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310 Forms of Argument

unproved; for you cannot derive it from any concrete expression of a third

unproved. Nor is it a schematic representation of any other valid form of

argument.

As for the quasi-disjunctive syllogism, there is not and there could not be a

sequence of matrixes which corresponds to its Galenic circumscription. Any

quasi-disjunctive syllogism which the circumscription captures will ﬁt some

pertinent set of matrixes or other; but there is no pertinent set of matrixes

which every quasi-disjunctive syllogism ﬁts. The circumscription corresponds

to a set of sets of matrixes—an inﬁnite set of sets—and not to a single set.

Insofar as the circumscription determines a particular syllogistic form, there

is at least one form which cannot be represented schematically—I mean,

which cannot be represented by a single schema.

Is that an argument against schemata? Not in itself; for it shows at most

that you can do some things with circumscriptions which you cannot do

with schemata; and perhaps those are things which a logician does not

want, or does not need, to do. Nonetheless, you might reasonably suspect

that the higher degree of generalization which circumscriptions offer is a

mark in their favour, from a logical point of view. After all, the inﬁnitely

many quasi-disjunctive schemata are all special cases of a general form, and

the general form can only be represented by a circumscription. Again, and

connectedly, you might think that the circumscription of the quasi-disjunctive

syllogism has an explanatory value which the innumerable schemata do not

possess: a quasi-disjunctive syllogism is valid in virtue of the fact that it

ﬁts the circumscription, not in virtue of the fact that it ﬁts this or that

quasi-disjunctive schema.

However that may be, there is a further question about schemata which

oughttoberaised.Istartoutfromafamiliarcomplaintmadebothby

Galen and by Alexander against the Stoic logicians: the Stoics, they say,

pay too much attention to forms of expression or φωναί and too little

to the items which they signify—to the πράγματα or σημαινόμενα; they

identify and classify arguments by appeal to their most superﬁcial features.

Now the circumscriptional way of identifying inferential forms neither says

nor implies anything about the manner in which such forms might be

expressed. The circumscription of a disjunctive form, for example, may

specify that one of the premisses be a disjunction; but it will leave open

the question of how a disjunction may be expressed, or may best be

expressed. On the other hand, schematic representations by their very nature

indicate—schematically—some particular manner of expression or other.

Schemata and Matrixes 311

The schematic representation of a disjunctive syllogism, say, will necessarily

contain some such matrix as

Either P or Q,

and that matrix ﬁxes a certain form of expression.

In that case, does not a schematic representation of an argument essentially

determine a certain form of linguistic expression? And if that is so, then did

not Galen and Alexander have philosophical or logical reasons for preferring

circumscriptions to schemata? (For they wanted to determine the form of

an argument without thereby ﬁxing any particular means of expressing it.)

Moreover, will there not be another—and a more straightforward—way in

which schemata and circumscriptions fail to pair off?

The argument runs like this. The circumscription of the ﬁrst unproved

states that from a conditional and its antecedent you may infer its consequent.

The mode of the ﬁrst unproved is this:

If the 1

,the2

;the1

: therefore the 2

So consider the following argument:

He’ll be in Oxford now provided that Eurostar wasn’t late. So since

Eurostar wasn’t in fact late, he’ll be there.

Is that a ﬁrst unproved? According to the circumscription, the answer is Yes.

For ‘He’ll be in Oxford …’ expresses a conditional proposition; ‘Eurostar

wasn’t in fact late’ expresses its antecedent; and ‘He’ll be there’ expresses

its consequent. But according to the schema, the answer seems to be No.

For if you start from the expression of the argument and replace expressions

by symbols, you will never arrive, however ingenious you are, at the matrix

which represents the mode of the ﬁrst unproved.

That is an unwelcome conclusion; for whatever difﬁculties may attend

the last four of the ﬁve unproveds, surely the mode of the ﬁrst unproved

ought to correspond exactly to its circumscription? One way of securing an

exact correspondence calls on the Stoic account of conditional assertibles.

According to Sextus,

non-simple assertibles are those … which are composed from a repeated assertible

or from different assertibles, and in which a connector or connectors govern. Take

for the moment what they call the conditional. That is composed from a repeated

assertible or from different assertibles by way of the connector ‘εἰ’or‘εἴπερ’.

(M viii 108–109)⁴⁰

⁴⁰ καὶ δὴ οὐχ ἁπλᾶ μὲν ἐστιν ἀξιώματα τὰ ἀνώτερον προειρημένα, ἅπερ ἐξ ἀξιώματος

διφορουμένου ἢ ἐξ ἀξιωμάτων διαφερόντων συνέστηκε καὶ ἐν οἷς σύνδεσμος ἢ σύνδεσμοι

312 Forms of Argument

If that account is right, and if it is taken au pied de la lettre, then the

Eurostar argument does not contain a conditional assertible—for neither

the connector ‘εἰ’ nor the connector ‘εἴπερ’ appears anywhere in it. So

the Eurostar argument is not, after all, a ﬁrst unproved according to the

circumscription; and schema and circumscription correspond.

But that makes matters worse rather than better; for it limits conditional

assertibles—and hence ﬁrst unproveds—to items which contain one of a

pair of Greek words. Well, no doubt the text in Sextus should not be taken

au pied de la lettre, and its linguistic chauvinism was surely unintended: we

may decently suppose that when the Stoics said ‘the connector ‘εἰ’or‘εἴπερ’’

what they meant was ‘the connector ‘εἰ’or‘εἴπερ’, or any translation of those

words in other languages’.

Even so, why limit conditional assertibles to assertibles which contain

those particular connectors? After all, there are numerous ways, in most

natural languages, of expressing conditional notions—why privilege one of

them? The limitation may have the effect of bringing the schemata and the

circumscriptions into line; but it does so by pushing on the wrong item.

There is a further point. The passage from Sextus, if it is taken strictly,

implies that complex assertibles are ontological hybrids: they consist of simple

assertibles, which are incorporeal items, together with connectors, which are

sounds and therefore corporeal items. That is curious; and in fact most

philosophers do not care for such hybrids. But perhaps the Stoics did?

According to some scholars, the Stoics construed simple assertibles, such as

Dio walks

as compounded from an incorporeal predicate, to walk, and a corporeal

case, namely Dio himself. And if that is right, then why should they have

jibbed at introducing another form of hybridization in the case of compound

assertibles? But it must be said that no Stoic text ever explicitly recognizes—let

alone discusses or defends—hybridization; and it may be deemed preferable

toavoiditifitcanbeavoided.

In the case of compound assertibles it is readily avoided. For example, why

not say that an assertible is a conditional if and only if in saying it you may say

that if something is the case then something is the case? Or that an assertible

is a conditional if and only if it may be said by uttering a sentence of the form

If P, then Q?

ἐπικρατοῦσιν. λαμβανέσθω δὲ ἐκ τούτων ἐπὶ τοῦ παρόντος τὸ καλούμενον συνημμένον. τοῦτο

τοίνυν συνέστηκεν ἐξ ἀξιώματος διφορουμένου ἢ ἐξ ἀξιωμάτων διαφερόντων καὶ διὰ τοῦ εἴ

ἢ εἴπερ συνδέσμου.

Schemata and Matrixes 313

Those two explanations of conditionality make use of the connector ‘if’;

and to that extent they may be regarded as natural elaborations of the Stoic

explanation which Sextus offers. On the one hand, the connector ‘if’ (or its

Greek translation) is invoked in the explanation of conditional assertibles.

On the other hand, conditionals are not language-bound, nor are assertibles

inevitably hybrids.

The matrix ‘If P, then Q’ may now be taken to represent a conditional

assertible inasmuch as an assertible is conditional if and only if it can be

expressed by an instance of the matrix. So the sentence

He’ll be in Oxford now provided that Eurostar wasn’t late

expresses a conditional assertible; for the assertible which it expresses may be

expressed by a sentence which is an instance of the matrix ‘If P, then Q’—say

by the sentence

If Eurostar wasn’t late, then he’ll be in Oxford now.

And the Eurostar argument is a ﬁrst unproved not only according to the

circumscription but also according to the schematic representation. For the

argument ﬁts the mode

If the 1

,the2

;the1

: therefore the 2

inasmuch as an appropriate replacement of the symbols by sentences will

produce a sequence of sentences which might be used to express the argument.

A schematic representation of a syllogism is expressed by means of a

sequence of matrixes. An argument is represented by a schematic representa-

tion if and only if it could be expressed by means of a sequence of sentences

which are instances of that sequence of matrixes. A schematic representation

ﬁxes the form of an argument by reference to a form of expression; but what

countsisnothowtheargumentisinfactexpressed,buthowtheargument

may or might be expressed.

Were the Stoics to have taken that line, or something like it, then they

would have been able to bring circumscriptions and schemata together—at

least in the case of the ﬁrst unproved. But that line is more or less the line

which Galen and Alexander urge us to take—and which the Stoics, according

to them, signally failed to take. Were Galen and Alexander wrong about, or

unfair to, the Stoics? Most scholars think that they were not. Indeed, several

scholars have thought that Galen and Alexander were quite right in their

report and quite wrong in their evaluation: the Stoics did indeed attend to

expressions rather than to meanings, or to syntax rather than to sense—and

that is just what a good formal logician ought to do. The Stoics, according to

that view, did indeed hold that an argument is a ﬁrst unproved if and only if

314 Forms of Argument

it is expressed by some substitution of sentences for symbols in the schematic

formula

If the 1

,the2

;the1

: therefore the 2

In the same way, an argument in the modern propositional calculus is an

example of modus ponens ifandonlyifitisexpressedbysomesubstitutionof

sentences for letters in the schema

P ⊃ Q, P: therefore Q.

So the Eurostar example is not, after all, a ﬁrst unproved—nor is it a case of

modus ponens.

Then what on earth is it?

SUBSYLLOGISTIC ARGUMENTS

Well, comes the answer, it is what the Stoics called a subsyllogistic argument.

Subsyllogistic arguments are not syllogisms; but each subsyllogistic argument

is related in a determinate manner to a particular syllogism, and it is that fact

which explains its validity.

In truth, we know little enough about subsyllogistic arguments—indeed,

the word ‘subsyllogistic’ is found exactly twice in the surviving texts on Stoic

logic. In one of them, Galen refers to

the arguments which are called subsyllogistic, being uttered by way of expressions

which are equipollent with syllogistic arguments.

(inst log xix 6)⁴¹

The text is in a rotten state, and Galen offers no illustrative example.

But the context of his remark implies that the Stoics—and in particular

Chrysippus—had named and discussed the things.

The second passage is more expansive. Alexander is discussing Aristotle’s

treatment of the second ﬁgure predicative mood Baroco. Aristotle says this:

Again, if M holds of every N and does not hold of some X, it is necessary that N does

not hold of some X. … And if M holds of every N and not of every X, there will be a

syllogism that N does not hold of every X—the proof is the same.

(APr 27a36-b2)⁴²

⁴¹ οἱ δὲ ὑποσυλλογιστικοὶ κληθέντες ἐν ἰσοδυναμούσαις λέξεσι τοῖς συλλογιστικοῖς

λεγόμενοι.

⁴² πάλιν εἰ τῷ μὲν Ν παντὶ τὸ Μ, τῷ δὲ Ξ τινὶ μὴ ὑπάρχει, ἀνάγκη τὸ Ν τινὶ τῷ Ξ μὴ

ὑπάρχειν· ... καὶ εἰ τὸ Μ τῷ μὲν Ν παντὶ ὑπάρχει τῷ δὲ Ξ μὴ παντί, ἔσται συλλογισμὸς ὅτι

οὐ παντὶ τῷ Ξ τὸ Ν· ἀπόδειξις δ᾿ ἡ αὐτή.

Subsyllogistic Arguments 315

There seem to be two syllogisms there—two Barocos, as it were. Alexander

explains that the second version, which substitutes ‘not of every’ for ‘not of

some’

is of the kind which the later thinkers call subsyllogistic inasmuch as it assumes

something which is equipollent with the syllogistic premiss and infers the same

conclusion from it. For ‘not hold of all’ has replaced ‘not hold of some’, with which

it is equipollent. But they do not call such items syllogisms since they attend to

language and expression whereas Aristotle, who, where the same things are meant,

looks to the meanings and not to the expressions, says that the same syllogism is

inferred when the language of the conclusion is transformed in this way—provided

that the conjugation is syllogistic in the ﬁrst place.

(in APr 84.11–19)⁴³

Take the following two schemata:

(A) B holds of every A (B) B holds of every A

B does not hold of some C B does not hold of every C

A does not hold of some C A does not hold of every C

Aristotle appears to suggest that those two schemata represent two dis-

tinct syllogistic forms—after all, he gives ﬁrst the one and then the

other, with no apology for repeating himself. But Alexander explains

that in fact—and hence in Aristotle’s opinion—they are two different

ways of representing one and the same syllogism, namely Baroco. Alex-

ander also notes that, according to ‘the later thinkers’—and he has no

doubt got the Stoics in mind—schema (B) represents a subsyllogist-

ic argument, which is different from the syllogistic argument which (A)

represents.

Did some Stoics—perhaps some imperial Stoics—comment on the two

versions of Baroco? That is perfectly possible; but Alexander does not explicitly

say that they did, and he probably means to say not that the Stoics had in

fact found two different arguments in (A) and (B) but rather that that is just

the sort of silly thing they would do.

⁴³ τοιοῦτός ἐστιν ὁ ὑποσυλλογιστικὸς ὑπὸ τῶν νεωτέρων λεγόμενος ὁ λαμβάνων μὲν τὸ

ἰσοδυναμοῦν τῇ προτάσει τῇ συλλογιστικῇ ταὐτὸν δὲ καὶ ἐκ ταύτης συνάγων· τῷ γὰρ τινὶ

μὴ ὑπάρχειν τὸ μὴ παντὶ ὑπάρχειν ἰσοδυναμοῦν μετείληπται. ἀλλ᾿ ἐκεῖνοι μὲν οὐ λέγουσι

τοὺς τοιούτους συλλογισμοὺς εἰς τὴν φωνὴν καὶ τὴν λέξιν βλέποντες, ἀλλὰ ᾿Αριστοτέλης

πρὸς τὰ σημαινόμενα ὁρῶν, ἐφ᾿ ὧν ὁμοίως σημαίνεται, οὐ πρὸς τὰς φωνάς, τὸν αὐτόν φησι

συνάγεσθαι συλλογισμὸν καὶ ἐν τῇ τοιαύτῃ τῆς λέξεως ἐν τῷ συμπεράσματι μεταλήψει, ἂν ᾖ

συλλογιστικὴ ὅλως συμπλοκή.

316 Forms of Argument

Two further texts are customarily added to the subsyllogistic dossier. First,

there is another passage from Alexander’s commentary on the Analytics.

When the same thing is meant primarily by different expressions and is taken in the

same way, the syllogism will be the same … Now that is Aristotle’s view about changes

in expression; but the later thinkers attend to expressions and not to meanings, and

they deny that you get the same thing when you exchange the terms for equipollent

expressions. For although ‘If the 1

,the2

’meansthesameas‘The2

follows the

’, they say that the argument is syllogistic if the expression is taken like this:

If the 1

,the2

; but the 1

: therefore the 2

but that

The 2

follows the 1

; but the 1

: therefore the 2

is not syllogistic but concludent.

(in APr 373.18–35)⁴⁴

Alexander does not say that the later thinkers called the second of those two

arguments subsyllogistic; but surely they should have done and no doubt

they did.

Finally, in his account of Stoic logic, Diogenes reports that there was

a distinction within the class of valid or concludent arguments between

arguments which are syllogistic and arguments which are not syllogistic but

merely concludent; and as an example of a non-syllogistic but concludent

argument he offers this:

It is false that it is day and it is night; but it is day: therefore it is not night.

(vii 78)⁴⁵

Diogenes does not introduce the term ‘subsyllogistic’; but, again, he surely

might have done so—no doubt the Stoics recognized the argument as a

subsyllogistic partner of a third unproved syllogism.

The general drift of all that is clear enough; but it is surprisingly difﬁcult

to frame a clear and distinct idea of what a subsyllogism is. One ﬁrst attempt

at doing so runs like this: A sequence of sentences

⁴⁴ ὥσθ᾿ ὅταν ταὐτὰ σημαίνηται ὑπὸ διαφόρων λέξεων προηγουμένως καὶ ὁμοίως λαμ-

βάνηται, ὁ αὐτὸς ἔσται συλλογισμός. ... ᾿Αριστοτέλης μὲν οὖν οὕτως περὶ τῶν κατὰ τὰς

λέξεις μεταλήψεων φέρεται· οἱ δὲ νεώτεροι ταῖς λέξεσιν ἐπακολουθοῦντες οὐκέτι δὲ τοῖς

σημαινομένοις οὐ ταὐτόν φασι γίνεσθαι ἐν ταῖς εἰς τὰς ἰσοδυναμούσας λέξεις μεταλήψεσι τῶν

ὅρων. ταὐτὸν γὰρ σημαίνοντος τοῦ εἰ τὸ πρῶτον τὸ δεύτερον τῷ ἀκολουθεῖ τῷ πρώτῳ τὸ

δεύτερον, συλλογιστικὸν μὲν λόγον φασὶν εἶναι τοιαύτης ληφθείσης τῆς λέξεως εἰ τὸ πρῶτον

τὸ δεύτερον, τὸ δὲ πρῶτον, τὸ ἄρα δεύτερον, οὐκέτι δὲ συλλογιστικὸν ἀλλὰ περαντικὸν τὸ

ἀκολουθεῖ τῷ πρώτῳ τὸ δεύτερον, τὸ δὲ πρῶτον, τὸ ἄρα δεύτερον.

⁴⁵ ψεῦδός ἐστι τὸ ἡμέρα ἐστὶ καὶ νύξ ἐστι· ἡμέρα δὲ ἐστιν· οὐκ ἄρα νύξ ἐστιν.

Subsyllogistic Arguments 317

,…,P

: therefore Q

expresses a subsyllogistic argument if and only if there is a sequence of

expressions

,…,R

: therefore S

which expresses a syllogism and is such that each R

is either the same as

or equipollent with the corresponding P

and S is either the same as or

equipollent with Q.

Thusthesequenceofexpressions

Itisfalsethatitisdayanditisnight;itisday:thereforeitisnotnight

expresses a subsyllogistic argument inasmuch as

It is not the case that it is day and it is night; it is day: therefore it is night

expresses a syllogism (a third unproved syllogism) and ‘It is false that it is day

and it is night’ is equipollent with ‘It is not the case that it is day and it is

night’, while ‘It is day’ and ‘It is night’ are common to the two sequences.

What arguments are thereby identiﬁed as subsyllogistic will depend on

the way in which equipollence is construed. The ancient texts give little

away—indeed, ancient texts use the terminology of equipollence without

ever feeling the need to explain what they mean, and often enough what

they say remains crucially indeterminate. The notion is expressed by way of

theverb‘ἰσοδυναμεῖν’ or of the verbal phrase ‘ἴσον δύνασθαι’, where the

‘-pollence’ of ‘equipollence’ is carried by ‘-δυναμεῖν’or‘δύνασθαι’. Thus

equipollence amounts to equality of δύναμις. Since one common meaning of

‘δύναμις’ is ‘meaning’, equipollence might well be taken to signify equality

of meaning—or synonymy. In fact, it is quite clear that, sometimes at least,

equipollence amounts to synonymy; and scholars generally suppose that, in

the context of subsyllogistic arguments, equipollence is synonymy.

So the ﬁrst attempt at deﬁning a subsyllogistic argument may be lightly

revised, thus: A sequence of sentences

,…,P

: therefore Q

expresses a subsyllogistic argument if and only if there is a sequence of

expressions

,…,R

: therefore S

which expresses a syllogism and is such that each R

is either the same as

or synonymous with the corresponding P

and S is either the same as or

synonymous with Q.

That deﬁnition probably requires a triﬂing qualiﬁcation—and also a non-

triﬂing qualiﬁcation. The triﬂing qualiﬁcation is needed in order to avoid

the conclusion that every syllogism is a subsyllogistic argument. For to every

318 Forms of Argument

sequence which expresses a syllogism there corresponds, trivially, a sequence

of expressions which expresses a syllogism and the elements of which are the

same as or synonymous with the sequence; for every sequence so corresponds

to itself. So let us write—say—‘… if and only if there is a different sequence

of expressions … ’ rather than simply ‘… if and only if there is a sequence of

expressions … ’.

The non-triﬂing qualiﬁcation is suggested by the nomenclature of the

items under examination; for the word ‘subsyllogistic’—or more precisely, its

Greek parent—suggests that we are dealing with something not quite up to

snuff, something not on the level of a genuine syllogism. But why think that a

subsyllogistic argument, as it has been thus far explained, is inferior to a syllo-

gism rather than merely a different syllogism? The notion of synonymy does

not help to answer the question inasmuch as synonymy is a symmetrical rela-

tion: the fact that a subsyllogism and its syllogistic counterpart are expressed

by synonymous formulas leaves them on the same level; and we need to ﬁnd

some asymmetry, some way of setting subsyllogisms below syllogisms. The

asymmetry has been discovered in a certain theory of linguistic degeneration.

Degeneration was a notion much loved by the ancient grammarians.

Language, they observed, was liable to change; language, they pessimistically

observed, tended to lose its pristine purity and present itself in dirty and

degenerate guises. The degeneration might be a matter of orthography, of

accentuation, of grammatical form, of construction, … . But the crucial fact

about degeneration was this: if X is a degenerate form of Y, then X has the

same meaning as Y. Degeneration does not occur when an expression changes

its sense but when a sense changes its expression. As Apollonius puts it,

every expression, in whatsoever way it has degenerated, nevertheless keeps its own

meaning—nor does it change the order it imposes if it imposes an order.

(conj 224.11–13)⁴⁶

Or again, and with illustrations:

Complete items retain their meaning even when they are mutilated and truncated;

for degeneration affects the sounds and not the meanings. The mutilated form ‘δῶ’

means ‘house’; ‘ἐθέλω’ with its epsilon truncated means the same as before; …

(adv 158.13–16)⁴⁷

⁴⁶ πᾶσα λέξις ὁτιδήποτε παθοῦσα ἔχει καὶ τὸ ἴδιον δηλούμενον, καὶ εἴ τινος τάξεως τύχοι,

ταύτης πάλιν οὐ μετατίθεται.

⁴⁷ τὰ μέντοι ἐντελῆ ὄντα καὶ ἀποκοπτόμενα καὶ ἀφαιρούμενα φυλάσσει τὸ δηλούμενον·

τῶν γὰρ φωνῶν τὰ πάθη καὶ οὐ τῶν σημαινομένων. ἀποκοπὲν τὸ δῶ σημαίνει τὸ δῶμα,

ἀφαιρεθὲν τὸ ἐθέλω τοῦ ε τὸ αὐτὸ σημαίνει, ...

Subsyllogistic Arguments 319

Degeneracy is asymmetrical: if X is a degenerate form of Y, then Y is not a

degenerate form of X. But degeneracy also guarantees synonymy. In other

words, it is just the thing which subsyllogistic arguments are looking for.

There is a speck of evidence that there was a formal connection between

subsyllogistic arguments and linguistic degeneration. It is found in the

sentence at the very end of Galen’s Introduction to Logic which mentions

subsyllogistic arguments. The sentence is corrupt, and scholars do not agree

even on its general sense. On one restoration, it runs like this:

Of merely concludent arguments, we have shown that some do not form a genus

of syllogisms of their own but are expressed in degenerate language—sometimes by

way of a coherent transposition, and in the case of the arguments which are called

subsyllogistic by being uttered by way of expressions which are equipollent with

syllogistic arguments.

(inst log xix 6)⁴⁸

If that restoration is roughly right, then subsyllogistic arguments are—

according to Galen—one of two types of argument which are expressed de-

generately.

It would be rash to rest a whole theory on those withered words of

Galen. On the other hand, it is difﬁcult not to be attracted by the suggested

connection between subsyllogisticality and degeneration—if only because it

is difﬁcult to ﬁnd anything else which is in the least attractive.

So the idea is this. A syllogism—a genuine syllogism—must be decked

out in pristine expressions. An argument degenerately expressed may indeed

say the same as a syllogism—but it will be subsyllogistic. And we may ﬁnally

explicate subsyllogistic arguments as follows:

A sequence of sentences

,…,P

: therefore Q

expresses a subsyllogistic argument if and only if there is a distinct sequence

of expressions

,…,R

: therefore S

which expresses a syllogism and is such that each P

is either the same as or

a degenerate synonym of the corresponding R

and Q is either the same as

or a degenerate synonym of S.

⁴⁸ ἐδείχθησαν γὰρ καὶ τούτων ἔνιοι μὲν οὐκ ἴδιόν τι γένος ὄντες συλλογισμῶν ἀλλὰ διὰ

πεπονθυίας λέξεως ἑρμηνευόμενοι, ποτὲ μὲν κατ᾿ ἀκολουθοῦσαν ὑπέρθεσιν, οἱ δὲ ὑποσυλλο-

γιστικοὶ κληθέντες ἐν ἰσοδυναμούσαις λέξεσι τοῖς συλλογιστικοῖς λεγόμενοι.