Barnes Jonathan. Truth, etc

Подождите немного. Документ загружается.

380 The Science of Logic

that conclusion. He was thinking of Aristotle’s deﬁnition of perfection or

completeness:

I call perfect a syllogism which needs nothing else apart from the items assumed in

order for its necessity to be apparent.

(APr 24b22–24)²⁶

Surely ‘nothing else’ means ‘no other term (apart from its constitutive three)’?

And surely you can make the necessity of Cesare apparent—surely Aristotle

does make the necessity of Cesare apparent—without importing any extra

terms?

That was Ammonius’ train of thought; and it is, of course, tempting to

suppose that it was also Boethus’ train of thought. In other words, Boethus

took Aristotle’s own deﬁnition of perfection to show that all the canonical

syllogisms of chapters 4–6 are perfect or complete.

Now it is true that Aristotle’s proofs of the second and third ﬁgure

syllogisms, which proceed by conversion or by reduction to the impossible,

do not introduce any terms which are not already present in their premisses.

A proof may introduce extra terms; and in fact Aristotle’s proofs by ecthesis

do so—but proofs by ecthesis are not needed in non-modal syllogistic.

Nonetheless, Ammonius has misunderstood Aristotle.

For when Aristotle says ‘nothing else’ he does not mean ‘no other term’.

Here is the text in a fuller form:

I call perfect a syllogism which needs nothing else apart from the items assumed in

order for its necessity to be apparent, imperfect one which needs one or more items,

which are necessary by way of the existing terms but which are not assumed in the

premisses.

(APr 24b22–26)²⁷

The ‘one or more items’ which an imperfect syllogism needs cannot be terms;

for a term cannot be ‘necessary by way of the existing terms’. What, then,

are they? What Aristotle has in mind emerges more clearly from a later

passage:

²⁶ τέλειον μὲν οὖν καλῶ συλλογισμὸν τὸν μηδενὸς ἄλλου προσδεόμενον παρὰ τὰ εἰλημμένα

πρὸς τὸ φανῆναι τὸ ἀναγκαῖον.

²⁷ τέλειον μὲν οὖν καλῶ συλλογισμὸν τὸν μηδενὸς ἄλλου προσδεόμενον παρὰ τὰ εἰλημμένα

πρὸς τὸ φανῆναι τὸ ἀναγκαῖον, ἀτελῆ δὲ τὸν προσδεόμενον ἢ ἑνὸς ἢ πλειόνων, ἃ ἔστι μὲν

ἀναγκαῖα διὰ τῶν ὑποκειμένων ὅρων, οὐ μὴν εἴληπται διὰ προτάσεων.

Perfection 381

It is clear too that all the syllogisms in this ﬁgure [i.e. in the second ﬁgure] are

imperfect; for they are all perfected when certain items are introduced—items which

either inhere in the terms from necessity or are laid down as hypotheses, i.e. when

we show something through the impossible.

(APr 28a4–7)²⁸

The ‘certain items’ which are introduced for the perfecting of second ﬁgure

syllogisms are, of course, the items which those syllogisms need in order for

their necessity to be apparent. The items are of two sorts: either they ‘inhere

in the terms’ or else they are ‘laid down as hypotheses’. The disjunction refers

to the two methods of proof which Aristotle in fact deploys. The second

disjunct explicitly refers to reduction to the impossible, in which Aristotle

starts by hypothesizing that the contradictory of the conclusion holds. The

ﬁrst disjunct, less evidently, refers to proofs by conversion. Neither method

of proof introduces—or can introduce—any term which is not already in

the premisses of the syllogism. So the extra items which imperfect syllogisms

need are not terms.

What exactly are they? They are sometimes characterized as operations—

the operations of converting and of hypothesizing. But the text suggests rather

that the items are propositions. Certainly, hypotheses are propositions and

not operations; and it is easier to think of a proposition than of an operation

as being inherent in certain terms. But the question is not very important:

what counts is that the items are not terms.

Thus Ammonius—or rather, the sentence from the Ammonian comment-

ary which I quoted—is wrong about completeness or perfection. Whether

Boethus made the same error is a question which is better postponed. For the

moment, it is enough to state that the best reason for thinking that ‘τέλειος’

means ‘complete’ turns out to be a bad reason: it seems as though ‘perfect’

and ‘imperfect’ rather than ‘complete’ and ‘incomplete’ are the words we

want.

But what about Aristotle’s claim that the items in the second and third

ﬁgures are potential syllogisms? When Aristotle says that

there will be no perfect syllogism of any sort in this ﬁgure, but there will be a potential

one when … ,

²⁸ δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς εἰσὶν οἱ ἐν τούτῳ τῷ σχήματι συλλογισμοί· πάντες γὰρ

ἐπιτελοῦνται προσλαμβανομένων τινῶν, ἃ ἢ ἐνυπάρχει τοῖς ὅροις ἐξ ἀνάγκης ἢ τίθενται ὡς

ὑποθέσεις, οἷον ὅταν διὰ τοῦ ἀδυνάτου δεικνύωμεν.

382 The Science of Logic

modern scholarship takes ‘potential one’ to be elliptical for ‘potential perfect

syllogism’ rather than for ‘potential syllogism’. The text does not mean that

there are potential but not actual syllogisms in the second ﬁgure: it means

that there are potentially perfect but not actually perfect syllogisms there.

That is hardly what the Greek suggests; but perhaps it is right.

If it is right, it is certainly odd—it is rather as if you were to describe

a theorem as a potential axiom. But that particular oddity is in any event

unavoidable. For Aristotle says that

every syllogism is perfected by way of the ﬁrst ﬁgure and reduced to the universal

syllogisms in that ﬁgure.

(APr 41b3–5)²⁹

Cesare, for example, is imperfect; but by means of Celarent it is perfected.

Now the verb ‘to perfect’ means ‘to make perfect’, so that when Cesare is

perfected, it becomes perfect—it becomes a perfect syllogism. So Cesare is,

now, perfect. Indeed, all predicative syllogisms are perfect—and have been

for a couple of millennia: Aristotle perfected them for us long, long ago.

Perhaps most of them were imperfect before Aristotle got to them. But when,

in 360 ad, the Emperor Julian decided for Boethus, he was right.

Of course, that is not what Aristotle means at all. If a syllogism is once

perfect, then it is always perfect; and if it is once imperfect, then it is always

imperfect. Aristotle’s use of the verb ‘to perfect’ is, at best, misleading. To

perfectasyllogismisnottomakeitperfect:itistoproveitonthebasisofa

perfect syllogism.

A syllogism is perfect if and only if its necessity is directly apparent—if and

only if its validity is evident. Perfect predicative syllogisms were later described

as unproved inasmuch as they do not need to be proved; and when Alexander

refers to syllogisms which are ‘perfect and immediately knowable and not

needing a proof ’ (in APr 6.23–25), he takes those three characterizations to

be distinct from but equivalent to one another. The Stoics did not use the

word ‘perfect’ in order to characterize their own unproved syllogisms. But

they might as well have done so; for according to Sextus, the Stoic unproveds

do not need to be proved ‘inasmuch as in their case the fact that they lead

to a conclusion is immediately evident’ (M viii 223). The unproveds are

unproved inasmuch as their validity is evident—inasmuch as they are, in the

Peripatetic jargon, perfect.

²⁹ … ἅπας τε συλλογισμὸς ἐπιτελεῖται διὰ τοῦ πρώτου σχήματος καὶ ἀνάγεται εἰς τοὺς ἐν

τούτῳ καθόλου συλλογισμούς.

Perfection 383

The unproved syllogisms do not need to be proved. There are no naked

needs: needs are needs for something or other. If you need something, then

you need it in order to do so-and-so; and if you don’t need something,

then you can do so-and-so without it. Imperfect syllogisms need something

which perfect syllogisms don’t need: they need and don’t need it for what

end or purpose? A syllogism which does not need a proof does not need one

inasmuch as it is immediately knowable. A syllogism which needs a proof,

then, presumably needs to be proved in order to be known—that is to say,

in order for its validity to be known.

Thus according to Aristotle and the orthodox Peripatetics, if you are to

know that Cesare (say) is valid, then you must have a proof that Cesare is

valid; but you may know that Celarent (say) is valid without having any proof

that Celarent is valid. Celarent, in other words, has the property of perfection,

a property which Cesare lacks. And it is a good thing that Celarent does have

that property. For—still according to the Peripatetic orthodoxy—Celarent

is primary and cannot be proved. So that were Celarent not also perfect, then

its validity could never be known.

There is a complication to the story which I shall mention only to set

aside. In his account of hypothetical syllogistic Sextus introduces a distinction

among unproved syllogisms to which I have already alluded:

Again, you should know that of the unproveds some are simple and some non-simple.

Simple are those for which it is immediately clear that they conclude, i.e. that the

conclusion is co-introduced with the assumptions. … Non-simple are those which

are constructed from the simple and which need analysis into them in order that it

may be recognized that they too conclude.

(M viii 228–229)³⁰

The distinction between simple and non-simple unproveds is found in no

other passage; and some scholars have supposed that Sextus has made a

careless mistake—when he says ‘of the unproved, some are simple … ’, he

should have said ‘of syllogisms, some are simple … ’.³¹

Perhaps there is no more than a Sextan error on display. But if

Sextus is reporting aright, then the Stoics—some Stoics—distinguished

³⁰ ἔτι χρὴ γινώσκειν ὅτι τῶν ἀναποδείκτων οἱ μέν εἰσιν ἁπλοῖ, οἱ δὲ οὐχ ἁπλοῖ. ὧν ἁπλοῖ

μέν εἰσιν οἱ αὐτόθεν σαφὲς ἔχοντες τὸ ὅτι συνάγουσιν, τουτέστι τὸ ὅτι συνεισάγεται αὐτῶν

τοῖς λήμμασιν ἡ ἐπιφορά. ... οὐχ ἁπλοῖ δέ εἰσιν οἱ ἐκ τῶν ἁπλῶν πεπλεγμένοι καὶ ἔτι χρείαν

ἔχοντες τῆς εἰς ἐκείνους ἀναλύσεως ἵνα γνωσθῶσιν ὅτι καὶ αὐτοὶ συνάγουσιν.

³¹ This suggestion comes from Susanne Bobzien.

384 The Science of Logic

between simple unproveds, which are immediately knowable, and non-

simple unproveds, which need something in order for their validity to be

grasped. They need not to be proved but to be analysed.

What is an analysis? Sometimes the term ‘analysis’ serves to designate a

proof;but—unlessSextusisinerror—itcannotdosohere—forinthat

case some unproved items would need proof. So what is an analysis? Various

guesses have been made. For example, a non-simple unproved might be

an unproved the logically pertinent parts of which are themselves complex

assertibles. A little later—it is a passage which I have already quoted in a

different context—Sextus sets down the following mode:

If the ﬁrst and the second, the third

But not the third

Thereforenottheﬁrstandthesecond

He says that it is a mode of the second unproved. (See M viii 236.) Perhaps

it is non-simple, and perhaps you need to analyse it by grasping the structure

of its complex premiss? Or again, a non-simple unproved might be a set or

sequence of simple unproveds, the conclusion of each member of the set

(except the ﬁnal member) being a premiss for one of its successors. Such

a compound syllogism would not need to be proved; but it would need

to be articulated or to have its simple components marked off one from

the next.

However that may be, it is plain enough—and undisputed—that when a

syllogism needs a proof, then it needs it in order that its validity be known

or recognized. And when a simple syllogism does not need a proof, that is

because its validity is immediately knowable.

EVIDENCE

Euclid mastered geometry, and he possessed knowledge of many of the truths

of that science. He grasped the truth of the theorems of geometry insofar

as he proved them on the basis of the axioms of geometry. And he grasped

the axioms of geometry … in some other way. So it is for any Aristotelian

science. So, mutatis mutandis, is it for the science or quasi-science of logic.

In predicative syllogistic, you know that the imperfect syllogisms are valid

insofar as you prove them to be valid on the basis of the perfect syllogisms.

And you know that the perfect syllogisms are valid … in some other way.

Evidence 385

That is familiar, and no doubt it is even true. But it is scarcely illuminating

until we are told something about the ‘other way’. The point is entirely

general and applies to the axioms of any science: the axioms are known; they

are not known on the basis of a proof; and so they are known in some other

way. But in what other way? Or perhaps rather: in what other ways? For

there is no reason to think that there is only one other way, or that all the

axioms of all the sciences are grasped in the same fashion—indeed, it must

seem initially plausible to imagine that the axioms of the science of botany

are grasped in a different way from the axioms of the science of arithmetic.

So in what other way (or ways) are the axioms or quasi-axioms of logic

grasped? The Stoic logicians must surely have addressed the question; but

there is virtually nothing about it in the surviving texts. Aristotle addressed

the question; but he did so unostentatiously, and his followers devoted less

time to the matter than they might have done. Galen too may be deemed to

have addressed the question, even if he did so obliquely. But Galen’s answer

is controversial—and it is also bedevilled by textual uncertainties. I shall start

with Aristotle.

Aristotle says that the necessity, or the validity, of perfect syllogisms is

apparent or evident. Later philosophers will say the same thing about axioms

in general: they are known inasmuch as they are plain or clear or patent or

evident; or inasmuch as they are self-evident, or (to stay closer to the Greek

phrase) evident from themselves. Equivalently, axioms and axiom-like items

are said to be known from themselves, or known immediately or directly—so

that perfect predicative syllogisms, as Alexander put it, are ones which are

‘immediately knowable’ or ‘knowable from themselves’ (in APr 6.24). And

again, such items are often said to be self-warranting or self-justifying: in

Galen’s words:

none of the other syllogisms is unproved, nor warranted from itself.

(inst log viii 3)³²

There is an extensive lexicon. But—at least at ﬁrst glance—it is an unsatis-

factory lexicon.

First, appeals to evidence, or to self-evidence, seem strangely vacuous. ‘The

answer’s a lemon.’—‘How do you know that?’—‘Why, it’s self-evident.’

That hardly explains how I know that the answer’s a lemon: it merely repeats,

with a mallet, that I do know. Nor does the answer ‘By intuition’ fare

³² τῶν δὲ ἄλλων οὐκέτ᾿ οὐδεὶς ἀναπόδεικτός ἐστιν οὐδ᾿ ἐξ ἑαυτοῦ πιστός.

386 The Science of Logic

any better. True, Aristotle’s translators make him speak of intuitions; true,

some modern philosophers invoke our moral intuitions, and some modern

philosophers appeal to our linguistic intuitions. But suppose I say: ‘I intuit

that the answer’s a lemon’: what on earth can I mean? Either I invoke an

imaginary faculty or capacity of the soul, or else I say in a fancy way ‘I

know—but not by way of a proof’.

Secondly, evidence or self-evidence is surely a relative and a subjective

matter: what is evident to you may be obscure to me, and vice versa; the

validity of Barbara might be obscure to a tiro and the validity of Bocardo

evident to an expert. How can the science of syllogistic be constructed on

such subjective sand? (A similar question may, of course, be asked of any

science.)

Thirdly, if Barbara—say—is evident, then on what does its evidence rest?

What feature of Barbara ensures that it is evident—or at least evident to

some people—whereas Baroco is not evident at all? To appeal to self-evidence

without even hinting at the locus or the basis of the evidence is to support

the science of syllogistic by magic and incantation.

Commentators on Aristotle’s syllogistic have discussed such issues at length;

and they have proposed various interpretations of evidence or self-evidence.

Here I restrict my attention to Aristotle’s own proposal.

THE DICTUM DE OMNI ET NULLO

In order to grasp what that proposal is, we must start from a passage at the

beginning of the Prior Analytics. The opening sentence of the work announces

a programme. (The ﬁrst item on the programme is dealt with at once and in

half a sentence; and the programme itself—like the programme which intro-

duces the de Interpretatione —covers not the whole work but only its preface.)

First we must say what our inquiry is about and with what it is concerned—it is

about proof and concerned with probative knowledge. Next, we must deﬁne what a

proposition is, what a term is, what a syllogism is, what sort of syllogism is perfect

andwhatimperfect;afterthat,whatitisforthistobe(ornotbe)inthatasina

whole, and what we mean by being predicated of every or of none.

(APr 24a10–15)³³

³³ πρῶτον εἰπεῖν περὶ τί καὶ τίνος ἐστὶν ἡ σκέψις, ὅτι περὶ ἀπόδειξιν καὶ ἐπιστήμης

ἀποδεικτικῆς· εἶτα διορίσαι τί ἐστι πρότασις καὶ τί ὅρος καὶ τί συλλογισμός, καὶ ποῖος τέλειος

The dictum de omni et nullo 387

Aristotle addresses the last two items on the programme in the following

passage:

For one thing to be in another as in a whole and for the other to be predicated of

all of the one is the same thing. We say that it is predicated of every when you can’t

take anything of which the other will not be said. And in the same way for of none.

(APr 24b25–30)³⁴

The elucidations of ‘of every’ and ‘of none’ which Aristotle there offers are

traditionally known as the dictum de omni et nullo.Thedictum is not lavish

with its words—and the de nullo part is a mere gesture.

Aristotle speaks of one item’s being in another item as in a whole and of

one item’s being predicated of all of another item. Should we take the word

‘another’ seriously? If we do, then Aristotle says nothing about what it is, or

might be, for something to be in itself as in a whole and nothing about what it

is, or might be, for something to be predicated of all of itself. Similar questions

arise in other Aristotelian contexts, and they sometimes have their importance.

Consider, for example, Aristotle’s explanation of relations or relativity in

the Categories.Hesaysthat

items are said to be relative to something when they are said to be what they are of

other items or in relation to some other item.

(Cat 6a36–37)³⁵

There we ﬁnd the neutral phrase ‘relative to something’, followed by references

to ‘other items’ or ‘some other item’. If we insist on the introduction of the

word ‘other’ in the second clause, and suppose that it corrects the neutral

‘something’, then we shall conclude that Aristotle formally excludes the

possibility that an item might be related to itself. Admirers are relational

items—provided that they aren’t self-admirers.

Now it is true that Aristotle does not discuss such things—he never

explicitly wonders whether an item may stand in a given relation to itself; and

it is also true that some later philosophers made a meal of the matter. (‘I was

καὶ ποῖος ἀτελής, μετὰ δέ ταῦτα τί τὸ ἐν ὅλῳ εἶναι ἢ μὴ εἶναι τόδε τῷδε, καὶ τί λέγομεν τὸ

κατὰ παντὸς ἢ μηδενὸς κατηγορεῖσθαι.

³⁴ τὸ δὲ ἐν ὅλῳ εἶναι ἕτερον ἑτέρῳ καὶ τὸ κατὰ παντὸς κατηγορεῖσθαι θατέρου θάτερον

ταὐτόν ἐστιν. λέγομεν δὲ τὸ κατὰ παντὸς κατηγορεῖσθαι ὅταν μηδὲν ᾖ λαβεῖν καθ᾿ οὗ θάτερον

οὐ λεχθήσεται· καὶ τὸ κατὰ μηδενὸς ὡσαύτως.—After ‘λαβεῖν’ the archetype of our MSS had

‘τοῦ ὑποκειμένου’: Alexander did not read the formula, which is an evident gloss.

³⁵ πρός τι δὲ τὰ τοιαῦτα λέγεται ὅσα αὐτὰ ἅπερ ἐστὶν ἑτέρων εἶναι λέγεται ἢ ὁπωσοῦν

ἄλλως πρὸς ἕτερον.

388 The Science of Logic

thinking of myself.’—‘How can that possibly be? Surely there must be one

item which is doing the thinking and another which is being thought of?’)

But I cannot think that Aristotle intended to exclude self-relativity—and

excluded it tacitly and without a word of explanation.

The word ‘other’—or some equivalent word—is often inserted where

strictly speaking it is out of place. When Aristotle explains afﬁrmations and

negations in the de Interpretatione what he actually says is this:

An afﬁrmation is an assertion of something of something, a negation is an assertion

of something from something.

(Int 17a25–26)³⁶

Very often, allusions to and paraphrases of that deﬁnition will say that an

afﬁrmation says one thing of another or predicates one item of another. The

‘another’ is not warranted by Aristotle’s deﬁnition. But it is natural to insert

it; for ‘something of something’ is stilted and ‘one thing of another’ is normal.

The insertion is done without malice and without meaning—and usually it

is not even noticed.

So, I suppose, it is with the word ‘other’ in the passage on relative items;

so too with dictum de omni et nullo: Aristotle means to be explaining what it

is for something to be (or not to be) in something as in a whole; he does not

mean to suggest that ‘x is in y as in a whole’ might mean one thing when x and

y are different and another thing (or nothing at all) when they are identical.

To be in something as in a whole is the same as to be wholly subjected to it;

or, more clearly, B is in A as in a whole if and only if A is predicated of every

B. The terminology of wholes and parts which Aristotle uses sporadically

throughout the Analytics is not without its interest; and in some contexts

it may be not without its signiﬁcance. But in the context of the dictum it

demands no special scrutiny, and I shall say nothing about it.

The dictum has two parts, each of which is itself a dictum. First, then, the

dictum de omni:

We say that it is predicated of every when you can’t take anything of which

the other will not be said.

Aristotle’s formulation is ungainly—at least in translation; and what he

means will be clearer if we replace his pronouns with a couple of Aristotelian

letters, thus:

³⁶ κατάφασις δέ ἐστιν ἀπόφανσις τινὸς κατὰ τινός, ἀπόφασις δέ ἐστιν ἀπόφανσις τινὸς

ἀπὸ τινός.

The dictum de omni et nullo 389

We say that A is predicated of every B when you can’t take any B of which

A will not be said.

That formulation may be simpliﬁed in four ways. First, and uncontroversially,

the verb ‘say’ in ‘will not be said’ is a synonym for ‘predicate’, so that ‘will

not be said’ may be replaced by ‘will not be predicated’. Secondly, and

equally uncontroversially, the future tense in ‘will not be said’ is an inferential

future—anidiomwhichislesscommoninEnglishthaninGreek,andwhich

in the context of the dictum is naturally replaced by an English present tense.

Thus ‘ … is not predicated’ may replace ‘ … will not be said’.

The third point concerns the word ‘can’t’ and is perhaps disputable. ‘Can’t’

is a modal word. Nonetheless, I think that the modal element, which is very

lightly indicated in the Greek, is not to be taken seriously: ‘you can’t take’

means no more than ‘you won’t take’. The reason for so construing the text

is this: were the modal element taken seriously, then Aristotle would not be

able to distinguish, as he does distinguish, between ‘A is predicated of every

B’ and ‘A must be predicated of every B’.

Finally, and again not wholly uncontentiously, it seems to me that what

counts for the dictum is not whether you do or don’t take a B of which A

isn’t said but whether there is or isn’t such a B to be taken. Were we to take

the taking seriously, then we should have to allow that A might be predicated

of every B even when there were some Bs of which A didn’t hold—provided

that, for some reason or other, we did not or would not or could not take

those particular Bs. And that is not on the Aristotelian cards.

As a result of those four little pedantries, the dictum de omni can be

rephrased in the following fashion. What we say is that

A is predicated of every B if and only if there isn’t any B of which A is not

predicated.

Alexander surely so understood the dictum. This is his paraphrase:

Since every predicative proposition is composed of a subject term and a predicate, the

predicated term is truly said of all the subject when you cannot take any of the subject

of which the predicate will not be said. For example, animal of every man—for it is

not possible to take a man of whom animal will not be said.

(in APr 24.30–25.2)³⁷

³⁷ ἐπεὶ γὰρ πᾶσα πρότασις κατηγορικὴ ἐξ ὑποκειμένου ὅρου ἐστὶ καὶ κατηγορουμένου,

τότε λέγεται ὁ κατηγορούμενος κατὰ παντὸς τοῦ ὑποκειμένου ἀληθῶς ὅταν μηδὲν ᾖ λαβεῖν

τοῦ ὑποκειμένου καθ᾿ οὗ οὐ ῥηθήσεται τὸ κατηγορούμενον· οἷον τὸ ζῷον κατὰ παντὸς

ἀνθρώπου—οὐδένα γὰρ λαβεῖν ἔστιν ἄνθρωπον καθ᾿ οὗ τὸ ζῷον οὐ ῥηθήσεται.