Barnes Jonathan. Truth, etc

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520 When is a Syllogism not a Syllogism?

Alexander’s three classes of argument correspond, at least roughly, to the ﬁrst

two Stoic unproveds (which depend on conditional propositions), the last

two unproveds (which depend on disjunctions), and the third unproved. But

Alexander suggests—implicitly but unmistakably—that the third unproved

may not be a genuine syllogism at all, or at any rate that it may not be

a genuine syllogism distinct from the hypothetical syllogisms which he has

already mentioned.

How might anyone think that a third unproved was either not a syllogism

at all or else somehow the same syllogism as one of the other four unproveds?

A text from Galen’s Introduction to Logic suggests an answer:

That not one syllogism by way of a negated conjunction is useful for proof … has

been proved elsewhere. … The Chrysippeans think that a third unproved concludes

from a negated conjunction and one of its members to the opposite of the other,

as in

Dio is not both in Athens and at the Isthmus;

and we have shown that this too is useful for many proofs in ordinary life and in the

law-courts.

(inst log xiv 3–4)⁸¹

In the ﬁrst sentence, Galen appears to say that arguments by way of

negated conjunctions are not probative; in the second, that Chrysippus’ third

unproved is such an argument; and in the third, that the third unproved

argument is useful for proofs. And that is self-contradictory.

The resolution of that little puzzle is found a few sentences later in the text.

There Galen reminds us that he has earlier distinguished between complete

conﬂict and partial conﬂict. (A set of propositions is in complete conﬂict if

and only if at most one of its members can be the case at any given time and

at least one of its members must be the case at any given time. A set is in

partial conﬂict if and only if at most one of its members can be the case at

any given time.) And he comments:

⁸¹ ὅτι δὲ δι᾿ ἀποφατικοῦ συμπεπλεγμένου συλλογισμὸς εἰς ἀπόδειξιν χρήσιμος οὐδὲ εἷς

ἐστι, ... ἀποδέδεικται δι᾿ ἑτέρων· ... τρίτον οὖν ἀναπόδεικτον τῶν περὶ τὸν Χρύσιππον

ἡγουμένων ἐξ ἀποφατικοῦ συμπεπλεγμένου καὶ θατέρου τῶν ἐν αὐτῷ τὸ ἀντικείμενον τοῦ

λοιποῦ περαίνοντα, ὡς ἐπὶ τῶν τοιούτων οὐχὶ καὶ ᾿Αθήνησίν ἐστι καὶ ᾿Ισθμοῖ ∆ίων, καὶ

τόνδ’ ἐπεδείξαμεν εἰς πολλὰς τῶν κατὰ τὸν βίον ἀποδείξεις εἶναι χρήσιμον ἄχρι καὶ τῶν

δικαστηρίων.—The text is desperate in places; but although what I print is highly conjectural, it

is scarcely to be disputed that it conveys the right sense.

Negated Conjunctions 521

It is in this latter sort of case that the syllogism we have described is useful: it has

the same expression as Chrysippus, but it is constituted not on a conjunction but on

conﬂicting items.

(inst log xiv 6)⁸²

In other words, the arguments which Galen has shown to be useful for proofs

may look just like third unproveds; but they are not third unproveds—for in

fact they are not built around negated conjunctions. No argument constituted

on a negated conjunction can be probative, and Chrysippus’ third unproved

is such an argument. So the third unproved is not a syllogism.

Take, for example, the argument:

Dio is not both in Athens and at the Isthmus.

Dio is in Athens.

Therefore Dio is not at the Isthmus.

Chrysippus takes that to be a third unproved, its ﬁrst premiss being a negated

conjunction. Galen observes that such an argument might well be used to

prove something—say, in a court of law. We are perhaps likely to agree both

with Chrysippus and with Galen, and to draw the conclusion that the third

unproved is useful for proof. Galen, convinced that the third unproved is

probatively valueless, concludes that the argument is not a third unproved

inasmuch as it does not invoke a negated conjunction.

If the ﬁrst premiss of the argument is not a negated conjunction, then what

is it? Well, it is a proposition of partial conﬂict. Earlier, Galen has observed that

in the case of partial conﬂict, the Greeks customarily speak like this:

Dio is not both in Athens and at the Isthmus.

(inst log iv 3)⁸³

And he has just explained that

when objects have a nature of the sort we have just described, it indicates a complete

conﬂict, whereas the other nature indicates a partial conﬂict, in respect of which

we say:

If Dio is in Athens, Dio isn’t at the Isthmus.

(inst log iv 1)⁸⁴

⁸² ἐν τούτοις οὖν τοῖς πράγμασιν ὁ εἰρημένος συλλογισμὸς χρήσιμός ἐστι, τῇ μὲν αὐτῇ λέξει

χρώμενος ᾗ Χρύσιππος, οὐ μὴν ἐπὶ συμπεπλεγμένῳ συνιστάμενος ἀλλ᾿ ἐπὶ τοῖς μαχομένοις.

⁸³ ἐπὶ μὲν οὖν τῆς ἐλλειπούσης μάχης ἐν ἔθει τοῖς ῞Ελλησίν ἐστιν οὕτω λέγειν· οὐκ ἔστιν

᾿Αθήνησί τε καὶ ᾿Ισθμοῖ ∆ίων.

⁸⁴ ἡ μὲν οὖν τοιαύτη φύσις τῶν πραγμάτων τελείαν ἐνδείκνυται τὴν μάχην, ἡ δὲ ἑτέρα

τὴν ἐλλιπῆ, καθ᾿ ἣν ὧδέ πως λέγομεν· εἰ ᾿Αθήνησίν ἐστιν ∆ίων, οὐκ ἔστιν ᾿Ισθμοῖ ∆ίων.

522 When is a Syllogism not a Syllogism?

The argument which Chrysippus wrongly takes to be a third unproved may

be expressed equally well as follows:

If Dio is in Athens, then he is not at the Isthmus.

Dio is in Athens.

Therefore Dio is not at the Isthmus.

And that argument is evidently a ﬁrst unproved. The sentence

Dio is not both in Athens and at the Isthmus,

which Chrysippus takes to express a negated conjunction, in fact expresses a

conditional assertible.

So where Alexander insinuates that the third unproved may not be a new

sort of syllogism, Galen implies that those genuine syllogisms which appear

to be third unproveds are in fact ﬁrst unproveds. Two minds with but one

thought.

The thought seems peculiar—or worse. After all, what is a negated

conjunction but a proposition of the form

Not (P

and P

and … and P

So how can

Dio is not in Athens and at the Isthmus

fail to be a negated conjunction? Surely, it negates

Dio is in Athens and at the Isthmus,

and surely that is a conjunction?

But Galen’s position, eccentric though it seems, is not insane. Galen

agrees—of course—that the sentence ‘Dio is in Athens and at the Isthmus’

is conjunctive; and he agrees—of course—that the sentence ‘Dio is not in

Athens and at the Isthmus’ is a negated conjunction so far as its expression

is concerned. But he claims that, quite generally, it is not the mode of

expression but the meaning of a sentence which determines whether it

formulates a negated conjunction or a conditional or a disjunction or

whatever.

That last and general claim is true—and uncontroversial. Evidently, it

is the sense of a sentence which determines what it says or what you can

say in uttering it. For example, it is not the presence of the sound ‘eh’ in

an appropriate position between a couple of Greek sentences which makes

the complex sentence express a disjunctive proposition. Sounds may express

different senses. A Greek sentence which mouths ‘eh’ between two sentences

expresses a disjunction only if the sound ‘eh’ represents the word ‘ἤ’andthe

Negated Conjunctions 523

word ‘ἤ’ functions as a disjunctive connector. What holds for ‘eh’ in Greek

holds also for—say—‘and’ in English. The sentence

You invite her again and I go to Bath

may have a conjunctive sound to it—it may be, as Galen puts it, conjunctive

so far as its expression goes; but it is most likely to express not a conjunctive

but a conditional proposition. Such facts are familiar, and they underlie

Galen’s apparently eccentric view of negated conjunctions.

But if the foundations are solid, what about the structure which Galen

here builds upon it? Galen’s underlying principle may be true; but is not his

particular view on negated conjunctions mistaken? He says that when the

Greeks want to express a partial conﬂict, then they often produce a sentence of

the form ‘Not both P and Q’; and in that case they have expressed a conditional

proposition. Allow—for the sake of argument—that he is right. It does not

follow that whenever a Greek utters something of the form ‘Not both P and

Q’ he has thereby expressed a conditional proposition. Nor, of course, does

Galen assert anything so wild: he does not deny the existence of negated

conjunctions, and he does not deny that a sentence of the form ‘Not both

P and Q’ may express—indeed, is admirably adapted to express—a negated

conjunction. What he denies is that negated conjunctions are of any interest

to a logician or of any use in logic. But whyever does he deny that? And why,

in particular, does he deny that the Stoic third unproved is a syllogistic form?

The background to his strange denial is sketched earlier in the Introduction.

After he has discussed predicative propositions, Galen goes on to remark that

there is another kind of proposition with which we make assertions not about the

holding of things but about what is the case something being the case, and what is the

case something not being the case. Let such propositions be called hypothetical …

(inst log iii 1)⁸⁵

In Galen’s view, there are just two sorts of hypothetical proposition—or at

any rate, just two sorts in which hypothetical syllogistic has an interest. And

in the next sentence he explains that such propositions are either conditionals

or disjunctives. The two types are characterized in somewhat different ways

in different parts of Galen’s œuvre; but the commonest—and perhaps also

⁸⁵ γένος ἄλλο προτάσεών ἐστιν ἐν αἷς τὴν ἀπόφανσιν οὐ περὶ τῆς ὑπάρξεως ποιούμεθα

τῶν πραγμάτων, ἀλλὰ περὶ τοῦ τίνος ὄντος τί ἐστι καὶ τίνος οὐκ ὄντος τί ἐστιν· ὑποθετικαὶ

δὲ ὀνομαζέσθωσαν αἱ τοιαῦται προτάσεις ...

524 When is a Syllogism not a Syllogism?

the clearest—characterization speaks in terms of implication and conﬂict, of

entailment and exclusion, of ἀκολουθία and μάχη. Thus Galen will speak

approvingly of

the men of old, logicians, scientists, practised in distinguishing the true and the false,

knowing how to determine what is implied and what is in conﬂict, trained from

youth upward in demonstrative method.

(MM x 10)⁸⁶

Those GOM knew all there was to know about logic; and that summa can be

summed up in the phrase ‘what is implied and what is in conﬂict’.

If we say ‘what is the case, something being the case’, then we state an

implication. (And to do so we might well use the conditional connector

‘If … , then … ’.) If we say ‘what is the case, something not being the case’,

then we state a conﬂict. (And to do so we might well use the disjunctive

connector ‘Either … or … ’.) The questions which concern hypothetical logic

are two: Does X imply Y? Does X exclude Y? The logical relations with which

hypothetical syllogistic is concerned are two: implication and exclusion. So

insofar as Stoic logic appeals to conjunctions, it appeals to relations which are

not logical—or to connexions which are not relations at all.

Galen’s view here is not idiosyncratic. On the contrary, it was more or

less commonplace outside the Stoa. True, it is not discussed elsewhere in the

way in which Galen discusses it. But it is taken for granted by Alexander,

and also by later commentators on Aristotle; and it also shows up in the

rhetorical tradition in which there are two bases for argument—implication

and conﬂict, ἀκολουθία and μαχή, consequentia and repugnantia.

The foundation of the view, at any rate in Galen’s mind, was the

familiar thesis that an argument-form which is not useful for proofs is

not a syllogistic form. Galen claims that he has ‘shown elsewhere’ that the

third unproved—and, more generally, any argument based upon a negated

conjunction—is useless for proof. That is why Chrysippus’ third unproved,

and any congeners which it may have, are not syllogisms.

The elsewhere was no doubt somewhere in the lost work On Proof ;but

no other text, so far as I know, sheds any further light on the matter, and if

we want to recover Galen’s argument against third unproveds we must rely

on pure speculation.

⁸⁶ ... ἄνδρας παλαιοὺς, διαλεκτικοὺς, ἐπιστημονικοὺς, ἀληθὲς καὶ ψευδὲς διακρίνειν

ἠσκηκότας, ἀκόλουθον καὶ μαχόμενον ὡς χρὴ διορίζειν ἐπισταμένους, ἀποδεικτικὴν μέθοδον

ἐκ παίδων μεμελετηκότας, ...

Negated Conjunctions 525

And not for the only time, pure speculation turns up trumps. The premisses

of a proof, we know, must be prior to the conclusion, and prior in several

ways. In particular, they must be prior in knowledge. That is to say, we must

be able to come to know the premisses without already having come to know

the conclusion—for how otherwise could the proof be a cognitive syllogism,

a syllogism which breeds knowledge?

Consider, then, a genuine negated conjunction—say:

Unlike David Hume, John Locke wasn’t both a philosopher and an

historian.

How might we come to discover the truth on that particular matter? There

is no logical relation of implication or exclusion between being a philosopher

and being an historian, so that we cannot maintain, on entirely general or

conceptual grounds, that history and philosophy won’t marry and thence

infer the negated conjunction about Locke. All we can do is consider the

two elements in the negated conjunction one after another: industrious

research will establish that Locke was a philosopher, and further research

will prove that he wasn’t a historian. Then we may afﬁrm that he wasn’t

both a philosopher and an historian. Suppose that we then put forward the

following argument, on the basis of the negated conjunction:

Locke wasn’t both a philosopher and an historian.

Locke was a philosopher.

Therefore Locke wasn’t an historian.

That argument may be logically impeccable in the sense that if its two

premisses are true then its conclusion must be true. But it is not a proof, and

it cannot be a proof. For in order to establish the truth of its complex premiss

we had ﬁrst to establish the truth of its conclusion.

Quite generally, it must seem, we can only come to know that not

both P and Q (if that is a genuine negated conjunction and not a cryptic

conditional) by ﬁrst coming to know either that P and that not-Q or else

that Q and that not-P. But the conclusion of a third unproved is either

not-P or not-Q. Hence the negated conjunction, which is the premiss

of such an argument, cannot be established without ﬁrst establishing its

conclusion; and for that reason the third unproved cannot be used to prove

its conclusion. Given Galen’s logical utilitarianism, it follows that the third

unproved is not a syllogistic form—and that it should not be found in a text

on logic.

526 When is a Syllogism not a Syllogism?

The argument which I have just rehearsed has, I hope, a plausible air to

it. Certainly, whenever I have put it forward, it has been well received. True,

there is no hint of the argument—or of any other argument—in Galen’s

texts: to ascribe it to Galen is pure speculation. But it is an easy argument to

dream up, and it is not at all easy to dream up any other argument.

Well, although the argument has often been smiled upon, in fact it is a

rank bad argument. For it is quite untrue that the only way to establish a

negated conjunction—a genuine negated conjunction—is to establish one

of the conjuncts and disestablish the other. There are dozens of other ways in

which you might come to learn the truth of a negated conjunction. A witness

afﬁrms that there was only one man in the dime-store at the time of the

hold-up. The witness is a pretty reliable sort. Without knowing, or needing

to know, anything whatever about the whereabouts of Smith and Wesson, I

inferthatitisnotthecaseboththatSmithwasthereandthatWessonwas

there. Or again, I know that the second letter in the answer to 1 across must

be the same as the ﬁrst letter in the answer to 2 down: I infer at once that it is

not the case both that the answer to 1 across is ‘aardvark’ and that the answer

to 2 down is ‘zebu’. Et caetera.

In sum, the argument which pure speculation assigned to Galen is a

rotten argument. Shall we conclude that pure speculation has failed, and

that Galen’s argument must have been something quite other? No—or at

least, not without more ado. On the one hand, Galen was no slouch at

logic—there is enough of his work still with us to vindicate that claim. And

so, in looking to interpret his arguments, and to supply their missing pieces,

we shall, in principle, be looking for a semi-respectable bit of reasoning. On

the other hand, Galen was not infallible: there are errors and oversights in his

extant texts, and it would be silly to suppose that the lost On Proof contained

no mistakes. So in interpreting Galen, the respectable arguments which we

look for need not always be true arguments.

I incline to think that, in the present case, it is on the whole plausible

to ascribe to Galen a bad argument—a bad argument which nevertheless

has appealed, at least temporarily, to many of my intelligent and informed

colleagues.

ENVOI

A book wants a moral. That last story suggests one. We are all better at

logic than Galen was—we are better at logic than Chrysippus was, and than

Envoi 527

Aristotle was. Not, of course, that we are more inventive than the glorious

dead—far, far from it. And I suppose that, in the whole history of logic,

only two men have ever equalled Aristotle in logical creativity. Nonetheless,

you and I are better logicians than Aristotle was inasmuch as we are often

capable of spotting the mistakes which he made and are relatively unlikely to

repeat them. That is not gasconade: it would be a scandal were things not

so. But it is something which it is peculiarly easy to forget; and forgetting

it, we think to squeeze delicious subtleties from dry texts and we let our

imaginations picture a sophisticated interior where our eyes see only a very

modest fac¸ade. We make Parmenides as profound as Dummett, and Plato as

reﬁned as Strawson. It is generous to do so, and it is clever. But it is not—it

is not often—right to do so; and generosity and cleverness, no less than

their unattractive opposites, have distorted many interpretations of ancient

philosophical texts—among them, no doubt, several of the interpretations

which you have just ﬁnished reading.

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Onomasticon

The onomasticon catalogues pretty well all the ancient ﬁgures who are used

or mentioned in the book. Dates are very often approximate.

Alexander of Aphrodisias: ﬂoruit 210 ad, held the chair in Peripatetic

philosophy at Athens; commentator on Aristotle.

Ammonius: ﬂoruit 500 ad, Alexandrian Platonist, commentator on Aristotle.

Anaxagoras of Clazomenae: ﬂoruit 425 bc, philosopher and rationalist.

Andronicus of Rhodes: end of 1

century bc, Peripatetic philosopher, editor

of and commentator upon Aristotle’s works.

Apollonius Dyscolus: 2

century ad, Alexandrian scholar—the leading

Greek grammarian.

Apuleius of Madaurus: 2

century ad, novelist, rhetorician and Platonist.

The essay On Interpretation is probably not his work.

Archelaus: ﬂoruit endof5

century bc, philosopher, pupil of Anaxagoras.

Aristotle: 384-322 bc, pupil of Plato and master of those who know.

Aemilius Asper:2

century ad, Latin grammarian and critic. (The surviving

Art of Grammar is not authentic.)

Anicius Manlius Severinus Boethius: 480–524 ad, the last of the Romans—

Platonist thinker who planned to translate Aristotle into Latin.

Boethus of Sidon: end of 1

century bc, Peripatetic philosopher and

commentator on Aristotle.

Carneades of Cyrene: 214-129 bc, head of the sceptical Academy and its

most celebrated controversialist.

Chaeremon:1

century ad, Egyptian priest and Stoic philosopher, tutor to

the Emperor Nero.

Flavius Sosipater Charisius:4

century ad, Latin grammarian.

George Choeroboscus of Byzantium: 9

century ad, churchman and

grammarian.

Chrysippus of Soli, in Cyprus; 280-207 bc, third head of the Stoic school.

Cicero: 106-43 bc, Roman statesman, orator, and philosopher.

Cleanthes: 300-230 bc, second head of the Stoic school.