Aristotles aim in this work is to set out the structure of an epistêmê, i.e., a structured body of scientific knowledge, or a science, for short. For Aristotle, the sciences are independent of one another; they are not arranged hierarchically. There is no universal science under which all the special sciences fall. Epistêmê is often (and correctly) translated as knowledge (and epistasthai as know). But we must distinguish it from two other verbs that also can be translated as know: gignoskein and eidenai. See the Glossary in Selections (under knowledge) for more on this distinction.
A science is portrayed as a deductive system. It begins with undemonstrated first principles (archai). Some of these are common axioms, shared by all sciences; some are axioms that are proper to the science in question. Axioms cannot be deduced from anything more basic.
Archai are first principles, things that cannot be otherwise. Aristotle sets out some characteristics of archai at 71b19-25:
Note that Aristotle distinguishes (71b32) two senses in which one thing can be better known than or prior to another. (a) Better known (or prior) in itself (by nature). It is universals, which are further from perception, that are better known or prior in themselves. (b) Better known (or prior) to us. It is particulars, which are closer to perception, that are better known or prior to us.
It is not always clear whether Aristotle has basic propositions (i.e., a class of necessary truths) or primitive concepts in mind by archai. But even if he sometimes thinks of archai as concepts, it is the definitions of those concepts that will ultimately figure in demonstrations. So for our purposes we can think of the archai as basic propositions: axioms. (More about archai and how we know them can be found below.)
3. Scientific Truth and Proof
A science is thus a deductive system. The truths of a science are the theorems of such a system - the propositions that we can deduce from the archai by means of syllogistic reasoning. Scientific proof or demonstration (apodeixis) consists in deriving a conclusion syllogistically from more basic truths (i.e., ones closer to the archai).
4. Explanation and Demonstration
Explanation is a mirror-image of demonstration. To explain why it is the case that p is to deduce p from more basic principles. (Cf. the Hempelian covering law model of explanation: to explain something is to subsume it under a general law. The differences: the Hempelian explanandum is an event, the Aristotelian explanandum is a fact. On the covering law model, to explain an event, E, is to find a law, L, such that from L + the initial conditions, I, one can deduce the occurrence of E.)
5. The Connecting Term Model
Aristotles Connecting Term (CT) model of explanation has the following features:
AaB & BaC
Example: Why are whales (C) warm-blooded (A)? Because: necessarily, mammals (B) are warm-blooded, and necessarily, whales are mammals.7. Some illustrative passages:
8. Apparent problems with the Connecting Term Model
a. Notice the importance of the necessity operator in CT. Without it, it would be easy to think of counter-examples to the CT model. Suppose we see a bus full of Husky basketball players, and we want to explain why they are all tall. Since we have this syllogism:
All the Husky basketball players are on the bus.
Everyone on the bus is tall.
Therefore, all the Husky basketball players are tall.
Obviously, their being on the bus does not explain why they are tall.
b. So we must rule out accidental connections. But Aristotles requirements for CT do this, since he requires that the middle term be necessary (i.e., necessarily connected to both of the extremes).
c. But it is still possible to have necessary connections that are irrelevant. Cf. geometry.
9. Explanations and Necessity
As we have seen above (§5), an explanandum in an Aristotelian science is a necessary truth. That is, we can prove that C belongs to A only if C belongs necessarily to A.
The reason: since the archai are all necessary truths (axioms and definitions), anything we deduce from them will also be a necessary truth. The facts that get explained in an Aristotelian explanation are not contingent facts.
Similarly, as Aristotle points out, the middle term that explains why a predicate attaches to a subject must also attach necessarily to the subject. B explains the fact that C belongs to A only if B belongs necessarily to A.
10. Things that cannot be explained
An important feature of the connecting term model of explanation: it holds that some things cannot be explained (i.e., demonstrated):
Aristotle insists on indemonstrable first principles because he feels that without them, he has only three alternatives:
1) infinitely long explanations
2) circular reasoning
These are all unacceptable. But the argument for skepticism seems powerful. How can you know anything at all? If to know something is to be able to prove it (the “knowledge requires proof” assumption), and to prove it is to be able to deduce it from something that you know (the “proof requires deducibility” assumption), this seems to yield the result that you cant know anything. The argument Aristotle sets out at 72 b 7-14 can be put something like this:
|1. You know something only if you can prove it.||Assumption|
|2. To prove something is to deduce it from something else that you know.||Assumption|
|3. Suppose you know something, viz., that p1.||Assumption|
|4. Then you can prove that p1.||3, 1|
|5. So you can deduce p1 from something else that you know, say p2.||4, 2|
|6. Then you can prove p2, etc.||5, 1|
|7. If there are no first principles, then the proof of p1 is infinitely long.||1–6|
|8. But no proof is infinitely long.||Assumption|
|9. So there is some first principle pn such that
(a) we prove p1 by deducing it from from pn, but (b) we cannot prove pn.
|10. So we do not know that pn.||9, 1|
|11. So we cannot prove p1.||10, 2|
|12. So we do not know that p1.||11, 1|
(12) contradicts (3). So there is no knowledge.
Aristotle accepts assumption (8):
Cf. 72b10: it is impossible to go through an infinite series. Indeed, Aristotle argues at length in I.19-22 that explanations cannot be infinitely long.But there may still be a way out of the argument. One might reject (7) by claiming that it does not follow from (1)–(6). That is, one might hold that there are no first principles but our proof of p1 still comes to an end.
How would this be possible? If the premise pn from which our proof begins is itself something that can be proved. In that case we could know that pn and our alleged non-knowledge of pn would not infect all of its consequences.
But from what would we deduce pn? The only possibility seems to be: something else that we can also prove. That is, we must allow circular reasoning .
So Aristotle addresses this response at 72b25: if circular reasoning is acceptable, then something will have to be prior to and more familiar than itself , which is impossible.
So Aristotles response to the skeptical argument must be to reject one of the other two assumptions:
That is, Aristotle accepts (2) (proof requires deducibility) but rejects (1) (knowledge requires proof ). He thinks that first principles can be known, but not proven. This leaves him with the problem of explaining what this kind of knowledge is; not all knowledge can consist in deducibility from other knowledge.
[The beginnings of foundationalism in epistemology.]
11. Demonstrations and Causes
We know that every demonstration (apodeixis) is syllogism whose premises (and conclusion) are necessary. But not every such syllogism is a demonstration. In order for a syllogism to be a demonstration (apodeixis), the middle term must be explanatory. In Aristotles terms, it must be an aition -- a cause or explanation.
(Cf. 85b22 [not in Selections]: A demonstration is a deduction that reveals (lit., shows, deiktikos) the explanation.)
That is, the middle term must explain the connection between the two terms it connects; it must provide not merely the that but the why. Aristotle sets this out in I.13, an important chapter. (This chapter is not included in Selections. You may read it here.)
78a22: Understanding the fact and the reason why differ. (This is better than the older Mure translation: Knowledge of the fact differs from knowledge of the reasoned fact.)
Aristotle means to distinguish between knowing that something is the case (hoti) from knowing why something is the case (dioti). The idea is that you may be able to establish syllogistically that something is the case without being able to explain why it is the case. Aristotles example makes clear what he means:
Why do the planets not twinkle? Because they are near. Near is a middle term that provides the why it explains why the planets dont twinkle. For Aristotle, this takes the form of a syllogism:
Every planet is near.
Everything near is a non-twinkler.
Therefore, every planet is a non-twinkler.
[Note how Aristotle chooses the values for A, B, and C in such a way that the syllogism is Barbara his favorite syllogism.]
But this is also a valid syllogism:
Every planet is a non-twinkler.
Every non-twinkler is near.
Therefore, every planet is near.
Here, the middle term is non-twinkler. But although the syllogism proves that the planets are near, it does not explain why the planets are near. The reason is that being near is a cause of non-twinkling, whereas non-twinkling is not a cause of being near.
This leaves open the question of what makes a middle term explanatory, which Aristotle does not answer here.
Another idea that Aristotle develops in this chapter is that, to be explanatory, the middle term must be positioned between the other two, and not outside. What does he mean?
His example: you cant explain why a wall doesnt breathe by saying that its not an animal. His reason is that this explanation would work only if you can explain why something does breathe by saying that it is an animal. But clearly you cant do this, since not all animals breathe.
The best you can do is this second figure syllogism (Camestres):
Everything that breathes is an animal AaB No wall is an animal AeW Therefore, no wall breathes \ BeW
This is, as we know, valid; but Aristotle claims it is not an explanation of why walls dont breathe. The reason he gives is that the middle term is in the wrong position. I.e., this is a second figure syllogism, not a (proper) first figure one. Is this just a bias on his part?
No: I take it that the correct explanation why a wall doesnt breathe has to be one that would explain why any non-breather doesnt breathe. And the explanation that Aristotle criticizes could not explain why snails, e.g., dont breathe.
The correct explanation (although Aristotle doesnt say so) might be that walls dont breathe because they dont have lungs. The relevant syllogism would be:
Nothing lungless breathes BeL Every wall is lungless LaW Therefore, no wall breathes \ BeW
Now we have a first figure syllogism (Celarent), with the middle term correctly positioned. And we have an explanation that explains the non-breathing of any non-breather.
Aristotle seems to be insisting that an adequate CT explanation must give both necessary and sufficient conditions.
That is, to explain the fact that C belongs to A, we need a connecting term, B, such that Bs belonging to A is both necessary and sufficient for Cs belonging to A:
[Thus, animal provides only a necessary, but not a sufficient condition, for breathing. Everything that breathes is an animal, but not every animal breathes.]
This leads to the doctrine (often attributed to Aristotle) of the commensurate universal:
The universal, B, that connects the predicate C to the subject A in an explanatory way must be co-extensive with C.
[Note that our example satisfies this condition: the terms lungless and non-breather are co-extensive.]
12. Questions about the CT Model
Here are some important questions about the connecting term model of explanation.
Of these three questions:
Re (2): Aristotles view of a science seems wildly off:
a. Science is empirical; Aristotles version is a priori.
b. Science explains contingent facts; in Aristotelian science, all propositions are necessary.
c. Science is inductive; Aristotelian science is deductive. That is, in real science, we begin with observations and work toward general principles. Aristotelian science seems to go the other way round.
d. Whats more, when Aristotle is actually doing science (e.g., in the Physics, or De Caelo), as opposed to philosophizing about it, he doesnt proceed in the deductive manner that is prescribed in APst.
But these discrepancies can be minimized by the following considerations:
Re (3): When does a middle term give the why (and so yield a demonstrative syllogism) and not merely the that?
The problem Aristotle faces is that not every middle term truly and necessarily interposible between an attribute, C, and a subject, A, explains why C belongs to A.
There may be a sequence of middle terms:
every one of which necessarily links C to A. I.e., for every one of the Bs there is a (valid) syllogism with necessary premises:
But only one of these is, presumably, the explanation of Cs belonging to A. Aristotle seems to want (although he doesnt make this very clear or explicit) to maintain a uniqueness thesis about causes:
And only the unique B that is the cause can explain why C belongs to A the others can only be used to show that it does.
Aristotle makes some suggestions about how to find the right middle, B:
1) It must be the first subject to which C belongs (73b33)
2) Whatever C belongs to, it belongs to qua B (i.e., in so far as it is B).
1) First subject: B is the first subject to which C belongs (alt: C belongs primitively to B) iff C belongs to every B and there is no M (distinct from both B and C) such that necessarily, CaM and MaB. That is, the connection between C and B must be unmediated.
Example: being a 180o figure belongs primitively to triangle, but not to isosceles.
2) Qua: C belongs to x qua B iff C belongs to x and it is only in so far as x is a B that C belongs to x. (I do not know how to put only in so far as into canonical notation.)
Same example: suppose that x is an isosceles triangle. Then being a 180o figure belongs to x qua triangle, but not qua isosceles.
It seems to be a consequence of the uniqueness thesis, with its requirement that the explanatory middle is the term qua which the attribute belongs to the subject, that the explanatory middle and the predicate of the explanandum be convertible.
That is: if B is the explanatory middle linking C to A, then the proposition CaB is convertible. That is, the proposition BaC will also be true. (This is the notorious doctrine of the commensurate universal, for the terms B and C in this case are said to be commensurate universals. Commensurate universals are universals which are necessarily coextensive.)
13. Problems with convertibility
If the statement linking the explanatory middle to the term of which it is predicated in the explanandum is convertible, then it will become difficult to establish which of the two commensurate universals is the cause. For consider the following two syllogisms:
(I) is supposed to establish B as the cause of Cs belonging to A. But if B and C are necessarily co-extensive, it will be hard to show that (II) hasnt just as much right as (I) to be a demonstrative syllogism. That is, how will one show that B is the first subject to which C belongs, rather than the other way around? Why isnt C the first subject to which B belongs? After all, B belongs to every C, and there is no middle term between the two.
One might hope that the qua condition will help, but that is a very obscure notion. The idea would be this: C belongs to x qua B, but not the other way around. It is qua triangle that x is a 180o figure; it is not qua 180o figure that x is a triangle.
But why not? One wants to say that triangle and 180o figure are different properties. But this raises a familiar can of worms: different properties can certainly be co-extensive (if it just happens that they apply to exactly the same instances), but how can they be necessarily co-extensive? It just happens that they necessarily have the same instances?
At any rate, Aristotle cannot avoid convertibility in at least some cases. One cannot hope to find a narrower term than triangle to cite as the cause of somethings being a 180o figure. For all triangles are 180o figures.
And in the most important case of all, convertibility is crucial: that is, the case in which the explanatory proposition is a definition.
Consider the following example (Barnes, Aristotle, p. 34):
Why do cows have horns? Because they are deficient in teeth (the matter that would have gone into teeth goes into horns, instead). Why are they deficient in teeth? Because they have four stomachs (and dont need to chew their food). Why do they have four stomachs? Because they are ruminants (they chew their cud). Why are they ruminants? Because they are cows there is no further feature, apart from their being cows, which explains why cows are ruminants; the cause of a cows being a ruminant is just its being a cow.
That is, at some point, a question of the form Why does C belong to A? will be answered by appealing to a middle term, B, that is the definition of A. (When one then asks, But why does B belong to A? the answer is: Thats what A is. This is what Hintikka (Ingredients . . . p. 59) calls a discussion stopper.)
In our example, ruminant is only part of the definition of a cow; but if one were to give the full definition, it would be convertible with the term its a definition of. Definitions are the first principles at which explanatory chains end; so every explanatory chain is going to feature a commensurate universal.
1. Discussion stoppers
How can Aristotle make definitions play such a crucial role in a science? How can definitions be discussion stoppers? Doesnt this make science a priori? Doesnt it reduce scientific questions to verbal ones? Are substantive scientific questions really, on Aristotles account, questions of meaning?
No, Aristotle has not reduced science to lexicography.
2. Some Texts
So, one cannot do science armed with nothing but a dictionary. Understanding what Aristotle means by definition will help us as we examine his treatment of the topic of definitions in APst. II. Well start with chapters 1 - 2. (These chapters are not included in Selections. You may read them here.)
Aristotle begins APst. II by listing four questions we can ask, corresponding to four kinds of things we can know:
1) the that (hoti)
2) the why (dioti)
3) whether it is (ei esti)
4) what it is (ti esti)
Presumably, these are questions that can be asked about a subject: i.e., a kind, a species something that would show up as a term in a syllogism:
1a) Is it the case that S is P?
2a) Why does P belong to S?
3a) Are there Ss (= does S exist?)
4a) What is S?
Aristotle gives these examples:
1b) Is the sun eclipsed?
2b) Why is the sun eclipsed?
3b) Do centaurs exist? Is there a god?
4b) What is a god? What is man?
Aristotle clearly thinks that our even asking question (2) presupposes that we have an affirmative answer to question (1). Likewise for the second pair: asking (4) presupposes an affirmative answer to (3).
In the case of the first pair [(1) and (2)], this seems obviously correct: there can be no explanation of solar eclipses unless there are solar eclipses; in general, there can be no reason why S is P unless S is P.
But the case of (3) and (4) seems quite different: surely one can ask what some kind is without knowing whether there are any things of that kind. Consider the following questions:
4c) What is electromagnetism?
4d) What is a dodo?
4e) What is a unicorn?
4f) What is self-deception?
Here one might say not only that these questions can be answered without first answering their corresponding type-(3) existence questions, but that the dependence is the other way around. That is, it seems plausible to insist that none of the questions:
3c) Does electromagnetism exist?
3d) Are there dodos?
3e) Do unicorns exist?
3f) Is there such a thing as self-deception?
can even be meaningfully raised unless some prior answer to its corresponding type-(4) what is it question has been provided. [Cf. Socrates on the primacy of the what is it? question.] For example, how can one even try to determine whether there is such a thing as self-deception unless one has been told what it is that he has been asked to determine the existence of?
In a way, this is right but what it shows is that Aristotles what is it? question is not the one that gets presupposed by existence questions. The what is it? question that does get presupposed is a request for a nominal definition, that is, a question that has the force of:
what do you mean by . . . ?
If I suggest to you that you should go out and try to determine whether snarks exist, you have every right to complain that you cannot reasonably be expected even to try to answer unless you have been told what a snark is, i.e., what I mean by snark.
3. Definitions and Essences
Aristotles what is it? question is not a request for a the meaning of a name or word. This is what he says (in a passage not included in Selections):
92b26-32. . . . 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 account whatever, so that we would all talk definitions and the Iliad would be a definition.
Gloss: if a definition involves nothing more than one word abbreviating a longer string of words (the longer string providing the definition of the other), then we could have definitions of things that dont even exist; indeed, any string of words could be considered a definition.
The definitions Aristotle is talking about are not merely formulas that we arbitrarily and stipulatively attach to linguistic expressions (names). His what is it? question is not one of the meaning of an expression in a language but about the nature or essence of something.
A definition, then, is a formula that states the essence of something. (Top. I.5, 101b38) and is given by genus and differentia (Top. VI.4, 141b26). E.g., man is a rational animal.
One term (man) is the definiendum; the other term (rational animal) is the definiens. The definiendum term denotes some natural kind, or species (in this case, humankind): a universal. The definiens denotes the essence of that kind.
An essence is thus the ontological counterpart of the definiens in a definition. The terms in the definiens (the genus and differentia) are in themselves (i.e., without qualification) better known and prior to those in the definiendum, although the definiendum may be better known and prior to us (cf. Top. VI.4, 141a26- b34).
4. Nominal and Real Definitions
[See Robert Bolton, Phil. Rev. 1976, re APst. II.7-10]
But signifies (sêmainei) here does not mean connotes or gives the sense of; rather, it means denotes or refers to. It is a word-thing, not a word-word, relation.
On this interpretation, Aristotle is not contrasting the (nominal) definitions of words with the (real) definitions of things (i.e., natural kinds in the world). Rather, both nominal and real definitions are of things.
The difference between them is this: whereas the definiens of a nominal definition only signifies (i.e., refers to, or picks out) an essence, the definiens of a real definition displays (i.e., reveals, or articulates) the essence:
Some examples from Aristotle:
These correspond (in order) to the three sorts of definition Aristotle describes at 94a10:
The idea is this: the nominal definition of thunder is a certain kind of noise in the clouds. This does not just give us the meaning of the word thunder, which would still leave us with the question whether there is such a thing as thunder. Rather, we begin by observing a certain familiar kind of noise, and we apply the name thunder to that. So thunder exists. The question remains: what is thunder? We cannot answer this question by repeating the nominal definition. For what we want to know is: what is the real nature of that noise we call thunder? For Aristotle, the question amounts to this: what is the explanation of the noise? I.e., the question what is (the real definition of) thunder? amounts to this: what is the cause of x? (where we replace x with the definiens of the nominal definition of thunder).
According to the meteorological theory Aristotle accepts, the noise in the clouds is caused by the extinguishing of fire in the clouds. So that is the real nature of thunder, and the correct answer to both the questions What is thunder? and Why does it thunder?
Now we can see why nominal definitions are the conclusions of demonstrations (cf. 94a13): The major premise of a demonstration will be a real definition of a natural kind, e.g., Thunder is the extinguishing of fire in the clouds. The minor premise will link the definiens of the real definition to the definiens of the nominal definition, e.g., the extinguishing of fire in a cloud is (produces?) a certain kind of noise. The conclusion is the nominal definition of the kind in question, e.g., Thunder is some sort of noise in the clouds.
The upshot is this: we identify the kinds we want to investigate, and whose natures we wish to discover, by their observable characteristics. We then try to discover their essences, i.e., the non-manifest (perhaps unobservable) causes of those observable characteristics. (How we do this is not clear in Aristotles account.) Presumably, an adequacy condition on the real definitions we arrive at as first principles is that they should yield our observations (i.e., our nominal definitions) as deductive consequences.
To put it another (albeit somewhat anachronistic) way: the real definition of a natural kind is the best explanation of the presence of the observable features of that kind. We arrive at a real definition not as the conclusion of a deductive inference (demonstration), but as an inference to the best explanation.
5. Essence and Existence
As the passage at 92b26-32 makes clear, Aristotle makes the following claim about essences:
[Existentialists like to say that their view that existence precedes essence is a reversal of the essentialists notion that essence precedes existence. In passages like this, Aristotle sounds more like an existentialist.]
This means that we can inquire into the nature of tigers (and ask Aristotles question What is a tiger?) because there are tigers whose nature can be studied. What about dodos? There are no dodos (any more) does that mean that there is no such thing as the nature or essence of a dodo, and that we cant raise the question What is a dodo? I think not. There are no longer any dodos the species is extinct but at least there have been dodos. So if we take exists to mean now exists or has existed, we can allow for inquiry into the essence of dodos.
But how can one study the nature of unicorns if there are none and never have been any? Clearly, this is the sort of case Aristotle means to rule out. For the centaur a mythical beast, not an extinct one occurs in Aristotles examples of type-(3) existence questions, but does not turn up in the corresponding type-(4) essence question. One can ask whether there are centaurs, but when one gets a negative answer, one is blocked from going on to ask what a centaur is, in Aristotles favored sense of that question.
The biologist can tell you what a tiger is, or what a dodo is, but he cant tell you what a unicorn is. Cf. this passage (not in Selections):
92b4-8: Anyone who knows what a man or anything else is must 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).
(Cf. Kripke, Naming and Necessity, pp. 24, 156-7.)
So its not as if essences are conceptual roadblocks to scientific investigation. Rather, knowledge of essences is what should emerge from a scientific investigation:
It is the business of a science to discover the essence of the (natural) kinds of things that actually exist.
On this account, Aristotle is not making science an a priori enterprise when he characterizes it as a search for essence, culminating in definitions. Rather, he is making questions about definition and essence depend on facts that are independent of us (as conceptualizers), that is, upon the way things actually are. This is not to make science a priori, but to make essence and definition (in a way) empirical things to be discovered, not stipulated.
6. Objections to Aristotelian essences
According to Aristotle, a things essence belongs to it of necessity, and yet it is the job of a science to discover essences. To this it may well be objected that the way things actually are is a contingent matter. But Aristotle treats the truths of a science as necessary they cannot be otherwise. So how does this square with the account above, according to which the discovery of essence is an empirical matter?
In making the discovery of necessary truths an empirical matter, Aristotle is certainly far from the tradition of logical empiricism, according to which necessity is a matter of convention and stipulation. But it is in just this respect that he is most up-to-date. Recently Kripke has given a brilliant (and basically Aristotelian) account along just these lines. A scientific study may be empirical a posteriori although the truths it discovers may be necessary in the highest degree. (Cf. his Naming and Necessity.) And Putnams ideas about the meaning of natural kind terms (they arent in the head but out in the world, determined not by the qualitative properties we use to identify instances but by the actual characteristics of their stereotypes) are similarly Aristotelian in character.
What Aristotle actually provides us by way of a procedure for discovering essences is terribly abstract. On the one hand, there is the discussion of the discovery of archai in II.19 (and definitions are archai). But much of the earlier part of book II concerns the discovery of definitions. II.13 gives a general characterization of the procedure:
He assumes that what is being defined is a species, S, and that we already know what its genus, G, is. We are to discover a list of attributes, A1, A2, A3, . . . ., each of which
(i) is included within the genus (i.e., belongs exclusively to things in that genus), and
(ii) belongs to every member of S.
The definition of S is the conjunction of such As:
He gives the following example: we give the definition of triplet by stating its genus (number), and listing its attributes: odd, prime (in the sense of not being divisible by 2), and prime (in the less familiar sense of not being the sum of any two numbers).
[Why is 3 prime in this sense? For 3 = 2 + 1. Answer: For Aristotle, 1 is not a number.]
Aristotle concludes: this, then, is precisely what a triplet is: a number that is odd, prime, and prime in this way.
This conjunction of attributes (odd, prime, and prime in this way), is what Aristotle elsewhere calls a differentia. The entire definition (a triplet is a number that is odd, prime, and prime in this way) clearly constitutes a classic Aristotelian definition per genus et differentiam.
The mathematical example makes the method seem non-empirical, but I think that is misleading. Aristotle spends the remainder of the chapter arguing against Platos method of division (an a priori method) as a way of arriving at definitions.
Instead of working our way down from universals (as in Platos method) Aristotle proposes to work our way up from particulars.
1. Features of First Principles
In APst I.2, Aristotle tells about the required characteristics for first principles (archai). First principles must be:
2. Primitive and Immediate
(b) and (c) amount to the same thing:
Primitive: a principle is primitive iff there is no prior principle from which it can be derived.
Immediate: a connection (between terms) is immediate iff there is no middle term that explains their connection.
These are equivalent conditions, given Aristotles conceptions of proof and explanation. (The proposition to be proved is of subject-predicate form; a proof is a deduction from principles; a deduction requires a middle term linking subject to predicate.)
So according to (b) and (c), first principles are non-demonstrable; they cannot be proved.
3. Knowledge of First Principles
But (f) requires that these principles must be knowable, in some sense. Therefore there must be a kind of knowledge that does not consist in demonstration.
Aristotles term for demonstrative knowledge is epistêmê. So we must have a familiarity or acquaintance with or apprehension of first principles that is not epistêmê.
This raises two questions:
Aristotle deals with these two questions in the notoriously difficult final chapter of APst, II.19. He first raises them at 99b18. The first part of II.19 deals with the problem of acquisition; the last part of the chapter (100b5ff) identifies the mental state involved in it.
4. Difficulties in APst II.19
There are a number of difficulties in II.19. They include:
Why doesnt it turn up in II.19, where it might be put to some use? Other references to this:
APst I.2, 71b34-72a5; APr 68b35-7; Phys. A.1; Metaph. Z.3, 1029b3-12; Top. Z.4
5. Answer to question (4a): the coherence of Aristotles account
There is nothing rationalistic about II.19s account of the acquisition of first principles. Our knowledge of first principles is not demonstrative (i.e., it is not deduced from other knowledge), but it is not innate either. Nor is it gained by a direct mental grasping of something that is self-evident. Rather, we move from individual cases, beginning with perception, to a grasp of the universal, by means of a process Aristotle calls epagôgê. This is usually translated as induction, although not every interpreter thinks that what he has in mind is what we understand as inductive inference. Still, the process he describes can be made to fit a standard pattern of inductive inference, although in places the fit is not quite right.
(If there is anything rationalistic involved, it would be in the role of nous, which we will discuss later.)
This is a four-stage process, as Aristotle describes it. (Compare his similar description in Metaph. A.1.)
This is the way, Aristotle says, that universals come to be in the mind. But what are these universals? A and E propositions? Or universal concepts? This brings us to our next question.
6. Answer to question (4b): concepts or propositions?
Aristotle seems to be indifferent to the question of whether he is describing the acquisition of basic concepts or basic propositions. And this is not surprising, for we can look at the matter either way. Heres what his story would look like on the two understandings (following Barnes, pp. 259-60).
a. Propositional knowledge:
This process is repeated for other swans, until
I.e., you grasp that all the swans you have ever seen (i.e., in your experience) have been white.
This is not demonstrative knowledge, of course, since you have not proved that all swans are white.
b. Concept acquisition:
This process is repeated for other swans, until
Aristotle might well have thought that there are no important differences between these two accounts. For the concept of a swan that you grasp, according to (b), just consists in the definition of a swan. In some sense, everything important there is to know about swans is contained in their definition. (If not there explicitly, it is deducible from the definition.) So Aristotle would be unlikely to distinguish sharply between concept acquisition, on the one hand, and propositional knowledge, on the other. To have a concept just is to grasp certain basic propositions.
So sequence (a) can be thought of as one part of sequence (b). That is, sequence (b) can be thought of a collection of sequences of the same type as (a). Having the concept of X is just knowing the various components of the definition of X.
7. Answer to question (4c): what is nous?
The state of mind that grasps these universals Aristotle calls nous. This is often translated intuition; and this leads to a problem. (This is part of the reason why Fine & Irwin translate nous as understanding.)
How can an empirical, inductive process, proceeding from perception of individual cases to knowledge of universals, be carried out by a rationalistic faculty of intuition? This is a long-standing interpretative problem with II.19.
Barnes (Aristotles Posterior Analytics) suggests that there is no incompatibility if we take induction and nous to be answers to two different questions viz., the ones Aristotle starts the chapter with.
i. How do first principles become familiar? I.e., how do we acquire first principles?
Answer: by epagôgê, the (inductive?) process Aristotle describes.
ii. What state (hexis) is it that grasps or apprehends first principles? (It cant be epistêmê, so what is it?)
Trouble arises only if one supposes that nous is the faculty that we use in acquiring knowledge of first principles. According to Barnes, nous is, rather, the state of mind (an alternative to epistêmê) we come to be in when we grasp first principles.
Nous is not the method of acquiring knowledge of first principles; it is the kind of apprehension we get, a kind of understanding that is not demonstrative.
8. Concepts and Definitions
Whether we accept this or not, Aristotles account of concept acquisition, for all its obscurities, fits in well with what he says earlier in APst II about definitions.
Aristotles idea is that essence and definition are a matter of scientific discovery. And among the most important of the archai that will be grasped by nous as the result of induction are definitions.
Aristotle has already argued earlier in Book II (chs. 3-7) that definitions cannot be proved. II.19 now helps us to see how we can acquire them: inductively, by a study of particular cases. We arrive at a definition of (e.g.) tiger by examining individual tigers. This examination leads us (through memory and experience) to a grasp of the definition of tiger; an understanding of what it is to be a tiger.
This knowledge may come late in the process of discovery; but when we come to get out a finished (e.g., biological) science, the definition of tiger would be a first principle - a basis for our explanations of the various traits and features of tigers.
This account of understanding of first principles (as the result of induction) fits well with the idea of interpreting APst as describing the presentation of a completed science, not a scientific methodology.
In the discovery mode, first principles come at the end: they are what we discover late in the process.
In the presentation mode, we state the definitions (first principles) first; for they form the basis of the rest of what we know when we know that science.
9. Grasping Universals
The final statement of how we grasp universals is given at 100a15-b6.
Questions about this passage:
Undifferentiated thing: The usual answer is infimae species, i.e., lowest level universals, such as man, horse, etc. (thus Ross, Fine & Irwin, Barnes). After all, Aristotle goes on to say that this is the first universal in the soul.
But there are problems with this line. (1) The process is supposed to begin with a perception of a particular, so one would expect the first thing in the soul to be an awareness of a particular. (2) Aristotle describes a 2-stage process: (i) abstracting a lowest-level universal from particulars, and (ii) abstracting a higher-level universal from lower-level universals. If these two stages are symmetrical, each should begin at a lower level of abstraction than the one it ends up at. (I.e., just as we abstract animal from man and horse, so we should abstract man from Callias and Socrates.
What happens if we take the undifferentiated things to be particulars? We take Aristotle to be saying this: when we are first aware of a particular, we do not differentiate it from others of its kind. In so doing, we are grasping its universality, not its particularity - we perceive it as a thing of a kind (although we do not have, as yet, a very good grasp of the kind).
That is why Aristotle says that although we perceive particulars, perception is of the universal. We perceive Callias, but when Callias first appears in the soul, he does so as a man, that is, as a thing of a kind.
First universal: We have, in effect, answered our second question. The very first mental grasp of a particular (one of the undifferentiated things) is, ipso facto, a first-level grasp of a universal.
Making a stand: A military stand, as opposed to a rout, is orderly. If all we had was a confused collection of perceptions, we would have no grasp of anything universal. But when a number of perceptions make a stand, they are recognized as being of several things of the same kind. They can then yield a grasp of a universal, which Aristotle describes as a new starting point, i.e., a principle. (The word in Gk., archê, means both things.)
Has no parts and is universal: Presumably, a highest level univeral a category. If so, Aristotle is using this single kind of process to explain how we acquire all universals, from man to animal to substance.
First things: these are presumably the lowest-level universals infimae species.
10. Final comments:
What Aristotle describes is a process that we have a capacity to undergo; it is not a method of discovery. As he says, the souls nature gives it a potentiality to be affected in this way (100a14). The mind is so constituted as to be able to grasp fundamental features of the world; the world is so constituted as to be be graspable in this way by the mind.
Thus, he leaves two crucial questions unanswered: