfied). So X and Z form indeed a partition of X + Z.
Because the parts X and Z are mutually separable, they indeed form a partition of X + Z into mutual separable parts. So  of Definition 8 is violated, implying that X + Z is not atomic. And thus X + Z is not substantial, implying further that  of Definition 12 is violated, so X + Z is not a Mereo-totality, and by implication, not a Substance.
So it is already  from Definition 8 which decides that the complex of the two contiguous billiard balls is not a Mereo-totality (and, consequently, not a genuine Substance).
So a whole, consisting of contiguous parts would not be a Substance, according to SMITH' definitions.
But this could, according to me, imply that multicellular organisms are not Substances. So if we nonetheless admit them to be Substances, because they seem to be even the primary instances of Substance, then mereotopology alone cannot decide this. This is to be expected because the relation "belongs to", in the sense that some entity belongs to some other entity, is not a purely mereotopological concept. We must stipulate that the interstices between organic tissue cells belong to the organism. So these interstices are then themselves also parts of that organism. And then the boundary between the boundary of a cell and the interstice (dividing it from another cell) could, it is true, still be interpreted as a bona fide boundary, but this boundary is boundary dependent towards both sides, implying that each cell is not separable from the interstice, and vice versa, so condition  of Definition 8 is satisfied, and then each cell becomes indeed just a substantial part of the organism. It also implies that an organism is not a contiguum but a, albeit heterogeneous, continuum [It is a continuum in the sense that, starting from one point (location) within the entity, we can reach every other point of that entity, without ever having (temporarily) to leave the confines of that entity. So it is a continuum by way of 'belonging to', that is, it is (only) a metaphysical continuum. It is not a mathematical continuum, because it has actual parts (if the latter indeed allow themselves to be interpreted as parts)].
In other cases we must stipulate that some cavities do not belong to the organism, by means of a functional argument, i.e. not by means of a mereotopological argument.
We can detect this clearly in the discussion of SMITH & BROGAARD, 2000, p. 25, where they discuss the question whether some items like the amnionic cavity, or the digestive canal, do or do not belong to the organism in question.
Definition 12, while basing itself on the Definitions 8, 9, 10 and 11, pretends to define a Mereo-totality (and by implication, a genuine Substance).
We should, however, investigate what then precisely is being defined. Is it the (Mereo-totality insofar as it is the) carrier-only, or is it the full-fledged Mereo-totality? And if (it defines just) the carrier-only, what then is the nature of such a carrier-only? Is it contiguous or continuous, and if the latter, is it homogeneous or heterogeneous?
In order for an entity X to be a Mereo-totality, Definition 12 demands X to be substantial, and that means, according to Definition 9 , that X must be atomic and not specifically dependent on any other entity. So X is the substantial part of the Substance-Accident composite, which means that X (as finally defined in Definition 12) should be the carrier-only (A substance-accident composite is not atomic) ( The mereototality-as-carrier-only is only generically dependent on its accidents).
This carrier-only (X) should then have the following three important characteristics :
(1). There is no partition of X into mutually separable parts (Definition 8 ). This means that the carrier-only must be a complete unity, and this in turn means that it is not a contiguum (like the two billiard balls, or the quartzite pebble), even not a tight contiguum.
A partition of non-separable parts is admitted though.
(2). X does not involve inherence (Definition 8 ).
(3). X is not a substantial part (Definition 12 ), i.e. it is not a proper part of a carrier-only.
These are then the characteristics of the Mereo-totality insofar as it is the carrier-only, that is, the historical individual.
A fully-fledged Mereo-totality, on the other hand, is the here-and-now individual, having all its accidents with it, and these accidents are fully determined.
If there exist in such a fully-fledged Mereo-totality partitions into mutually separable parts, and if such partitions are replaceable, then they are genuine determinations, that is, they are then mere accidents. And that means that these parts are not parts at all (because accidents are not parts) : they are virtual parts, or, equivalently, they are actual determinations. So also such a fully-fledged Mereo-totality is still a continuum.
So if partitions of mutually separable parts are just replaceable determinations (meaning that these 'parts' are not parts at all, then a fully-fledged Mereo-totality, having these partitions, also complies with Definition 12 .
It is perhaps useful to reproduce Definition 12 in its fully analysed form :
( X is a Mereo-totality ) =
X is atomic Definition 8 ) :
Let's proceed to the next Definition.
( X is an accidental ) = (  X is atomic and  X is specifically dependent on some individual Y. )
( X is an accident of Y ) = (  X is an accidental of Y,  Y is a Mereo-totality,  there is no substantial proper part Z of Y such that X is an accidental of Z. )
A given snow crystal happens to be sectored-plated.
A given crystal has a diameter of 1 cm.
A given snow crystal happens to be tabloid-shaped ( In snow crystals the habitus is not intrinsic, because it can vary with external conditions (such as ambient temperature).
( X is an atom ) = ( X is either a Mereo-totality (or Substance) as carrier-only, or X is an Accident. )
A historical individual of a crystal species ( = carrier-only).
Sector-plateness (of a snow crystal) ( = accident).
Actual size (of a crystal) (say, a diameter of 1 cm) ( = accident).
The relation of specific dependence is the bond which holds atoms together in molecules of different sorts. Such molecules are the referents of simple empirical judgments, such as "Socrates is tanned", "John is kissing Mary", etc.
( X is closed under specific dependence ) = ( no part of X is specifically dependent on any entity discrete from X. )
"no part of X" can also mean X itself.
So X can be a Mereo-totality as carrier-only, a Substance as carrier-only (See Remark), or a Mereo-totality (Substance)-Accident composite. It cannot be an accidental, implying that it cannot be an Accident.
Remark : A carrier-only is only generically dependent on its being determined by a (complete) set of types of (replaceable) determinations. Ontologically it does not matter precisely which determination (here, which Accident) represents each type.
Although definition 16 is meant to rule out accidentals, on closer scrutinity it turns out not to do so. An accidental is, just like the carrier-only, not specifically dependent (on a carrier), it is only generically dependent on its having a carrier. Thus something being an accidental is not ruled out by that Definition, as given by SMITH.
If however we hold (with SMITH) that the dependence of Accidents on their carriers (substrates) is specific, i.e. a one-sided specific dependence, then accidentals and Accidents are indeed ruled out by something that is closed under specific dependence. Indeed on p. 4 of the Internet version of his article SMITH states :
"My headache, for example, is specifically dependent on me (as also on my head)."
Of course in this case my headache is not just dependent on one or another (human) substrate, but is dependent on a certain individual, namely me. It is an individual Accident. However, this is because the Accident headache is qualified by "my". But "my" is extrinsic to a headache as such, so it should not be included in a term which purports to signify an Accident. Headache as such can occur in several 'species' of head, and -- referring to a genuine (and complete) Substance -- in several species of 'animal'.
An Accident needs an appropriate substrate in order to exist, but this still leaves open several possibilities for a substrate, and thus an Accident (and also an accidental) is just generically dependent on a substrate. If it were specifically dependent, then it is either a per se proprium (such as Space Group PLUS Chemical Composition in crystals), or a 'per accidens' proprium (such as RATIONAL or CAPABLE OF LAUGHING in humans), but such a 'per accidens' proprium is only in a very special and very weak sense per accidens. For all intents and purposes it is a per se determination, and consequently not a genuine accidens (but a proprium). So it seems that indeed all true accidents are only generically dependent on a substrate (just like a substrate is only generically dependent on [its having] accidents).
Although I will not once and for all rule out the possible asymmetry between, on the one hand, the dependence of a carrier-only (substrate) on Accidents, which is generic, and, on the other hand, the dependence (specific) of an Accident on its carrier, I, for the time being, and preferably, will stick to the generic nature of the dependence of Accidents on their having a carrier (substrate), and should change the relevant Definition accordingly.
With respect to the example of a complex molecule, namely the molecule John is kissing Mary, I would like to state the following :
'kissing' as such is not exclusively dependent on John (and 'being kissed' as such is not exclusively dependent on Mary).
If the act of kissing can only occur in one particular species (of organisms), then it is specifically dependent on a carrier (substrate).
So generally we must say that accidents are either specifically or generically dependent.
Consequently, we must always take a possible generic dependence of accidents into account, and thus the asymmetry must be indicated by upward and downward dependency.
As a result of the foregoing I will propose to amend Definition 16, in order to properly exclude accidentals and consequently Accidents.
The dependence relation between an Accident and its having a carrier is an asymmetric relation in the following way :
The carrier-only is upwardly dependent on its having Accidents (The dependence is a generic one). The Accident is downwardly dependent on its having a carrier (The dependence is also of a generic nature). So if we want to rule out Accidents we should exclude downward dependence. If we do so then we exclude accidentals and Accidents, but admit Mereototality(Substance)-Accident composites and also carriers-only.
So we define :
(X is closed under downward dependence) = (no part of X is downwardly dependent on any entity discrete from X)
A historical individual of a crystal species is closed under downward dependence. We here have a carrier-only.
A here-and-now individual of a crystal species is closed under downward dependence. We here have the fully-fledged substance.
The next definition (defining a molecule) should rule out carriers-only :
( X is a molecule ) = (  X is closed under downward dependence,  X has discrete parts,  all atoms Y, Z which are parts of X are connected, directly or indirectly, by relations of specific or generic dependence. )
In  we have replaced "specific" in SMITH' definition with "downward".
In  we have added "or generic". All those dependencies are meant to be ontological, which means that they are not referring to efficient causality.
 rules out Accidents (i.e. it rules out that X is an Accident-only).
 rules out that X is a carrier-only (  of Definition 8).
So the definition rules out atoms-only.
 admits ontological wholes.
Example : A here-and-now individual of a crystal species is a molecule (in the mereotopological sense).
A molecule X can also for example be : "John is kissing Mary". John is a part of X, Mary is a part of X. They are moreover discrete parts. But these parts are not downwardly dependent on any entity discrete from X, i.e. discrete from "John is kissing Mary". So  of Definition 17 is satisfied at least with respect to the parts John and Mary.
When we consider the Accident "is kissing" also as a part of X (but, see below), then also this part is not downwardly dependent on any entity discrete from X. So X, i.e. "John is kissing Mary", satisfies Definition 16a, and by implication  of Definition 17.
 of Definition 17 is also satisfied, because John and Mary are discrete parts of X.
John is an atom of X, Mary is an atom of X, and "is kissing" is an atom of X. Moreover John, Mary and "is kissing" are all the atoms of X. These atoms are connected by relations of specific or generic dependence. So  of Definition 17 is satisfied. They are, as has been said, not downwardly dependent on any entity discrete from "John is kissing Mary". So "John is kissing Mary" satisfies Definition 17.
An analogous argument applies to (simple molecules like) "Socrates-tanned", "Socrates is running", etc.
If we consider a simple molecule, like Socrates-tanned, then the atom being-tanned is connected with the atom Socrates by a relation of specific (NOTE 2) or generic dependence, but, they cannot be considered as parts, because Substance and Accident are not related to each other as parts to a whole. But above, in connection with a complex molecule, we said that John is a part of X, Mary is a part of X, and also "is kissing" is a part of X. But they cannot be genuine parts either, they are atoms, constituting the molecule X, but not in the way that parts constitute a whole. So John as well as Mary should be interpreted as carriers-only, connected by the dependent atom "is kissing" (not considering all the other Accidents).
"is kissing" inheres in John, and, in the form of "is being kissed" it inheres in Mary. Inherence does not involve parts, which means that no parts are created or connected by it.
So it would according to me be more appropriate when we replace the mereotopological notion of "are parts of " in the definition by the metaphysical notion of "belong to".
But despite of this, we must distinguish between a simple molecule such as Socrates-tanned and a complex molecule such as John-is-kissing-Mary, because in the latter case we have not one, but two genuine subsistent individuals (connected by the relational accident 'is kissing'). So here we are, in a sense, entitled to call 'John' and 'Mary' p a r t s of the molecule.
 states that X has discrete parts, where the term "parts" in this context refers again to atoms. So we should replace the content of  by : "X comprises more than one atom", to express the composite nature of the molecule, in which the atoms could be parts in the case of complex molecules (like "John is kissing Mary"), however, only in the sense that John and Mary are discrete subsistent individuals (connected by the Accident "is kissing" into a molecule), but just atoms in the case of simple molecules, like "Socrates-tanned", because Socrates is not a part of Socrates-tanned.
So we shall reformulate the definition of a molecule as follows :
( X is a molecule ) = (  X is closed under downward dependence,  X comprises more than one atom,  all atoms Y, Z which belong to X are connected, directly or indirectly, by relations of specific or generic dependence. )
In  we have replaced "parts of" by "belong to", because not all (types of) molecules involve parts in this respect.
So, while a molecule - being a subsistent entity -- does not have any specific or generic dependence directed outwards, it has them within.
Example : A here-and-now individual of a crystal species is a molecule (in the mereotopological sense).
It is possible that dependent atoms may themselves serve as carriers for further dependent accident-like entities of a higher order. For example the individual redness of my bruise is dependent on the bruise itself, which is in turn dependent on me (SMITH, 1997). Such dependencies can form longer or shorter chains, but always of finite length, because the dependence must stop somewhere. It terminates with one or more independent atoms.
Critical Comment on Definition 17 (and 17a).
When we consider the phenomenon "John is kissing Mary", we must admit that there are two genuine individuals involved. In fact we see an interaction between two independent systems, namely John and Mary. This is quite different from a simple molecule like "Socrates runs". In the latter case we have one system finding itself in a certain condition or state.
Perhaps it is better to intepret "John is kissing Mary" as John, who happens to kiss (Mary), and Mary, who happens to be kissed (by John), i.e. as two molecules, temporarily united into a higher-order molecule.
It is clear that further investigation into these matters is necessary.
"Substances, as we have seen, may have substantials as proper parts. Accidents, correspondingly, may have accidentals as proper parts. Both substances and accidents may also, however, have essential parts, parts whose destruction leads necessarily to the destruction of the whole. Jim's individual humanity is an essential part of Jim. Hue, saturation and brightness are essential parts of the accident that is Jim's individual white or whiteness. Pitch, timbre and loudness are essential parts of Jim's present whistle."
Let us define :
( X is an essential part of atom Y ) = (  X is an individual proper part of Y, and  no part of X is substantial or accidental or a boundary, and  Y is necessarily such that it cannot exist unless X exists. )
 does not allow X = Y, i.e. X is not identical to a Mereo-totality (carrier-only), nor to an Accident.
 implies that such an essential part, or a proper part thereof, is, again, not a Mereo-totality, nor a spatial part (like a substantial or a boundary) of such a Mereo-totality, and that it is also not an occurrent entity, and, finally, that it is not a determination of a substantial, i.e. it is not a part of an Accident. Although an essential part is a proper part of the atom, it pervades the whole atom. This should be clear from the above examples, and is compatible with the atoms being continua. SMITH continues :
"Standardly one and the same atom can be partitioned into essential parts in a variety of different ways, each one of which captures some aspect of the atom's structure. The idea is that the internal structure of every atom could be represented exhaustively by a family of distinct complex partitions, representing cuts through reality of different sorts and on different levels, in which all essential parts would be eventually displayed."
SMITH ' example is :
Jim's individual humanity is an essential part of Jim
Here X = ' Jim's individual humanity '
Y = ' Jim '
As such the example is in fact ill-chosen, because being a human ( = having the form ' humanity ' ) is per accidens with respect to Jim : ' This individual, here named Jim, happens to be a human '.
On the other hand, Jim, as representing the species [pretending there is but one species of humans] Homo, necessarily possesses the form ' humanity ', while Blacky, as representing the species Canis [pretending all dogs and no other animals belonging to Canis], necessarily possesses the form ' caninity '.
Now, in the case of Jim, so formulated, ' humanity ' is a proper part of Jim, because ' humanity ' as such does not involve individuation, that is to say, all individuation conditions are excluded by (the notion of) ' humanity ' , while in ' Jim ' they are not. So  of Definition 18 is satisfied. Here ' Jim ' is a historical individual, and thus a(n) (individual) carrier-only, and consequently an atom.
"A part of (the form) ' humanity ' ", as it figures in  of Definition 18 , can mean either a proper part of it or the whole.
As to this clause  :
As regards  :
The historical individual representing the species Homo, is necessarily such that it cannot exist unless (the form) ' humanity ' exists.
So  of Definition 18 is also satisfied.
We can indeed legitimately say that (the form) ' humanity ' is an essential part of the historical individual (which is the carrier-only, and thus an atom) representing the species Homo.
A Mereo-totality can be specifically dependent on some essential part.
Examples (concerning true substances, here : crystals) :
This type of dependency is not a dependency between two ' partly overlapping' entities, because '(S + C)[crystal species] ' ( = X in Def.18) is totally contained in the historical individual ( Y ) of that crystal species as depicted here :
Therefore it is not a specific dependency as was the type defined in Definition 4a , where these entities only partly overlap (while both have parts discrete from some parts of the other). So with something having essential parts, it is not a matter of inherence. An essential part of a Mereo-totality (conceived as carrier-only, and, equivalently, as a historical individual) is indeed a proper part of that Mereo-totality, because the individuation conditions (that are present in the historical individual crystal -- to stick with our example) are missing in '(S + C)[crystal species] '.
And here, that is in specific dependency just like that, the dependence is also the other way around :
It is important to detect two main types of specific or generic dependency, namely as regards formal content, or as regards actual or possible existence. Perhaps these two types are identical.
While these cases represent a specific dependency (but with one dependent totally included within the other), Mereo-totalities can also be generically dependent (but, again, with one dependent totally included within the other), as the next examples demonstrate :
So while such contents, as they are in themselves, are not dependent one any given crystal (while any given crystal is, qua its content, dependent on its Space Group PLUS its Chemical composition), they are, in order to exist, dependent on the existence of one or more individuals of a certain crystal species.
Let us analyse these specific and generic per se dependencies (as they occur) in crystals with respect to Definition 18 (as we did for ' humanity ' ).
For specific dependency :
'(S + C)[crystal species] ' is -- as a form or nature, residing in the phenotypical domain of a historical individual of a given crystal species -- an individual proper part of this historical individual, because in contrast to that historical individual, it lacks the individuation conditions. So  of Definition 18 is satisfied.
As to  of Definition 18 :
As regards  :
The historical individual representing the given crystal species, is necessarily such that it cannot exist unless (the form) ' (S + C)[crystal species] ' exists.
So  of Definition 18 is also satisfied.
We can indeed legitimately say that (the form) ' (S + C)[crystal species] ' is an essential part of the historical individual (which is the carrier-only, and thus an atom) representing the given crystal species.
The same goes for generic dependency.
' SPACE GROUP ' is -- as a form or nature, residing in the phenotypical domain of a historical individual of a given crystal species -- an individual proper part of this historical individual, because in contrast to that historical individual, it lacks the individuation conditions. So  of Definition 18 is satisfied.
As to  of Definition 18 :
As regards  :
The historical individual representing the given crystal species, is necessarily such that it cannot exist unless (the form) ' SPACE GROUP ' exists.
So  of Definition 18 is also satisfied.
We can indeed legitimately say that (the form) ' SPACE GROUP ' is an essential part of the historical individual (which is the carrier-only, and thus an atom) representing the given crystal species.
Other essential parts of a given crystal species are its promorph ( = stereometric basic form [See Second Part of Website : Basic Forms ] ) and intrinsic shape (if the latter can be defined for some given crystal species). Both involve generic dependence.
We now have characterized essential parts.
So now we have established that a true essential part is neither an accidens, nor a proprium.
Discussion of Essential Parts.
The carrier-only ( ' Jim ' ) contains the genotypical and phenotypical essential parts, and in addition to them it contains the individuation conditions (making it an (historical) individuum). So the carrier-only is more than its essential parts, and the latter indeed form an individual proper part of this carrier-only, so  of Definition 18 is satisfied.
But its being a proper part of the carrier-only does not prevent it from invading all of the carrier-only.
An instructive analogue in number theory is the following :
The set of rational numbers (integers and true fractions) is a proper subset of the set of (so-called) real numbers ( = all numbers that allow to be expressed as decimal expansions, and thus including the rationals, but in addition to them also containing the irrational numbers [such as pi , SQUARE ROOT , SQUARE ROOT , etc.] ). Although the set of rational numbers is a proper part of the set of real numbers, the rational numbers are all strewn between the irrational numbers, in such a way that they can be found everywhere in the set of real numbers, that is we always will find a rational number in any part of this set no matter how small, despite the fact that if we want to index (label) the real numbers (with the help of integers of which we have an infinity at our disposal) we can do it for the rationals, but we will run out of labels when indexing the set of real numbers. So in the set of real numbers there exist other numbers in addition to rational numbers. Nevertheless the rationals are everywhere in this set. There are no subareas where no rationals are present, and no subareas where no irrationals are present. So the set of rational numbers does not inhere in the set of real numbers. The same can be said of essential pars. They do not inhere in that something of which they are essential parts.
Let us dwell a little longer on this point.
In contrast to such an all-pervading content, as Jim's humanity (which as such is not confined to a certain part of Jim), a per accidens determination (Accident), which as such is one-sidedly specifically or generically dependent, does not pervade the whole Totality.
When we look to some determination as it is in itself, we cannot decide whether it is, or is not, an all-pervading determination, and thus whether it is, or is not, an essential part. Only the given actual context can decide on this issue : The blue color of my bruise, although per se for such a bruise, is accidental with respect to my body (which is a true Totality). On the other hand, the blue of a crystal of copper sulfate (CuSO4 . 5H2O) is all-pervading, i.e. it is everywhere present in the crystal. It cannot be removed without destroying the essence (specific identity) of the crystal. Because ' blue ' also occurs elsewhere, it is not a specific, but a per se generic proprium for a copper sulfate crystal, that is, a phenotypic per se determination of it, and as such an essential part (here we, of course mean the material configuration in the crystal that is responsible for the blue color when we look to such a crystal exposed to daylight). On the other hand, qua what it is (all by itself), ' blue ' is a quality, and thus a predicament (and in a predication it can in some cases refer to an essential determination of something, in other cases to an accidental determination of something).
Definition 18 excludes accidentals, and by implication Accidents. So it is presupposed that a Mereo-totality is not only (going to be) determined by Accidents, but that there are also determinations of another nature (than Accidents) involved. These are the "essential parts". They display a dependence relation that obtains in the opposite direction : While Accidents are dependent on the carrier-only, the latter is dependent on its essential parts.
So the difference between accidental determinations (that come to determine a carrier-only) on the one hand, and the essential parts of the carrier-only, on the other, turns on a difference of dependence relations only.
Both are, however, determinations of a (full-fledged) Mereo-totality.
Now it could be alternatively asserted that a full-fledged Mereo-totality is dependent on one or another complete set of determinations, i.e. a set in which all types of determinations are represented. The "one or another" means that this dependence is of a generic nature only.
Could such a complete set, that -- it is true -- changes with time, be equated with "an essential part" of the full-fledged Mereo-totality? Stated more precisely, could the here-and-now full-fledged Mereo-totality depend for its existence entirely on the here-and-now complete set of determinations, so that we can dispose of the introduction of a new kind of determination - "essential parts" -- after all?
If this would indeed be the case, we could say that any determination is dependent on the carrier-only, but (then we must also say) that the complete set of here-and-now determinations (whatever this set is here or there, now or then) is essential, not for the carrier-only, but for the very being of the full-fledged Mereo-totality, and through this for the co-existence of the carrier-only.
Let me elaborate on this a little further.
In order for an observable thing to exist, a complete set of determinations, in which all types of determination (like Quantity, Quality, Relation, etc.) are represented, must be 'presupposed'. Only on completion, this set, all by itself, creates the carrier-only, i.e. this complete set IS the carrier-only, which is now equivalent to the full-fledged Mereo-totality. This Mereo-totality is indeed a totality of Accidents.
The distinction between the carrier-only and Accidents, only becomes apparent when we consider one Accident only, or when we consider an incomplete set of Accidents. Such a single Accident, or an incomplete set of Accidents, inheres in the complete set of Accidents.
But maybe this is not so.
In Part One of the Essay on Determinations (See Non-classical Series in First Part of Website ) I distinguish between the "phenotypic domain" and the "genotypic domain" of a Substance, and by utilizing these concepts we can speculate further :
With respect to determinations of a Mereo-totality a complete set of types is ontologically presupposed. It is a general precondition for something to be a Mereo-totality. For many determinations their being a pre-condition does not imply their ultimate specification, as long as we consider the Mereo-totality, not as a here-and-now entity, but as an entity considered over the whole time span of its individual existence, i.e. as an individual historical entity. Some of these determinations are wholly extrinsic, like the location of the Mereo-totality, and its point in time. Others need to be actualized (and thus be ultimately specified) by extrinsic agents, but only when we consider the Mereo-totality as a here-and-now entity.
But some of the determinations i n t r i n s i c a l l y must come in (their) ultimate specifications already when the Mereo-totality is considered as just an individual historical entity : As such, i.e. as completely specified, they are generated by the (particular) dynamical law (while their general type was already ontologically presupposed).
So while the complete set of types of determinations is just a general precondition for any Mereo-totality to exist, and thus is not a carrier-only or subject (see directly below), those determinations that are directly and totally generated by the dynamical law are the essential parts belonging to the Mereo-totality's phenotypical domain. And such determinations turn out to be all-pervading determinations, that is, they are everywhere present in the given substantial being ( The Mereo-totality should now be moreover a genuine Substance) ( NOTE 85c ). The particular dynamical law itself is also an essential part of the (particular species of) Substance, but a part residing in the Substance's genotypical domain.
Just above we said that because the complete set of types of determinations is just a general precondition for any Mereo-totality to exist, this set is not a carrier-only. Let us explain this.
First of all, the complete set of types of determinations (accidents) is a general precondition for the fully-fledged Mereo-totality to exist. Therefore it is (also) a general precondition for the carrier-only to co-exist. But as g e n e r a l precondition (for the Totality to exist) it doesn't c a r r y anything. It doesn't carry its special over-forming, but is (only) c o n t r a c t e d to it (i.e. the 'general' is contracted, resulting in the 'special'). And thus our general precondition is not a carrier-only. We do not say : "animal est homo" (in the sense of : carries homo, like we do say "man is white", in which 'white', where this is actually the case, is carried by a human body in the form of some local material structure). Likewise we also do not say "color is red" (because the color of something can also be blue, yellow, etc.). 'Red' is not carried by 'color', but color can be contracted to 'red'.
The dynamical law is thus considered to represent the genotypical domain of the Substance in question, and is equated with the Essence of that Substance. This Essence remains the same under accidental changes during which the (complete) set of types also remains the same, but the latter is, as has been said, nevertheless not the carrier-only, not the subject, because it is only a general precondition for a Mereo-totality to exist. The relation General-Specific is not a Carrier-Carried relation.
As has been said, some determinations appear to be necessary determinations (they persist during the whole time-span of the existence of their Substance). They are necessary representatives of the (complete) set (of types) of determinations. For example : The complete collection of actual symmetries, i.e. the total symmetry, of a crystal, which is called its Space Group, is one of the determinants for the crystal's identity. We could say : it is an essential part of the crystal ( I consider a crystal to be a genuine Substance). If this Space Group is replaced by another, then the crystal's identity is changed. So the Space Group indeed appears to be an essential part (See also the Essay on Crystals and Metaphysics in the Non-classical Series of documents in First Part of Website ).
If we, for example, replace the Space Group R3*c ( The * is normally written as a score above the preceding numeral) of the mineral Calcite (CaCO3 ) by another Space Group, say Pmcn -- under conservation of chemical composition -- then we obtain the mineral Aragonite (CaCO3 ), while a replacement of one crystal habit ( = the general shape ) of Calcite by another (say, a transition from prismatic to scalenohedral) is not supposed to give rise to another mineral species, i.e. the crystals remain calcite crystals.
Or, when we see a (profound) change in the chemical composition when we go from one crystal to another having the same symmetry (the same Space group), we assume that we have to do with disparate substances (in the metaphysical sense) differing qua whatness, despite their identical symmetries.
From this it should be clear (but see below) that the distinction between necessary determinations on the one hand, and just occurrent determinations (Accidents) on the other, seems more or less arbitrary, for it seems merely stipulated (and not shown) that the Space Group belongs to the co-determinants of a crystal's specific identity. But why not its habit (instead)?
Further, it is possible that atomic or ionic constituents (atoms, in a chemical sense, or electrically charged atoms) of a crystal are being replaced (substituted) by other (specifically different) constituents of approximately the same size. By stipulation the crystal then changes identity (except when we enter the possibility of certain substitutions of atomic species into the definition of chemical composition), because chemical composition is stipulated (and not demonstrated) to be a co-determinant of the crystal's specific identity, even though (as in this case) the (conceptual or real) transition does not involve any space-time discontinuity : it still appears to be the same individual.
So the "essential parts" seem to be of a more or less arbitrary nature.
Because in Metaphysics (when conceived as a general ontology) one often sticks to the same example, namely the human individual, say, Socrates, things appear to be all together too clear and easy : Socrates undergoes a brain damage, which changes his psychological character, but still remains Socrates .... or does he? That is just a matter of stipulation.
The dynamical systems approach on the other hand (developed on this Website, especially in First Part of Website ), solves this problem, by recognizing the fact that every intrinsic whole is generated by one dynamical system, governed by one dynamical law. This law can be identified with the Essence or Identity of such a whole, and can be considered to be the very essential part of that whole, residing in its genotypical domain. In the case of crystals the Space Group + Chemical Composition are then direct effects of such a law, and so are also essential parts, but residing in the phenotypical domain. In this there is no stipulation involved. The only assumption made is that an intrinsic whole is generated (or admits to be considered as such) by ONE dynamical system (involving ONE dynamical law). If there is a multitude of dynamical systems involved in this generation, I assume that these systems are coordinated in some way so as to constitute in fact ONE dynamical system after all. Where this is not the case, i.e. where such an assumption cannot be made, we have to do with an aggregate.
Of course it could be the case that all dynamical systems in the Universe are coupled and interdependent, implying that there is in fact only ONE dynamical system in the Universe, namely the Universe itself. We then come closer, in a way, to some eastern philosophies, that do not admit of any discreteness of objects, and also not of any fundamental distinction between subject and object. In this case a Substance-Accident Metaphysics becomes untenable.
But for the time being I cling to the approach which presupposes the existence of discrete intrinsic things (wholes), each generated (or interpretable as (so)generated) by ONE dynamical system. And wherever a local multitude of dynamical systems, which are not mutually coupled, is involved, we have to do with an aggregate.
As remarked earlier, sometimes one and the same single dynamical system generates not one product, but a multitude of it. The original elements of the dynamical system are, as it seems, not organized into one structured more or less continuous whole, but are scattered and dipersed in a stronger or weaker degree. Should the result of such a dynamical system be interpreted as one single entity or being, on the basis of the fact that it is produced by one single dynamical system, despite the fact that this supposedly single entity seems to be more of a multitude of more or less mutually independent and separate entities? Or should that result be interpreted as just an aggregate? Here it is indeed that dynamical systems theory is of no further help anymore. And it is here that our ontological investigation will benefit most of Mereotopology. The science of Mereotopology directly investigates the product, not its history. It evaluates the internal coherence of parts in such a product, in order to determine whether we have to do with a single thing or a multitude of it.
Essential parts of given individual accidents do not pose a problem. Such an essential part is either that what is signified by its definition (where the accident is expressed by an abstract term) or by that what is signified by a part of that definition.
After having discussed the Predicables , we will now, in the next document, continue with the VIA PRAEDICATIONIS and discuss the Predicaments.
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