Barnes Jonathan. Truth, etc

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480 When is a Syllogism not a Syllogism?

Gall and safﬂower attract phlegm.

What attracts phlegm is phlegm-like.

Therefore gall and safﬂower are phlegm-like.

It is, now, a proof of necessity insofar as we have (now) reason to think that

its conclusion holds not contingently but necessarily.

If that is so, then Galen’s logical theorem should not be designed to

construct a new syllogism from an old syllogism, an apodictic syllogism from

a non-modal syllogism. Rather, it should allow us to infer that the conclusion

of the old non-modal syllogism, namely

Gall and safﬂower are phlegm-like,

in fact holds of necessity. Is there any such theorem? Yes. This will do the

trick:

If an argument from a given set of premisses to a given conclusion is valid,

then if the each of the premisses of the argument holds of necessity, the

conclusion too holds of necessity.

Or schematically:

If ‘P

,…,P

: therefore Q’ is valid, and necessarily P

, and necessarily

, … , and necessarily P

, then necessarily Q.

When that theorem is applied to the gall and safﬂower syllogism, it does not

produce a second syllogism: it allows us to infer that

Gall and safﬂower are phlegm-like

holds of necessity—given, of course, that we have established that the two

premisses of the argument hold of necessity.

That interpretation of the passage from Simple Drugs will seem forced. It is

forced. But I think that any coherent interpretation of the text—whether or

not it hopes to reconcile what is there said with what al-Farabi reports about

On Proof —will be forced.

It may also be urged that there are really no substantive differences between

the Modal Theorem and the theorem which I have just put on the table. To

simplify things a little, the Modal Theorem says that if

X: therefore Y

is a syllogism, then so is

Necessarily X: therefore necessarily Y;

and the other theorem says that if

X: therefore Y

is a syllogism, then if

Problematic Syllogisms 481

Necessarily X

then

Necessarily Y.

Tweedledum and Tweedledee—only a pedant will discern any difference.

Perhaps. But Tweedledum and Tweedledee are two distinct items; and there

is some distance between them as the crow ﬂies.

Nevertheless, I can’t say that I’m enamoured of that way of dealing with

Simple Drugs.It’sjustthatIcanﬁndnobetterwayofreconcilingitwiththe

report from al-Farabi.

So why not reject the report? After all—to return to the question which

was earlier shelved—why ever should Galen have rejected modal syllogistic?

PROBLEMATIC SYLLOGISMS

al-Farabi gives no hint. But it has proved tempting to guess that Galen’s

rejection of problematic syllogistic had something to do with Theophrastus’

views on the subject.

In Aristotle’s problematic syllogistic, possible negations convert; that is

to say—it is an unusual use of the word ‘convert’—they are equivalent to

possible afﬁrmatives. Thus according to Aristotle, A holds possibly of no B

ifandonlyifAholdspossiblyofeveryB.AlatecommentaryonthePrior

Analytics informs us that

Theophrastus and Eudemus—and also the Platonists—do not want the possible

negation to convert to the possible afﬁrmation since in that case we shall have

abandoned the sort of possible which holds for the most part, which is what the

argument is about. For the syllogisms are advanced by those arts which are concerned

with the possible which holds for the most part—if it is possible that no man has six

ﬁngers for the most part, then it is possible that every man (and that some man) has

six ﬁngers for the lesser part.

([Ammonius], in APr 45.42–46.2)⁴⁷

That is far from pellucid—whatever does it mean to say that no man has

six ﬁngers for the most part? The other late texts which report the views of

⁴⁷ οἱ δ᾿ ἑταῖροι αὐτοῦ Θεόφραστος καὶ Εὔδημος καὶ ἔτι οἱ Πλατωνικοὶ οὐ βούλονται

ἐνδεχομένην ἀπόφασιν ἀντιστρέφειν πρὸς ἐνδεχομένην κατάφασιν, ἐπειδὴ οὐ μένει τὸ ὡς ἐπὶ

τὸ πολὺ ἐνδεχόμενον, περὶ οὗ ὁ λόγος. οἱ γὰρ συλλογισμοὶ ὑπὸ τεχνῶν προβάλλονται αἳ περὶ

τὸ ὡς ἐπὶ τὸ πολὺ ἐνδεχόμενον ἔχουσιν· εἰ γὰρ ἐνδέχεται μηδένα ἄνθρωπον ἑξαδάκτυλον

εἶναι ὡς ἐπὶ τὸ πολύ, ἐνδέχεται πάντα ἄνθρωπον καὶ τινὰ ἄνθρωπον ἑξαδάκτυλον εἶναι ἐπ᾿

ἔλαττον.

482 When is a Syllogism not a Syllogism?

Theophrastus and Eudemus shed no light; and Alexander says nothing on

the subject.

Nonetheless, one point is reasonably plain—and it is the point which

matters here. It is this. Aristotle thought that there were two senses of

‘possible’ and hence, in principle, two sorts of problematical syllogism.

The sciences which are concerned with contingent matters (medicine, for

example, or navigation) employ ‘possible’ in only one of the two senses—and

so problematical syllogistic may reasonably limit itself to that sense of

‘possible’. Now in that sense,

A holds possibly of every B

will be true provided that A holds naturally of every B, or A holds for the

most part of every B. But once possibility is so understood, the conversion

rules which Aristotle proposes for problematic propositions will not work. In

particular, the conversion from

A holds possibly of no B

A holds possibly of every B

must be dropped; for it is perfectly plain that

A holds naturally of no B

does not entail

A holds naturally of every B.

Now Aristotle’s problematic syllogistic depends on his conversion rules.

Hence it cannot be invoked by those sciences which deal with what holds for

the most part.

Galen knew Theophrastus’ logical works, on some of which he had

written commentaries. Presumably he had read and digested Theophrastus’

criticisms of Aristotle’s problematic syllogistic; no doubt he saw that they

made problematic syllogistic useless; and perhaps—utilitarian as he was—it

was for that reason that he rejected the study of problematic arguments.

That is a plausible story; but I suppose that it is false. First, the Theo-

phrastan criticism does not for a moment imply that problematic syllogistic

is useless for the sciences. Rather, it suggests that Aristotle’s problematic

syllogistic is incapable of performing the scientiﬁc services for which it was

allegedly designed. The moral to be drawn is not: Abandon problematic

syllogistic. It is: Develop a better problematic syllogistic. After all, that is

what Theophrastus set himself to do.

Secondly, al-Farabi indicates that Galen rejected apodictic as well as

problematic syllogistic. True, Theophrastus had criticized certain aspects of

Moods and Matter 483

Aristotle’s apodictic syllogistic, just as he had criticized certain fundamental

aspects of his problematic syllogistic. But if it is just about conceivable

that someone might have taken the Theophrastan criticisms of problematic

syllogistic to indicate that the subject had better be abandoned in toto,it

is quite unimaginable that anyone—let alone Galen—would have thought

to abandon apodictic syllogistic on account of Theophrastus’ criticisms of

Aristotle.

So the question remains: Why did Galen reject modal syllogistic?

MOODS AND MATTER

Near the beginning of his commentary on the Analytics,Alexandermakesa

few preliminary remarks about modal operators and modal propositions; and

he raises the question of why the things come into the Prior Analytics at all:

It seemed to me worth asking why, when in this book he is discussing syllogisms and

ﬁgures, he also takes into consideration material differences among propositions—for

the fact that something holds in this way or in that way is a material difference, and

such differences among propositions will therefore seem pertinent not to something’s

being a syllogism simpliciter but rather to its being a syllogism of such-and-such a

sort—a probative syllogism, say, or a dialectical syllogism.

(in APr 27.27–28.2)⁴⁸

The propositions

Every triangle has an angle-sum of 180

◦

and

Every child ought to honour his parents

have a different modal status: the former is necessary, the latter merely reput-

able. But that—according to Alexander—is a material difference between

the propositions: it may help to determine whether a certain syllogism is

probative or not, but it has no bearing on the question whether a certain

argument is a syllogism or not. From the point of view of predicative syl-

logistic, there is no difference between the two propositions. So why does

Aristotle interest himself in modalities at all?

⁴⁸ ἄξιον δὲ ἔδοξεν ἐπισκέψεως εἶναί μοι τί δήποτε περὶ συλλογισμῶν καὶ σχημάτων τὸν

λόγον ἐν τούτοις τοῖς βιβλίοις ποιούμενος παραλαμβάνει καὶ τὰς τῶν προτάσεων κατὰ τὴν

ὕλην διαφοράς· ὑλικαὶ γὰρ διαφοραὶ τὸ οὕτως ἢ οὕτως ὑπάρχειν· ἤδη γὰρ δόξουσιν αἱ

τοιαῦται διαφοραὶ τῶν προτάσεων οὐχ ἁπλῶς πρὸς συλλογισμὸν συντελεῖν, ἀλλὰ πρὸς τὸ

τοιοῦτον ἢ τοιοῦτον αὐτὸν εἶναι, ἀποδεικτικόν, ἂν οὕτως τύχῃ, ἢ διαλεκτικόν.

484 When is a Syllogism not a Syllogism?

Alexander offers what appears to be rather a confused answer to his

question. He begins like this:

Or perhaps such differences among propositions are in general necessary to the

method and study of syllogistic? After all, the conversions do not come about in the

same ways when propositions differ in respect of the moods we have described …

(in APr 28.2–5)⁴⁹

But the conversions do ‘come about in the same ways’; for, so far as

conversions go, the propositions

Every triangle has an angle-sum of 180

◦

and

Every child ought to honour his parents

behave in exactly the same ways.

There are indeed differences in conversion, according to Aristotle, ‘in

respect of the moods’; but the differences are not between the two propositions

which I have just illustratively cited but rather between the two related

modalized propositions:

Necessarily every triangle has an angle-sum of 180

◦

and:

Possibly every child ought to honour his parents.

For example, the latter proposition—according to Aristotle—entails that

Possibly no child ought to honour his parents,

whereas the former does not entail

Necessarily no triangle has an angle-sum of 180

◦

Such differences are not, as Alexander would have put it, merely material:

they hold in general between propositions of the form

Necessarily every B is A

and

PossiblyeveryBisA.

It is one thing to ask what is the modal status of a proposition—does

what it states hold necessarily or in point of fact or as a possibility? (Or,

of course, not at all.) It is another thing to ask what modal operator the

proposition carries—does it have the form ‘Necessarily so-and-so’ or ‘Possibly

so-and-so’ or neither? In Alexander’s terms, the former question asks after

a material aspect of the proposition, the latter after a formal aspect. A

⁴⁹ ἢ αἱ διαφοραὶ αἱ τοιαῦται τῶν προτάσεων πρὸς τὴν συλλογιστικὴν μέθοδόν τε καὶ

πραγματείαν ἀναγκαῖαι καθόλου· τῷ τε γὰρ μὴ ὁμοίως γίνεσθαι τὰς ἀντιστροφὰς τῶν κατὰ

τοὺς προειρημένους τρόπους διαφερουσῶν προτάσεων ...

Moods and Matter 485

little earlier in his commentary Alexander had in fact made the pertinent

distinction:

He claims that propositions take necessity and non-modality and possibility not

from what underlies them and is meant by them but rather from the adjunct which

is added and co-predicated and which says that this holds of that from necessity, or

that it holds, or that it can hold.

(in APr 27.1–5)⁵⁰

Modal logic—Aristotle’s modal syllogistic—concerns propositions which

predicate something necessarily or possibly of something; it concerns pro-

positions which say that A holds M-ly of B. Modal syllogistic works with

modalized propositions, and it is not concerned with the modal status of the

propositions which are or may be modalized.

Since he has set down that point fairly clearly, it is strange that, a page or

so later, Alexander should wonder why Aristotle deals with material aspects

of propositions; for Aristotle does not do so. Nonetheless, what Alexander

says—in passing—on the modal status of propositions is pertinent to the

question of the applicability of modal syllogistic—and hence to the question:

Why did Galen reject modal logic?

The modal status of a proposition, Alexander says, may help to determine

whether a syllogism in which it appears is probative or dialectical or of

some other character. Now proofs or probative syllogisms must, on a strict

interpretation of a strict Aristotelian view, have premisses which are true and

necessary. What matters, in other words, is their modal status. Modalization

is not the issue. The apodictic proposition

Necessarily every triangle has an angle-sum of 190

◦

could not, despite its modal operator, feature in a proof—after all, it is false.

The non-apodictic proposition

Every triangle has an angle-sum of 180

◦

can—and does—ﬁgure in proofs; for it is true, and it is necessary.

In general, if the proposition that such-and-such is to feature in a strict

Aristotelian proof, then it must be the case that necessarily such-and-such. It

does not follow, and it is not true, that if the proposition that such-and-such is

to feature in a strict Aristotelian proof, then it must have the form ‘Necessarily

so-and-so’ or say that necessarily so-and-so.

⁵⁰ οὐ γὰρ ἀξιοῖ τὰς προτάσεις ἀπὸ τῶν ὑποκειμένων αὐταῖς καὶ δηλουμένων ὑπ᾿ αὐτῶν

λαμβάνειν τὸ ἀναγκαῖον καὶ τὸ ὑπάρχον καὶ τὸ ἐνδεχόμενον, ἀλλὰ ἀπὸ τῆς προσθήκης τῆς

προστιθεμένης καὶ προσκατηγορουμένης τῆς λεγούσης ὅτι τόδε τῷδε ἐξ ἀνάγκης ὑπάρχει ἢ

ὑπάρχει ἢ ἐνδέχεται ὑπάρχειν.

486 When is a Syllogism not a Syllogism?

Surely that cannot be right? After all, Aristotle invented apodictic syl-

logistic in order to satisfy the probative demands of those sciences whose

propositions hold true by necessity; and apodictic syllogisms are constructed

from modalized propositions, from propositions of the form ‘Necessarily

so-and-so’. So surely the apodictic proposition

Necessarily every triangle has an angle-sum of 180

◦

is more appropriate to scientiﬁc proofs than the non-modal proposition

Every triangle has an angle-sum of 180

◦

No, it isn’t. First, consider actual scientiﬁc practice. Pythagoras’ theorem

states that the square on the hypotenuse is equal to the sum of the

squares on the other two sides: it does not state that necessarily the

square on the hypotenuse etc. Euclid’s Elements contains no modalized

propositions: all its theorems are, if true, then necessary; but none has the

form ‘Necessarily so-and-so’. Secondly, the propositions which constitute

a strict Aristotelian proof must hold of necessity. Hence the modalized

proposition

Necessarily every triangle has an angle-sum of 180

◦

can feature in such a proof only if its own modal status is one of necessity; in

other words, only if

Necessarily necessarily every triangle has an angle-sum of 180

◦

Aristotle never considers such doubled modalities—nor has he any scientiﬁc

reason to do so.

In other words, the sciences—even where they deal in necessities—do

not use apodictic propositions. They therefore have no use for apodictic

syllogisms. In just the same way, the sciences do not use problematic pro-

positions. There are, to be sure, sciences of the contingent; and despite the

ofﬁcial deﬁnition in the Posterior Analytics which requires necessity of every

component proposition in a proof, Aristotle does in fact recognize—and

in the Posterior Analytics—that some scientiﬁc proofs start from contingent

premisses and arrive at contingent conclusions. Such proofs, he thinks, deal

with what happens ‘for the most part’.

Thus, for example, it is a truth, and a contingent truth, that vines shed their

leaves in autumn: they do not do so necessarily, but they do so by nature and

for the most part. Let us follow Aristotle in imagining that that contingent

truth is a theorem of the science of botany. Now the theorem is not:

Contingently, vines are lose their leaves in autumn.

It is:

Vines lose their leaves in autumn.

Repetitive Arguments 487

What goes for botany goes for medicine, and for navigation; and no doubt

for gymnastics too.

When Alexander and al-Farabi urge the contrary thesis, stating that the

theorems of such sciences are problematic propositions, they take themselves

to be bowing both to Aristotle and to the truth. They are certainly not bowing

to the truth; and although their reading of Aristotle is not whimsical—indeed,

it is based ﬁrmly on some passages in the Analytics—it is not in the end

the best overall interpretation of Aristotle’s view of the nature of scientiﬁc

proof.

The conclusion which Galen and Alexander should have drawn from such

considerations is plain: modal arguments, pace Aristotle, have no probative

utility; so they are not syllogisms; and they ought to be excluded from the

purlieus of logic. Alexander did not draw that conclusion, which would

doubtless have been embarrassing for an Aristotelian commentator who held

an imperial chair in Peripatetic philosophy. Indeed, so far as I have noticed,

Alexander nowhere hints that there may be something fundamentally odd

about Aristotle’s modal syllogistic. Galen, I suppose, drew the conclusion.

That is to say, Galen recognized that the sciences do not use modal pro-

positions, and he therefore concluded that modal syllogistic is superﬂuous.

That is why he does not mention modality in his Introduction to Logic;

and that is why no modal syllogism every appears in any of his voluminous

writings—perhaps not even in Simple Drugs.

REPETITIVE ARGUMENTS

A utilitarian logician should have no time for modal syllogistic. But if modal

syllogisms cannot be used to prove anything, that is not because of any

formal deﬁciencies on their part but rather because of the nature of scientiﬁc

theorems. Other would-be syllogisms were rejected by the utilitarians on

purely formal grounds.

Aristotle’s deﬁnition of the syllogism is found in three slightly different

versions. In the Analytics it looks like this:

A syllogism is a saying in which, certain items being laid down, something other than

the items laid down comes about by necessity from the fact that they are the case.

(APr 24b18–20)⁵¹

⁵¹ συλλογισμὸς δέ ἐστι λόγος ἐν ᾧ τεθέντων τινῶν ἕτερόν τι τῶν κειμένων ἐξ ἀνάγκης

συμβαίνει τῷ ταῦτα εἶναι.

488 When is a Syllogism not a Syllogism?

In a syllogism, certain items—the premisses—are laid down, and something

other—the conclusion—results. The ancient commentators read the deﬁn-

ition minutely, and they gave each word a weight. Thus Alexander has a

long discussion of the phrase ‘something other’—or rather, he has two long

discussions of it, one in his commentary on the Analytics and the other in his

commentary on the Topics.

The passage in the commentary on the Analytics —in the state in which

we have it—is not a smooth piece of writing. Commentaries in general

are rough beasts which shamble from one lemma to the next. Moreover,

they are functional items, not literary compositions; and their value is

instrumental: the inventor of the instrument may himself produce later

models; and an owner of the instrument may adapt it to his own uses.

Certainly, many surviving ancient commentaries show signs of such a

history. The best known case of the phenomenon is provided by Alexan-

derhimself:wehavetwoversionsoftheearlierpartsofhiscommentary

on the Metaphysics, one of them (at least by and large) from his own

hand and the other from the pen of some later professor. I suspect that

parts of the commentary on the Analytics—among them the discussion of

‘something other’—have been similarly written and rewritten. The com-

parable passage in the commentary on the Topics contains essentially the

same material, here abbreviated and there expanded; and it runs more

smoothly.

The passage in the commentary on the Analytics begins in the following

way:

He was absolutely correct to add that the conclusion must be other than the items

laid down. For it is useless—it destroys all syllogistic utility—to infer what is agreed

and laid down. No syllogistic utility comes from

If it is day, it is light

But it is day

Therefore it is day

—nor in general from what the more recent philosophers call non-differently

concluding arguments. … For a syllogism is a tool, and it is taken up for some useful

purpose—for showing something. So what is not useful is not a syllogism at all.

(in APr 18.12–22)⁵²

⁵² πάνυ δὲ καλῶς τὸ δεῖν ἕτερον τῶν τεθέντων εἶναι τὸ συμπέρασμα προσέθηκεν· ἄχρηστον

γὰρ καὶ συλλογιστικῆς χρείας φθαρτικὸν τὸ τὸ ὁμολογούμενον καὶ κείμενον ἐπιφέρειν. τὸ

γὰρ χρειῶδες τοῦ συλλογισμοῦ οὐ παρέχεται τὸ εἰ ἡμέρα ἐστίν, φῶς ἐστιν· ἀλλὰ μὴν ἡμέρα

ἐστίν· ἡμέρα ἄρα ἐστί, καὶ ὅλως οἱ λεγόμενοι ὑπὸ τῶν νεωτέρων ἀδιαφόρως περαίνοντες.

Repetitive Arguments 489

The ‘more recent philosophers’—Alexander is referring to the Stoic logi-

cians—studied certain sorts of argument which Aristotle excluded from the

domain of syllogistic. Perhaps their exclusion was an unforeseen consequence

of Aristotle’s deﬁnition? Alexander does not think so: he notices—in a

passage which I have already cited—that Aristotle explicitly repudiates any

discussion of the indeterminate sort of possibility inasmuch as such possibility

is not subject to scientiﬁc study; and then he adds:

Hence it is clear that, in the case of those items which are useless for proofs and on

which he said nothing and the more recent philosophers said their say, he omitted

them too because of their uselessness and not because of his own ignorance. I mean,

for example, the duplicated arguments …

(in APr 164.27–30)⁵³

Aristotle’s explicit attitude to certain useless modal arguments shows that he

excluded the useless Stoic arguments not from ignorance but from policy: he

excluded them because they are not syllogisms, because they do not satisfy

one of the conditions which, according to his deﬁnition, every syllogism must

satisfy.

A modern reader may wonder not merely whether Aristotle presciently

rejected a number of arguments which the Stoics were later to promote

but also whether the phrase ‘something other’ will bear the weight which

Alexander puts upon it—and he may well be inclined to hope that it

will not bear the weight and that Aristotle had no particular qualms about

acknowledging arguments which conclude to one of their own premisses.

(After all, a modern logician has no such qualms.) In the dictum de omni et

nullo Aristotle offers to explain what it is for one item to be in another as

in a whole. But—or so I have urged—the phrase ‘in another’ is not to be

pressed: he means to say what it is for something to be in something as in a

whole. In the same way, when he says that the conclusion of a syllogism must

be ‘something other’, perhaps the word ‘other’ is not to be pressed? Perhaps

Aristotle doesn’t really mean to exclude arguments which conclude to one of

their own premisses?

But he does, and Alexander’s interpretation is correct. Had Aristotle simply

said that the conclusion must be ‘something other’—had he written ‘from

... ὄργανον γὰρ ὁ συλλογισμὸς καὶ χρείας τινὸς χάριν καὶ δείξεως παραλαμβανόμενον, ὥστε

τὸ μὴ χρήσιμον οὐδὲ συλλογισμός.

⁵³ ὅθεν δῆλον ὅτι καὶ ταῦτα περὶ ὧν αὐτὸς μὲν οὐκ εἴρηκε, λέγουσι δὲ οἱ νεώτεροι ἀχρήστων

ὄντων πρὸς ἀπόδειξιν, δι᾿ ἀχρηστίαν οὐ δι᾿ ἄγνοιαν παρέλιπεν, οἷοί εἰσιν οἱ διφορούμενοι

λόγοι ...