Commentary Aristotle Posterior Analytics
Lecture 2
Whether the definition which signifies the quod quid of a thing can be demonstrated a36. Let us now state how essential nature is revealed and in what way it can be reduced to demonstration; what definition is, and what things are definable. And let us first discuss certain difficulties which these questions raise, b1. beginning what we have to say with a point most intimately connected with our immediately preceding remarks, namely the doubt that might be felt as to whether or not it is possible to know the same thing in the same relation, both by definition and by demonstration.
b3. It might, I mean, be urged that definition is held to concern essential nature and is in every case universal and affirmative; whereas, on the other hand, some conclusions are negative and some are not universal; e.g., all in the second figure are negative, none in the third are universal.
b7. And again, not even all affirmative conclusions in the first figure are definable, e.g., ‘every triangle has its angles equal to two right angles’. An argument proving this difference between demonstration and definition is that to have scientific knowledge of the demonstrable is identical with possessing a demonstration of it: hence if demonstration of such conclusions as these is possible, there clearly cannot also be definition of them. If there could, one might know such a conclusion also in virtue of its definition without possessing the demonstration of it; for there is nothing to stop our having the one without the other.
b14. Induction too will sufficiently convince us of this difference; for never yet by defining anything—essential attribute or accident—did we get knowledge of it.
b16. Again, if to define is to acquire knowledge of a substance, at any rate such attributes are not substances. It is evident, then, that not everything demonstrable can be defined.
b19. What then? Can everything definable be demonstrated, or not? There is one of our previous arguments which covers this too. Of a single thing qua single there is a single scientific knowledge. Hence, since to know the demonstrable scientifically is to possess the demonstration of it, an impossible consequence will follow:—possession of its definition without its demonstration will give knowledge of the demonstrable.
b23. Moreover, the basic premises of demonstrations are definitions, and it has already been shown that these will be found indemonstrable; either the basic premises will be demonstrable and will depend on prior premises, and the regress will be endless; or the primary truths will be indemonstrable definitions.
b28. But if the definable and the demonstrable are not wholly the same, may they yet be partially the same? Or is that impossible, because there can be no demonstration of the definable?
There can be none, because definition is of the essential nature or being of something, and all demonstrations evidently posit and assume the essential nature — mathematical demonstrations, for example, the nature of unity and the odd, and all the other sciences likewise.
b33. Moreover, every demonstration proves a predicate of a subject as attaching or as not attaching to it, but in definition one thing is not predicated of another; we do not, e.g., predicate animal of biped nor biped of animal, nor yet figure of plane — plane not being figure nor figure plane.
b38. Again, to prove essential nature is not the same as to prove the fact of a connexion. Now, definition reveals essential nature, demonstration reveals that a given attribute attaches or does not attach to a given subject; but different things require different demonstrations — unless the one demonstration is related to the other as part to whole. I add this because if all triangles have been proved to possess angles equal to two right angles, then this attribute has been proved to attach to isosceles; for isosceles is a part of which all triangles constitute the whole. But in the case before us the fact and the essential nature are not so related to one another, since the one is not a part of the other.
So it emerges that not all the definable is demonstrable nor all the demonstrable definable; and we may draw the general conclusion that there is no identical object of which it is possible to possess both a definition and a demonstration. It follows obviously that definition and demonstration are neither identical nor contained either within the other: if they were, their objects would be related either as identical or as whole and part.
So much, then, for the first stage of our problem.
Let us now state how essential nature. After showing that every question is in some sense a question of the middle, which is the 'quod quid' and the 'propter quid', the Philosopher now begins to make manifest how the middle becomes known to us.
And it is divided into two parts.
In the first part he shows how the 'quod quid' and 'propter quid' are related to demonstration.
In the second part he shows how the 'quod quid' and 'propter quid' should be investigated, at: so let us now discuss (97a22; [527]).
The first part is divided into two parts.
In the first he manifests how the 'quid est' is related to demonstration.
In the second he manifests how the 'propter quid' which signifies a cause is related to demonstration, at: we think we have (95a20; [490]).
First, he states his proposal.
Second, he pursues his proposal, at: beginning what we have (91b1; [420]).
He says therefore first (91a36) that since every question for whose solution a demonstration is adduced is a question of the middle which is 'quid' and 'propter quid', our first task is to declare how the 'quod quid' is made known to us, namely, is it through demonstration or division or some other way. Furthermore, we must point out how to reduce to the 'quod quid' those items of a thing that are apparent. And because a definition is a statement which signifies the 'quod quid', it is necessary also to know what a definition is and why certain things are definable. Consequently we shall proceed in these matters in the following way.
First, by raising difficulties.
Second, by establishing the truth (94a1; [470]).
Then (91b1) he pursues his proposal in the order stated, at: beginning what we have.
First, he proceeds dialectically by raising difficulties.
Second, by determining the truth, at: we must now start afresh (94a1; [470]).
First, he proceeds by disputing about the definition which signifies 'quod quid'.
Second, about the 'quod quid' which is signified by the definition, at: the next step is to raise (92a12; [431]).
Concerning the first he does three things.
First, he inquires disputatively whether there is definition of all things of which there is demonstration.
Second, whether conversely there is demonstration of all things of which there is definition, at: what then? (91b19; [425]).
Third, whether there can be definition and demonstration of the same thing, at: but if the definable (91b28; [427]).
First, he states his intention.
Second, he pursues his proposal, at: the doubt that might be felt (91b1).
He says therefore first (91b1) that when a number of items are planned for discussion, one should take his start from the item which is most fruitful in settling the “had,” that is, the subsequent, problems.
This would mean that in our case we should begin with the fact that someone might wonder whether the same thing and according to the same aspect could be known through definition and demonstration.
Then, at: it might, I mean (91b3) he proves that there is not definition of all things of which there is demonstration. This he does in four ways.
The first of these is the following: a definition is indicative of the 'quod quid'; but anything which pertains to the 'quod quid' is predicated both affirmatively and universally: therefore, a definition contains or signifies only those things which are predicated affirmatively and universally. But not all syllogisms demonstrate affirmative universal conclusions: in fact some are negative, as, for example, all those in the second figure; and some are particular, as for example, all those in the third figure. Therefore, there is not definition of all things of which there is demonstration.
Second, at: and again, not even (91b7), he shows the same thing when he says that there cannot even be definition of all things which are concluded through affirmative syllogisms—which occurs only in the first figure, as when it is demonstratively syllogized that every triangle has three angles equal to two right angles. Now, the reason for this statement, namely, that there cannot be definition of all things thus syllogized, is that to know something demonstratively is nothing else than to have a demonstration. From this it follows that if the science of all these things is had only through demonstration, there is not definition of them. For things which have a definition are made known through their definition. Otherwise it would follow that a person who does not have a demonstration of these things would have scientific knowledge of them, for the simple reason that nothing hinders a person who knows the definition from not having at the same time the demonstration—although the definition is a principle of the demonstration. For not everyone who knows the principles knows how to deduce the conclusion by demonstrating.
Third, at: induction too will sufficiently (91b14; [423]), he shows the same thing through an induction from which a conviction can arise sufficient for admitting the above-mentioned conclusion. He says, therefore, that demonstration is concerned with things which are per se in something, as is clear from what has been established in the first book. But a definition never gives anyone a knowledge of those items which are per se in a thing, much less of those which are in it per accidens—not that there cannot be definitions of accidents which are in a thing per se or per accidens, as is stated in Metaphysics VII; but because no one has ever given the definition of that which is to be in a thing per se or per accidens, which the syllogism concludes.
Fourth, at: again, if to define (91b16), he shows the same thing with a reason, namely, that the definition makes the substance known:
first of all, because substance is defined in a primary way, whereas accident is defined secondarily by means of a definition which adds something alien, as is stated in Metaphysics VII;
second, because an accident is not defined save insofar as it is signified after the manner of a substance by employing a noun. But the things upon which demonstrations bear are not substances nor things signified after the manner of substances, but things signified after the manner of accidents, namely, after the manner of something inhering in some thing. Hence he concludes that it is not possible that there be a definition of everything of which there is demonstration.
Then, at: what then? (91b19) he inquires whether conversely there is demonstration of everything of which there is a definition.
And he shows that there is not for two reasons, the first of which has already been touched upon above. For it seems that of one thing, precisely as it is one, there is one science, that is, one way of knowing it. Hence if that which is demonstrable is truly known scientifically in virtue of the fact that a demonstration is had of it, then something impossible follows, if it can also be scientifically known through a definition: for the one having the definition would be knowing something demonstrable without having the demonstration—which seems absurd. And this reason was the second of the four previously presented.
Then, at: moreover, the basic premises (91b23; [426]), he gives the second reason. For definitions are principles of demonstrations, as was established in Book I; but the principles are not demonstrable—otherwise it would follow that there are principles of principles and that demonstrations would proceed to infinity, which is impossible, as we have shown in the first book. Hence it follows that definitions are not demonstrable, being as they are first principles in demonstrations. Consequently, there is not demonstration of all things of which there is definition.
Then, at: but if the definable (91b28) he inquires whether it is possible for there to be definition and demonstration of the same thing in some cases even though not of all.
And with three reasons he shows that there is not.
The first of these reasons is that a definition is manifestive of the 'quod quid' and of the substance, that is, of the essence, of a thing. Demonstrations, on the other hand, do not manifest this but suppose it: thus in mathematical demonstrations it is supposed from arithmetic what unity is and what odd is; and the same applies in other demonstrations. Therefore, there is not demonstration and definition of the same thing.
Then, at: moreover, every demonstration (91b33) he presents the second reason and it is this:
In the conclusion of a demonstration something is predicated of something either affirmatively or negatively; but in a definition nothing is predicated of anything: thus in the definition that man is a two-footed animal, neither 'animal' is predicated of two-footed nor 'two-footed' of animal. Likewise in the definition that a circle or a triangle is a plane figure, neither plane is predicated of figure nor figure of plane. For if the parts of a definition were to be joined to one another, the resulting predication would have to be understood in a manner which suits a definition, namely, in quod quid. But this we do not observe. For the genus is not predicated quod quid of the difference, nor the difference of the genus. Therefore, there is not definition and demonstration of a same thing.
Then, at: again, to prove essential (91b38) he gives the third reason and says that it is one thing to manifest the 'quod quid' and another the 'quia', as is clear from the difference between the questions listed above; but the definition shows the 'quid' of something, whereas a demonstration shows either affirmatively or negatively that something is or is not so of something. But we see that for different things different demonstrations are required, unless those two different things are related as whole and part: for in that case there would be one and the same demonstration concerning both. Thus, the demonstration that a triangle has three angles equal to two right angles applies also to isosceles, which is related to triangle as part to whole. But that is not the case in these two things, namely, in the 'quia' and in the 'quid'; for neither is a part of the other.
Therefore, it has been established that there is not demonstration of everything of which there is definition, nor conversely. Furthermore, it has been possible from this to conclude that these are not of the same thing, and that definition and demonstration are neither the same nor is one in the other as a subjective part in its whole—otherwise it would be necessary that even those things of which they are be related after the manner of whole and part in such a way, namely, that everything definable would be demonstrable, or vice versa—which was disproved above.
Finally, he summarizes and concludes that so far we have proceeded by opposing.