Barnes Jonathan. Truth, etc

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5. The Science of Logic

LOGIC AS AN ARISTOTELIAN SCIENCE?

In his Elements Euclid ﬁrst sets down certain primary truths or axioms and

then deduces from them a number of secondary truths or theorems. Before

ever Euclid wrote, Aristotle had described and commended that rigorous

conception of a science for which the Elements was to provide a perennial

paradigm. All sciences, in Aristotle’s view, ought to be presented as axiomatic

deductive systems—that is a main message of the Posterior Analytics.Andthe

deductions which derive the theorems of any science from its axioms must

be syllogisms—that is a main message of the Prior Analytics.

What, then, is the status of logic itself? More precisely, what is the status of

syllogistic or of the theory of deduction? Surely it must be a science? Surely in

that case it ought to be axiomatized àlaEuclid? At any rate, surely Aristotle

thought of his own syllogistic as a science?

Well, he never says in so many words that logic is a science. Indeed, he

never uses the formula ‘syllogistic science’, nor any near equivalent. In the

Sophistici Elenchi, it is true, you once ﬁnd the expression ‘syllogistic art’ (Soph

El 172a35). But what that means is indicated by a later remark:

We undertook to discover a certain syllogistic capacity …

(Soph El 183a37–38)¹

And that remark in turn refers back to the opening sentence of the Topics:

The aim of this study is to discover a method on the basis of which we shall be able

to syllogize …

(Top 100a18–19)²

So the phrase ‘syllogistic art’ refers to a capacity or a method and not to

a science.

There is something comparable in the Rhetoric. In one passage Aristotle

refers to ‘the analytical science’:

¹ προειλόμεθα μὲν οὖν εὑρεῖν δύναμίν τινα συλλογιστικήν ...

² ἡ μὲν πρόθεσις τῆς πραγματείας μέθοδον εὑρεῖν ἀφ᾿ ἧς δυνησόμεθα συλλογίζεσθαι ...

LogicasanAristotelianScience? 361

What we said earlier is indeed true: rhetoric is composed of the analytical science and

of the political science concerned with character.

(Rhet 1359b8–11)³

Isn’t ‘the analytical science’ the science purveyed in the Analytics?Nodoubt;

but the phrase ‘analytical science’ is misleading; for the earlier passage to

which Aristotle refers is this:

Rhetoric is as it were an offshoot of dialectic and of the study of character, which

may properly be called politics. … Neither of those studies is a science of the

facts belonging to some determinate kind: rather, they are capacities for producing

arguments.

(Rhet 1356a25–33)⁴

So ‘analytical science’ in the later passage is a variant on ‘dialectical capacity’;

and a dialectical capacity is contrasted with a genuine science.

The English nouns ‘syllogistic’ and ‘dialectic’—and ‘logic’ too—began

life as Greek feminine adjectives. The Greek adjectives are often used without

any accompanying noun. Sometimes in Aristotle, and often in later texts,

there is no need to supply a silent noun: ‘ἡ συλλογιστική’ and the like

mean ‘syllogistic’ and the like. But if you insist on discovering an appropriate

noun, you have a choice: ‘ἐπιστήμη’ or ‘science’ is one option; but there

are also ‘τέχνη’or‘art’,‘μέθοδος’or‘method’,‘δύναμις’ or ‘capacity’, and

‘πραγματεία’ or ‘discipline’. I should guess that ‘μέθοδος’ is the most likely

supplement, and ‘ἐπιστήμη’ the least likely, in those Aristotelian passages in

which a supplement can plausibly be made.

If Aristotle does not afﬁrm that syllogistic is a science, neither does he

expressly deny it. He shilly-shallies. His later followers made up their minds:

logic is not a science. Alexander—with his eye on the passage from the

Rhetoric which I have just cited—has this to say:

Each of the sciences is concerned with a determinate genus of things, and it shows

and apprehends, by way of principles appropriate to that genus, those items which

are appropriate to the genus and hold of it in their own right. These disciplines [i.e.

dialectic and rhetoric] are not like that. For neither of the two has a single genus as

³ ὅπερ γὰρ καὶ πρότερον εἰρηκότες τυγχάνομεν ἀληθές ἐστιν, ὅτι ἡ ῥητορικὴ σύγκειται

μὲν ἔκ τε τῆς ἀναλυτικῆς ἐπιστήμης καὶ τῆς περὶ τὰ ἤθη πολιτικῆς ...

⁴ συμβαίνει τὴν ῥητορικὴν οἷον παραφυές τι τῆς διαλεκτικῆς εἶναι καὶ τῆς περὶ τὰ ἤθη

πραγματείας, ἣν δίκαιόν ἐστι προσαγορεύειν πολιτικήν. ... περὶ οὐδενὸς γὰρ ὡρισμένου

οὐδετέρα αὐτῶν ἐστιν ἐπιστήμη πῶς ἔχει, ἀλλὰ δυνάμεις τινὲς τοῦ πορίσαι λόγους.

362 The Science of Logic

subject; and whatever they show, they do not show it by way of principles appropriate

to and belonging to the essence of that about which they argue.

(in Top 3.28–4.2)⁵

Dialectic fails to be an Aristotelian science for two reasons: ﬁrst, it is not

concerned with a single class or genus of things—as geometry is concerned

with points, or zoology with animals; and secondly, it does not argue on the

basis of appropriate ﬁrst principles—as arithmetic argues from the axioms

concerning numbers, or botany from those concerning plants. The second

reason does not apply to the probative use of syllogistic, or to ‘apodeictic’;

but the ﬁrst reason holds of syllogistic in general, and there can be little doubt

but that syllogistic, in Alexander’s eyes, was not a science.

But Alexander’s argument seems to be muddled. Allow that any science

must be concerned with some determinate genus of items: then has syllo-

gistic—or dialectic—some such uniﬁed subject-matter? Alexander argues

that it has not, inasmuch as dialectical arguments—or syllogisms in general—

may bear on any subject whatsoever. It is true—and Aristotle says—that

dialectic or syllogistic is ‘topic neutral’, that it can be applied in any and every

ﬁeld of inquiry. But topic neutrality is not to the point. Syllogistic is the study

of syllogisms; and so long as syllogisms form a uniﬁed and a determinate genus

of items, syllogistic may perfectly well be an Aristotelian science. Arithmetic

may be applied in calculations about any items whatsoever, and like logic it

is topic neutral. Nonetheless, it has a subject-matter of its own—numbers,

or units. Similarly for syllogistic.

But if Alexander’s argument is bad, his conclusion might for all that be

true; and true or false, it was, I suppose, an orthodoxy among the ancient

Peripatetics.

THE STRUCTURE OF PREDICATIVE SYLLOGISTIC

Yet if syllogistic was not a science, there were certain close and evident

resemblances between the structure of an ideal or Euclidean science and the

structure of Aristotle’s syllogistic.

⁵ οὐ γὰρ ὡς τῶν ἐπιστημῶν ἑκάστη περί τι γένος ἀφωρισμένον οὖσα τὰ οἰκεῖα ἐκείνῳ τῷ

γένει καὶ καθ᾿ αὑτὰ ὑπάρχοντα δείκνυσί τε καὶ λαμβάνει διὰ τῶν οἰκείων ἀρχῶν ἐκείνῳ τῷ

γένει, οὕτως καὶ αὗται. οὔτε γὰρ ἕν τι γένος τὸ ὑποκείμενον αὐτῶν ἑκατέρᾳ, οὔτε περὶ οὗ

ἂν τὸν λόγον ποιῶνται, διὰ τῶν οἰκείων ἐκείνῳ καὶ ἐν τῇ οὐσίᾳ ὄντων αὐτοῦ δεικνύουσιν ἃ

δεικνύουσιν.

The Structure of Predicative Syllogistic 363

Chapters 4–6 of the ﬁrst Book of the Prior Analytics contain the formal

exposition of Aristotle’s predicative syllogistic—more precisely, of his non-

modal predicative syllogistic. There Aristotle distinguishes between ‘perfect’

(or ‘complete’) syllogisms on the one hand and ‘imperfect’ (or ‘incomplete’)

syllogisms on the other. The perfect syllogisms are the four canonical

syllogisms of the ﬁrst ﬁgure: Barbara, Celarent, Darii, Ferio. All the syllogisms

of the second and third ﬁgures are imperfect.

Aristotle shows that the imperfect syllogisms are valid, and he does so

by reducing them to, or analysing them into, the perfect syllogisms. He

sometimesexplicitlyusestheword‘prove’ofsuchreductiveoranalytical

manoeuvres; and it is in any event plain that he took himself to be proving

the validity of the imperfect syllogisms. For example, the following passage is

evidently offered as a proof of the validity of Darapti, the ﬁrst mood of the

third ﬁgure:

When both P and R hold of every S, P will hold by necessity of some S. For since the

afﬁrmative converts, S will hold of some R, so that since P holds of every S and S of

some R, it is necessary that P hold of some R—for a syllogism comes about through

the ﬁrst ﬁgure.

(APr 28a18–22)⁶

That argument might be set out formally as follows:

(1) P holds of every S premiss

(2) R holds of every S premiss

Therefore (3) S holds of some R 2, conversion

Therefore (4) P holds of some S 1, 3, Darii

That is a proof of Darapti on the basis of Darii. Darapti, then, has the

status of a theorem in Aristotelian syllogistic—or at least, a status comparable

to the status of a theorem—and Darii looks set to have the status of an

axiom.

That in itself is enough to establish the Euclidean, or quasi-Euclidean,

structure of the syllogistic. But there are two further developments which

should be mentioned. The ﬁrst occurs in chapter 23 of the Prior Analytics,

where Aristotle purports to establish that

⁶ ὅταν καὶ τὸ Π καὶ τὸ Ρ παντὶ τῷ Σ ὑπάρχῃ, ὅτι τινὶ τῷ Ρ τὸ Π ὑπάρξει ἐξ ἀνάγκης·

ἐπεὶ γὰρ ἀντιστρέφει τὸ κατηγορικόν, ὑπάρξει τὸ Σ τινὶ τῷ Ρ· ὥστ᾿ ἐπεὶ τῷ μὲν Σ παντὶ τὸ

Π, τῷ δὲ Ρ τινὶ τὸ Σ, ἀνάγκη τὸ Π τινὶ τῷ Ρ ὑπάρχειν· γίνεται γὰρ συλλογισμὸς διὰ τοῦ

πρώτου σχήματος.

364 The Science of Logic

every proof and every syllogism necessarily comes about through the three ﬁgures we

have described.

(APr 41b1–3)⁷

What exactly Aristotle proposes to establish and what (if anything) he actually

does establish, are matters of controversy. But it is indisputable that he thinks

he has shown, at the very least, that a vast number of complex predicative

syllogisms can be reduced to, or proved on the basis of, those syllogisms

whose validity is recognized in chapters 4–6—and hence on the basis of the

perfect syllogisms of the ﬁrst ﬁgure.

There are no examples offered in the chapter; but it is easy to guess

what sort of thing Aristotle has in mind. Consider the following complex

predicative form:

A holds of every B, B holds of every C, C holds of every D: so A holds of

every D.

Is that form valid? Well, it is not a syllogistic form of the sort recognized in

chapters 4–6—if only because it has too many premisses. But it can readily

be shown to be valid on the basis of those forms, thus:

(1) A holds of every B premiss

(2) B holds of every C premiss

(3) C holds of every D premiss

Therefore (4) A holds of every C 1, 2, Barbara

Therefore (5) A holds of every D 3, 4, Barbara

Just as the earlier proof established the validity of Darapti on the basis of

Darii, so this argument establishes the validity of the complex predicative

form on the basis of Barbara.

The second of the two developments I adverted to is set out in chapter 7 of

the Prior Analytics. For there Aristotle proves—and this time he really does

prove—that all the syllogisms of chapters 4–6 can be established on the basis

of the ﬁrst two syllogisms of the ﬁrst ﬁgure, Barbara and Celarent. Since the

imperfect moods of the second and third ﬁgures have already been reduced to

the perfect moods of the ﬁrst ﬁgure, the thesis of chapter 7 will be established

once it is shown that the other two perfect moods, Darii and Ferio, can be

proved on the basis of Barbara and Celarent. This is what Aristotle says:

⁷ … πᾶσαν ἀπόδειξιν καὶ πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τριῶν τῶν προειρη-

μένων σχημάτων.

The Structure of Predicative Syllogistic 365

It is possible to reduce all the syllogisms to the universal syllogisms in the ﬁrst

ﬁgure. … The particular syllogisms in the ﬁrst ﬁgure are indeed perfected through

themselves, but it is also possible to show them by way of the second ﬁgure by

bringing them to the impossible. Thus if A of every B and B of some C, then A of

someC;forifofnone,andofeveryB,BwillholdofnoC—weknowthisbyway

of the second ﬁgure. Similarly with the proof in the privative case: if A holds of no

B and B of some C, then A will not hold of some C. For if it holds of every C and

of no B, then B will hold of no C—that was the middle ﬁgure. Thus since all the

syllogisms in the middle ﬁgure are reduced to the universal syllogisms in the ﬁrst

ﬁgure and the particular syllogisms in the ﬁrst ﬁgure to the syllogisms in the middle

ﬁgure, it is evident that the particular syllogisms in the ﬁrst ﬁgure will be reduced to

the universal syllogisms in the ﬁrst ﬁgure.

(APr 29b1–19)⁸

Universal syllogisms are syllogisms with universal conclusions, particular

syllogisms syllogisms with particular conclusions. Thus Darii and Ferio are

reduced to Barbara and Celarent. For they are proved by way of certain

second ﬁgure syllogisms which are in turn proved by way of Barbara and

Celarent.

The intermediate proof for Darii can be set out like this:

(1) A holds of every B premiss

(2) B holds of some C premiss

(3) A holds of no C hypothesis

Therefore (4) B holds of no C 1, 3, Camestres

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

That proves Darii on the basis of Camestres—which has already been proved

on the basis of Celarent. Aristotle might as well—or better—have proved

Darii directly from Celarent, like this:

(1) A holds of every B premiss

(2) B holds of some C premiss

⁸ ἔστι δὲ καὶ ἀναγαγεῖν πάντας τοὺς συλλογισμοὺς εἰς τοὺς ἐν τῷ πρώτῳ σχήματι καθόλου

συλλογισμούς. ... οἱ δ᾿ ἐν τῷ πρώτῳ, οἱ κατὰ μέρος, ἐπιτελοῦνται μὲν καὶ δι᾿ αὑτῶν, ἔστι

δὲ καὶ διὰ τοῦ δευτέρου σχήματος δεικνύναι εἰς ἀδύνατον ἀπάγοντας· οἷον εἰ τὸ Α παντὶ τῷ

Β,τὸδὲΒτινὶτῷΓ,ὅτιτὸΑτινὶτῷΓ·εἰγὰρμηδενί,τῷδὲΒπαντί,οὐδενὶτῷΓτὸΒ

ὑπάρξει· τοῦτο γὰρ ἴσμεν διὰ τοῦ δευτέρου σχήματος. ὁμοίως δὲ καὶ ἐπὶ τοῦ στερητικοῦ ἔσται

ἡ ἀπόδειξις. εἰ γὰρ τὸ Α μηδενὶ τῷ Β, τὸ δὲ Β τινὶ τῷ Γ ὑπάρχει, τὸ Α τινὶ τῷ Γ οὐχ ὑπάρξει·

εἰ γὰρ παντί, τῷ δὲ Β μηδενὶ ὑπάρχει, οὐδενὶ τῷ Γ τὸ Β ὑπάρξει· τοῦτο δ᾿ ἦν τὸ μὲσον σχῆμα.

ὥστ᾿ ἐπεὶ οἱ μὲν ἐν τῷ μέσῳ σχήματι συλλογισμοὶ πάντες ἀνάγονται εἰς τοὺς ἐν τῷ πρώτῳ

καθόλου συλλογισμούς, οἱ δὲ κατὰ μέρος ἐν τῷ πρώτῳ εἰς τοὺς ἐν τῷ μέσῳ, φανερὸν ὅτι καὶ

οἱ κατὰ μέρος ἀναχθήσονται εἰς τοὺς ἐν τῷ πρώτῳ σχήματι καθόλου συλλογισμούς.

366 The Science of Logic

(3) A holds of no C hypothesis

Therefore (4) C holds of no A 3, conversion

Therefore (5) C holds of no B 1, 4, Celarent

Therefore (6) B holds of no C 5, conversion

Therefore (7) A holds of some C 2, 6, impossibility

The last steps in those two proofs, which rely on the type of inference nor-

mally called reduction to the impossible, require some explanation. It must

be admitted that Aristotle never explains how a reduction to the impossible

works, and that his followers have some pretty confused things to say on the

matter; but in principle a satisfactory explanation can be provided.

So Aristotle’s syllogistic has the structure of an axiomatized deductive

science: even if, as Alexander urges, it is not itself a genuine science, it makes

a good job of imitating one. The theorems, or quasi-theorems, of syllogistic

include all the imperfect syllogisms which Aristotle considers and also the two

particular perfect syllogisms. But they also include many more—inﬁnitely

many more—syllogisms. For the three-premissed argument which I set out

a few pages ago is a theorem, and so too are its four-premissed siblings,

and its ﬁve-premissed siblings, … And in addition, there are any number of

other three-premissed and four-premissed and ﬁve-premissed … predicative

syllogisms.

Historians of logic sometimes speak as though Aristotle’s syllogistic were a

strictly ﬁnite system embracing between fourteen and twenty-four argument

forms. In other words, they take chapters 4–6 of the ﬁrst Book of the Prior

Analytics—together with a few additions—to constitute the sum of (non-

modal) predicative syllogistic. That is at best misleading; for the syllogisms

set out in chapters 4–6 constitute only a fraction of the syllogistic system.

Historians of logic sometimes speak as though the basis of the system—the

set of axioms, or quasi-axioms, of the syllogistic—consists of the four perfect

syllogisms of the ﬁrst ﬁgure. Thus when later commentators speak of

Aristotle’s logical system they follow him in saying that all syllogisms have

been reduced to the perfect syllogisms of the ﬁrst ﬁgure: given those four

syllogisms—or better, given Barbara and Celarent—the whole syllogistic

may take to the air. That too is at best misleading. It is true that all the

other predicative syllogisms can be proved on the basis of a set of axioms

the only predicative syllogisms among which are Barbara and Celarent. But

it does not follow from that, and it is not in fact the case, that all the other

predicative syllogisms can be proved on the basis of Barbara and Celarent. It

The Structure of Predicative Syllogistic 367

is true, in other words, that the only syllogisms you need to take as axiomatic

are Barbara and Celarent. It is not true that the only axioms you need are

Barbara and Celarent.

And that should have been obvious enough. After all, in chapter 3 of the

ﬁrst Book of the Prior Analytics, Aristotle himself sets out the conversions

on which the majority of the proofs in chapters 4–6 will depend. Thus the

proof of Darapti which I cited depends not only upon Darii but also upon

the conversion rule which allows you to infer from something of the form ‘A

holds of every B’ to the corresponding item of the form ‘B holds of some A’. It

is equally evident that Aristotle’s syllogistic depends on some rule or principle

of reduction to the impossible; and that rule itself presupposes certain

further logical principles—principles of contradiction and principles of

subordination, among them the thesis that a universal afﬁrmative proposition

entails the corresponding particular afﬁrmative proposition.

The upshot is this. Aristotle’s predicative syllogistic is, or can be recon-

structed as, an axiomatized deductive system the axioms (or quasi-axioms)

of which are two syllogistic forms, certain principles of conversion and of

subordination, a principle of reduction to the impossible, and a rule of

exposition or ecthesis. And the theorems (or quasi-theorems) are certain

derived principles of conversion and subordination—and an inﬁnite number

of syllogisms.

That structure was implicitly acknowledged by the ancient logicians. For

example, in later texts a perfect syllogism is frequently called ‘primary’

or ‘πρῶτος’, and also ‘unproved’ or ‘ἀναπόδεικτος’. So Galen remarks

that

there are three ﬁgures for predicative propositions and several syllogisms in each of

them, … some of which are unproved and primary …

(inst log viii 1)⁹

And Alexander speaks in the same voice—for example, he observes that

we shall reduce imperfect syllogisms ‘into one of the perfect and unproved

syllogisms in the ﬁrst ﬁgure’ (in APr 24.4–5).¹⁰ Now in the Posterior Analytics

Aristotle had stated that any science must

⁹ τριῶν οὖν ὄντων σχημάτων ἐν ταῖς κατηγορικαῖς προτάσεσι καθ᾿ ἕκαστον αὐτῶν

γίγνονται συλλογισμοὶ πλέονες ... ἔνιοι μὲν ἀναπόδεικτοι καὶ πρῶτοι ...

¹⁰ … εἴς τινα τῶν ἐν τῷ πρώτῳ σχήματι τῶν τελείων καὶ ἀναποδείκτων.

368 The Science of Logic

depend on primary unproveds, because otherwise, not having a proof of the items,

you will not know them. For to know (non-accidentally) something of which there

is a proof is to have a proof of it.

(APst 71b26–29)¹¹

The axioms or ﬁrst principles of any science are primary and they are

unproved. The perfect syllogisms of Aristotelian syllogistic were later charac-

terized as primary and unproved. Evidently, such a characterization intimates,

and was meant to intimate, that the perfect syllogistic forms are axioms, or

quasi-axioms, for the science, or quasi-science, of syllogistic.

The perfect forms are quasi-axioms, the imperfect forms are quasi-

theorems: why the cautious ‘quasi’? Why not speak boldly of axioms and

theorems? The axioms of a science are the fundamental truths on which

the remaining truths of the science depend. Axioms are propositions, and

so too are theorems. Concrete syllogisms are not propositions: they are

sets or sequences of propositions. Afortiori, syllogistic forms are not syllo-

gisms: neither Darii nor Darapti is a proposition, nor even a sequence of

propositions—so Darii cannot be a genuine axiom nor Darapti a genuine

theorem.

But that is needlessly pedantic, both on an historical and on a philosophical

count. For ﬁrst, some ancient theorists took expressions of the form ‘X, but

Y: so Z’ to constitute a single saying; and when Aristotle characterized a

syllogism as a λόγος in which certain things are thus-and-so, they will readily

have understood the word ‘λόγος’ to mean ‘saying’ rather than ‘argument’.

Secondly, even if syllogisms are not propositions or sayings, why should

that disqualify them from forming part of an axiomatized science? Euclid’s

Elements contains numerous theorems which are not propositions. Indeed,

the very ﬁrst theorem of the ﬁrst Book of the Elements is this:

On a given ﬁnite straight line, to construct an equilateral triangle.

(i i)¹²

That is a construction; and if constructions may count as theorems, then why

may not syllogisms or argument forms do so? Thirdly, it is a simple matter to

construct a syllogistical science the axioms and theorems of which are honest

to God propositions. For example, you might take the circumscriptions of

¹¹ ἐκ πρώτων δ᾿ ἀναποδείκτων, ὅτι οὐκ ἐπιστήσεται μὴ ἔχων ἀπόδειξιν αὐτῶν· τὸ γὰρ

ἐπίστασθαι ὧν ἀπόδειξις ἔστι μὴ κατὰ συμβεβηκός, τὸ ἔχειν ἀπόδειξίν ἐστιν.

¹² ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι.

‘Primary Unproveds’ 369

the perfect forms as your axioms, and then derive the circumscriptions of

the imperfect forms as theorems. Or you might use schemata in one way or

another—‘Every syllogism which is an instance of the schema S is valid’, …

‘PRIMARY UNPROVEDS’

Aristotle afﬁrms that the principles on which scientiﬁc proofs are based must

be ‘primary unproveds’ (APst 71b26). At ﬁrst glance, that phrase seems

pleonastic: ‘primary’ and ‘unproved’ may not be synonymous expressions;

but is not an item primary if and only if it is unproved?

If an item is primary, is it unproved? Yes. An item is primary if and only if

there is no item prior to it; or better: a member of a set of items is primary in

the set if and only if no item in the set is prior to it. The type of priority which

Aristotle here has in mind is priority in knowledge, or epistemic priority.

Roughly speaking, to say that x is prior in knowledge to y is to say that x may

feature as a premiss in a proof of y (whereas y cannot feature as a premiss in

a proof of x). It follows that if x is primary, then it cannot be proved; for

there is nothing prior to x which might feature as a premiss in a proof of x.

Or better: if x is primary in a set S, then there is no proof of x which uses a

member of S as a premiss.

If an item is unproved, is it primary? The answer to that question depends

on how we construe the word ‘unproved’—or rather, on how we construe the

Greek word which I have used it to translate. That word is ‘ἀναπόδεικτος’.

The word is a member of a large family of negative verbal adjectives. The

members of the family are all formed from an ‘alpha privative’, a verbal

root, and a termination in -τος: they have the construction ‘ἀ-verb-τος’.

They are all systematically ambiguous, having two chief senses: ‘ἀ-verb-τος’

may mean either ‘unverbable’ or ‘unverbed’. The ambiguity was noticed by

Aristotle, and by the Stoics after him; it is remarked upon by Galen, and

by several other ancient authors. Any Peripatetic or any Stoic who used the

word ‘ἀναπόδεικτος’ in his logical theory will presumably have realized that

his words might be taken in different ways; he will presumably have intended

them to be taken in one determinate way; and—if he was sensible—he will

have indicated the way in which he intended them to be taken.

The ancient texts do indeed show that there was some explicit comment

upon the sense of ‘ἀναπόδεικτος’ in logical contexts. Yet the indications

which they offer are surprising. According to Sextus,