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330 Forms of Argument

a point to using numerals rather than letters. Letters already had a r

ole in

Peripatetic logic where they represented terms; and a Stoic might have chosen

numerals in order to mark the fact that his symbols differ syntactically from

Peripatetic symbols. (Thus a modern logician will generally use the letters

‘P’, ‘Q’, ‘R’, … for sentences and ‘F’, ‘G’, ‘H’, … for predicates.) He might

have done—but I doubt if he did. Rather, the use of ordinal numerals to

express modes is explained by the link between modes and argumodes.

‘A HOLDS OF EVERY B, ...’

Apuleius, who does not use schemata at all in his presentation of predicative

syllogistic, nonetheless passes a remark on the Peripatetic use of schematic

letters:

Thus in the Peripatetic fashion, by the use of letters, the ﬁrst unproved [i.e.

Barbara]—with the order of its premisses and their parts inverted but their force

unchanged—is this:

A of every B, and B of every C: therefore A of every C.

They begin with the predicate, and therefore with the second premiss. This mood,

when it is put together in what they take to be the reverse way, is this:

Every C is B; every B is A: therefore every C is A.

(int xiii [212.4–10])⁵⁵

There is a comparable text in Alexander:

Aristotle uses ‘of every’ and ‘of no’ in his exposition because the validity of the

arguments is recognizable by way of these formulas, and because the predicate and

the subject are more recognizable when things are stated in this way, and because ‘of

every’ is prior by nature to ‘in as in a whole’ (as I have already said). But syllogistic

usage is normally the other way about: not ‘Virtue is said of every justice’ but the

other way about—‘Every justice is virtue’. That is why we should exercise ourselves

in both types of utterance, so that we can follow both usage and Aristotle’s exposition.

(in APr 54.21–29)⁵⁶

⁵⁵ ut etiam Peripateticorum more per litteras ordine propositionum et partium commutato sed vi

manente sit primus indemonstrabilis: A de omni B, et B de omni C, igitur A de omni C. incipiunt a

declarante atque ideo et a secunda propositione. hic adeo modus secundum hos pertextus retro talis est:

omne C B, omne B A, omne igitur C A.

⁵⁶ χρῆται δὲ τῷ κατὰ παντὸς καὶ τῷ κατὰ μηδενὸς ἐν τῇ διδασκαλίᾳ ὅτι διὰ τούτων

γνώριμος ἡ συναγωγὴ τῶν λόγων, καὶ ὅτι οὕτως λεγομένων γνωριμώτερος ὅ τε κατηγορού-

μενος καὶ ὁ ὑποκείμενος, καὶ ὅτι πρῶτον τῇ φύσει τὸ κατὰ παντὸς τοῦ ἐν ὅλῳ αὐτῷ, ὡς

προείρηται. ἡ μέντοι χρῆσις ἡ συλλογιστικὴ ἐν τῇ συνηθείᾳ ἀνάπαλιν ἔχει· οὐ γὰρ ἡ ἀρετὴ

‘A Holds of Every B, ...’ 331

In other words, there are—at least—two ways of representing universal

afﬁrmative propositions of the sort which feature in predicative syllogisms.

You may say something of the form

AissaidofeveryB

(or ‘A holds of every B’, or ‘A is predicated of every B’, or simply ‘A of every

B’); and you may say

Every B is A.

Similarly for the three other styles of Aristotelian predication.

Apuleius indicates that the Peripatetics somehow prefer the former mode

of expression, regarding the latter mode as inverted. Alexander remarks—no

doubt more accurately—that the former mode is preferred by Aristotle in his

exposition of the syllogistic, whereas the latter mode is normal syllogistic usage.

He means that everyone—himself and Aristotle included—will standardly

use the second mode of expression (or, I suppose, something more or less like

it) in the presentation of actual syllogisms.

Why the double usage? Alexander recognizes that there is something to

explain. The forms of expression which we are said by Alexander to use in

our syllogistic practice—items of the form ‘Every A is B’, and the like—are

perfectly ordinary Greek. (True, Alexander’s example, ‘Every justice is virtue’,

is not exactly household Greek … ) On the other hand, sentences of the form

‘A holds of B’ are rare and abnormal, and sentences of the form ‘A is predicated

of B’ are—as Porphyry says—an Aristotelian invention. So the question for

Alexander is this: Why did Aristotle, in his exposition of syllogistic theory,

use ‘A holds of B’ or ‘A is predicated of B’ rather than the quotidian

‘B is A’?

Alexander provides three answers to his question. Presumably they are

collaborative rather than competitive. The ﬁrst is pretty dubious, and the

third is pretty ethereal; but the second has some force:

The predicate and the subject are more recognizable when things are stated this way.

At least this much is true: if I say something of the form

AispredicatedofB

I thereby explicitly mark ‘A’ as a predicate and I implicitly mark ‘B’ as a

subject. So the idea is something like this: in order to grasp the structure

of a predicative piece of syllogizing, you need to determine what items in it

λέγεται κατὰ πάσης δικαιοσύνης, ἀλλ᾿ ἀνάπαλιν πᾶσα δικαιοσύνη ἀρετή. διὸ καὶ δεῖ κατ᾿

ἀμφοτέρας τὰς ἐκφορὰς γυμνάζειν ἑαυτοὺς ἵνα τῇ τε χρήσει παρακολουθεῖν δυνώμεθα καὶ τῇ

διδασκαλίᾳ.

332 Forms of Argument

function as subjects and what as predicates; and the locution ‘A is predicated

of B’—and perhaps, by a sort of natural extension, ‘A holds of B’—makes

the determination child’s play.

Perhaps it does. But then why use it only in setting out syllogistic

theory—why not use it also in actual syllogizing? Apuleius seems to present

a couple of matrixes—a couple of quartets of matrixes—for predicative

sentences, and Alexander in effect does the same. One of the matrixes brings

out the structure of a predicative syllogism more perspicuously than the other

does. And yet in syllogistic practice the more perspicuous matrix is scarcely

ever exempliﬁed. Aristotle and his successors will normally say ‘A holds of

every B’ or ‘A is predicated of every B’ when they are engaged in syllogistic

theory—when, for example, they are engaged in proving the validity of a

given form of argument. But Aristotle will say, with the rest of us,

Every nice girl loves a sailor

rather than anything like

Item which loves a sailor holds of every nice girl.

So too will Alexander, and Uncle Tom Cobbley. Surely that is odd behaviour

on the part of logicians who believe that ‘A is predicated of every B’ is the

most perspicuous matrix for universal afﬁrmative sentences?

Perhaps it would be if they did—but they don’t. The alleged oddity in

Peripatetic linguistic behaviour depends on the claim that ‘A is predicated of

every B’ was, in Peripatetic eyes, the most perspicuous matrix for universal

afﬁrmatives. But no Peripatetic ever actually says that ‘A of every B’ is a

perspicuous matrix; for no ancient logician ever talks about matrixes as such.

Why, then, suppose that they really took it to be a perspicuous matrix? Well,

they certainly took it to be perspicuous, and isn’t it a matrix?

No: ‘A of every B’ is not a matrix for universal afﬁrmative propositions.

It is not a matrix at all. Consider one of Aristotle’s principles of conversion:

E-style propositions convert, if no bird sings then no singing item is a bird, if

you interchange subject and predicate in a true E-predication then the result

is a true E-predication. Or, using Aristotelian letters:

IfAispredicatedofnoB,thenBispredicatedofnoA.

Isn’t that a sentential matrix, a matrix which schematically represents the

principle of conversion? No, it is not a matrix. For no replacement of

the symbols ‘A’ and ‘B’ in it will produce an English sentence. Exactly

the same holds of the Greek and the Latin versions of the thing. And the

reason is simple and syntactical: any replacement of ‘A’ and of ‘B’ must

be at once a singular term (in order to precede ‘is predicated’) and also a

‘A Holds of Every B, ...’ 333

general term in order to follow ‘no’). And no term is both singular and

general.

That consideration is powerful; but it is not decisive. A partisan of

Porphyrean predication might make an honest matrix out of the thing by

adapting it to say:

If A is predicated of nothing of which B is predicated, then B is predicated

of nothing of which A is predicated.

There, both ‘A’ and ‘B’ are singular terms; and no doubt the matrix is a

schematic representation of the principle of E-conversion. But ‘A is predicated

of nothing of which B is predicated’ is not a matrix for a universal negative

sentence: if you replace the letters by words, what you get is a singular

afﬁrmative sentence, not a universal negative sentence.

There is another and non-Porphyrean way of getting a matrix out of the

matter. Aristotle’s usage has a peculiarity which I have so far overlooked:

whereas in English (and also in Latin) the syllogistic letters are merely letters

and we write ‘A of every B’, ‘B of no C’, and the like, in Greek Aristotle

invariably preﬁxes a deﬁnite article and writes ‘τὸ Α’, or perhaps ‘τὸ ἄλφα’,

rather than the plain ‘Α’or‘ἄλφα’. Why so?

Well, the phrase ‘τὸ ἄλφα’ is, among other things, a name for the letter

alpha. The grammarians will say things like

there are twenty-four letters, from alpha to omega;

([Dionysius Thrax], 6 [9.2])⁵⁷

and in doing so they use ‘τὸ Α’ to name the letter alpha. But that does not

help us; for when Aristotle writes ‘The A of …’ he evidently does not mean

to say that the Greek letter alpha is predicated of … Then perhaps the phrase

‘τὸ ἄλφα’ is elliptical, and we must supply some noun with it? In Greek

geometrical texts you will often ﬁnd similar phrases: you will also ﬁnd ‘τὸ

ἄλφα σημεῖον’ (‘the point A’), and it is evident that ‘τὸ ἄλφα’ is elliptical

for ‘the point A’. But in the case of Aristotle’s syllogistic, there is no possible

noun which might be understood or supplied with the letters.

Nonetheless, it seems at least to be clear that the letters ‘A’, ‘B’, ‘C’, … do

not have the syntax of singular terms. Perhaps ‘τὸ Α’ is a singular term; but ‘A’

itself is not—rather, it has the syntax of a common noun, it is an item which,

preﬁxed by ‘is a’ will make a one-placed verbal formula. That grammatical

⁵⁷ γράμματά ἐστιν εἰκοσιτέσσαρα ἀπὸ τοῦ α μέχρι τοῦ ω.

334 Forms of Argument

observation is conﬁrmed by a further feature of Aristotelian usage. When

he gives a schematic presentation of the principle of E-conversion, what

he actually says—or rather, what, in all probability, he actually writes—is

this:

εἰ οὖν μηδενὶ τῶν Β τὸ Α ὑπάρχει, οὐδὲ τῶν Α οὐδενὶ ὑπάρξει τὸ Β.

If the A holds of none of the Bs, then the B will hold of none of the As.

(APr 25a15–16)

Here he uses singulars (‘the A’, ‘the B’) and plurals (‘the Bs’, ‘the As’) side

by side; and so it is elsewhere in the Analytics often enough. (How often

the plural occurs is unclear: the manuscripts often vary between a singular

and a plural—they do so in the passage I have just quoted; and there is

often no way of deciding which reading to accept.) It is plain that there is

no signiﬁcant difference between the plural and the singular turn: ‘holds of

some B’ is given indifferently by ‘τινι τῶν Β ὑπάρχει’andby‘τινι τῷ Β

ὑπάρχει’.

In the formula ‘The A holds of none of the Bs’, both Greek letters are

syntactically on a level. Then why not take the formula to be a matrix? The

initial answer is simple: you can’t take the formula to be a matrix because no

replacement of letters by appropriate expressions produces a sentence.

The stone holds of none of the men

The justice holds of none of the vices

The item which hates itself holds of none of the philosophers.

Such monsters may be said to have the syntax of sentences; but they are not

sentences—they are nonsense. To be sure, we can understand them; but, as

I have already said, nonsense is sometimes as easy to understand as sense.

That is not the end of the business: there is yet a further linguistic fact to

be exploited. Aristotle sometimes uses his syllogistic letters as part of complex

formulas—he will sometimes write something of the form ‘that on which

the A is’ or ‘ἐφ’ ᾧτὸΑ’. There must be ﬁfty or more occurrences of such

items in the Prior Analytics. The ﬁrst is this:

οἷον εἰ τὸ μὲν Α εἴη κίνησις, τὸ δὲ Β ζῷον, ἐφ᾿ ᾧ δὲ τὸ Γ ἄνθρωπος.

E.g. if the A were motion, the B were animal, and that on which the C is were man.

(APr 30a29–30)

In that sentence there is evidently no interesting difference between the

simple ‘the A’ and the complex ‘that on which the C is’; and in fact wherever

‘A Holds of Every B, ...’ 335

the complex formula is found it is plainly equivalent to the simple one. The

complex formula sometimes takes a plural form, ‘those items on which the

A is’, where we ﬁnd the genitive rather than the dative case (‘ἐφ’ ὧν’rather

than ‘ἐφ’ οἷς’)—for example, APr 44a13.

Why does Aristotle sometimes use the complex formula when the simple

formula will do just as well and twice as rapidly? It seems reasonable to

hypothesize that the longer expression was the original, and that the shorter

expression came into being as an easy abbreviation. So ‘the A’ means ‘that

on which the A is’—and ‘the As’ means ‘those items on which the A is’. But

what on earth is the meaning of ‘that on which the A is’?

It is commonly supposed that Aristotle learned his use of letters from the

geometers. (I shall return to the supposal.) Expressions of the sort ‘that on

which the A is’ do not—so far as I have observed—occur in Euclid’s Elements;

but items very like them are found in earlier geometrical texts. Simplicius

quotes a long passage of Eudemus in which he discusses Hippocrates’ attempt

to square the circle. Eudemus uses plenty of letters; and sometimes—not

always—he uses them in the style ‘that on which A is’. Here is an example:

Let there be a circle the diameter of which is that on which AB is, and its centre that

on which K is. And let that on which CD is cut in half and at right angles that on

which BK is.

(Simplicius, in Phys 64.11–17)⁵⁸

Eudemus also uses the plural formula ‘ἐφ’ ὧν Α’ (e.g. 67.22–24). He does

not here preﬁx his letters with deﬁnite articles. Nonetheless, it is clear that

there can be no difference of sense between his ‘that on which K is [ἐφ᾿ ᾧ

Κ]’ and the Aristotelian ‘that on which the K is [ἐφ᾿ ᾧ τὸ Κ]’; and in fact

Aristotle too sometimes drops the deﬁnite article.

It is plain that the phrase ‘that on which K is’ means ‘the item which is

labelled with the letter K’. I suppose that, among the Greek geometers, the

cumbersome expression ‘that on which A is’ was original; and that it was later

dropped in favour of the simpler ‘A’. In any event, the cumbersome expression

indicates how the simple expression is to be understood. It indicates how it is

to be understood in geometrical texts—and also, surely, in the Prior Analytics.

After all, Aristotle uses the same type of expressions outside the Analytics.Itis

frequent in the Physics, and in the other works on natural philosophy; and it

⁵⁸ ἔστω κύκλος οὗ διάμετρος ἐφ᾿ ᾗ ΑΒ, κέντρον δὲ αὐτοῦ ἐφ᾿ ᾧ Κ· καὶ ἡ μὲν ἐφ᾿ ᾗ Γ∆ δίχα

τε καὶ πρὸς ὀρθὰς τεμνέτω τὴν ἐφ᾿ ᾗ ΒΚ.

336 Forms of Argument

occasionally crops up elsewhere—in the Ethics and in the biological works.

And often it is used in geometrical or quasi-geometrical contexts.

In other words, and after all, the phrase ‘τὸ ἄλφα’ does there designate

the ﬁrst letter of the Greek alphabet. Not that Aristotle wants to say that the

ﬁrst letter of the alphabet is predicated of the second: rather, he wants to say

that the items to which the ﬁrst letter is attached is predicated of the item to

which the second letter is attached.

With that in mind, let us return for a last time to the question of matrixes.

Take the formula:

The item to which the letter A is attached holds of none of the items to

which the letter B is attached:

is that a matrix? Of course not—it is no more a matrix than is, say,

The shelf to which the letter A is attached holds the dictionaries.

That is not a matrix: it is a complete sentence, and it contains no symbols

at all. Similarly for the syllogistic formula: it is a sentence, not a matrix. It

has, of course, a subject–predicate form (among other forms); but although

it somehow stands for or represents a universal negative predication, it is not

itself a universal negative predication.

Or is it a sentence? Perhaps it has the syntactical form of a sentence; but

whatever does it mean? does it mean anything at all? Consider this sentence

or quasi-sentence:

The item to which the expression ‘stone’ is attached holds of none of the

items to which the expression ‘man’ is attached.

What does it mean? Either it is nonsense or else it is jargon. So no doubt it is

jargon; and no doubt it is a jargon form of

None of the items of which ‘man’ is true is an item of which ‘stone’ is true.

That is intelligible, and it is more or less English.

But what is the point of it? Suppose you were offered this formula as a

schematic presentation of the ﬁrst Stoic unproved:

The 2

follows the 1

;the1

: therefore the 2

That, you might say, is nonsense—indeed, it is hardly even grammatical.

But isn’t that too strict? Why not take the formula as a piece of jargon—as

the jargon version of, say,

If the 1

,the2

;the1

: therefore the 2

Peripatetic Letters 337

Perhaps the formula might be so understood; but what could the purpose or

point of the jargon possibly be? Well, it serves as an indication of a certain

semantic structure; it indicates that the ﬁrst premiss of a Stoic ﬁrst unproved

is a conditional assertible.

In a similar way, the point of writing the Peripatetic jargon is to bring

a certain semantic structure to the fore, to indicate that the items which

are involved in a certain syllogism have this or that predicative structure.

My imagined Stoic jargon in fact has little point; for the structure which it

allegedly serves to indicate will, in most ordinary uses of natural language, be

sufﬁciently clear without its help. But that is not so with predicative structure;

and an earlier chapter has noticed that the predicative structures of sentences

are not always written on their surfaces.

Suppose that all that is true: it still leaves at least one thing unexplained;

for if we can now understand a jargon phrase like

The item to which the expression ‘stone’ is attached holds of none of the

items to which the expression ‘man’ is attached,

we cannot yet understand an item like

The item to which the letter A is attached holds of none of the items to

which the letter B is attached.

We know to which items the expression ‘stone’ is attached—it is attached

to all stones and to nothing else. But to what item or items is the letter A

attached? What, in other words, are these Peripatetic letters really up to?

PERIPATETIC LETTERS

Alphas and betas and gammas crop up in the Analytics in various different

contexts, and we are not obliged to suppose that they always have the same

status. True, Aristotle never indicates that they have different statuses; but

then he never comments on their status at all.

In some passages at least, the letters have a determinate sense; and it is

natural to think of them as merely abbreviatory devices—like the numerals

in the Stoic argumodes. Here is an example:

There will be a proof by way of this—e.g. that the planets are near by way of their

not twinkling. Let planets be that on which C is, not twinkling that on which B,

338 Forms of Argument

being near that on which A. Now it is true to say B of C; for planets do not twinkle.

And A of B; for what does not twinkle is near (suppose that that has been grasped by

induction or perception). Therefore it is necessary that A holds of C, so that it has

been proved that the planets are near.

(APst 78a29–36)⁵⁹

The three letters there stand in for the three predicates in the proof (one of

which, it may be remarked, is negative). But although the letters have a ﬁxed

sense, it is at best misleading to refer to them as abbreviatory devices. After all,

far from shortening the argument, the letters double its length; for Aristotle

gives both a literal and a verbal version of the proof. The point of the formula

‘It is true to say B of C’ is not to say the same thing in fewer letters as ‘Planets

donottwinkle’;rather,itistobringoutthepertinentstructureof‘Planets

do not twinkle’ in a clear and brief fashion.

Passages of this sort, in which Greek letters stand in for Greek predicative

expressions, are numerous, especially in the Posterior Analytics. But what we

rightly think of as the characteristic use of letters in the Analytics seems to be

quite different. For example, a little later in the Posterior Analytics, where he

is discussing certain features of negative proofs, Aristotle remarks that

again, if B holds of every A and of no C, A holds of none of the Cs.

(APst 82b14–15)⁶⁰

There the letters do not appear to stand in for any particular predicative

expressions. It is not merely that there are no suitable expressions in the

vicinity. Also, and more importantly, it is plain that Aristotle wants to say

something about terms in general and not about a particular determinate

triad of terms. In other words, his letters here seem to function like the letters

in an algebraical formula such as:

x + y = y + x.

That formula expresses the thought that addition is commutative. It would

be foolish to wonder what particular numbers the letters ‘x’ and ‘y’ there

stand for or designate.

⁵⁹ ἔσται διὰ τούτου ἡ ἀπόδειξις, οἷον ὅτι ἐγγὺς οἱ πλάνητες διὰ τοῦ μὴ στίλβειν. ἔστω ἐφ᾿

ᾧ Γ πλάνητες, ἐφ᾿ ᾧ Β τὸ μὴ στίλβειν, ἐφ᾿ ᾧ Α τὸ ἐγγὺς εἶναι. ἀληθὲς δὴ τὸ Β κατὰ τοῦ Γ

εἰπεῖν· οἱ γὰρ πλάνητες οὐ στίλβουσιν. ἀλλὰ καὶ τὸ Α κατὰ τοῦ Β· τὸ γὰρ μὴ στίλβον ἐγγύς

ἐστι· τοῦτο δ᾿ εἰλήφθω δι᾿ ἐπαγωγῆς ἢ δι᾿ αἰσθήσεως. ἀνάγκη οὖν τὸ Α τῷ Γ ὑπάρχειν, ὥστ᾿

ἀποδέδεικται ὅτι οἱ πλάνητες ἐγγύς εἰσιν.

⁶⁰ πάλιν εἰ τὸ μὲν Β παντὶ τῷ Α, τῷ δὲ Γ μηδενί, τὸ Α τῶν Γ οὐδενὶ ὑπάρχει.

Peripatetic Letters 339

That the syllogistic letters typically serve to introduce a generality was

noticed by Alexander.

He presents the position by way of letters in order to indicate to us that the

conclusions do not come about because of the matter but because of the ﬁgure

and the particular combination of propositions and the mood—it is not because

the matter is such-and-such that so-and-so is syllogistically inferred but because

the conjugation is such-and-such. So the letters show that the conclusion will be

thus-and-so universally and always and in the case of every assumption.

(in APr 53.28–54.2)⁶¹

The letters indicate universality. They indicate that the argument does not

turn upon its particular matter—that it does not depend for its validity on

the fact that it contains those terms rather than these.

Alexander repeats the point elsewhere. No doubt he inherited it from

his predecessors. Certainly he bequeathed it to his successors. Here is

Philoponus:

Having shown by way of examples how each of the propositions converts, next—lest

anyone should think that his account of the conversions is eased on its way by the

matter of the chosen examples or by anything else, and lest it be unclear whether there

are not perhaps some examples for which the stated conversions do not work—for

that reason he now sets down universal rules, using letters rather than terms so that

we may each of us take whatever matter we wish in place of the letters, the thesis

having been shown universally and without the use of matter by way of the letters.

(in APr 46.25–47.1)⁶²

But if Philoponus knows that Aristotle’s letters express universality, he does

not explain how they manage to do so.

Thereisanexplanation,ofsorts,inAlexander.Ishallcitealongpassage:

in parts it is obscure, and so too is the Aristotelian text on which Alexander

is commenting. Here, ﬁrst, is Aristotle:

⁶¹ ἐπὶ στοιχείων τὴν διδασκαλίαν ποιεῖται ὑπὲρ τοῦ ἐνδείξασθαι ἡμῖν ὅτι οὐ παρὰ τὴν

ὕλην γίνεται τὰ συμπεράσματα ἀλλὰ παρὰ τὸ σχῆμα καὶ τὴν τοιαύτην τῶν προτάσεων

συμπλοκὴν καὶ τὸν τρόπον· οὐ γὰρ ὅτι ἥδε ἡ ὕλη συνάγεται συλλογιστικῶς τόδε, ἀλλ᾿ ὅτι ἡ

συζυγία τοιαύτη. τὰ οὖν στοιχεῖα τοῦ καθόλου καὶ ἀεὶ καὶ ἐπὶ παντὸς τοῦ ληφθέντος τοιοῦτον

ἔσεσθαι τὸ συμπέρασμα δεικτικά ἐστιν.

⁶² δείξας ὅπως ἑκάστη τῶν προτάσεων ἀντιστρέφει διὰ παραδειγμάτων, ἵνα μή τις οἰηθῇ

διὰ τὴν ὕλην τῶν παραληφθεισῶν προτάσεων ἢ δι᾿ ἕτερόν τι εὐοδῆσαι αὐτῷ τὸν περὶ τῶν

ἀντιστροφῶν λόγον, ἄδηλον δὲ εἶναι μή πώς ἐστί τινα παραδείγματα ἐν οἷς αἱ εἰρημέναι

ἀντιστροφαὶ χώραν οὐκ ἔχουσι, διὰ τοῦτο ἐνταῦθα καθολικοὺς κανόνας παραδίδωσι τὰ

στοιχεῖα παραλαμβάνων ἀντὶ τῶν ὅρων, ἵνα ἕκαστος οἵαν βούλοιτο ὕλην ἀντὶ τῶν στοιχείων

παραλαμβάνοι, δειχθέντος καθολικῶς τε καὶ ἀύλως ἐπὶ τῶν στοιχείων τοῦ λόγου.