Namely on the basis of our ability to know at all.
If there exists, departing from some first syllogism (having as its conclusion : 'this thing is a dog ', for example the syllogism :
All dogs, and only dogs, can bark
This thing can bark
Therefore, this thing is a dog ),
a last syllogism, the maius of which thus is the term ' substance ' :
All substances and only substances xxxxxx
This thing xxxxxx
Therefore, this thing is a substance,
then the number of syllogisms can be finite (but need not necessarily be so).
This is so because the sequence or chain seems to be (1) denumerable (which means, that when we know some syllogism of the sequence, we know what the next syllogism is [such a denumerable sequence can still be infinite] ), and (2) because, moreover, there is in this sequence not only a first but also a last syllogism, resulting in the fact that this denumerable sequence can be countable (that is, exhaustively denumerable) and consequently finite. If, in addition to (1) and (2), the principle, according to which the syllogisms mentioned in (2) are first and last syllogisms, is the same as that according to which the syllogisms can be indexed, resulting in a denumerated sequence, then the number of syllogisms between the first and last is finite. And, for syllogisms, it is evident that the principle governing (1) and (2) is indeed the same, namely (in both cases it is) the degree of generality. But the finiteness of the number of syllogisms (potentially existing) between the first and the last all depends on whether indeed the chain of syllogisms (ascending from dog to substance, or descending from substance to dog ) is denumerable.
(There exist sequences, having a first and last element, but nevertheless are not denumerable, and which are thus infinite, for example the interval [0,1] in the set of real numbers, which interval is the set of all real numbers between 0 and 1, and these included. In this set the numbers are ordered with respect to magnitude and in this ordering 0 is the first element (lowest magnitude) and 1 is the last element (highest magnitude). Nevertheless this sequence (that is, the set of numbers) contains infinitely many elements (infinitely many numbers).
So if the chain of syllogisms would be 'continuous' (as is the sequence of real numbers from 0 to 1), then, in spite of the existence of a first and last syllogism, the number of syllogisms of our sequence would not be finite.). And because it seems that one always can insert a new middle term (and thus a new syllogism) between two consecutive ones already existing (in contrast to merely add one in the sense of attaching it to the end of the series), it appears that the series of syllogisms, insofar as it appears in its mentioned ordering from less general to more general, is not denumerable (because we do not know the immediately next middle term : for having successfully inserted a middle term [that was 'forgotten' to be placed in the sequence] between two consecutive existing ones, one can always insert yet another middle term between the inserted term and the previous term, and so on and so on), resulting in the fact that we never reach the last term ( 'substance ' ), and thus can never present a complete definition. However, insertion of more and more middle terms might be done after the manner of rational numbers (or even the algebraic numbers), and then the set is denumerable ( NOTE 201-1 ), for although we can never in these (rational) numbers determine the qua magnitude immediately next number, the set can nevertheless be indexed (with the whole numbers 1, 2, 3, 4, 5, etc.) on the basis of some other method, and thus rendered denumerable (and indeed it it is shown that the rational numbers are denumerable [in contrast to real numbers] ). But because here the indexing does not correspond with the magnitude of the numbers, the existence of a qua magnitude first and last number does not mean that the series is finite (Those rational numbers, as they generally have their positions on the number line, that lie between, say, 1/8 and 1/3, form a set consisting of an infinite number of elements, despite the fact that we have a qua magnitude first (1/8) and a qua magnitude last (1/3) number). And thus there may exist infinitely many syllogisms (which are denumerable) between the first and the last one after all.
Summarizing all this we can -- departing from the existence of a qua degree of generality first and last syllogism -- say the following :
If the sequence or chain of syllogisms is denumerable, and when the indexing of this denumeration corresponds to the degree of generality, then the series is countable ( = exhaustively denumerable) and thus finite.
If, on the other hand, the indexing of the denumeration does not follow the degree of generality, then the series is, it is true, denumerable, but not per se countable, and thus not per se finite.
Although we can generate an ascending or descending sequence of syllogisms (ascending or descending with respect to the degree of generality), the sequence is not therefore necessarily denumerable, because it might be that we can insert middle terms indefinitely.
In the ascending sequence of natural numbers (positive integers) we can always insert a number between two chosen numbers of this sequence, but this has its limit (meaning that the above 'always' must be qualified) : At some point we arrive at, say, the numbers 2 and 3, and between them nothing can be inserted anymore (also not by taking the mean, because this does not yield a positive integer, but a rational number instead : 5/2.
In the case of rational numbers, on the other hand, we can always insert numbers, that is, always, without qualification : we can always keep on inserting numbers between any two numbers (by taking their mean for example. And taking a mean of two rational numbers always yields a rational number).
So if we indeed can insert again and again middle terms (see NOTE 203 below), then the number of possible middle terms of the series is INFINITE, and with it the number of syllogisms. And this in turn means that any given substance term, such as ' dog ', cannot completely be defined.
However, it appears evident that the scale of (degree of) generality (which orders a chain of syllogisms) is not continuous : It is hard to believe that degrees of generality could be so close to each other as to differ only infinitesimally. The extension of middle terms (representing syllogisms), that is, their domain of signification, is a set of individuals. The extension of a more general middle term is larger than that of a less general middle term, that is, the set of individuals (of which [individuals] that middle term can be predicated) is larger ( For instance the set of vertebrate animals is larger than that of mammals). And it is now clear that the size of such a set cannot increase in a continuous way, that is, by infinitesimal increments. On the contrary, it increases by adding individuals. So we cannot insert an intermediate set between two sets whose sizes have come to differ by only one individual (one element). So repeated insertion of middle terms between two given middle terms (on the basis of degree of generality) must eventually come to an end. And this means that the series of middle terms, and with them of syllogisms, must be denumerable, and they are so on the basis of the degree of generality. And having now a first and last syllogism, where 'first' and 'last' are also based on the degree of generality, we have a countable and thus finite series of syllogisms (say, between one involving ' dog ' and one involving ' substance ' ).
So if we want to define, say, the term ' dog ', we have to ascend, till we arrive at ' substance ' (which is the highest genus of this sequence), mentioning all the in-between genera and differentiae, and this we can actually do because the sequence is finite. And, now generalizing, this means that a complete definition of any given substantial term can indeed be given.