by Aristotle
[35] 3 · Now, that everything we seek is a search for a middle term is clear; let us now say how one proves what a thing is, and what is the fashion of the reduction, [90b1] and what definition is and of what, first going through the puzzles about them. Let the start of what we are about to say be whatever is most appropriate to the neighbouring arguments.
A man might puzzle over whether one can know the same thing in the same respect by definition and by demonstration, or whether that is impossible.
For definition seems to be of what a thing is, and what a thing is is in every case [5] universal and affirmative, but deductions are some of them negative and some not universal—e.g. those in the second figure are all negative and those in the third not universal.
Next, there is not definition even of all the affirmatives in the first figure—e.g. that every triangle has angles equal to two right angles. The argument for this is that to understand what is demonstrable is to have a demonstration; so that since [10] there is demonstration of such things, clearly there will not also be definition of them—for someone might understand them in virtue of the definition without having the demonstration; for nothing prevents him from not having them together.
An induction, too, is sufficiently convincing; for we have never yet become aware of anything by giving a definition—neither of anything belonging in itself nor [15] of any accidental.
Again, if definition is becoming familiar with some substance, it is evident that such things are not substances.
So it is clear that there is not definition of everything of which there is demonstration.
Well then, is there demonstration of everything of which there is definition, or not?
Well, one argument is the same in this case too. For of one thing, as one, there [20] is one mode of understanding. Hence, if to understand what is demonstrable is to have a demonstration, something impossible will result; for anyone who has the definition without the demonstration will understand.
Again, the principles of demonstrations are definitions, and it has been proved [25] earlier that there will not be demonstrations of these—either the principles will be demonstrable and there will be principles of the principles, and this will go on ad infinitum, or the primitives will be non-demonstrable definitions.
But if the objects of definition and demonstration are not all the same, are some of them the same? or is this impossible? For there is no demonstration of that of which there is definition. For definition is of what a thing is and of substance; but [30] all demonstrations evidently suppose and assume what a thing is—e.g. mathematical demonstrations assume what a unit is and what odd, and the others similarly.
Again, every demonstration proves something of something, i.e. that it is or is not; but in a definition one thing is not predicated of another—e.g. neither animal of [35] two-footed nor this of animal, nor indeed figure of plane (for plane is not figure nor is figure plane).
Again, proving what a thing is and that it is are different. So the definition makes clear what it is, and the demonstration that this is or is not true of that. And [91a1] of different things there are different demonstrations—unless they are related as a part to the whole (I mean by this that the isosceles has been proved to have two right angles if every triangle has been proved to be so; for one is a part and the other a whole). But these things—that it is and what it is—are not related to one another in [5] this way; for neither is part of the other.
It is evident, therefore, that neither is there demonstration of everything of which there is definition, nor is there definition of everything of which there is demonstration, nor in general is it possible to have both of the same thing. Hence it [10] is clear that definition and demonstration are neither identical nor the one included in the other; for then their underlying subjects would be similarly related.
4 · Now so much for these puzzles; but is there deduction and demonstration of what a thing is, or is there not, as the argument just now supposed?
For deduction proves something of something through the middle term. But [15] what a thing is both is proper to it and is predicated in what it is. And these necessarily convert; for if A is proper to C it is clear that it is also proper to B and this to C; so that all are proper to one another. And if A belongs to every B in what it [20] is, and B is said universally of every C in what it is, necessarily A is said of C in what it is. But if you do not assume them in this double way, it will not be necessary for A to be predicated of C in what it is (if A holds of B in what it is, but of what B is said of B does not hold in what it is). But both these will contain what it is; therefore B [25] too will hold of C in what it is.
Thus if both contain what a thing is and what it is to be it, what it is to be it will be prior in the case of the middle term. And in general if one can prove what a man is, let C be man, and A what man is—whether two-footed animal or something else. If, then, it is deduced, it is necessary for A to be predicated of every B, and there will [30] be an intermediate account other than this,35 so that this too will be what man is. So you assume what you have to prove; for B is what man is.
We must inquire in the case of two propositions and of what is primitive and immediate; for there what we are saying becomes especially evident.
[35] Now those who prove through conversion what soul is, or what man is, or anything else that there is, postulate the point at issue—e.g. if someone were to claim that soul is what is explanatory of its own being alive, and that this is a number that moves itself; for it is necessary to postulate that soul is just what is a [91b1] number that moves itself, in the sense of its being the same thing.
For it is not the case that if A follows B and this C, A will be what it is to be C, but it is true36 to say only A will be C—even if A is just what is some B and is [5] predicated of every B. For what it is to be an animal is predicated of what it is to be a man (for it is true that every case of what it is to be a man is what it is to be an animal, just as every man is an animal), but not in the sense of their being one thing.
If, then, you do not assume in this way, you will not deduce that A is what it is to be C and its substance; and if you do assume in this way, you will already have [10] assumed what is what it is to be C, viz. B. Hence it has not been demonstrated; for you have assumed the point of issue.
5 · But neither does the method of division deduce, as we said in our analysis of the figures.37 For it nowhere becomes necessary for the object to be that if these are the case—just as someone who is giving an induction does not demonstrate. For [15] one must not ask the conclusion, nor must it be the case by being granted; but it is necessary for it to be the case if those are the case, even if the answerer denies it.
Is man an animal or inanimate? If38 he assumed animal, he has not deduced it. Again, every animal is either terrestrial or aquatic: he assumed terrestrial. And that man is the whole—a terrestrial animal—is not necessary from what he has said, but [20] he assumes this too. It makes no difference whether he does this in many steps or in few; for it is the same. (Indeed those who proceed in this way actually make non-deductive use even of what can be deduced.) For what prevents all this from being true of man yet not making clear what a man is or what it is to be a man? [25] Again, what prevents you from positing something additional, or from abstracting something, or from passing over something in its substance?
Now these points are ignored; but it is possible to solve them if one assumes everything in what the thing is, and makes the division consecutive by postulating what is primitive, and leaves nothing out. [This is necessary if everything falls into [30] the division and nothing is omitted; and this is necessary, for it must already by atomic.]39
But nevertheless there is no deduction in it; but it makes us familiar with what the thing is, if at all, in some other fashion. And this is nothing absurd; for neither, presumably, does someone who gives an induction demonstrate, but he nevertheless makes something clear. And someone who states the definition as a
result of the [35] division does not state a deduction. For just as in the case of conclusions without middle terms if someone says that if these are the case it is necessary that this is the case, it is possible to ask why; so too this is possible in the case of divisional definitions. What is man? An animal, mortal, footed, two-footed, wingless. Why (at [92a1] each additional posit)? For he will say, and prove by the division as he thinks, that everything is either mortal or immortal. But a whole argument of this sort is not a definition, so that even if it were demonstrated by the division, that does not make the definition a deduction. [5]
6 · But can one actually demonstrate what a thing is in respect of substance, but do so on a supposition, by assuming that what it is to be something is the property composed from the things in what it is, and that these alone are in what it is, and that the whole is proper to it? For this is what it is to be that thing.
Or do you again assume what it is to be the thing in this case too? For it is [10] necessary to prove it through the middle term.
Again, just as in a deduction you do not assume what being deduced is (for the proposition on which the deduction depends is always whole or part), so too what it is to be something must not be in the deduction, but this must be separate from what is laid down. And if anyone disputes whether something has been deduced or not, we [15] meet him by saying that “that is what a deduction is”; and if anyone says that what it is to be it has not been deduced we can say that “Yes it has; for that is what we supposed what it is to be something is.” Hence it is necessary for something to have been deduced without assuming what deduction is or what it is to be something.
[20] And even if you prove it from a supposition—e.g. if being bad is being divisible, and for things which have a contrary being their contrary is being contrary to what they are,40 and the good is contrary to the bad, and the indivisible to the divisible—therefore being good is being indivisible.
For here too you prove by assuming what it is to be something, and you assume [25] it in order to prove what it is to be it.—Yet something different.—Granted; for in demonstrations too one assumes that this is true of this—but not itself, and not something that has the same account and converts.
And in both cases—if you prove in virtue of a division and if you produce a deduction in this way—there is the same puzzle: why will man be a two-footed [30] terrestrial animal and not animal and terrestrial? For from the assumption there is no necessity for what is predicated to become a unity, but it might be as if the same man were musical and literate.
7 · Well now, how will a definer prove a thing’s substance or what it is?
[35] For neither, as in demonstration, will he make it clear from what is agreed to be the case because necessarily if these are the case something else is (for this is demonstration); nor, as in induction, will he show through the particulars, which are clear, that everything is thus since nothing is otherwise (for in induction you do not [92b1] prove what a thing is, but that either it is or it is not).
Now what other way is left? For you will hardly prove it by perception or by pointing with your finger.
Again, how will you prove what a thing is? For it is necessary for anyone who [5] knows what a man or anything else is to know too that it is (for of that which is not, no one knows what it is—you may know what the account or the name signifies when I say goatstag, but it is impossible to know what a goatstag is). But if you are to prove what it is and that it is, how will you prove them by the same argument? [10] For both the definition and the demonstration make one thing clear; but what a man is and that a man is are different.
Next, we say it is necessary that everything that a thing is should be proved through demonstration, unless it is its substance. But being is not the substance of anything; for what is is not a genus. Therefore there will be a demonstration that it [15] is. And that is what the sciences as a matter of fact do; for the geometer assumes what triangle signifies and proves that it is. So when you define what it is, what will you prove? Triangle?41 Then you will know by definition what it is, but you will not know if it is. But that is impossible.
It is evident too from the present fashions of definition that definers do not [20] prove that a thing is. For if it is in fact what is42 equidistant from the middle, why should what has been defined be? and why is this a circle? For one might say that it was a definition of mountain-copper. For definitions do not in addition make clear either that what is said is possible, or that it is that of which they say they are definitions, but it is always possible to say “Why?” [25]
If, therefore, the definer proves either what a thing is or what its name signifies, then if a definition has nothing at all to do with what a thing is, it will be an account signifying the same as a name. But that is absurd.
For, first, there would be definitions even of non-substances, and of things that are not—for one can signify even things that are not.
Again, all accounts would be definitions; for one could posit a name for any [30] account whatever, so that we would all talk definitions and the Iliad would be a definition.
Again, no demonstration would demonstrate that this name makes this clear; nor then do definitions make this clear in addition.
From this, then, it is evident that definition and deduction are not the same, [35] and that deduction and definition are not of the same thing; and in addition, that definition neither demonstrates nor proves anything, and that you can become aware of what a thing is neither by definition nor by demonstration.
8 · We must inquire again which of these points is correctly argued and [93a1] which not correctly; and what a definition is; and whether there is in some way demonstration and definition of what a thing is, or in no way at all.
Since, as we said, to know what something is and to know the explanation of the fact that it is are the same—the argument for this is that there is some [5] explanation, and this is either the same thing or something else, and if it is something else it is either demonstrable or non-demonstrable—if, then, it is something else and it is possible to demonstrate it, it is necessary for the explanation to be a middle term and to be proved in the first figure; for what is being proved is both universal and affirmative.
Well, one way would be the one just examined—proving what a thing is [10] through another definition. For in the case of what a thing is, it is necessary for the middle term to state what the thing is (and in the case of what is proper it must be proper). Hence you will prove the one but you will not prove the other instance of what it is to be the same object. Now that this way will not be a demonstration was said earlier (but it is a general deduction of what the thing is). [15]
But let us say in what way a demonstration is possible, speaking again from the beginning. Just as we seek the reason why when we grasp the fact—sometimes they actually become clear together, but it is not possible to become familiar with the reason why before the fact—it is clear that similarly we cannot grasp what it is to be something without grasping the fact that it is; for it is impossible to know what a [20] thing is if we are ignorant of whether it is. But as to whether it is, sometimes we grasp this accidentally, and sometimes when grasping something of the object itself—e.g. of thunder, that it is a sort of noise of the clouds; and of eclipse, that it is a sort of privation of light; and of man, that he is a sort of animal; and of soul, that it is something moving itself.
[25] Now in cases in which we know accidentally that a thing is, necessarily we have no hold on what it is; for we do not even know that it is, and to seek what it is without grasping that it is, is to seek nothing. But in the cases in which we grasp something, it is easier. Hence in so far as we grasp that it is, to that extent we also have some hold on what it is.
So in cases in which we grasp something of what the thing is, let it be first like [30] this:—eclipse A, moon C, screening by the earth B. So to ask whether it is eclipsed or not is to seek whether B is or not. And this is no different from seeking whethe
r there is an account of it; and if this is, we say that that is too. (Or: of which of the contradictory pair does the account hold—of its having two right angles or of its not having them?)
[35] When we discover it, we know at the same time the fact and the reason why, if it is through immediates; if not, we know the fact but not the reason why. Moon, C; eclipse, A; not being able to produce a shadow during full moon though there is nothing evident between us, B. Then if B—not being able to produce a shadow [93b1] though there is nothing evident between us—belongs to C, and A—being eclipsed—to this, then it is clear that it is eclipsed but not yet why; and we know that an eclipse is but we do not know what it is.
When it is clear that A belongs to C, then to seek why it belongs is to seek what [5] B is—whether screening or rotation of the moon or extinction. And this is the account of the one extreme, i.e. in this case of A. For an eclipse is a screening by the earth.
What is thunder? Extinction of fire in cloud. Why does it thunder? Because the fire in the cloud is extinguished. Cloud C, thunder A, extinction of fire B. Thus B [10] belongs to C, the cloud (for the fire is extinguished in it); and A, noise, to this; and B is indeed an account of A, the first extreme. And if again there is another middle term for this, it will be from among the remaining accounts.
[15] We have said, then, how what a thing is is grasped and becomes familiar, hence no deduction and no demonstration of what a thing is comes about—yet it is clear through deduction and through demonstration. Hence without a demonstration you cannot become aware of what a thing is (in cases where the explanation is something else), yet there is no demonstration of it (as we said when we went [20] through the puzzles).
9 · Of some things there is something else that is their explanation, of others there is not. Hence it is clear that in some cases what a thing is is immediate and a principle; and here one must suppose, or make apparent in some other way, both [25] that they are and what they are (which the arithmetician does; for he supposes both what the unit is and that it is); but in those cases which have a middle term and for which something else is explanatory of their substance, one can, as we said, make them clear through a demonstration, but not by demonstrating what they are.