Aristotelian Metaphysics
Theory of Natures

Ontology of the individual thing

Part V

Substance and Accidents ( The Thing and its Determinations)
The Definition

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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.

Via Diffinitionis


  1. The definition, formally viewed.
  2. Definition and proof.
  3. The three kinds of definition.
  4. Natural Science is induction-deduction and verification.
  5. Metaphysics.
           Definability of substance.
  6. Definability of the accidents.
  7. Definitiones sunt sicut numeri.

a.   The Definition, formally viewed.

" Asking quid est [...] ends up in knowledge (scire) per viam diffinitionis " (AERTSEN, J.A., Natura en Creatura, 1982, p.63).
In all this we must realize that 'scire' according to Aristotle in Posteriora Analytica, and St Thomas in loco NOTE 167 ),  is discursive knowledge in a more strict sense, that is to say, a knowledge through proof (deductive demonstration), which accordingly distinguishes itself from knowledge through definition. This latter is intuitive knowledge.
It has been especially Ockham who has taken pains to not identify without argument our way of knowing with objective reality (that is, reality which is independent of being [or going to get] known) NOTE 168 ),  and it appears therefore not superfluous to have a look at the definition through a 'terministic' magnifying glass.
In the proposition  ' homo est animal rationale '   ' homo '  is the definiendum ( = that what has to be defined),  and  ' animal rationale '  is the definiens, that is, the definition.
The definition is, as is the definiendum, a sign which points to something in reality NOTE 169 ),  where the definiens indicates the precise place where that for which the definiendum stands must be subsumed (that is, in what class of things). Subsumption into classes is a typical act of the human mind, that is, a certain way of knowing. Also St Thomas is well aware of this, for example in De Ente et Essentia, Cap.II, from line 194 :  The parts of the definition are concepts which together form the concept of the whole definition. These partial concepts signify explicitly and implicitly aspects in reality which complement each other. At other places in his work St Thomas speaks about parts of the definition as being 'rationes intelligibiles' (intelligible contents) (De spir. creat. q.1. 3. ad 3, and also S.G. II, 58, cited in AERTSEN, Hoofdstuk II, noot 103).
Here St Thomas clearly distinguishes between the ontological and logical dimension.
The genus as 1st intention, that is, not as genus, but a given genus, is a sign which has a larger range than that of the difference and also than that of the species, and only in this logical sense we can speak of 'part' and 'whole', not in the metaphysical sense.
Ockham in Summa totius logicae I, Cap.26 distinguishes between : The real definition designates the individual nature without connotation of something extrinsic.
There are two ways to present such a definition NOTE 170 ) : In the natural definition the words and corresponding concepts are signs for parts of the thing to be defined. For example, the natural definition of the species  ' human (being) '  reads :  "a substance composed of body and intellective mind" (and not :  ' rational animal ' ).
The metaphysical definition wants to present the definiendum in the definiens as UNUM (i.e. the definition must express a unity), in conformity with the interest of Metaphysics when viewing a thing, not as a changeable and changing thing, but as ens unum ( = a being that is one). This is the definition with genus and difference. The terms of such a metaphysical definition accordingly always are, in contrast to those of the natural definition, presented in the nominative case and not in oblique cases NOTE 171 ).  The genus (for example 'animal' )  is an absolute term which precisely signifies what can be signified by the definiendum, while the difference in its concrete form is a connotative term (for example  ' rational ',  that is, the rational {thing} ) :  This term signifies this same individual nature ( that for which the definiendum stands) and co-signifies a certain essential aspect distinguished by us, while this essential aspect ( = further determination of te Essence) is directly ( = in an absolute way) signified by the abstract counterpart of the concrete term (for instance by the abstract term  ' rationality ' ) NOTE 172 ).

The metaphysical definition of the term  ' homo '  ( = human being)  is :  animal rationale ( = rational animal).
In this the term  ' homo '  signifies the Essence, which is present in every individual from the range of the term  ' homo '.  The term  ' animal '  signifies a class (a collection) of organisms, while the term  ' rationale '  takes care that the signification is limited to a special part of this class. The term  ' animal rationale '  then, as does the term  ' homo ',  signifies the Essence, but now in an analyzed way (See also next NOTE ).
Homo is the species, animal the genus, and rationale is the difference.

In my conception of the Essence (of a given intrinsic being) as dynamical law, the elements of the definition, namely genus and difference, should refer to aspects of this dynamical law. If this law for  human being  would read, say, :

Xn+1 = 3Xn2 + 5Xn + 7  (See Part III)

then the genus (lying immediately above) could be, for example,

Xn+1 = aXn2 + 5Xn + 7  (that is, we have replaced the definite coefficient  "3"  of Xn2  by the indeterminate coefficient  "a" ),

and the difference is then :  {3}  (another difference, say, {6}, would determine a different species of that genus).

The definition of the term   ' homo ',  namely  animal rationale,  can be predicated in  a  per se  manner of the term   ' homo ' :  homo est animal rationale.
The term   ' homo '  now stands for  3Xn2 + 5Xn + 7.
The term  ' animal '  (assuming [for the sake of argument] that this is the genus directly lying above  homo )  could then stand for  aXn2 + 5Xn + 7.
The term  ' rationale '  could then stand for  {3}.
In al this we take the term  ' rationale '  not as a determination (which in fact it is), but stipulatively let represent the difference by the term  ' rationale '.
All these matters were already extensively discussed earlier, namely in VIA PRAEDICATIONIS (Part III) (press LINKS) :
c1     Meaning and Extension.  Distinction of Predicables revisited.
c2     Again, the distinction between  Difference  and  Proprium.  Genotype and Phenotype.  The  per accidens  nature of  Determinations.

Of course nobody knows what precisely the Essence of Socrates (as an individual of the species Man) is, also not when we say that that Essence is  " human being "  or  " humanity ".  The analyzed (verbal) expression of the Essence, namely the definition, and thus  animal rationale,  already gives some more information, especially as a result of bringing in the term  ' rationale '.  Nevertheless things remain vague. Also my considerations about the Essence as dynamical law, including the fictitious example of such a dynamical law, do not contribute anything in this respect. But this is in fact not mandatory, because metaphysics is not about what precisely is the Essence of Socrates (and with it of Plato, Peter, etc.), or, in other words, how precisely the relevant dynamical law reads  ( This is a task of natural science). In metaphysics it is only about what Essence-as-such is, that is, what Essence qua Essence is. The same holds for the definition. As regards the goal of metaphysics, my considerations about the Essence as dynamical law are certainly productive, because, firstly, we, thanks to these considerations, have found out that  a  ' philosophy of essence '  is still relevant and worthy of study even today (that is in the light of recent knowledge about things and processes), and, secondly, because my considerations have teached us something about the general nature of  ' to be an Essence ' :  it has the nature of a law.

Nominal Definition

Only terms in the category of Substance can, because they are absolute, have real definitions. The other terms (the accidents), and also the transcendental terms, are connotative and thus signify no  per se  unum at all :  Something is primarily signified and something else secondarily (For Ockham Quality is an accident, while all other categories -- except Substance -- are transcendentals). Also terms such as  ' phoenix '  which do not signify anything in extramental reality, do not have a definitio quid rei ( = real definition), because with respect to all these cases we either have to do with an incomplete essence, or no essence at all. For these cases one uses a nominal definition, that is to say, a definition that indicates what the word means, not what the thing (essentially) is, and such a definition often proceeds by bringing up accidental aspects NOTE 173 ).
With respect to the definition we must realize that TERMS such as  ' definition ',  ' genus ',  ' difference ',  ' proprium ',  are terms of second intention, they do not signify things but (other) terms, which latter are either also terms of second intention, or are terms of first intention. So the term  ' difference '  means :  the second part of the definition, while the term  ' genus '  means :  the first part of the definition. And the term  ' definition '  means :  the analyzed ( = diffentiated into parts) term which can be predicated per se of the term that has to be defined (definiendum).
But this is, of course, still insufficient.
In our discussions about  genus  and  difference  we have determined to what entities the instances of these terms NOTE 174 ), figuring in one or another 'concrete' definition (that is, one or another given definition), should refer,  that is, what their supposita should be, or, equivalently, for which entities in extramental reality the terms of a given definition should stand. So in such a given definition we have to do with terms of first intention.

b.  Definition and Proof.

After having established some formal aspects of the definition as such, which, for an important part are borrowed from Ockham, we are able to investigate more sharply a number of expositions by Aristotle (and St Thomas in loco) in order to delimit this 'via' more clearly. We determine the position of the definition in the proof ( = deductive demonstration). For then we can also see clearly in what way metaphysics lies at the foundation of science ('physics'), because after the completion of the via diffinitionis the via demonstrationis follows, in which, as regards the subject of investigation (the object we nowadays say), we leave the strict domain of metaphysics.
At the beginning of the Posteriora Analytica Aristotle concludes that discursive thinking must be based on something which itself is not discursive thinking, otherwise discursive knowledge would have no fundament and thus be itself impossible. To this necessary non-discursive thinking belongs defining. The definition, namely, is always only middle term and not a conclusion of the syllogism (so the conclusion -- obtained knowledge -- is achieved through the definition) NOTE 175 ).  It is a term in both premises.
In the business of scientific enquiry it is proved that an attribute necessarily belongs to the subject, whereby it is presupposed :  (1) the existence of the subject, and (2) what the subject is (definitio quid rei), and (3) also the nominal definition of the attribute (which here is a more or less external description of it) NOTE 176 ).  By way of induction one finds that the attribute is in fact connected with the subject :  One finds the attribute connected with the subject in all inspected cases (for example in all salt crystals that were investigated [by using, say, X-ray diffraction methods] the total symmetry turned out to be according to the Space Group Fm3m ),  and generalizes these findings (this is what induction is). And having now generalized, we can now prove with respect to any newly appearing case that the aforementioned attribute necessarily belongs to the subject (of that new case), that is, by way of deduction one finds that the attribute is necessarily connected to the subject, and this in turn means that one finds the  ' why ',  one finds the cause, and the latter is signified by the middle term of the deductive demonstration (syllogism) :

All  S  are A
X  is an  S
Therefore  X  is  A

Here S is middle term.  And  X is A  because X is an S, so S signifies the cause.

An actual example of this syllogism :

All mammals are hairy
A cat is a mammal
Therefore a cat is hairy

Here mammal is the middle term. And a cat is hairy because it is a mammal, so (being a) mammal is the cause of being hairy, or, equivalently, the term  ' mammal '  signifies the cause.

In  In II Post. An., lectio 1, St Thomas shows that if we want to demonstrate that a given necessary attribute (proprium) necessarily belongs to a given subject, we need the definition ( = (DP) in the next example) of this attribute as middle term, and at the same time we must prove that the definition of this attribute belongs to the subject, in virtue of which we at the same time find that the attribute ( = (P)) (itself also) belongs to the subject ( = (S)) (see also next Figure to prepare) :

1st Syllogism (See St Thomas, In Post. An. II, lectio 1) (middle term in bold type) :


A figure with external angle equal to the two other internal angles (taken together) ((DP))  IS  a figure with the three internal angles (taken together) equal to 180 degrees ((P)).


A triangle ((S))  IS  a figure with external angle equal to the two other internal angles (taken together) ((DP)).



A triangle ((S))  IS  a figure with the three internal angles (taken together) equal to 180 degrees ((P)).

The definition proper of the attribute (proprium) (DP) ( = middle term) is :
A figure with external angle equal to the two other internal angles (taken together).

The attribute (proprium) (P) is :
A figure with the three internal angles (taken together) equal to 180 degrees.

The whole definition (that is, as predication of the definiens of the definiendum) of this attribute (proprium) then reads :
(P)  is  (DP).

The subject (S) is :
A triangle.

The Conclusion then reads :
(S)  is  (P),

that is, the subject  IS  the attribute (which in turn means :  the attribute necessarily inheres in the subject, or, more ontologically expressed, and now with respect to a corresponding material reality :  the phenotypic essential part (proprium) necessarily inheres in (is carried by) the substrate ( However, because the proprium is an essential part it resides (according to my revised theory) wholly within the carrier (of replaceable determinations), as is, for instance, the case with the  Space Group PLUS Chemical Composition  (which is a proprium) in crystals. If, on the other hand, we consider the other kind of proprium, namely a proprium which as phenotypic feature is the result of a very long causal path leading back to the genotypical cause, such as the ABILITY TO LAUGH in man, then, in contrast to the aforementiond kind of propria, such a proprium can be erased or replaced without the Essence being destroyed. And then it behaves as a replaceable determination, and consequently does not reside (wholly) inside the substrate (i.e. does not, qua all its content, coincide with part of the substrate), but differs qua content (or at least a large part of its content) from that of (parts of) the substrate, and so truly inheres in the substrate (as we understand the connection between replaceable determinations and their substrate)).

So in a formal scheme our syllogism reads :

( DP )  is  ( P )     ( DP = middle term)
( S )  is  ( DP )
( S )  is  ( P )

Well, according to St Thomas (Ibid., nr.415), the maior is a proposition in which a definiendum (P) is being affirmed of a definiens (DP), that is to say, as such we should take it for the time being, because he says :  "Quod est quasi definitio passionis" (translation :  "Which is, as it were, a definition of the property" ),  and because of this the maior cannot be proved, for between two concepts with the same range a middle term cannot be found.
The minor, on the other hand, must be demonstrated, and this should be done by means of the definition (DS) of the subject (S), which latter is the triangle :

2nd Syllogism (middle term in bold type) :


Every figure closed by three straight lines ((DS))  IS  a figure with external angle equal to the two other internal angles (taken together) ((DP)) NOTE 177 ).


A triangle ((S))  IS  a figure closed by three straight lines ((DS)).



A triangle ((S))  IS  a figure with external angle equal to the two other internal angles (taken together) ((DP)).

Definition of  ' triangle ' :  Every figure closed by three straight lines (DS).

Subject is :  a triangle (S).

A figure with external angle equal to the two other internal angles (taken together) is (DP), that is the definition of the proprium, figuring in the 1st syllogism.

Summarized in scheme we thus have :

 DS   is  ( DP )     ( DS is middle term)
( S )  is  ( DS )
( S )  is  ( DP )

The minor of the previous (1st) syllogism is now proved NOTE 178 ).

All this clearly shows that in a demonstration or proof (which in most cases consists of more than one syllogism) the definition of the subject, indicating the whatness of the subject (the subject is that of which something is said in the conclusion), must be presupposed, and not only the definition of the attribute (unde et in principio dictum est quod oportet praecognoscere quid est, non solum the passione, set etiam de subiecto NOTE 179 )).
It is this definition of the given proprium which has been found to be the cause (that is, signifying the cause) of the state of residing-in of the proprium in the subject (that is, the residing-in of that what is signified by the proprium term  in  that [namely the substrate] what is signified by the subject term). This definition of the given proprium is, as is also the mentioning of the appropriate subject, indicated in the minor of the first syllogism :
( DP )  is  ( P ) (maior),  ( S )  is  ( DP ) (minor),  ergo  ( S )  is  ( P ) (conclusion)
NOTE 180 ).
In the last part of nr.417 St Thomas is, according to me, a little imprecise in his distinctions :  He there says that  quid est  and  propter quid  aim to the same answer, as to what Aristotle expounds in the second Capitel of the IInd Book of his Posteriora Analytica with respect to states of affairs, for instance the lunar eclipse or the concording of high and low tones. Well, St Thomas says that quid est refers to esse simpliciter NOTE 181 ),  but it is states of affairs that are at issue, therefore it is about esse secundum quid NOTE 182 ).  In my opinion, for states of affairs applies :  Quia (is  S  P ? ) NOTE 183 )  asks for an esse secundum quid and ends up in propter quid, which yields as answer the cause (signified by the middle term), and this cause is now also the quid est of that state of affairs.
A regards substances (things), according to me, the following applies :  Quid est presupposes an affirmative answer to the question  si est  (an est), where quid est asks, not for the cause of the esse simpliciter, but for the cause of the being-such, or, said differently :  Quid est does not ask whether there is a cause (that is, no existential question), but what that cause is, and we have seen that it is the Form (See Part II, VIA  QUAESTIONIS,  f. )  ( Within Aristotle's philosophy the Form is also cause of the esse simpliciter).
So only in the case of Substance the quid est refers, albeit indirectly, to the esse simpliciter NOTE 184 ).

Definitions (for instance, rational animal) are, among other reasons, not demonstrable because they are not enunciations (nothing in them is assered), not catallel (such as a conclusion of a demonstration is), and even it they, when connected with their definienda, become enunciations, for example, man is a rational animal, there is still no catallel structure ( = dual structure), because the definiendum stands for precisely the same as the definiens ( = definition), resulting in the fact that we still cannot speak of 'predicating something of something else'. So it is especially because of this that definitions are not demonstrable, and not because they are supposed to be just conventional, for within the framework of Aristotelian epistemology they are not conventional.
If an Essence would be demonstrable (that is, when the attribution of the definiens to the definiendum could be concluded ),  then it would apply that quid est would boil down to propter quid, that is to the question for the cause.
Because in the case of Substance (that is, of any given Substance) the cause of being-such is identical to the Essence, no demonstration why a given substance is such and such is possible, because there is no duality (catallelism) NOTE 185 ).  So also a demonstration on the basis of equivalence (see NOTE 180) is not possible, as can be done in the case of demonstrating the necessary attribution of a proprium to a subject (NOTE 180). Here (in the case of a proprium) surely a catallel structure is definitively present. With (necessary) properties it is such that the (material) cause of their whatness (being so and so) lies outside themselves, namely in the substance (that is, in their subject), resulting in the fact that there is duality, and therefore demonstrability :  Inductively the quia est (is  S  P ? )  is demonstrated, and deductively the propter quid (why is  S  P ? ),  whereby then the cause of the residing-in is signifyed by the middle term. And this cause is the definition of the attribute, thus an answer to the question quid est with respect to the attribute.

So we recognize for the case of properties two causes, which both are intrinsic with respect to the substance-attribute composite, namely (1) the material cause, and (b) the formal cause :

  1. With respect to the attribute-proper (which, in the context of demonstration is a proprium, and which ontologically is an accident) this material cause is extrinsic, that is, outside the true essence of the accident. It coincides with the subject of the accident or, equivalently, with the substance, and provides the explicit content for the genus of the definition of the property. This material cause is, in the demonstration, signified by the minus ( = extreme lesser term [extreme inferior term], and 'triangle' of the above example ) (See Part II, VIA  QUAESTIONIS,  b. ).
  2. Intrinsic with respect to the attribute is its formal cause, it coincides with the accidental form of the substance-accident composite, and provides the explicit content for the difference of the definition of the property, and is, in demonstration, signified by the middle term (definition proper -- that is, without appropriate subject -- of the attribute).
Summarizing things we, accordingly, can, for properties, set up the following correspondences :

Substance as subject
Material cause Genus Lesser term
Accidental form Formal cause Difference Middle term

The essence-proper of the property must be identified with that for which the middle term stands, but also the subject (substrate), signified by the lesser term, is, in a way, connected with this essence. As we had it above :

( DP )  is  ( P )     ( DP = middle term)
( S )  is  ( DP )
( S )  is  ( P )

In this way, Aristotle, can, at the end of Chapter 8, and in Chapter 9, of the IInd Book of the Posteriora Analytica conclude that one cannot know the essence of a property without demonstration (proof), but that it also cannot be proved :  The essence of the property is merely being revealed through the demonstration  ( 'proof ', 'demonstration', is in the present context always a demonstration of the necessary attribution of a property to a subject).

So there are two sorts of essences, and consequently also two sorts of (real) definitions :

c.  The three kinds of definition.

In Chapter 10 of  II Posteriora Analytica  Aristotle presents a number of types of definition. The definition is the expression (logos) of the whatness of something. We can distinguish three (main) forms :

  1. The definitio quid rei of something in the Category of Substance.
  2. The definitio quid nominis.
  3. The definitio quid rei of a property.
The definitio quid rei (real definition) of a (term of) substance or of a (term of) property indicates the essence of the definiendum, while a definitio quid nominis (nominal definition) is not a proper definition (St Thomas, in  In II Post. An., lectio 8, calls it :  Ratio expositiva significationis nominis ( = explaining expression of the meaning of the word)). The nominal definition does not directly refer to the essence of that which it so defines, but circumscribes it by pointing to accidental features. Only after a definitio quid nominis (whether or not explicitly given) one can ask further for being and essence of the entity in question.

In what follows we will attempt to give some examples, especially examples different from those traditionally given. This, to test the value of all the traditional distinctions and determinations, and further to delineate them more sharply. For these purposes the traditional examples are, according to me, not always suitable. Indeed, it turns out not to be easy to find appropriate examples, therefore I limit myself to just a few of them. And among them we prefer a specific class of examples :  The consideration of crystals has turned out very instructive (that's of course why we have devoted so much attention to them (First and Fourth Part of Website)). It should be added, that in spite of this the 'human example' is also very instructive, if handled properly, without prejudice!, and interesting, because we know ourselves relatively well, and because 'substantiality' and 'essence' seem to manifest themselves very clearly in this case. Nevertheless caution is called for.

1.  Example of a definitio quid rei of a (given) substance :

The definitio quid rei of a crystal (that is, what the term  'crystal '  essentially means).

(A crystal is)  a  p e r i o d i c a l l y  o r d e r e d  b o d y.

Where  'ordered body '  is the genus, and  ' periodically '  the difference.

Explanation of this definition :
There are also ordered (physical) bodies which are not periodic (that is, they do not have a strictly repetitive structure), for example organisms :  Their body is morphologically either not ordered at all, or ordered, but in the latter case the structure is not throughout repetitive (where 'througout repetitive' means :  consisting of some morphological unit that is repeated all over in the body also with respect to its orientation), but refers to a morphological center (which either is a point, line [axis] or plane). Surely, in morphologically ordered organisms parts can be repeated (as we see in starfishes, earthworms, and centipedes), but these repeated parts either do not all have an identical and parallel orientation with respect to each other, but every such part is directed to one morphological center, which is itself is not repeated (starfishes), or those parts are not exactly identical (earth worms, centipedes, tapeworms). Sometimes a form of (exact) periodicity can be found in organisms, for example in blue algae (cyanophytes) where identical cells are strung together. But this is a repetition in only one dimension. Along the other two dimensions there is no repetition at all. Exact repetition in two dimensions (only) can be encountered in certain bryozoans. But in all these cases, however, we have to do with colonies consisting of relatively independent organisms, that is, we do not have to do with genuine substances, but with aggregates. Each member of such an aggregate is a genuine substance, and lacks a periodic structure. The difference between periodic and non-periodic order can be made graphic (in two dimensions) by considering the pattern of old-fashioned wallpaper (is a periodic pattern) or that of many drawings of Escher (also periodic), and that of a persian carpet (non-periodic).

The treelet of Porphyrius in which this definition has its place looks as follows :

2.  The just given definitio quid rei (real definition) of a crystal can still better be grasped if we also try to give the definitio quid nominis (nominal definition) of it :

(A crystal is) a body bounded by flat faces.

This is, of course, what one always had understood a crystal to be. But as it is the case with almost all concepts (also those of 'substance', 'accident', 'cause', 'essence'), a characterization of something by a concept quickly turns out to be insufficient, or even wrong, as soon as one considers the given case more closely :  While science progresses further, the concepts become more and more theoretical (that is to say, much more indirect, much more hidden) than one originally imagined them to be :  It turns out that they can be unearthed in their pure form (that is, as truly signifying their object) only after laborious analysis and observations (See also further down, Section g.) NOTE 187 ).
Well this, then, also happened with the concept  ' crystal ' :  because the  ' possession of flat faces'  turned out not at all to be fundamental for crystals, and already in the 17th century Steno discovered that it was the (constancy of) angles that is fundamental for the crystal, and later one found that this was an effect of a very specific internal periodic structural pattern. And only then one began to understand precisely what a crystal in fact essentially is :  A transition took place (as seen with hindsight) from the definitio quid nominis to the definitio quid rei (and of course such processes are never completed).
In fact it is already apparent that  ' possession of flat faces '  cannot perhaps be fundamental :  There are, namely, also crystal species that have curved faces, such as Diamond and Siderite (an iron mineral). Further, in nature, crystal faces are seldom developed (for example in many natural rocks).
All the observations of the crystal, which we have just described, lead to the unearthing of the essence and are as acts of sense and reason, comparable with the very graphic example which Aristotele gives in Chapter 19 of the IInd Book of the Posteriora Analytica, of the army which was fleeing and yet in the end putting itself in order again.

The example of a definitio quid rei which we have just given was about the definition of the term  ' crystal '.  And there it was left open whether the term referred to a common salt crystal, a sugar crystal, a mica crystal, an ice crystal, or to whatever sort of crystal. It was a definition of a generic term, namely the term  ' crystal '.  And this definition is correct. The term  ' crystal '  stands for a class of substances, not for a certain kind (species) of substance. That's why the term  ' crystal '  does not signify one or another essence, and if it does, then it does so only in an incomplete way. Completely signifying one or another essence only happens when a specific term is defined. Let us give an example :
Common salt can be found in nature as the mineral Halite. With respect to the definitio quid rei of (the term) halite crystal we can first remark the following :  We already know that it is a crystal (that is to say, that the term  ' halite crystal '  stands for a  part of the set of things, which each for themselves can already be indicated by the term  ' crystal ' ),  and thus halite crystal is a (logical) subsumpt of crystal (that is, halite crystal is logically subsumed under crystal ).  We must now define what halite crystal (essentially) is, that is to say, we must define the term  ' halite crystal '.  This we could do by stating its Space Group symmetry together with its Chemical Composition. The Space Group NOTE 188 )  of Halite is  F 4 / m 3* 2 / m,  and the Chemical Composition is expressed by the chemical formula :  NaCl.  The crystal consists of as many sodium (Na) components as chlorine (Cl) components.
So here we have, then, a precise characterization of a Halite crystal NOTE 189 ),  but a characterization in terms of parts which are ordered in a certain way (implying the  F 4 / m 3* 2 / m  symmetry, while the precise distances between the components among each other are implied by the being-sodium (Na) and the being-chlorine (Cl) -- in the crystal as positively charged Na ions and negatively charged Cl ions -- of the crystal components). And this is, consequently, not a metaphysical definition (genus-difference definition) but a natural definition (matter-form definition), where the crystal components are the matter, and their (periodic) arrangement, with the  F 4 / m 3* 2 / m  symmetry, the form. But this 'matter-form' definition is perhaps yet not a genuine matter-form definition, in any case not one that refers to genuine parts, because the ordering itself is not a part (as it is so with body and soul, when the soul is taken to be something independently existing. Now the putting together of body and soul results in man). Following this lead, we could interpret the Chemical Composition, namely NaCl, as the genus, and the Ordering of it as the difference. But perhaps it is better to take  Crystal  as genus, and Space Group (symmetry) PLUS Chemical Composition (S + C) as the difference. And so the definitio quid rei of  ' halite crystal '  would then read :

A crystal [ = genus]  that consists of Na+ ions and Cl - ions (in equal numbers) which have arranged themselves into a three-dimensional lattice having a symmetry according to the Space Group  F 4 / m 3* 2 / m  [ = difference].

True, the difference is here still described in terms of  "consists of .....",  but this can in principle be transformed linguistically into something that ends with  " " (as in  ' rational ' ),  resulting in :  A crystal [ = genus] that  is [ = difference],  or (equivalently) : crystal  (as in :  A rational animal, which is the metaphysical definition of man ).
This metaphysical definition of  ' halite crystal '  is, according to me, correct, but it is a definition that (still) refers to the phenotypic domain.
But a genuine metaphysical definition should refer to the genotypic domain, thus to the domain of the true Essence, namely the dynamical law (which is in the present case the law governing the crystallization of sodium chloride (NaCl) either from a melt or from a solution).
How, then, would the true metaphysical definition of  ' halite crystal '  read?
The metaphysical definition must be a term consisting of two parts (genus and difference) which together signify the Essence of any halite crystal whatsoever. The first part of such a definition should refer to a determinable aspect of the Essence, while the second part should refer to a subsequent determination of that determinable aspect. This Essence is the relevant crystallization-law which is 'applied' (by the NaCl-crystallization system itself, because the law is inherent in the elements of the system) resulting in the generation of one or more halite crystals. However, in the case of the crystallization of sodium chloride, as in the majority of the other cases, the crystallization law, operating in such a case, is not precisely known. But if it were known, it could be formulated mathematically. And, as had already been shown ealier by using (fictitious) examples, in such a mathematical formulation  a  genus  and  a  difference  can be distinguished, and so it can represent the metaphysical definition of  ' halite crystal '.
The  genus + difference  ( = definition) can then be predicated  per se  of  (the term signifying) the dynamical law (halite-crystallization law) NOTE 190 ),  and with it of (the term standing for)  ' halite crystal '  ( "A halite crystal is ... " ).
The term  halite crystal  is the species, and is equivalent to the term signifying the relevant dynamical law (the halite-crystallization law), so the term signifies the Essence. This is the intentional meaning of the term. In the extensional meaning the term stands for the collection of all halite crystals.
One single halite crystal (and thus a particular halite crystal pointed to with the finger) is an instantiation of (the species) halite crystal (It is a member of the just mentioned collection of all halite crystals, or, equivalently, it is an individuation NOTE 191 ), of the corresponding Essence [and that individuation/particularization is an actually generated halite crystal as a result of the operation of the halite-crystallization law] ).
If this individual crystal would have a proper name (like 'Socrates' is a proper name of some individual human being), say  ' Halo ',  then we can say :

Halo is a halite crystal "

And, as we saw, we can then give a definio quid rei of the term halite crystal.
In all this it is clear that we must, in almost all cases of attempts to give a genuine metaphysical definition of whatever (complete) being, resort to all kinds of external features, that is, determinations, and that we are thus forced to stay in the phenotypic domain, because the relevant dynamical law (which resides in the genotypic domain) is in most cases unknown.
Thus in the actual practise of defining (that is, in a particular scientific context) it is not very meaningful to say that the term halite crystal (thus as a species that is to be defined) refers to the dynamical law (that is, to the Essence), and (not very meaningful) that that dynamical law can then be analysed into a generic and a differential aspect, when we do not know that dynamical law. We can, alternatively, signify the Essence by a term referring to the specific proprium (for a halite crystal we have done this above, by indicating its Space Group symmetry and its Chemical composition). Or we can use a matter-form definition, and to give such a definition is generally less difficult.
However, for metaphysics it is sufficient to refer to the dynamical law just as  ' the dynamical law '  (that is, without actually formulating one such law), because in metaphysics it is not about one or another given Essence (in the sense of :  not about the Essence of halite crystals [while it is about 'one or another given Essence' in general] ),  but (metaphysics is) about the Essence as Essence (that is, Essence, only insofar as it is Essence), where knowledge of particular Essences, such as the Essence of halite crystals, is not necessarily demanded, although such knowledge would certainly be instructive in that sense that, as a result of actually knowing one or more such Essences, the concept  ' Essence '  would become less abstract for us.

With respect to the definitio quid nominis and quid rei of properties a same attitude as pictured above is necessary, but here things are more difficult, because the essence of a property, in virtue of the typical relation the property has with the subject, is much harder to characterize. Here we should order properties under the heading 'states' or 'conditions' :  A substance is a product of a Totality-generating dynamical system, and enters, according to the course of the system, into different states, successively following one after the other in time. And in addition to it there are perturbations of the system, that is, the influence of contingent external factors. We will give examples at some other occasion.

The three forms of definition, given by Aristotle, relate to demonstration as follows (See SEIDL, H., Aristoteles  ZWEITE ANALYTIKEN.  Mit Einleitung, Übersetzung und Kommentar herausgegeben von Horst Seidl, 1984 ) :

Having shown the location of the definition in demonstration, it is perhaps useful to give an example (see d.) of the way this discursive thinking is actually professed in science leading to scientific results, and only then (after having given this example) to return to metaphysics (e.). For only by contrasting metaphysics with other intellectual activities (here, empirical science) we learn to apprehend what metaphysics in fact is, because to answer this has turned out not so easy!

d.  Natural Science is Induction-Deduction- and Verification.

As Ernst Haeckel clearly expounds in the Chapter concerning methodology of his controversial but fascinating Generelle Morphologie der Organismen (1866) (and of which Wilhelm Bölsche, Ernst Haeckel, in Männer der Zeit, 1900, once exclaimed :  " Philosophers should read this! " ),  the process of knowing in natural science always consists of induction and deduction : 

Namely :  Basing oneself on the results of a series of special experiences (repeated instances of observation) as regards some assumed type of behavior, a general law is assumed to be at work (induction), and this law is now applied (in the form of a prediction of the outcome) (deduction) to a case not yet investigated, i.e. which did not belong to the series of observations that led one to the assumption of that law to be at work. A new experience or experiment can verify (or fasify) this deduction. The more times such a verification turns out to be positive, and with no exception, the more certainty we obtain that this indeed is the law that actually operates. In other words, the more empirical verifications of predictions-done-by-applying-that-law we have obtained, the better that law is itself verified, that is, the more the induction becomes certain. Let us give an example :


All organisms consist of (one or more) cells (or, equivalently :  All organisms are cellular).
( This guarantees that also a unicellular being distinguishes itself from, say, a crystal, because the cell does not possess (whereas the crystal does) a periodically repeating structural pattern, but a pattern that is directed toward a morphological center. This premise could have been concluded in a previous syllogism, but is in the end always found by induction. It is the general law).


Sponges are organisms.
( This is, to be sure, still the incomplete definition  [ ' organisms ' ],  predicated of its definiendum, that is, predicated of the subject  [ ' sponges ' ].  'organisms' is middle term, the minor is found by induction



Sponges consist of (one or more) cells.
( found by deduction )  NOTE 193 ) 

Microscopical and optical enquiry of actual sponges can verify the conclusion (verification). The induction of the first premise (maior) has now become more strenghened, however (this strenghening) still depending on another induction that is expressed in the minor. We see, by the way, that natural science is a truly empirical science, it begins with observation (leading to induction) and ends with observation (experimental or observational verification). The progression of natural science always proceeds through a chain of such syllogisms, and thus we can continue and see that indeed progress has been made :


All sponges consist of (one or more) cells (or, equivalently :  All sponges are cellular).
( Conclusion of previous syllogism ).


Scytalia spec.'  was a sponge.
( Namely a not further identified fossil extinct species of sponge belonging to the group  ' Scytalia ',  which lived from the Jurassic until the Cretaceous, which means that no representatives of this group are living today. The features of the fossil convince us that it was a sponge.  In the present syllogism ' sponge '  is middle term ).



Scytalia spec.'  consisted of (one or more) cells.

We have now reached the conclusion of  ' the apparently cellular character '  of an organism that has left this fossil (for example, an impression in a rock, or a petrification), that is to say we are led to a conclusion concerning a state things which cannot be directly observed anymore (we cannot, ex hypothesi, observe the cellular nature in the fossil), and so this conclusion really is a step forward in our knowledge.
For the middle term it is sufficient to give a relevant part of the (total) definition NOTE 194 ).  From all this it is also clear that the deductive aspect is just a small element in the practice of natural science :  It is mainly observation that delivers scientific knowledge. Verification of the conclusion of the last given syllogism could be :  We find yet another fossil Scytalia, in which the cells are indeed visible (as a result of microscopic enquiry of that new fossil, which in one way or another was very well preserved) NOTE 195 ).
In physics one sets up, with respect to a given group of (observed) phenomena, a mathematical model, of which one hopes that it explains those phenomena, and one sees whether all kinds of predictions the model gives actually come true. If it does, then we have an indication that the model is correct. If it doesn't, then the model must be modified, or rejected. Such a prediction always is the outcome of a possible observation or experiment. The models -- which are thus in fact theories or hypotheses -- can often be very complex.

We have been somewhat elaborate with an example of the fundamental method of science, as to its inductive and deductive aspects, because examples are often scanty in the treatises on epistemology of Aristotle and St Thomas NOTE 196 ).  Further, often either the same example is put forward again and again, rendering the chance of a systematic error large, or sometimes (as would turn out to be so) inadequate examples are given. And totally in the spirit of St Thomas, we also say :  In philosophy (and, of course, also in natural science) it is not so much about the opinions of others, but about the truth. And this is a call for further elaborating on what was accomplished earlier.

e.  Metaphysics.  Definability of (a given) substance.

Having sketched the art of natural science, we can now better apprehend what metaphysics is about :
In contrast to natural science, in metaphysicis it is not about states of affairs, not about the attribution of a content (proprium) to a subject, but about this subject itself, in its generality, and then not first of all this subject considered as carrier of determinations (accidents), but as to its intrinsic relations, in which metaphysics, by considering the Essence qua Essence, in its retrogression down along the 'ex-structure' of the thing, deeper and deeper descends, or, if one wishes, ascends, perhaps all the way to the divine NOTE 197 ).
Natural science already presupposes that there are beings at all, and that they have, without qualification, content, and, consequently, just asks whether these beings, are, in given cases, for example, fishes or amphibians, and (asks) what their attributes are  ( It does not ask what it is to be a substance [that is, to what, in general, the subject term points] ).  Induction further results in the possibility to set up a law (i.e. a regularity) of causality between A and B, assumed to actually be at work between the two, which means that natural science is about extrinsic causes, namely the efficient cause (which is the only type of cause which is accessible to natural science). And this only with respect to states of affairs, not with respect to things.
In metaphysics it is about substance and its intrinsic causes, that is, the intrinsic causes of Being and Content. The fact that substance has, in addition to intrinsic causes, also an efficient and final cause, gradually becomes clear during metaphysical investigations, but these efficient and final causes are, according to me not belonging to the subject (we nowadays say :  object) of metaphysics.
And in metaphysics it is not, as it is in the other fields of knowledge, the via demonstrationis, but, among others, the via diffinitionis which should be entered NOTE 198 ).  This via, however, will only yield results (that is, one or another sort of knowledge) if substance is definable at all. We here mean that one or another given substance allows itself to be determined (we do not mean what the definition of a given substance is, and also not what the definition of Substance qua Substance is). Thus metaphysics is an enquiry into definiteness qua definiteness. A condition for this definability is that the definition can be known completely, that is, that there is no infinite regress, neither upwards (higher generality) nor downwards (to the more special).
Aristotle investigates this in the first Book of the Posteriora Analytica, Chapter 19-23 in connection with scientific proof (because this must start from the definition of the subject).
In the category of Substance we can think of a sequence of terms in which sequence we see, starting from a discrete term standing for a single individual, terms that become progressively more general the higher up we go along the sequence :  It is this side of the sequence which we can imagine, because only individuals are known to us as entities existing in extramental reality. The sequence can be depicted by a Venn-diagram in which the relative extensions [ = domains of significata] of the concepts (where concepts are terms, and are conventionally represented by words) are expressed :

The sequence begins with a last subject, that is, this subject cannot in turn become a predicate for yet another subject anymore. Can the sequence go up indefinitely? If it can, a complete definition is impossible (that is a definition containing not only the difference and proximate genus, but also all higher differences and genera). For example that of  ' man ' :  rational-animal,  rational sensitive-organism,  rational sensitive living-body, etc.  Now, Aristotle holds that the sequence does not go up indefinitely, and this he shows on the basis of the assumption that we are able to know at all.
So it is clear that with the question whether the via diffinitionis yields results the answer is already stipulated, so that we have to do with a petitio principii.
Aristotle  ( I Posteriora Analytica, Chapter 19) asks himself (1) whether, departing from a last subject, there are an infinite number of predicable terms (predicates) before we reach a first predicate (which we then would not reach), and (2) whether, departing from a first predicate, there are infinitely many subjects before reaching the last subject. He further asks, whether, departing from (two) fixed extreme terms  (for example :  ' this individual dog '  and  ' substance '  --- they can figure in the conclusion :  ' this individual dog NOTE 199 )  IS  a substance ' ),  there are, in this case (or are not), an infinite number of middle terms present between these two fixed extreme terms :

       ( NOTE 200 ).

For it is in principle possible to show that in this (partial) sequence (going from  ' dog '  to  ' substance ',  there must be a first and a last term. These we may thus presuppose, allowing to limit ourselves to the question whether the number of middle terms is infinite or not. To investigate this latter question is, however, identical to an investigation whether the demonstrations themselves are, or are not, infinite in number, for one can always prove that a more general term, can, in virtue of a (qua degree of generality) middle term, be predicated of a less general term. And if every term is an essential predicate, then we precisely have catched sight of our above described problem, the problem of the completeness of the definition.
And that these demonstrations themselves do not go into infinity has already been established at the beginning of the Posteriora Analytica NOTE 201 ).
The fact that there is no infinity involved in demonstrations and, consequently, also not in predicates, is (here succinctly summarized) imagined by Aristotle as follows :
If we depart from the assumption that scientific knowledge is possible, then automatically another type of knowledge (as principle of scientific knowledge) must be possible, and this is the definition, because definitions provide us with middle terms. And when we thus suppose that  ' X '  is some least general term, which disjunctively (that is, with  or, ..., or )  points to several individual substances, that is, a term, directly coming after the particular individuals, then this term (and also the more general terms) must be able to serve as a definiendum ( = that what must be defined), and this implies that the number of essential predicates must be finite (because knowledge is presupposed to be possible) :  finite in the downward direction, because this is presupposed, since we started out from a least universal term NOTE 202 ), and finite in the upward direction NOTE 203 ).
So the above reasoning (in the main text) purporting that, in the business of defining a given substantial term, the number of predicates is finite, and that thus complete definition is possible, has in fact not demonstrated this, but presupposed (this presupposition is that of Parmenides :  Being is intelligible). But, as we have seen in NOTE 201, we can demonstrate (departing from an assumption different from the assumption that knowledge is possible at all )  that complete definition of a given substantial term is possible NOTE 203a ).  Let us for this consider the chain of demonstrations (syllogisms), embodying the complete definition of a given substantial term, and thus containing a sequence of middle terms that starts from some first middle term ( = proximal definition [definiens] of the given substantial term to be defined completely) (see NOTE 203) and ends up at a last middle term ( = proximal definition [definiens] of the term immediately preceding the supreme term  ' substance '  and containing the latter as its genus). Into this sequence of middle terms we can insert new middle terms NOTE 203b ).  If, however, the present case is such that the repeated insertion of middle terms cannot go on indefinitely (rendering the sequence denumerable), and (because a denumerable sequence can still have an infinite number of elements between a first and a last element, as we saw (NOTE 201) for rational numbers) if this insertion takes place on the basis of the degree of generality, and if, finally, 'the first' and 'last' with respect to middle terms, are also based on degree of generality,  then the complete definition neatly corresponds to a sequence of qua magnitude increasing natural numbers (positive integers) in an interval, say the interval [1, 10000] NOTE 204 ),  from which it follows that then the number of middle terms, and, consequently, the number of syllogisms, between the given substantial term to be (completely) defined and the ultimate term  ' substance '  is  finite,  and thus there is only a finite number of (essential) predicates, resulting in a complete definition to be possible. And indeed, in NOTE 201 we did show that middle terms cannot be inserted indefinitely, so their number, when going from the (substantial) term-to-be-completely-defined all the way up to substance is finite, and a complete definition is possible.

When we want to present all this in terms of Dynamical Laws (which should then be signified by the [essential] predicates), we can do this in the following way :
Suppose that the Dynamical Law (which is the Essence of the thing generated by the dynamical system governed by that particular dynamical law) of the species DOG were a recursive cubic ( = of third degree) polynomial law NOTE 205 ),  say
Xn+1 = 3Xn3 + 5Xn2 + 2Xn + 81. Then we can have the following consideration :
From this particular Dynamical Law we can  ascend  to ever more general dynamical laws, until we have arrived at the most general cubic law :

Xn+1 = a1Xn3 + 5Xn2 + 2Xn + 81

Xn+1 = a1Xn3 + a2Xn2 + 2Xn + 81

Xn+1 = a1Xn3 + a2Xn2 + a3Xn + 81

Xn+1 = a1Xn3 + a2Xn2 + a3Xn + a4

But when we ascend from dog to mammal, vertebrate, chordate, etc., we will surely not have to do with only cubic dynamical laws. A whale (which is a mammal) for instance could perhaps embody a 5th degree dynamical law. A general dynamical law, as the general Essence of the supposita of the term  ' mammal '  (the general Essence of the individuals in the extension of the concept  ' mammal ' )  should provide for this. So when we want to further ascend towards more and more general polynomial laws, then we will, if we want to provide for everything, reach as end point the most general polynomial law, which looks as follows :

Xn+1 = a1Xnm + a2Xnm-1 + a3Xnm-2 + .  . . + ak-2Xn2 + ak-1Xn + ak

In a purely mathematical sense the number of terms of such an expression can, in principle, be infinite, but here (that is, in a context in which the expression is physically interpreted) this number is finite, namely  k  ( = m+1), and this should be so, because it is not very probable that there exist physical dynamical laws with infinitely many terms. The powers (m, m-1, m-2, etc.) are, by definition (for polynomials), always exclusively integers.
From this it is clear that every sequence of polynomials, starting with a particular polynomial (that is, with determined coefficients (for instance a1X = 3X), powers (for instance Xm = X4, and a determined constant (for instance a4 = 73)), and ascending toward the most general polynomial, is finite (because the number of terms in each expression is finite, so the process of successively determine the coefficients, powers and constant, always comes to an end (always reaches completion)).
The fictitious example, just given, at least suggests (but not, of course, demonstrates) that no infinity is involved in the genotypic domain, when we ascend from one or another completely specified dynamical law of a given Totality species (say the species mento  a maximally general dynamical law.
We must not confuse this with the number of in principle (mathematically) possible dynamical laws. So, mathematically, infinitely many cubic polynomial laws are possible, because in  Xn+1 = a1Xn3 + a2Xn2 + a3Xn + a4  the coefficients a1, a2, a3, and the constant a4, can vary over the set of real numbers. They thus can assume every value whatsoever from this set, and this set is infinitely large.
In reality, however, dynamical laws, especially those of organisms, cannot just like that assume every mathematically possible form. The corresponding Totalities (intrinsic things) generated by such laws must have a certain degree of stability in order to be able to exist at all.

This concludes our discussion about the possible ascent from special to general with respect to (the terms in the category of) Substance.
Do we also have such an ascent and descent with respect to the terms of each of the remaining categories, that is, with respect to the accidental categories?

f.  Definability of the accidents.

With the accidents, that is, with the replaceable determinations, we have entered the phenotypic domain of things NOTE 206 ).
Determinations are wholly or partially generated by the dynamical law (the Essence) and in this way form the Totality.
Despite the fact that metaphysics is only concerned with substance, we, precisely by studying accidents, obtain a good insight in what substance truly is NOTE 207 )  ( Also St Thomas devotes much attention to accidents :  In the problem of the Trinity he investigates Discrete Quantity and Relation, and in the problem of the Eucharist he investigates Quantity and Quality).
As regards the accidental categories (each for themselves) we might think of an ascent or descent, in which the subalternateness is always indicated by a predication  ( It must be so indicated, because the subalternateness is asserted ) :

The (white) thing IS a (colored) thing
( The thing, that happened to be white, IS ... )

and, ascending :

A (colored) thing IS a (qualitatively determined) thing
( A thing, that happened to be colored, IS ... )

But here accidents are predicated of accidents, and that is not a proper predication (that is, not a logically  per se  predication), and such a predication that is about the  accidental  (that is, the subject here stands for an accidental state of things) NOTE 208 )  cannot play a role in whatever process of gaining knowledge, it even drops out of the province of logic and so also out of that of philosophy. The same applies to a predication of a subject of an accident. So in such cases no subalternateness can be asserted  ( ' the white thing is a colored thing '  [here  ' white thing '  is asserted to be subsumed under  ' colored thing ',  but they are just accidental facts] ).
Thus, as regards the accidents, only the predication of an accident of a subject is relevant for us in the present context, for example  ' Socrates is tanned ',  or  ' men is able to laugh '.  But after this predication, we cannot, in ascending, proceed any further, because then the logical subject (of the new proposition) becomes an accident  ( ' tanned ' ). That is, we then should get :  ' tanned is ...'  which is equivalent to :  ' the (a) tanned thing is ...' ,  so the subject has become just an accidental fact, resulting in a non-proper predication.
Descending, we arrive (in fact we already are) at  substance  (Socrates, men), and then descend further to a last subject, standing for an individual thing (Socrates).
So in the accidental categories (each for themselves) there is no infinity involved, neither in the process of ascending, nor in that of descending, (no infinity involved) in the attribution of predicates respectively, subjects.
Nevertheless the following predication seems to be logically  per se  and also  per se  in reality (that is, in quid) :

Whiteness is a color

According to Ockham  ' whiteness '  is an absolute term, that is, a quality is signified primarily, without at the same time something else being connoted. In the context of this view, we here indeed have to do with a logically  per se  predication ( = proper predication), because now the subject term does not stand for some accidental fact.
However, we could ask whether things are seen in this way also by St Thomas :  For the time being our impression is that he solves the problem, the problem namely whether substance and accident (quality) are, or are not, in fact detached (or could be detached) from each other at the Eucharist,  in a different way than Ockham did :  According to St Thomas, quality can be detached from substance (that is, substance freed from quality), but not from quantity, which serves as carrier of quality :  So any quality is always in a carrier (subject), resulting in the fact that (for St Thomas)  whiteness  does not have a complete essence, and thus the fact that the term  ' whiteness '  (as abstract term) necessarily must (nevertheless) connote a subject NOTE 209 ).  Further is, as we know, the (possible) independence of accidents (of a substrate) less strongly conceived (that is, less completely) as it is conceived by (or as it is according to) Ockham :  Accidents, in so far as their being is concerned, always refer to substance. So when we say  ' whiteness is a color,  then, according to St Thomas,  ' color '  is not predicated of  a  per se unum  (because whiteness, although signified by an abstract expression, is, according to St thomas, not totally independent from a carrier),  implying that we, for the time being, can assess this predication as logically accidental (improper predication) NOTE 210 ).  So, again, according to St Thomas, as it seems, the accident as logical subject, whether expressed concretely or abstractly, entails an improper predication. And because the positing of A as subalternate of B is the same as the predication 'A is B', we cannot, in this interpretation, hold that  ' whiteness '  is a subalternate of  ' color '.  This entails the peculiar conclusion that we, when dealing with accidents, cannot, seemingly, speak about higher and highest genera. To solve this problem would bring us too far away from our main issues, so we will not go beyond the present aporematic treatment of these matters, a method of working characteristic of many schoolmen (scholastici). Clearly, the subaternateness in accidents must involve improper predications, and this is what distinguishes them from substances.

Summarizing things, we have the following :
Presupposing that knowledge is possible implies that also complete definition must be possible, and so in essential predication there is finitude in the downward direction as well as in the upward direction. Also as regards accidents there is such finitude.
If we, with respect to accidents (expressed abstractly), do predicate, that is, nevertheless predicate (for example  ' whiteness is a color ',  ' color is a quality ' ),  then we abstract wittingly from the carrier (substrate) NOTE 211 ).  And if we then (going against the Summa quotation) stipulate that such a predication is a genuine predication, then also here we have to do with a kind of essential predication, that is to say, in this case we can also speak of a first modus dicendi per se, or, equivalently, the predication of a definition (at any rate the genus) of a definiendum :  white [definiendum] is a color [genus] NOTE 212 ).  And for this to be possible, also here the sequence must be finite.
To exemplify such a (shortened) sequence within an accidental category, we come up with the following :

Socrates  is  white
(the) white (thing)  is  a color
color  is  a quality

Here  ' Socrates '  is a concrete term which stands for an individual substance NOTE 213 )  ( = last subject). The term  ' white ',  and also the term  ' the white thing '  (which is what  ' white '  means), is a concrete term which primarily signifies substance and secondarily an accident (something that is attached to that substance, while not belonging to it) (See Aristotle, I Posteriora Analytica, Chapter 22, 83a 5) NOTE 214 ).  However, the term  ' color '  is an abstract term (in contrast to the concrete term  ' colored ' ).  So, in ascending toward the more general, as represented by the three propositions just given, we suddenly switch over to terms which abstract from any carrier or substrate NOTE 215 ),  that is,  from  ' white '  (concrete)  to  ' color '  (abstract), after which (switching over) we can then ascend further to more and more general predicates.
(Conceded) possibility of defining accidents NOTE 216 )  forces us to hold that we will arrive at the highest genus in a finite number of steps, and this (highest genus) then would (for our example) be the genus Quality. The same applies to the remaining accidental categories.
Diagrammatically things look like this :

Because we can say instead of  ' Socrates is white '  also  ' man is white '  (for instance with respect to the color of the teeth), and then (going back) :  ' Socrates is a man ',  an accident can also  ' shoot off '  halfway from the substantial stem (this stem consists of the sequence :  Socrates, man, mammal, vertebrate, animal, ... substance) :

After the concrete term  ' white '  we switch to the abstract term  ' color '.  Of course we can do this already earlier, that is to say, directly continue with  ' whiteness '.  So after  ' man is white ' :  ' whiteness is a color '.
Indeed, in De Ente et Essentia, Chapter 6, from line 123, St Thomas holds that determinations (accidents) can only be placed in a category when they are signified abstractly. However, the following pecularity shows up :  In De Ente et Esentia an abstract term does not signify the carrier (substrate, subject), and it is because of this (that is, because not being connotative) that such a term can be placed into a category. But such a category implies the possibility of a sequence of terms with increasing generality, so that the more general term can be predicated of its subaltenate (for example :  a dog is an animal). But this is in turn possible not until we comply with the Summa quotation  ( See NOTE 209 ),  in which the abstract and concrete term are both connotative and therefore exhibit a certain degree of exchangeability (and thus allowing for the above mentioned 'switching over' from concrete to abstract NOTE 217 )),  making it then possible to ascend as follows :

In all this we suppose (in the sense of stipulation) that  being white  for Socrates as well as for man is an accidental determination, that is, not an essential quality, and that man is for Socrates a determinable (namely determinable all the way down to the individual) essence. Whether this really is the case, is immaterial with respect to the problem of the finite or infinite sequence of predicates.
That  white  as the color of the skin is accidental for man, is clear from the extensions of  ' man '  and  ' white ' :

There are people which are not white, but who we will nevertheless call  ' man ,  because we are convinced they are. In addition to this, there are also other things that are white.

That the whiteness of the teeth also can be asserted to be accidental, is already more difficult :

                ( NOTE 218 )

For this diagram is very much the same as that which expresses the predication of a determinable essence :

Man is an animal (animal = determinable essence).

And how things are with entities that can replace the essence (namely propria) is even more unclear :  Here the extensions precisely coincide, but this is also the case where we predicate the determinable essence [ = genus] PLUS essence-determining formal content [ = differentia] NOTE 219 ),  showing pretty well the inadequacy of the (exclusively) extensional approach. It becomes even more difficult if we apply all this to other objects, that is, objects different from man. And such application should be possible because all these assessments about substance, essence, accident, etc. are pretended by philosophers to be universally valid, as is evident by the fact that they do present examples other than man (despite a domination of the human example, because this domination is understandable because (1) we probably know ourselves best in these respects and because (2) we consider it to be the most important example, and presumably the most interesting domain of application of general findings) NOTE 220 ).

So Classical Metaphysics stipulates that  possessing white teeth  is  per accidens  with respect to the Essence of man (despite the fact that all humans possess white teeth, and by reason of the fact that a number of non-humans also possess white teeth). But it also stipulates that  to be an animal  is essential for man, despite the fact that also a number of non-humans are animals, that is to say, despite the fact that we have here to do with the same, or at least comparable, relative extensions of the relevant concepts (man, animal) as in the case of  possessing white teeth  (man, possessing white teeth) NOTE 221 ).  The reason for  to be an animal  to be essential for man (that is, in order that something can be a man) is that  to be an animal  figures in the definition of the term  ' man '  (namely as genus).
But we could just as well define :

Man is a white-teethed rational thing

That is, man (definiendum) is a white-teethed thing (genus) that is rational (difference).
The white-teethed thing might not need to be animal, if that something in virtue of which man is man can exist on another substrate, that is, being sustained or carried by another substrate (different from an animal substrate).
So the necessity of  animal  is just assumed. The (scientific) grounds for such an assumption, however, are not ill-based. Here it is important to realize that metaphysics cannot proceed totally without using certain results of natural science (albeit just to back up examples that it presents), implying that it can contain uncertainties, because natural science itself does.

As shown more than once, stipulation is also the case with respect to  ability to laugh ( = a proprium, and thus, according to classical metaphysics, nevertheless an accident), and to  rational  ( = essence-determining formal content) :
One says :

Ability to laugh  happens to be  always and exclusively  present in man.

However, it is not argued why  ability to laugh  is accidental (with respect to human nature) and  rational  essential.
Earlier we have looked into this problem in the context of the difference to be argued between  proprium  and  differentia,  and suggested a solution.
What then is the  "accidental"  in  "... why  ability to laugh  is accidental"?  What should this  mere concurrence  (coincidence) mean?
Coincidence can refer to the interaction of two entities, without the one entity necessarily entailing from its nature this interaction with the other entity (while the interaction took place nontheless) NOTE 222 ).  This might apply to the case of the  being white  of the teeth, but also to that of  ability to laugh.
Here we see that the existence of definite natures is always presupposed.
Is such an interaction, as just described, possible at all? Yes, it is possible as long as we indeed consider the accidental character of the event with respect to some definite (and specified) nature (with respect to other natures, or with respect to a broader context, it might not be accidental) NOTE 223 ).
Seen in a broad physical context of consideration, the World is almost certainly throughout deterministic (apart from some unimaginably small uncertainty as is implicit in the Second Law of Thermodynamics [See Fourth Part of Website ,  the documents on Thermodynamics] ),  but nevertheless in many cases not predictable by us as to its course, and the course of all kinds of partial processes such as, for instance, the weather.
Only in the submicroscopic domain (that is, the subatomic domain) it is doubtful whether also there such a thoroughgoing determinism reigns (According to Van MELSEN, A., Natuurfilosofie, 1955, such a determinism must be presupposed in order to evaluate the results of experiments in this domain, that is, the results of quantum mechanical experiments, because also the submicroscopical particles, and the fields associated with them, necessarily act according to their  nature).
However, the reality status of this submicroscopic world is not completely clear  ( See, among others, GRIBBIN, J., In search of Schrödinger's cat, QUANTUM PHYSICS AND REALITY, 1987 ).  It could be that the submicroscopical entities are incomplete or deficient beings.
(Therefore) on this website I limit myself to the world of macroscopic beings (that is beings with a size well above that of atoms). In this world, as has been said, total determinism rules, that is, a physical determinism, which refers to the course of physical processes. And in these processes there is no place for the  per accidens.
But in  a  metaphysical  consideration of the (macroscopic) world  beings insofar as they are beings  are central. All kinds of relations there are always, and exclusively seen  from  some given individual complete being (a Totality). Such a being is and remains the point of reference, and in such a context there is definitely room for  a  per accidens,  but this involves a different kind of  ' happens to (be) '  than the not-according-to-their-nature-interaction of two beings. It is  a  per accidens  with respect to the Essence, and that means  a  per accidens  relative to the Dynamical Law, the 'genotype' of the given Totality. Some formal contents are (totally deterministically) exchanged for others, while the Essence remains the same during such a transformation. They are then  per accidens  with respect to that Essence, because their being there depends on the point in time of observation, and that point in time is itself  per accidens  with respect to that Essence.
Precisely because of the distinction between on the one hand, Essence (of a given Totality) and the full Totality itself, on the other, such  a  per accidens  is possible. Something can be connected with the Totality (the given intrinsic being or thing), but might not necessarily be so connected, "necessarily" in the sense of :  always being present (and not just always observed to be so present) together with the Essence of that Totality.
Now we know (as far as observation goes) that  ability to laugh  and also  rational  are always present together with the Essence (the Essence of every human being), and one of them, namely  rational,  is caused by that Essence, while  ability to laugh  is perhaps only caused by the combination  Essence + something which is extrinsic to that Essence, implying it to be accidental with respect to that Essence.
In a broader context  ability to laugh  is surely not coincidentally.
According to our latest interpretation  a  proprium  (such as the ability to laugh) belongs to the phenotypical part of the carrier (whereas the [physically interpreted] dynamical law (Essence), which as such includes genus and difference [which latter is signified by the term  ' rational ' ],  belongs to the genotypical part of the carrier. It is symbolized in the next diagram. The green area indicates the position of propria, whereas true accidents belong in the dark blue area :

Further we still have the problem of how it is about that what is not essential with respect to human nature, but might be essential with respect to a given individual or group of individuals :  Curved nosed, for example, surely is not essential with respect to human nature, but might be essential with respect to a certain race (it then should figure in the definition of that race as differentia). The same applies to white insofar as it refers to skin color :  While to be white might be essential (in the philosophical sense of  per se )  to a subclass (this latter also definable on the basis of other features) of man (to judge from the fact that all members of this subclass, and only they, are white), it is not, as it seems, essential to human nature as such, because there do exist many as such unanymously recognized human beings which are black. To be essential, therefore, is not absolute but relative. Such a conclusion follows from the assumption that the essence is identical either in all human individuals, which taken together we call 'class' E, or in certain groups of them (subclasses), that is to say, the Essence E either is present in every individual of class E, or it is not, but then Essence J is present in all individuals of the (sub)class J, Essence G is present in all individuals of the (sub)class G, etc. In the latter case the 'species' (in the logical sense) of man turned out to be a mixture of several distinct species, and it is then not Essence E but a more general Essence (for which Essence J and Essence G are special cases) that is possessed by all individuals of class E. And if we then see a certain feature to be present in one or more individuals of such a (sub)class while absent in other individuals of that same (sub)class, then we conclude that this feature is accidental with respect to the essence that is common to the individuals of that (sub)class.
The counterpart of such an assumption would be the following :  Every human individual (that is, every individual entity that is commonly recognized as to be a so-called human being) has its own Essence, different qua content from that of any other individual, perhaps in such a way that those different essences cannot be subsumed under some higher essence (and this is in fact the nominalistic position). If we focus, in the case of organisms, on a more or less clearly visible aspect of the Essence (which Essence itself is the particular dynamical law), namely their  genome  (as such embodying the genetic make-up of a given organic individual), then it seems obvious that each organic individual, and then also each human individual, is a different -- also qua content -- substance (in the metaphysical sense), that is, each possessing qua content a, albeit similar, altogether different dynamical law (and thus Essence). The  per se  and  per accidens  character of some given feature of a human being then only refers to his own Essence, meaning that a given feature is  per accidens  if it does not intrinsically belong to the dynamical system as defined by its (physically interpreted dynamical law), but is only the consequence of some perturbation of the system (that is, a disturbance coming from outside the dynamical system). So as to the  per se  and  per accidens  nature of any given feature we must -- in this view -- take into account the very individual who carries this feature (and who embodies the particular dynamical system), because it is with repect to the carrying individual whether a given feature is  per se  or  per accidens. If it is present during all of the lifetime of this individual it is probably an essential feature, if not, then it might be  per accidens,  except when it appears and disappears in a regular way. In the latter case its sequence is  per se.
So it is clear that of any given formal content (say  white )  taken as such, we cannot say anything about its attribution, that is, whether this attribution is  per se  or  per accidens. Therefore in its definition its proper subject must always be mentioned, in order to determine or express the  per se / per accidens  nature NOTE 224 ).
And in the case of a(n) (potentially)  essential  predicate (such as  ' rational ' ),  in contrast to a mere essential determination (proprium) :  To a definiens ( = definition proper) always belongs a proper definiendum ( = that which is to be defined). For only then we can see whether it is an essential predicate NOTE 225 ).
With respect to the relation between individual and essence there are (thus) two possibilities (how, in reality things turn out to be in this respect) :
  1. There are more than one individuals possessing exactly the same (qua content) essence.
  2. Each individual possesses an Essence, different qua content from that of any other individual.
Possiblity  1  can be the case even when each human individual has an essence of his own, different from that of any other individual except in the case of monovular multiples.  In this latter case the members of such a multiple can be considered as possessing an identical Essence (identical qua content), if we base this on their respective genomes.

g.  Definitiones sunt sicut numeri NOTE 226 ).

In Chapter 2 of Natura en Creatura, 1982, p.70, AERTSEN discusses the interesting comparison of  definitions and (natural)  numbers.  He sums up four correspondencies :

  1. Unity
  2. Identity
  3. Discretion
  4. Order
It is especially the third correspondency that deserves attention, namely the discretion of Being. Because, if indeed, for example, man has originated from non-rational organisms, and ultimately even from inorganic beings, is there then not a continuous transition between, for instance, a non-rational sensitive organism and a rational sensitive organism, or between a non-sensitive organism and a sensitive orgamism? NOTE 227 ).
In what way can here differentiae still being discrete diversa? Said differently, our way of thinking always involves or invokes the  genus-difference-species-individuum  scheme or framework, but does there correspond to this logical discrete hierarchy an ontological order? Here we can only say something about discretion :  It seems that evolutionary theories demonstrate continuity instead of discretion, but although the transitions between natural things among themselves indeed turned out to be much more subtle and gradual than was assumed in Aniquity and Middle Ages, it has been shown that the assumption of absolute continuity is false. Light, for example, turned out not to be a continuous phenomenon, likewise physical matter, and even it is doubted that space is a continuous entity (for instance by Branislav PETRONIEVICS NOTE 228 ),  who even has created a special discrete geometry to describe this space). Also the genetic apparatus of organisms is discontinuous. There the ultimate units are the nucleotides (to be precise, their organic bases [in the chemical sense] :  Adenine, Guanine, Cytosine, Uridine, and Thymine). These units can, within the genetic context of encoding the structure of the protein to be synthesized, be taken as 'diversa', that is, they function as ultimate differences without carrier (in contrast to differre aliquo [ = differing in something] ).
From all this it is clear that the discretion is, it is true, maintained, but that the ontological interpretation of the differentiae of the definitions becomes much more theoretical NOTE 229 ) :  Just like we cannot concretely point to 'prime matter', we also cannot so point to (the significata of) differentiae anymore :  for the initially ascertained -- by the Greeks -- discretion in nature is being reduced to a very subtle level of quasi continuity by natural science NOTE 230 ).
The fourth correspondency between definitions and numbers refers to the hierarchy of Being. As far as we can see, St Thomas does nothing more than just stipulating this order of  ' perfect '  and  ' more perfect ',  and despite the fact that the world surrounding us seems to point to it, we do not, at St Thomas, find a serious attempt to indeed demonstrate it. The idea of  ' higher '  and  ' lower '  animals (like higher and lower numbers), for example, is dominant in biology, but it is hard to concretely demonstrate in virtue of precisely what an animal is higher, or is lower. It could be something like higher (or lower) complexity, resulting in the ability to execute more, and more differentiated, functions, but this is, according to me, not the same thing as being "nobilis" and "nobilior" (noble and more noble) NOTE 231 ).  And even complexity can sometimes decrease again during periods of evolution, as we can see for example in many organisms that have adopted a parasitic way of life.
An absolute criterion deciding about the  ' higher '  and  ' lower '  of organisms has, up to date, not been found, at least not in biology, no matter how much such a hierarchy forces itself upon us. The discovery of such a criterion is moreover hampered by the almost unintelligible wealth of organic species and forms, also with respect to patterns of behavior and way of life, making a general judgement about animals and their status much too premature. Many researchers doubt whether there have actually been long-term developmental tendencies in organic history, although roughly something like this seems to be apparent, but that could be misleading (just like the so-called  ' constancy of species ' has indeed been this for centuries). And precisely by already presupposing such developmental tendencies, one erected all kinds of  ' ancestral trees '  (that is, typological ancestral trees).
We should not, without further consideration, ontologize our way of description (here as regards  ' higher '  and  ' lower ' ),  that is, we cannot (thematically) say without qualification :  " The compositio of genus and differentia in the definition articulates this hierarchical segmentation of reality" (AERTSEN, p.73). This is so because, for example, the definition of  ' ape '  is not just  ' sensitive organism '  (that is, an organism possessing sense organs, namely an animal) NOTE 232 ) as if we (in so defining) would wish to indicate with this term that some link is missing in apes as compared to man :  Like man, apes form an endpoint (until now) of their own positive development, and it is only that we consider ourselves to be so special and know ourselves relatively well (while we in fact also today know very little of animals), that we are quick to attribute to animals (as contrasted with man) some sort of privative structure, resulting in the differentia ( = further determination) of  ' animal ',  in their case, to acquire a negative sense :  'animal irrationale'  ( Also the term  ' brutum '  is totally privative). There is, however, nothing that indicates that animals were to miss something what they should possess :  Animals differ from humans in the same way that animals differ among each other NOTE 233 ),  that is, they differ only by positive differences, whereby we must, for the time being, be very careful when it comes to 'evaluate' these differences. Of course we ourselves rate our rationality highly, but whether this is also that high in an objective cosmological context is not self-evident NOTE 234 ).
Despite these critical remarks, the idea of a hierarchy of Being remains an interesting one, which cannot, without qualification, be dismissed, and which incites us to think further.

To conclude, regarding the  definition  we want to consider the following :
Because only substance has a quod quid erat esse ( = essence), only substance (in the sense of :  any given substance) has a genuine definition. All other entities, such as accidents have only an incomplete essence, and therefore not a proper definition. This also applies to  ' matter ',  ' substantial form '  NOTE 235 ), and  ' nature '.
Because  nature  is in fact the significatum of the definiens ( = definition proper) of a definiendum ( = that what is to be defined) pertaining to whatever thing NOTE 236 ),  that is, because  nature  is  THE  DEFINITION  as such NOTE 237 ),  a definition of  ' nature '  means a definition of the definition, or, equivalently, the exposition of the properties and characteristics of the definition as such, because a consideration about the  definition as such  cannot be separated from a general consideration of the significata, that is, from  nature  (see previous note)  (See AERTSEN, p.292).
Nature is the definiens which stands for whatever thing, and is, consequently, as such undetermined, and therefore  ' nature '  has no definition.

In First Part of Website we have found that every real ontologically fully-fledged being (such as a crystal or an organism) is a Totality (of determinations) generated by a dynamical system, and that the Essence of this Totality is the dynamical law of that dynamical system. The  effect  of such a dynamical law is an integration of parts into one single whole. It is a totality-resultant of the system, not an aggregate-resultant. The dynamical law is immanent in the system elements, and thus follows from the  nature  of these elements. If these elements themselves are in turn each for themselves totality-resultants, then this is based on lower-level dynamical laws. Eventually we end up at last parts (ultimate parts), which each for themselves have a certain  nature,  and thus form the initial stage for dynamical laws at the next higher level and the next ones.
Nature  is an ultimate content, which cannot be reduced further anymore.
It is  form.
Substantial form must refer to matter.
Matter must refer to form.
Substance refers (within the real world), it is true, to the substantial form, but this latter does not lie outside substance.
Perhaps the substantial form refers to the SEPARATE FORM, residing in the ideal sphere of Being.

With all this we have come to the end of what we could call an  update of Aristotelian epistemology,  and we have seen how much this is tied up with Aristotelian metaphysics.
And in having discussed (1) the basic questions, (2) the predicables, (3) the predicaments, and (4) the definition, we have paved the way leading to LOGIC.  Indeed we will treat of logic in the next documents. We do this because of the following :  Although logic must itself be based on metaphysics, logic can nevertheless provide us with some metaphysical insights. But this must be a genuine logic, that is, a science which is about the formal elements, necessary for acquiring knowledge (in contrast to things like brains, sense organs, which are the material elements necessary for knowledge to be possible, and studied in psychology and biology). If we set up such a logic in such a way as to initially leave aside as many metaphysical presuppositions as possible, then we might end up with a logic that in fact demonstrates that the substance-accident metaphysics is the correct metaphysics. And this then is, of course, the reason why we will, in what follows, devote so much attention to logic.

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