Barnes Jonathan. Truth, etc

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340 Forms of Argument

You should not think that any absurdity results from the fact that a certain item is set

out; for we do not in the least make use of the fact that this is such-and-such—rather,

we are like the geometer who speaks of this foot line and this straight line and this

breadthless line although they are not so, but who does not use them as syllogizing

from them. … We use setting out as we also use perception, in the interest of the

learner.

(APr 49b33–50a2)⁶³

The text and the translation are in places disputed; and what Aristotle means

is, even in its most general outline, contested. But Alexander himself found

no general difﬁculties:

By ‘setting out’ he means making a diagram of the terms. Since in the exposition of

the syllogisms he used letters instead of terms and showed which conjugations were

syllogistic and which non-syllogistic by means of them, he now comments on the

fact, saying that we should not suppose that because of this way of taking or setting

out terms anything false or absurd results, as though it were the taking of the letters

which were responsible for something’s seeming to be shown or not shown to lead to

a conclusion—just as often things are shown to conclude to something because of

their matter, although they are not syllogistic.

‘For we do not in the least make use’ in taking the letters (and proof according to

the logicians depends on items being thus-and-so related to one another—for it is

when one is whole and another part)—we make no use of the kinship of the terms

to one another so as to show the conclusion by means of that (as for example that

this is a genus of that, or a property or a deﬁnition), as we would if we set down

the matter. For the letters themselves are taken merely as common signs of the terms

and they contribute nothing in themselves toward showing that the conjugation is

concludent or non-concludent. For just as a geometer, for the sake of clarity in

exposition, makes a diagram and says ‘Let this be a foot line’, or ‘Let this be straight’,

but does not assume that the foot line is a foot long or that the straight line is

straight and does not make any use of the diagrammed items in proving what is

before him—rather, he uses them as signs which contribute nothing and introduce

nothing to the demonstrandum (for he can prove what is before him just as well

without drawing these lines and without making any use of them—rather, he takes

them up in order that what he says may be easy to follow—so that the intellect,

being able as it were to repose upon them, may more easily follow); in the same way,

we set out letters, which themselves introduce nothing into the demonstrandum.For

⁶³ οὐ δεῖ δ᾿ οἴεσθαι παρὰ τὸ ἐκτίθεσθαί τι συμβαίνειν ἄτοπον· οὐδὲν γὰρ προσχρώμεθα τῷ

τόδε τι εἶναι, ἀλλ᾿ ὥσπερ ὁ γεωμέτρης τὴν ποδιαίαν καὶ εὐθεῖαν τήνδε καὶ ἀπλατῆ εἶναι λέγει

οὐκ οὔσας, ἀλλ᾿ οὐχ οὕτως χρῆται ὡς ἐκ τούτων συλλογιζόμενος. ... τῷ δ᾿ ἐκτίθεσθαι οὕτω

χρώμεθα ὥσπερ καὶ τῷ αἰσθάνεσθαι, τὸν μανθάνοντ᾿ ἀλέγοντες.

Peripatetic Letters 341

the inference does not depend on one of them’s being A and another B or C—the

same thing results if we use other letters instead of them.

This is not so in the case of

Every man is an animal.

Every item capable of laughter is an animal.

For from these it seems that it can be concluded that every man is capable of laughter.

But that is because of a certain relation of these particular terms to one another, not

because of the ﬁgure. For if other terms are taken in the same conjugation, nothing

is concluded—say in the case of

Every man is an animal.

Every horse is an animal.

But when letters are set out, it is not like that—as Aristotle indicated by his use of

different letters in the different ﬁgures. For in the case of letters, you cannot take

one as whole and one as part, as in the case of animate and animal (where one is

whole—animate extends further and is universal—and the other part); and again,

if something else so related to animal, which was taken as a part in the ﬁrst premiss

‘is taken as part to whole’—e.g. man (for man is a part of animal which was a part

of animate), it is from items which are thus related—that is, one of the terms being

predicated and one subject in the premisses—that proofs depend. For if items have

no kinship with one another, it is never possible to show that anything concludes

from them syllogistically, and letters have no such relation to one another. Hence it

is not because of the letters that something results or does not result. For that reason,

the proofs are done by means of them. And that could not be said were we to conduct

the proofs by means of the matter which we use in our syllogisms. For because of its

particularities, it often appears that something is concludent when it is not.

‘So that there is no syllogism’—he has made it clear that proofs done with such

letters are delineations of syllogistic modes, and not themselves syllogisms. For a

syllogism includes the matter about which something is proved.

(in APr 379.14–380.27)⁶⁴

⁶⁴ ἔκθεσιν μὲν λέγει τὴν τῶν ὅρων καταγραφήν. ἐπεὶ δὲ ἐν τῇ τῶν συλλογισμῶν παραδόσει

κέχρηται τοῖς στοιχείοις ἀντὶ τῶν ὅρων καὶ ἐπ᾿ αὐτῶν δέδειχε τάς τε συλλογιστικὰς συζυγίας

καὶ τὰς ἀσυλλογίστους, νῦν λέγει περὶ τούτου ὅτι μὴ παρὰ τὴν τοιαύτην λῆψιν τῶν ὅρων καὶ

ἔκθεσιν ὑπολαμβάνειν χρὴ ἄτοπόν τι καὶ ψεῦδος συμβαίνειν ὡς τῆς τῶν στοιχείων λήψεως

αἰτίας γινομένης τοῦ δοκεῖν δείκνυσθαί τι συνάγον ἢ μὴ δείκνυσθαι, ὡς πολλάκις δείκνυταί

τινα παρὰ τὴν ὕλην συνάγοντά τι οὐκ ὄντα συλλογιστικά.

οὐδὲν γὰρ προσχρώμεθα ἐν τῇ διὰ τῶν στοιχείων λήψει (ἐκ δὲ τῶν οὕτως ἐχόντων

πρὸς ἄλληλα ἡ κατὰ τοὺς συλλογιστικοὺς δεῖξις· ὅταν γὰρ τὸ μὲν ὡς ὅλον τὸ δὲ ὡς

μέρος) τῇ οἰκειότητι τῇ πρὸς ἀλλήλους τῶν ὅρων ὡς διὰ τούτου δεικνύναι τὸ συναγόμενον,

οἷον ὅτι τόδε τοῦδέ ἐστι γένος ἢ τόδε τοῦδέ ἐστιν ἴδιον ἢ ὁρισμός, ὥσπερ ἂν εἰ τὴν ὕλην

παρετιθέμεθα· τὰ γὰρ στοιχεῖα αὐτὰ μόνον σημεῖα κοινὰ τῶν ὅρων εἴληπται οὐδὲν παρ᾿

αὑτῶν συντελοῦντα εἰς τὸ ἢ συνακτικὴν δειχθῆναι τὴν συζυγίαν ἢ ἀσύνακτον. ὡς γὰρ ὁ

γεωμέτρης ὑπὲρ σαφηνείας τῆς κατὰ τὴν διδασκαλίαν καταγραφὴν ποιεῖταί τινα καὶ λέγει

342 Forms of Argument

Despite the numerous oddities in those paragraphs—some of which my

translation has glossed over—some things emerge clearly enough.

First, Alexander takes Aristotle to be talking about his use of letters in

proving logical theses. Secondly, he thinks that there is a parallel between

Aristotle’s use of letters and the geometers’ use of diagrams. Thirdly, he

supposes that the letters stand in no logical relations to one another, and

hence cannot be responsible for the validity of any inference. Fourthly, he

suggests that it is for that reason that letters may be used to prove universal

logical theorems.

The ﬁrst point is surely a wayward interpretation of what Aristotle says in

the passage under scrutiny. Nonetheless, Alexander may be offering a correct

interpretation of Aristotle’s use of letters, even if he attaches the interpretation

to an impertinent text. In any event, it is Alexander’s interpretation which

concerns me at the moment.

The second point may seem equally wayward: surely the comparison at

which Aristotle hints ought to be not between the logician’s use of letters and

the geometer’s use of diagrams but rather between the logician’s use of letters

ἔστω ποδιαία ἥδε ἢ ἔστω εὐθεῖα ἥδε οὔτε τὴν ποδιαίαν ποδιαίαν λαμβάνων οὔτε τὴν εὐθεῖαν

εὐθεῖαν, οὐδὲ τοῖς καταγεγραμμένοις προσχρώμενος δείκνυσιν αὐτῷ τὸ προκείμενον, ἀλλὰ

τούτοις σημείοις χρῆται οὐδὲν συντελοῦσιν οὐδὲ συνεισφέρουσι πρὸς τὸ δεικνύμενον (οὐδὲν

γὰρ ἔλαττον καὶ μὴ καταγράψας ταύτας μηδὲ προσχρησάμενος αὐταῖς δύναται δεῖξαι τὸ

προκείμενον, ἀλλ᾿ ὑπὲρ τοῦ εὖ παρακολουθῆσαι ἐν τοῖς λεγομένοις λαμβάνει ταῦτα ἵν᾿ ἔχουσά

πως ἡ διάνοια ἐπαναπαύεσθαι τούτοις ῥᾷον παρακολουθῇ), οὕτως καὶ ἡμεῖς τῶν στοιχείων

τὴν ἔκθεσιν πεποιήμεθα οὐδὲν ἡμῖν εἰς τὰ δεικνύμενα παρ᾿ αὑτῶν συνεισφερόντων. οὐ γὰρ

παρὰ τὸ τὸ μὲν Α αὐτῶν εἶναι τὸ δὲ Β ἢ Γ ἡ συναγωγή· τὸ γὰρ αὐτὸ γίνεται, κἂν ἄλλοις ἀντὶ

τούτων χρησώμεθα.

ὃ οὐ γίνεται ἐπὶ τοῦ πᾶς ἄνθρωπος ζῷον, πᾶν γελαστικὸν ζῷον· ἐκ γὰρ τούτων συνάγεσθαι

δοκεῖ τὸ πάντα ἄνθρωπον γελαστικὸν εἶναι. ἀλλὰ τοῦτο διὰ τὴν τῶν εἰλημμένων ὅρων σχέσιν

ποιὰν πρὸς ἀλλήλους, οὐ διὰ τὸ σχῆμα· ἄλλων γὰρ ὅρων ἐν τῇ τοιαύτῃ συζυγίᾳ ληφθέντων

οὐδὲν συνάγεται, ὥσπερ ἐπὶ τῶν πᾶς ἄνθρωπος ζῷον, πᾶς ἵππος ζῷον. ἐπὶ δὲ τῆς τῶν

στοιχείων ἐκθέσεως οὐχ οὕτως· ὃ ἔδειξε καὶ αὐτὸς ἄλλοτε ἄλλοις χρησάμενος καθ᾿ ἕκαστον

σχῆμα. οὐ γὰρ ἔστιν ἐπὶ τῶν στοιχείων ἵνα τὸ μὲν ὡς ὅλον τὸ δ᾿ ὡς μέρος τούτου ληφθῇ,

ὡς ἔχει τὸ ἔμψυχον καὶ τὸ ζῷον· τὸ μὲν γὰρ ὡς ὅλον ἐστίν (ἐπὶ πλέον γὰρ καὶ καθόλου

τὸ ἔμψυχον), τὸ δ᾿ ὡς μέρος. καὶ πάλιν ἂν ἄλλο τι πρὸς τοῦτο τὸ ζῷον ὃ ἦν ὡς μέρος

εἰλημμένον ἐν τῇ πρώτῃ προτάσει ὡς μέρος πρὸς ὅλον ληφθῇ, οἷον ὁ ἄνθρωπος (μέρος

γὰρ τοῦ ζῴου, ὃ ἦν μέρος τοῦ ἐμψύχου), ἐκ τῶν οὕτως ἐχόντων, τοῦτ᾿ ἔστιν ὡς τὸν μὲν

κατηγορεῖσθαι τὸν δὲ ὑποκεῖσθαι τῶν ὅρων ἐν ταῖς προτάσεσιν, αἱ δείξεις. ἐξ οὐδενὸς γὰρ

τῶν ἃ μὴ ἔχει πρὸς ἄλληλα οἰκειότητα οἷόν τέ τι δειχθῆναι συναγόμενον συλλογιστικῶς. τὰ

δὲ στοιχεῖα οὐδεμίαν τοιαύτην ἔχει σχέσιν πρὸς ἄλληλα. οὔκουν παρὰ ταῦτα ἢ συμβαίνει τι

ἢ οὐ συμβαίνει. διὸ καὶ ἐπὶ τοιούτων αἱ δείξεις, ὃ οὐκέτ᾿ ἐνῆν λέγειν εἰ ἐπὶ ὕλης ἡμῖν ἐφ᾿ ὧν

χρώμεθα τοῖς συλλογισμοῖς αἱ δείξεις ἐγίνοντο· παρὰ γὰρ τὴν ταύτης διαφορὰν πολλάκις

συνακτικόν τι φαίνεται οὐκ ὂν τοιοῦτον.

ὥστ᾿ οὐδὲ γίνεται συλλογισμός· ὅτι αἱ ἐπὶ τῶν τοιούτων στοιχείων δείξεις ὑπογραφαὶ

συλλογιστικῶν εἰσι τρόπων, οὐ μὴν ἤδη συλλογισμοί, ἐδήλωσεν· ὁ γὰρ συλλογισμὸς μετὰ

τῆς ὕλης ἐφ᾿ ἧς τι δείκνυται.

Peripatetic Letters 343

and the geometer’s use of letters? No doubt it ought to be; but Alexander’s

comparison is constrained by Aristotle’s text; and in fact Alexander indicates

two points of comparison. First, the logical letters, like the geometrical

diagrams, are not essential to the proofs: they are feather-bedding, they offer

the tired intellect something to repose on. Euclid does not need his diagrams,

and his proofs are not about the diagrams. Aristotle does not need his

letters. Secondly, the letters and the diagrams introduce—or rather, might

introduce—falsities into the presentation of the proofs; but those falsities do

not infect the proofs themselves—the proofs do not trade on the falsities.

When a geometer says something like ‘Let the line AB be a foot long’, it is

absurd—according to Alexander—to object that in fact the line is not a foot

long. For whether or not it is in fact a foot long is of no relevance to the

proof. The geometer does not use the line for its foot length. In the same

way—Alexander insinuates—when Aristotle says ‘Let A be said of every B

and of no C’, it is absurd to object that in fact A is not so said of B and C.

For whether or not in fact things are so is of no relevance for the proof. The

logician does not use the actual relations which hold among his terms.

At least, that appears to be one of the things which Alexander must have

had in mind in making his second point. But it is scarcely consistent with

the third point, according to which the logical letters in fact stand in no

relation to one another. Is A really said of every B, as Aristotle seems to assert?

No, of course not—the letters ‘A’ and ‘B’ do not stand in any pertinent

relation to one another, they are not items which could in principle stand

in a predicative relation to one another. And that very fact constitutes one

aspect of their utility. If a syllogistic form is presented by way of a concrete

argument—by a paradigm, as we might say—then you might wonder if its

validity did not depend upon the relations which hold among its concrete

terms. If a syllogistic form is presented schematically, by way of letters, then

you cannot coherently entertain such a suspicion.

The problem is this: Alexander appears to suggest, on the one hand, that

Aristotle’s letters mean nothing at all—that they are empty signs; and on the

other hand, that the letters have a sense but that their sense is not used in

the arguments in which they appear. It is hard to reconcile those two hands.

In addition, it is hard to see how exactly either of them might bear upon

the fourth of the four points—the claim that the letters serve to introduce

generality into the proofs.

A passage in Boethius’ essay On Predicative Syllogisms—which was ﬁrmly

based upon if not literally translated from a Greek original—brings out the

344 Forms of Argument

same difﬁculty. (The Latin in the only edition which I have seen is in parts

ungrammatical, and I have tacitly emended it; but I cannot say that I have

understood all those parts which are grammatically impeccable.)

Whenever we speak in such a way as to set down letters instead of terms, we do so for

the sake of brevity and concision. That which we want to show universally, we show

by means of letters. For with terms it is perhaps inevitable that some falsity will slip

in. But with letters we are never deceived. That is why we use letters for the purpose,

as though we were setting down terms. But in the letters themselves no truth and

no falsity may be found unless the collocation of the terms is ﬁxed and sound. So

whenever we want to show that one thing is predicated of all another, we set it out

thus: let the ﬁrst term be A and the second B and let A be predicated of every B. But

let this be so construed as if we have set down A for animal and B for man.

(syll cat 810cd)⁶⁵

If you use concrete terms, then you may be misled into taking something

to hold universally when in fact it holds only for a limited number of

terms. If you use letters, you avoid the danger. Letters inhibit the intrusion

of falsity. They do not do so because they introduce nothing but truth:

they do so because they introduce neither falsity nor truth. Since ‘A holds

of every B’ is neither false nor true, it cannot introduce falsity into an

argument.

That much seems clear in the text, and it corresponds to Alexander’s claim

that letters stand in no pertinent relations to one another. But then Boethius

adds that letters will not introduce truth and falsity unless there is a proper

combination of terms. Yet if letters, in themselves, exclude both falsity and

truth, then how can a proper combination of them introduce a truth? And if it

can introduce a truth, then why cannot it equally introduce a falsity? Finally,

Boethius says that when we say ‘A holds of every B’ we should construe that

as though it were ‘Animal holds of every man’. But if that is how we should

understand ‘A holds of every B’, then the use of letters is not, after all, a way

of escaping whatever dangers concrete terms may threaten—it is a way of

hiding the fact that you are courting those dangers.

⁶⁵ quotienscumque ita dicimus ut litteras pro terminis disponamus, pro brevitate hoc et compendio

facimus. id quod demonstrare volumus universaliter, per litteras demonstramus. nam fortasse in terminis

aliquid falsum ingerendum necesse sit: in litteris vero numquam fallimur, quoniam ad hoc utimur litteris

quasi terminos poneremus. in litteris vero ipsis, nisi terminorum coniunctio per se ﬁrma valensque fuerit,

neque veritas neque falsitas reperietur. quotiens igitur aliud de alio omni praedicari volumus ostendere,

sic ponimus: sit primus terminus A, secundus B, et praedicetur A de omni B. hoc autem ita accipito

tamquam si posuerimus A animal, B hominem.

Peripatetic Letters 345

Take again the Aristotelian sentence which I quoted a few pages ago:

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

(APst 82b14–15)

That is supposed to express something universal—and to express a universal

truth. But the letters it contains apparently designate nothing and apparently

have no sense; and in that case the sentence in which they feature has

no sense—and afortiorino truth-value. It is not a sentence at all, but a

schema for a sentence. So there is a dilemma: express the point in terms of

concrete propositions and you will not say anything universal; chip out the

concrete and you will not say anything at all. The dilemma underlies the

incoherence which appears to infect the halting explanations which Alexander

and Boethius propose for Aristotle’s use of syllogistic letters.

Or is there really a dilemma? When an algebraist writes

x + y = y + x,

he says something true and universal, and he does so despite the fact that his

‘x’ and his ‘y’ designate nothing and have no sense of their own. He does so,

we generally think, inasmuch as his letters are what the logicians call variables,

and insofar as a standing convention requires his formula to be understood as

though it were universally quantiﬁed. In other words, his formula is a short

form of the convenient barbarism:

For any x and any y, x + y = y + x.

What that barbarism expresses is peculiarly difﬁcult to express in decent

English—that is why the barbarism is a boon. But it is somehow clear what

the barbarism means, clear that it is a genuine sentence, and clear that it

expresses a truth.

Why not say the same for Aristotle’s syllogistical letters? Scholars have

done so. Thus when Aristotle writes

If B holds of every A and of no C, A holds of none of the Cs,

it has been urged that we should take the letters to be variables and the

sentence to carry, implicitly, a trio of universal quantiﬁers. In other words, it

is an abbreviated version of the barbarism:

For any A, B, and C: if B holds of every A and of no C, then A holds of

no C.

It is difﬁcult to express that proposition in intelligible English—or in

intelligible Greek. But it is clear what it means, and it is clear that what it

says is true. And there is no mystery about how the sentence manages to say

something universal in scope.

346 Forms of Argument

In many passages in the Analytics the right universal truths are found

in the text once the letters are construed as variables and read with a tacit

quantiﬁer. True, the notion of a variable (so far as I know) is not found in

ancient logic before the sixth century; and as for Aristotle, he never produces

any sentences like the barbarous ‘For any A, B and C …’. But that is only

to say that Aristotle did not trouble to explain—perhaps he was not able to

explain—his own use of syllogistic letters.

Should we then conclude that the letters which Aristotle uses so abundantly

in his Analytics sometimes stand in for determinate predicates and sometimes

function as variables? Perhaps; but there are also numerous texts—and

important and central texts—in which the Greek letters appear to be neither

determinate nor variable. Here is an example:

In the second ﬁgure, if the negative premiss is necessary, the conclusion too will be

necessary; but if the afﬁrmative, the conclusion will not be necessary. First, let the

negative be necessary, and let A be possible for no B, and let it hold simply of C.

Since the negative converts, B is possible for no A. But A holds of every C, so that B

is possible for no C—for C is under A.

(APr 30b7–13)⁶⁶

‘Let A be possible for no B’: what is the status of the letters in that sentence?

Does ‘A’ perhaps mean ‘planets’? or ‘plants’? or ‘planes’? Aristotle does not

tell us—and is that not because there is nothing to tell? Surely the letters do

not here stand for concrete predicate expressions? But neither can they be

construed as variables which carry unspoken universal quantiﬁers. After all,

Aristotle does not mean:

For any A and any B, let A be possible for no B.

His proof does not depend on any such bizarre supposition. Nor, even more

evidently, when he says ‘A holds of every C’ does he mean to afﬁrm that

everything holds of everything.

What is true of that text is true of all the texts in which Aristotle proves

the validity of a syllogistic form.

⁶⁶ ἐπὶ δὲ τοῦ δευτέρου σχήματος, εἰ μὲν ἡ στερητικὴ πρότασίς ἐστιν ἀναγκαία, καὶ τὸ

συμπέρασμα ἔσται ἀναγκαῖον, εἰ δ᾿ ἡ κατηγορική, οὐκ ἀναγκαῖον. ἔστω γὰρ πρῶτον ἡ

στερητικὴ ἀναγκαία, καὶ τὸ Α τῷ μὲν Β μηδενὶ ἐνδεχέσθω, τῷ δὲ Γ ὑπαρχέτω μόνον. ἐπεὶ

οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῷ Α οὐδενὶ ἐνδέχεται· τὸ δὲ Α παντὶ τῷ Γ ὑπάρχει,

ὥστ᾿ οὐδενὶ τῷ Γ τὸ Β ἐνδέχεται· τὸ γὰρ Γ ὑπὸ τὸ Α ἐστίν.

Geometrical Letters 347

Perhaps, nevertheless, the letters can be taken as variables provided that the

tacit quantiﬁers are given the broadest possible scope? We should not supply

‘For any A, B, C, …’ in front of each sentence of the passage I have just

quoted: rather, it should be supplied at the beginning of the whole paragraph,

its scope should extend to the end of the argument. Something of that sort

can indeed be done. But in order to do it successfully you must prepare the

ground in one of two ways: either you must recast Aristotle’s text in such

a way that the sequence of sentences with which he expresses his proofs is

replaced by a single complex sentence; or else you must develop a new way

with quantiﬁers which allows them to govern units of discourse of more than

one sentence in length.

You must do something like that, and you could do something like that.

But why bother? After all, Aristotle’s proofs—for example, the proof of

Darapti which I set down a few pages ago—seem to be logically respectable

in the form in which he sets them out. So ought there not to be some other

way of taking his letters?

GEOMETRICAL LETTERS

There are in principle several other ways of construing Aristotle’s logical

letters; but the way which will ﬁrst come into the mind of any commentator

on Aristotle takes its inspiration from geometry. Aristotle knew his geometry;

it is clear that he took some of his logical terminology from the geometers;

and—as I have already suggested—it is highly plausible to think that he

decided to use letters in his syllogistic because he knew how useful letters

were in geometry. And of course, even if that biographical conjecture is

mistaken, the geometrical use of letters might nevertheless illuminate the

logical use.

There are, to be sure, some differences between Aristotelian letters and

geometrical letters. For example, when Euclid writes ‘τὸ Α’ or ‘the A’, that is

elliptical: it stands for ‘τὸ Α σημεῖον’or‘thepointA’.Inthesameway‘ἡ

ΑΒ’ or ‘the AB’ is short for ‘ἡ ΑΒ γραμμή’ or ‘the line AB’. And so on. As

often as not, the noun to which the deﬁnite article belongs will be explicit in

the text. When Aristotle writes ‘the A’, there is no noun to be supplied. But

such differences are superﬁcial—at any rate, I shall now glance at Euclid’s

use of letters.

348 Forms of Argument

Doubtless, the Greek geometers used their letters in more ways than one;

but one way is particularly striking. Here is a simple example:

On the given ﬁnite straight line, to construct an equilateral triangle.—Let the given

ﬁnite straight line be AB. Then we must construct an equilateral triangle on AB. Let

a circle ABC have been drawn, with centre A and radius AB, and again let a circle

ACE have been drawn, with centre B and radius BA …

(Euclid, i i)⁶⁷

Here the individual letters pick out geometrical points. Pairs of letters pick

out lines, namely the lines bounded by the two points which the letters pick

out. And so on. Aristotle too uses both individual letters and groups of letters.

Usually, an individual Aristotelian letter will pick out a term, whereas a pair

of letters will pick out a proposition; and the formula ‘AB’ will normally pick

out a proposition the terms of which are indicated by ‘A’ and ‘B’.

Euclid introduces his letters without any explanation. So far as I know,

his ancient commentators do not discuss them—nor suggest that there is

anything about them which needs discussion. But they understood what was

going on. Here is Proclus:

They usually make the conclusion in a certain fashion double—for having shown

it for the given case they also infer it as a universal, running up from the particular

conclusion to the universal. For because they do not make use of the peculiarities of

the subjects but draw the angle or the straight line merely in order to put the given

item before our eyes, they think that the same thing which was concluded in that

case has been concluded for every similar case. Hence they transfer to the universal so

that we may realize that the conclusion is not particular, and they make the transfer

with reason inasmuch as in the proof they make use of the items set out not as these

items but as similar to the others. For it is not insofar as the angle has such and such

a degree that I divide it in two but simply insofar as it is a rectilineal angle.

(in Eucl 207.4–18)⁶⁸

Not everything is perfect in that paragraph; but the main lines are discern-

ible—and in addition, it is evident that there is at least a superﬁcial or

⁶⁷ ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι. ἔστω

ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ. δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον

συστήσασθαι. κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ὁ ΒΓ∆, καὶ πάλιν

κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ὁ ΑΓΕ.

⁶⁸ τό γε μὴν συμπέρασμα διπλοῦν εἰώθασι ποιεῖσθαί τινα τρόπον· καὶ γὰρ ὡς ἐπὶ τοῦ

δεδομένου δείξαντες καὶ ὡς καθόλου συνάγουσιν ἀνατρέχοντες ἀπὸ τοῦ μερικοῦ συμπεράσ-

ματος ἐπὶ τὸ καθόλου. διότι γὰρ οὐ προσχρῶνται τῇ ἰδιότητι τῶν ὑποκειμένων, ἀλλὰ πρὸ

ὀμμάτων ποιούμενοι τὸ δεδομένον γράφουσι τὴν γωνίαν ἢ τὴν εὐθεῖαν, ταὐτὸν ἡγοῦνται τὸ

ἐπὶ ταύτης συναγόμενον καὶ ἐπὶ τοῦ ὁμοίου συμπεπεράνθαι παντός. μεταβαίνουσι μὲν οὖν

Geometrical Letters 349

terminological similarity between what Proclus said there about geometry

and what Alexander said about logic in the long passage which I quoted

earlier.

Proclus says this: Euclid will generally ﬁrst prove a conclusion about the

particular lines or angles which he has set out; he will then infer the appropriate

universal proposition; and he will be justiﬁed in doing so inasmuch as his

proof does not depend on any peculiar features of the particular items with

which it is concerned.

In systems of natural deduction, as contemporary logicians call it, there is

generally a rule of universal generalization or universal introduction which

looks something like this:

Givenanargumentconcludingto‘F(a)’onthebaseofpremissesP

,…,

, infer ‘For any x, F(x)’, on the basis of the same premisses—provided

that the name ‘a’ occurs neither in ‘F’ nor in any P

The main clause in that rule corresponds to the ‘transfer’ from particular to

universal which Proclus describes—given a proof that some individual item

is thus-and-so, we may infer that everything is thus-and-so; given a proof that

F(a), we may infer that F(everything). The proviso in the rule corresponds to

the idea that the geometers’ proofs ‘do not make use of the items set out as

those items’. Proclus describes, in an approximate fashion, a rule of universal

generalization; and Euclid argued in accordance with such a rule.

The rule makes appeal to individuals; for when Proclus speaks of particulars

he means ‘individual’ by ‘particular’. That is to say, he supposes that ‘the

items set out’ are individual geometrical objects—lines and circles and so on.

Moreover, he takes the Greek letters to designate or introduce these objects:

the formula ‘AB’ in Euclid i i, for example, is a singular term which designates

an individual line.

That seems ﬁne: you prove that a certain individual so-and-so has a given

property; and because your proof does not at any point turn on the fact

that it is this individual so-and-so you have chosen rather than another one,

you advance a further step and infer that all so-and-sos have the property in

question. So in the Euclidean proof, ‘AB’ designates an individual line. That

is surely so—but then which individual line does it designate? The question

ἐπὶ τὸ καθόλου ἵνα μὴ μερικὸν ὑπολάβωμεν εἶναι τὸ συμπέρασμα. εὐλόγως δὲ μεταβαίνουσιν

ἐπειδὴ τοῖς ἐκτεθεῖσιν οὐχ ᾗ ταῦτά ἐστιν ἀλλ᾿ ᾗ τοῖς ἄλλοις ὅμοια χρῶνται πρὸς τὴν ἀπόδειξιν.

οὐ γὰρ ᾗ τοσήδε ἐστὶν ἡ ἐκκειμένη γωνία, ταύτῃ τὴν διχοτομίαν ποιοῦμαι, ἀλλ᾿ ᾗ μόνον

εὐθύγραμμος.