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Contents.

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.

• Real definition
  • Natural definition
  • Metaphysical definition
• Nominal definition

• Real definition (definitio qui rei), and
• Nominal definition (definitio quid nominis).

• Natural definition ( = Matter-Form definition in BOBIK, J.,   AQUINAS  On Being and Essence, 1970, p.95 ),  and
• Metaphysical definition (In BOBIK :  Genus-Difference definition).

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, :

X_n+1 = 3X_n² + 5X_n + 7  (See Part III)

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

X_n+1 = aX_n² + 5X_n + 7  (that is, we have replaced the definite coefficient  "3"  of X_n²  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  3X_n² + 5X_n + 7.
The term  ' animal '  (assuming [for the sake of argument] that this is the genus directly lying above  homo )  could then stand for  aX_n² + 5X_n + 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.

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.

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) :

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

2nd Syllogism (middle term in bold type) :

MAIOR : 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 ). MINOR : A triangle ((S))  IS  a figure closed by three straight lines ((DS)). ----------------------------------------------------------------------------------------- CONCLUSIO : 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 ).

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).

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

( 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 :

• Immediate essence, namely that of a given Substance, where the cause of its essence (within epistemology, or philosophy of science) is identical to this essence itself :  Its whatness (i.e. its formal content) ( NOTE 186 )  and its existence must be assumed in an immediate premise of demonstration (that is not preceded by a middle term anymore).

• Mediate essence, namely that of a property, where the material cause of that essence lies outside that property, which is rendered questionable (one can ask questions about it) thanks to the property's catallel structure, in virtue of which the residing-in of the property becomes demonstrable, while the essence, although fathomable in (the context of) demonstration, is not demonstrable (provable). That is to say that predication of the definiens (of the accident) of the definiendum can never be a conclusion, also not of an equivalence demonstration (a demonstration on the basis of equivalence), because the subject (the Substance) belongs to that essence, and we know that the what-is-it of this Substance cannot be demonstrated.

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.

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.

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 :

(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  "....al " (as in  ' rational ' ),  resulting in :  A crystal [ = genus] that  is   ...al [ = difference],  or (equivalently) :  A  ...al 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.

• The definitio quid rei of a substance corresponds to an immediate premise of demonstration ( NOTE 191a ).
• The definitio quid rei of a property corresponds to the complete demonstration (it is revealed through this demonstration). See the syllogism above .
• The definitio quid nominis of a property corresponds to the conclusion ( NOTE 192 )  (of that demonstration that reveals the definitio quid rei ).  See the syllogism above .  (A definitio quid nominis of a substance (that is, of the subject) does not play a part in the demonstration).

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 :

MAIOR : 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). MINOR : 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 ----------------------------------------------------------------------------------------- CONCLUSIO : 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 :

MAIOR : All sponges consist of (one or more) cells (or, equivalently :  All sponges are cellular). ( Conclusion of previous syllogism ). MINOR : ' 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 ). ----------------------------------------------------------------------------------------- CONCLUSIO : ' 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.

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 :

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
X_n+1 = 3X_n³ + 5X_n² + 2X_n + 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 :

X_n+1 = a₁X_n³ + 5X_n² + 2X_n + 81

X_n+1 = a₁X_n³ + a₂X_n² + 2X_n + 81

X_n+1 = a₁X_n³ + a₂X_n² + a₃X_n + 81

X_n+1 = a₁X_n³ + a₂X_n² + a₃X_n + 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 :

X_n+1 = a₁X_n^m + a₂X_n^m-1 + a₃X_n^m-2 + .  . . + a_k-2X_n² + a_k-1X_n + a_k

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 X^m = 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 men)  to  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  X_n+1 = a₁X_n³ + a₂X_n² + a₃X_n + 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?

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 :

• Socrates is a man
• Man is white (with respect to his teeth)
• White is a color
• etc.
• until arriving at :  ... is a quality

• (Socrates is) white [concrete]
• (whiteness [abstract] is a) color [abstract]
• (color [abstract] is a) quality [abstract]

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

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 :

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.

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

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.

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