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370 The Science of Logic

syllogisms are called ἀναπόδεικτοι in two ways: those which have not been proved,

and those which do not need proof inasmuch as in their case the fact that they lead

to a conclusion is immediately evident. We have often shown that it is the second

meaning of this appellation which holds of the items which Chrysippus lists at the

beginning of his ﬁrst Introduction to Syllogisms.

(M viii 223)¹³

The syllogisms to which Sextus alludes are ‘the celebrated unproveds of the

Stoics’, and Sextus says that they were called ‘ἀναπόδεικτοι’ insofar as they

do not need to be proved. He implies the same thing—without any explicit

reference to the ambiguity of ‘ἀναπόδεικτος’—intheOutlines of Pyrrhonism:

As far as redundancy is concerned, even the celebrated unproveds of the Stoics will

turn out to be invalid—and if they are destroyed, the whole of logic is overturned.

For those are the syllogisms which, they say, need no proof for their own constitution

but are probative of the fact that the other syllogisms conclude.

(PH ii 156)¹⁴

And the remark is conﬁrmed by Diogenes Laertius:

There are also some syllogisms which are unproved insofar as they do not need a

proof. They are given differently by different Stoics—but Chrysippus gives ﬁve. By

means of them every argument is put together.

(vii 79)¹⁵

There need be no doubt that the Stoic unproveds were so called—and so called

by the Stoics themselves—inasmuch as they were deemed not to need proof.

Nevertheless, the sense which Sextus ascribes to the word in its Stoic use is

surprising. Words of the form ‘ἀ-verb-τος’ do not in fact mean ‘not needing

to be verbed’; nor—the present case apart—does any ancient text suggest

that they do. Had some Stoic logician, wittingly or unwittingly, given a new

sense or ascribed an improper sense to the word ‘ἀναπόδεικτος’? Philosophers

do give new senses to old words; and as for the improper option, Chrysippus’

¹³ ἀναπόδεικτοι λέγονται διχῶς, οἵ τε μὴ ἀποδεδειγμένοι καὶ οἱ μὴ χρείαν ἔχοντες

ἀποδείξεως τῷ αὐτόθεν εἶναι περιφανὲς ἐπ᾿ αὐτῶν τὸ ὅτι συνάγουσιν. ἐπεδείξαμεν δὲ

πολλάκις ὡς κατὰ τὸ δεύτερον σημαινόμενον ταύτης ἠξίωνται τῆς προσηγορίας οἱ κατ᾿

ἀρχὴν τῆς πρώτης περὶ συλλογισμῶν εἰσαγωγῆς παρὰ τῷ Χρυσίππῳ τεταγμένοι.

¹⁴ ὅσον γὰρ ἐπὶ τῇ παρολκῇ καὶ οἱ θρυλούμενοι παρὰ τοῖς Στωϊκοῖς ἀναπόδεικτοι ἀσύνακτοι

εὑρεθήσονται, ὧν ἀναιρουμένων ἡ πᾶσα διαλεκτικὴ ἀνατρέπεται· οὗτοι γάρ εἰσιν οὕς φασιν

ἀποδείξεως μὲν μὴ δεῖσθαι πρὸς τὴν ἑαυτῶν σύστασιν, ἀποδεικτικοὺς δὲ ὑπάρχειν τοῦ καὶ

τοὺς ἄλλους συνάγειν λόγους.

¹⁵ εἰσὶ δὲ καὶ ἀναπόδεικτοί τινες τῷ μὴ χρῄζειν ἀποδείξεως, ἄλλοι μὲν παρ᾿ ἄλλοις, παρὰ

δὲ τῷ Χρυσίππῳ πέντε, δι᾿ ὧν πᾶς λόγος πλέκεται.

‘Primary Unproveds’ 371

Greek was notoriously patchy. And yet I wonder if the Stoics did not use

the word in one of its proper and standard senses—namely, in the sense

of ‘unproved’. The ﬁve Chrysippean unproveds are the ﬁve syllogistic forms

which the Chrysippean system does not prove. No doubt it does not prove

them because they do not need to be proved. But it does not follow that, in

calling them ‘ἀναπόδεικτοι’, the Stoics meant that they do not need to be

proved: they may simply have meant that they are not proved.

That is why I translate ‘ἀναπόδεικτος’ by ‘unproved’. If you don’t like

that, then you must opt for ‘not needing proof ’. Usually the word is translated

as ‘unprovable’ or ‘indemonstrable’, and scholars standardly speak of the ﬁve

Chrysippean indemonstrables where I speak of the ﬁve unproveds. Whether or

not the ﬁve items are in fact, or were held to be, unprovable, the standard trans-

lation is demonstrably inaccurate—and that is a pretty good reason to reject it.

What about Aristotle? The word ‘ἀναπόδεικτος’ certainly may, and

certainly sometimes does, mean ‘unprovable’, and that may well be its sense in

the phrase at APst 71b26 which I have so far translated as ‘primary unproveds’.

After all, in that passage Aristotle is out to show that the ﬁrst principles of

a science must be items which cannot be proved; and it is therefore rather

plausible to suppose that the thesis of unprovability is expressed by the word

‘ἀναπόδεικτος’. But it does not follow that when later authors describe

Aristotelian syllogisms as ἀναπόδεικτοι they mean that they are unprovable.

First, when Sextus distinguishes two senses in which ‘ἀναπόδεικτος’is

used in logical contexts, he does not even mention the sense of ‘unprovable’

or ‘indemonstrable’. To be sure, Sextus does not purport to list all the possible

senses of the word. Nonetheless, what he says strongly suggests that he has

given the senses of the word which are pertinent to the discussion of syllogisms.

Secondly, several passages which use ‘ἀναπόδεικτος’ of predicative syllo-

gisms certainly do not use it in the sense of ‘unprovable’. In a passage from

his Introduction to Logic which I have already quoted in a truncated form,

Galen remarks that

there are three ﬁgures among predicative propositions, and in each of them there

are several syllogisms, just as there are among hypotheticals: some are unproved and

primary, some need a proof.

(inst log viii 1)¹⁶

¹⁶ τριῶν οὖν ὄντων σχημάτων ἐν ταῖς κατηγορικαῖς προτάσεσι καθ᾿ ἕκαστον αὐτῶν

γίγνονται συλλογισμοὶ πλέονες ὥσπερ κἀν ταῖς ὑποθετικαῖς, ἔνιοι μὲν ἀναπόδεικτοι καὶ

πρῶτοι, τινές δ᾿ ἀποδείξεως δεόμενοι.

372 The Science of Logic

The last clause shows plainly that ‘ἀναπόδεικτος’ does not mean ‘unprovable’:

the word is contrasted with ‘needing a proof ’, and it therefore means either

‘not needing proof ’ or (as I quixotically prefer) ‘unproved’.

The same implication is found in a text with impeccably Peripatetic

credentials. Near the beginning of his commentary on the Prior Analytics,

Alexander notes that Aristotle will explain, among other things,

which of the syllogisms are perfect and immediately knowable and not needing a

proof, and which are imperfect and not unproved.

(in APr 6.23–25)¹⁷

There is the same contrast as in Galen, so that Alexander too means

‘ἀναπόδεικτος’ not in the sense of ‘unprovable’ but in the sense of ‘unproved’

(or ‘not needing proof ’).

Finally, let me cite a passage from Apuleius’ On Interpretation.Apuleius

is discussing predicative syllogistic. Like most later logicians, he ﬁnds nine

syllogisms in the ﬁrst ﬁgure—the four canonical syllogisms of the Prior

Analytics and ﬁve others which Theophrastus is said to have added:

Now of these nine moods in the ﬁrst ﬁgure, the ﬁrst four are called unproved—not

because they cannot be proved, like the measurement of the whole of the ocean, nor

because they are not proved, like the squaring of the circle, but because they are so

simple and so evident that they do not need a proof.

(int ix [205.21–206.5])¹⁸

This text is generally taken to show that Apuleius recognized three senses for

the Latin term ‘indemonstrabilis’; and it is inferred that his Greek copy-text

reported three senses for the Greek ‘ἀναπόδεικτος’. The second and the third

of Apuleius’ senses correspond to Sextus’ two senses. The ﬁrst is the sense

which we are surprised not to ﬁnd in Sextus, namely ‘unprovable’.

In fact, matters are not quite so simple. The received text of the passage

makes no sense: there is no easy correction, and the version which I have

given (and translated) is anything but certain. The corruption covers at least

the words which I have rendered as ‘like the measurement … or because’, and

it may be more extensive. That is to say, it is not even certain that Apuleius

¹⁷ … καὶ τίνες μὲν τέλειοι τῶν συλλογισμῶν καὶ αὐτόθεν γνώριμοι καὶ οὐ δεόμενοι

ἀποδείξεως, τίνες δὲ ἀτελεῖς καὶ οὐκ ἀναπόδεικτοι.

¹⁸ ex hisce igitur in prima formula modis novem primi quattuor indemonstrabiles nominentur, non

quod demonstrari nequeant, ut universi maris aestimatio, aut quod non demonstrentur, sicut circuli

quadratura, sed quod tam simplices tamque manifesti sint ut demonstratione non egeant.

Are any Predicative Syllogisms Primary? 373

noticed three rather than two uses of the word ‘indemonstrabilis’. But I shall

not insist on that point; and in any event it will rightly be urged that there

probably was a threefold distinction in the text. However, is there a threefold

distinction of senses? Apuleius does not say so: he says that the ﬁrst four

moods of the ﬁrst ﬁgure are called indemonstrabiles because they do not need

to be proved. There is no pressing need to gloss ‘because’ by ‘in the sense that’.

In any event, one thing is plain: Apuleius does not think that ﬁrst ﬁgure

predicative syllogisms are called ‘indemonstrabilis’inthesenseof‘unprovable’.

They are indemonstrabiles insofar as they do not need to be proved—whether

‘indemonstrabilis’ is taken to mean ‘not needing proof ’ or simply ‘unproved’.

So let me return to the question: If an item is unproved, does it follow that

it is primary? Evidently not. To be sure, if something is unprovable (relative

to a given set), then it is primary (relative to that set). But if something is

unproved it does not follow that it is unprovable; and if something does not

need to be proved it does not follow that it is unprovable.

That conclusion is hardly astounding. For Darii and Ferio are unproved

in chapters 4–6 of Book One of the Prior Analytics, and they do not need to

be proved. But they are not unprovable, nor are they primary—for they are

proved in chapter 7.

ARE ANY PREDICATIVE SYLLOGISMS PRIMARY?

Orthodox Peripatetics held that some predicative syllogisms are primary and

so unprovable. Boethus of Sidon, a contemporary of Strabo and a leading

Peripatetic of the time, disagreed. Galen reports that

some of the Peripatetics, among them Boethus, call the syllogisms which are based

on leading assumptions [i.e. certain hypothetical syllogisms] not only unproved but

also primary; but they are not prepared to call primary those unproved syllogisms

which depend on predicative propositions.

(inst log vii 2)¹⁹

The interpretation of the text is controversial—and Galen says nothing

more about Boethus’ view. But he plainly states that, according to Boethus,

¹⁹ καὶ μέντοι καὶ τῶν ἐκ τοῦ Περιπάτου τινὲς ὥσπερ καὶ Βόηθος οὐ μόνον ἀναποδείκτους

ὀνομάζουσι τοὺς ἐκ τῶν ἡγεμονικῶν λημμάτων συλλογισμοὺς ἀλλὰ καὶ πρώτους· ὅσοι δὲ ἐκ

κατηγορικῶν προτάσεών εἰσιν ἀναπόδεικτοι συλλογισμοί, τούτους οὐκ ἔτι πρώτους ὀνομάζειν

συγχωροῦσι.

374 The Science of Logic

no predicative syllogisms are primary: none has an axiomatic, or quasi-

axiomatic, status; each can be proved. Galen also suggests—though he does

not strictly imply—that Boethus recognized certain predicative syllogisms

as unproved and as not being in need of proof. So perhaps Boethus agreed

with Aristotle that Barbara and Celarent are unproved and do not need to be

proved. He disagreed with Aristotle inasmuch as he thought that they can be

proved.

How did he think to prove Barbara and Celarent? It is evident that an

item may be unproved relative to one set of items—and even unprovable

relative to one set of items—while being provable, and proved, relative to

another set. In other words, an axiom, or quasi-axiom, of one system may be a

theorem of another. So when Boethus claimed that Barbara and Celarent can

be proved, perhaps he meant not that they can be proved within predicative

syllogistic or relative to the set of predicative syllogisms but rather that they

can be proved in some larger system of logic and relative to a larger set of

syllogisms. More particularly, perhaps he meant that Barbara and Celarent

can be proved within hypothetical syllogistic?

That is exactly what Galen’s report has been taken to intimate: the primary

syllogisms which Boethus recognized in hypothetical syllogistic will have

served as axioms not only for the other hypothetical syllogisms but also for

Barbara and Celarent—and hence for the whole of predicative syllogistic.

And is it not pleasing to imagine that Boethus had in that way anticipated

Frege? For Frege in effect claimed—and demonstrated—that by means of

the ﬁrst unproved syllogism of Stoic logic he could, with the aid of a handful

of axioms, prove the validity of Barbara and Celarent.

Perhaps Boethus was a Fregean avant la lettre. But I doubt it: nothing

in Galen’s text really suggests as much; and there is another possibility

which the rest of our evidence about Boethus tends to favour. According

to that other possibility, Boethus thought that Barbara and Celarent could

be proved within predicative syllogistic or relative to the set of predicative

syllogisms.

How might that be? Take Celarent ﬁrst; and consider Aristotle’s proof of

Cesare, which runs like this:

(1) B holds of no A premiss

(2) B holds of every C premiss

Therefore (3) A holds of no B 1, conversion

Therefore (4) A holds of no C 2, 3, Celarent

Are any Predicative Syllogisms Primary? 375

There, Cesare is reduced to, or proved on the basis of, Celarent. Now it is

evident that Aristotle’s argument may be run in the opposite direction, not

from Celarent to Cesare but from Cesare to Celarent, thus:

(1) A holds of no B premiss

(2) B holds of every C premiss

Therefore (3) B holds of no A 1, conversion

Therefore (4) A holds of no C 2, 3, Cesare

If the ﬁrst argument proves Cesare on the basis of Celarent, does not the

second prove Celarent on the basis of Cesare? And so is not Celarent a

theorem? Or rather, may we not take Celarent as a theorem and Cesare as an

axiom rather than vice versa?

ThesametrickcannotbeturnedinexactlythesamewayforBarbara—if

only because no other syllogism has a universal afﬁrmative conclusion. But

the trick can be turned in a different way. Here is Aristotle’s proof of the

validity of Baroco:

(1) B holds of every A premiss

(2) B does not hold of some C premiss

(3) A holds of every C hypothesis

Therefore (4) B holds of every C 1, 3, Barbara

Therefore (5) A does not hold of some C 2, 4, impossibility

Baroco is there proved on the basis of Barbara, by reduction to the impossible.

The reduction to the impossible may be run in the opposite direction,

thus:

(1) A holds of every B premiss

(2) B holds of every C premiss

(3) A does not hold of some C hypothesis

Therefore (4) B does not hold of some C 1, 3, Baroco

Therefore (5) A holds of every C 2, 4, impossibility

One reduction takes us from Barbara to Baroco, the other from Baroco to

Barbara. If the ﬁrst argument is a proof of Baroco, is not the second a proof

of Barbara?

That you may argue in both directions in that sort of way is not an

arcane secret of predicative syllogistic: on the contrary, it is something which

Aristotle explicitly recognizes and elaborates. In the second Book of the Prior

Analytics he offers to prove, quite generally, that

376 The Science of Logic

everything which can be concluded directly will also be shown by way of the

impossible, and everything by way of the impossible directly, through the same

terms.

(APr 62b38–40)²⁰

He then proves the point, elaborating in effect a set of reciprocating or

counterpart arguments in which X is shown to be valid on the basis of Y and

Y is shown to be valid on the basis of X. Here is an example:

Again, let it have been shown in the middle ﬁgure that A holds of every B. Then the

hypothesis was that A does not hold of every B, and it was assumed that A of every

C and C of every B. For in that way the impossible will come about. And that is the

ﬁrst ﬁgure: A of every C and C of every B.

(APr 63a25–29)²¹

The passage is done in Aristotle’s best telegraphese; but from it you may

readily extract an argument for the validity of Barbara—the very argument

which I have just set down.

Aristotle argued for Baroco from Barbara and for Barbara from Baroco.

He argued for Cesare from Celarent, and he must have seen that there

was an argument from Celarent to Cesare. Two of those four arguments

he took to be proofs. Two he did not. Why not? There is no explicit

answer to that question in our texts; but the true answer is not far to seek.

According to Aristotle, Barbara and Celarent possess a property which Baroco

and Cesare lack: Barbara and Celarent are perfect, Baroco and Cesare are

not. In order to qualify as an axiom or quasi-axiom, a syllogism must be

perfect. Baroco and Cesare, then, cannot serve as axioms or quasi-axioms

for predicative syllogistic, and there are no proofs from them to Barbara and

Celarent.

But perhaps Boethus did take all four arguments to be proofs? Perhaps he

thought that the argument from Cesare to Celarent proved the validity of

Celarent, just as the argument from Celarent to Cesare proved that validity

of Cesare? Well, he could only have thought so if he took a different line

from Aristotle on perfection. And that is just what he did. According to

Ammonius,

²⁰ ἅπαν δὲ τὸ δεικτικῶς περαινόμενον καὶ διὰ τοῦ ἀδυνάτου δειχθήσεται, καὶ τὸ διὰ τοῦ

ἀδυνάτου δεικτικῶς διὰ τῶν αὐτῶν ὅρων.

²¹ πάλιν ἐν τῷ μέσῳ σχήματι δεδείχθω τὸ Α παντὶ τῷ Β ὑπάρχον. οὐκοῦν ἡ μὲν ὑπόθεσις

ἦν μὴ παντὶ τῷ Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Α παντὶ τῷ Γ καὶ τὸ Γ παντὶ τῷ Β· οὕτω

γὰρ ἔσται τὸ ἀδύνατον. τοῦτο δὲ τὸ πρῶτον σχῆμα, τὸ Α παντὶ τῷ Γ καὶ τὸ Γ παντὶ τῷ Β.

Are any Predicative Syllogisms Primary? 377

Boethus … held the contrary opinion to Aristotle on this matter [i.e. on the matter

of perfection]. His opinion was correct, and he proved that all the syllogisms in the

second and third ﬁgures are perfect. Porphyry and Iamblichus followed him, and so

too did Maximus … But Themistius the paraphrast held the contrary opinion—the

opinion which Aristotle had held. These two—Maximus and Themistius—holding

contrary opinions on the matter and establishing, as they thought, each his own view,

the Emperor Julian acted as arbiter and gave the vote to Maximus and Iamblichus

and Porphyry and Boethus. It seems that Theophrastus, who was a pupil of Aristotle

himself, held the contrary opinion to him on this point.

(in APr 31.13–23)²²

Certain aspects of that report may be doubted—for example, it may be

doubted if Theophrastus really disagreed with his master about perfection.

And in general, the Ammonian commentary on the Analytics is pretty

second-rate. But the gist of its report is conﬁrmed by an earlier text.

The text is an essay by Themistius which offers ‘to answer Maximus on

the reduction of the second and third ﬁgures to the ﬁrst’. The essay, which

survives only in Arabic translation, is in many places difﬁcult to understand.²³

But the central point is unambiguous. For Themistius describes Maximus’

position as follows:

He attempts to prove that the syllogisms in the second and third ﬁgures are perfect

in themselves and do not need to be proved or to be reduced to the ﬁrst ﬁgure.

(ad Max 180)

And more than once he declares that Boethus upheld the same position as

Maximus.

So Boethus held—according to Galen—that no predicative syllogisms are

primary. He also held—according to Ammonius and Themistius—that all

predicative syllogisms, or at least, all the fourteen canonical syllogisms, are per-

fect. Those two theses ﬁt snugly together. There is an argument from Cesare

²² ὁ δὲ Βοηθὸς ... ἐναντίως τῷ ᾿Αριστοτέλει περὶ τούτου ἐδόξασεν, καὶ καλῶς ἐδόξασεν

καὶ ἀπέδειξεν ὅτι πάντες οἱ ἐν δευτέρῳ καὶ τρίτῳ σχήματι τέλειοί εἰσιν. τούτῳ ἠκολούθησεν

Πορφύριος καὶ ᾿Ιάμβλιχος, ἔτι μέντοι καὶ ὁ Μάξιμος ... καὶ Θεμίστιος δὲ ὁ παραφραστὴς

τῆς ἐναντίας ἐγένετο δόξης τῆς καὶ τῷ ᾿Αριστοτέλει δοκούσης. τούτοις οὖν τοῖς δύο, τῷ τε

Μαξίμῳ καὶ τῷ Θεμιστίῳ, ἐναντία περὶ τούτου δοξάζουσιν καὶ κατασκευάζουσιν, ὡς ᾤοντο,

τὸ δοκοῦν αὐτοῖς διῄτησεν αὐτὰ ὁ βασιλεὺς ᾿Ιουλιανός, καὶ δέδωκεν τὴν ψῆφον Μαξίμῳ καὶ

᾿Ιαμβλίχῳ καὶ Πορφυρίῳ καὶ Βοηθῷ. φαίνεται δὲ καὶ Θεόφραστος ὁ ᾿Αριστοτέλους αὐτοῦ

ἀκροατὴς τὴν ἐναντίαν αὐτῷ περὶ τούτου δόξαν ἔχων.

²³ There is a French translation—in parts evidently unreliable—in A. Badawi, La transmission

de la philosophie grecque au monde arabe, Études de philosophie médiévale 56 (Paris, 1987

pp. 180–194.

378 The Science of Logic

to Celarent. Cesare is perfect. Therefore the argument proves the validity of

Celarent. Therefore Celarent is not primary. Similarly for Baroco and Barbara.

It remains to ask why Aristotle thought that perfection was limited to

the four canonical syllogisms of the ﬁrst ﬁgure—and how Boethus came to

disagree. And in order to tackle those questions, it will be necessary to know

what the property of perfection consists in.

But before I turn to that, I must add a caution to what I have just said about

Boethus. According to Ammonius, Boethus ‘proved that all the syllogisms in

the second and third ﬁgures are perfect’. Themistius, however, says this:

I think that Boethus claims that the moods in the second ﬁgure which do not need

proof are three and not four.

(ad Max 186)

Themistius does not say which three; but there will be no doubt that the

odd man out was Baroco—the only syllogism in the second ﬁgure which

cannot be proved by conversion. And although Themistius says nothing on

the subject, who will not guess that if Baroco was the exception in the second

ﬁgure, then there was also an exception in the third ﬁgure—namely, Bocardo?

In itself, the making of such exceptions is unexceptionable. But in the

present context it is unsettling. For if Barbara is to be proved within

predicative syllogistic, then the proof must be based either on Baroco or

else on Bocardo—no other predicative syllogism may serve as an axiom or

quasi-axiom for it. But if neither Baroco nor Bocardo is perfect, then Barbara

cannot be proved within predicative syllogistic.

What is to be inferred? Perhaps Boethus did not tie proof to perfection

in the Aristotelian way? Perhaps he thought that there was some other way

of proving Barbara, a way which did not rely on either Baroco or Bocardo?

Perhaps Ammonius is right and Themistius wrong, so that Boethus held

that all fourteen canonical syllogisms are perfect? I am mildly inclined to the

last of those possibilities. True, Themistius is, on this issue, a more detailed

and a more intelligent reporter than Ammonius. But—if we may trust the

Arabic text—Themistius does not say that Boethus ascribes perfection to

only three of the second ﬁgure syllogisms—he says: ‘I think … ’. In other

words, Themistius is here inferring rather than reporting.

PERFECTION

However that might be, what is perfection? Aristotle speaks of a syllogism’s

being perfect or imperfect, τέλειος or ἀτελής;andhewillsayofanimperfect

Perfection 379

syllogism that it can be perfected (he uses the verbs ‘τελειοῦν’and‘ἐπιτελεῖν’).

The Greek ‘τέλειος’ may mean ‘complete’ rather than ‘perfect’; and some

sentences in the Analytics have suggested that it was in fact completeness

rather than perfection which Aristotle had in mind.

Consider, for example, the following remark which occurs in Aristotle’s

introduction to the moods of the second ﬁgure:

There will be no τέλειος syllogism of any sort in this ﬁgure, but there will be a

potential one both when the terms are universal and when they are not universal.

(APr 27a1–3)²⁴

The remark is repeated, virtually verbatim,forthethirdﬁgure(APr

28a15–17). The contrast between ‘τέλειος’ and ‘potential’ readily suggests

the following notion: a syllogism which is τέλειος is already all there, while

something which is a potential syllogism is the makings of a syllogism but not

yet an actual syllogism. That pile of bricks and lumber next to the concrete-

mixer is a potential house: the thing next to it is a τέλειος house—not, or

not necessarily, a perfect house but a complete or completed house.

The Ammonian commentary on the Prior Analytics has a number of things

to say about perfection or completion, not all of them mutually consistent.

One of the things is this:

From what Aristotle himself says you can see that these syllogisms [i.e. the syllogisms

of the second and third ﬁgures] are not imperfect, as he thinks (for their terms are

perfect)—rather the terms in them are simply confused.

(in APr 33.19–21)²⁵

Cesare, for example, is not imperfect—or rather, it is not incomplete. For

it has a complete set of terms: its terms are perfect—or rather complete.

True, the terms are ‘confused’, so that you need to straighten them out by

conversion; but the terms are all there, and so Cesare lacks nothing.

Plainly, Ammonius—in this passage at least—took perfection to be a

matter of completeness. That is how he came to think that all syllogisms

are perfect or complete; for Cesare is not an incomplete or unﬁnished piece

of argument. And he thought that Aristotle himself should have accepted

²⁴ τέλειος μὲν οὖν οὐκ ἔσται συλλογισμὸς οὐδαμῶς ἐν τούτῳ τῷ σχήματι, δυνατὸς δ᾿ ἔσται

καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων.

²⁵ ὥστε καὶ παρὰ τῶν λεγομένων παρ᾿ αὐτῷ τῷ ᾿Αριστοτέλει λάβοις ἂν ὅτι οὐκ εἰσὶν

οὗτοι οἱ συλλογισμοὶ ἀτελεῖς, ὡς αὐτῷ δοκεῖ (τοὺς γὰρ ὅρους τελείους ἔχουσιν), ἀλλὰ μόνον

συγκεχυμένοι εἰσὶν ἐν αὐτοῖς οἱ ὅροι.