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280 Forms of Argument

is not so: every ﬁgure contains conjugations which yield no conclusion at

all, and every conjugation which yields a conclusion fails to yield at least

two other conclusions. The point is in any case evident; for the ﬁgure of

a syllogism is determined exclusively by the relation among its premisses,

whereas the validity of a syllogism is a relation between premisses and con-

clusion.

SYLLOGISTIC FORM AND SYLLOGISTIC MATTER

So the shapes or ﬁgures of the Analytics do not serve the distinction between

formal and material consequence; and nothing else in the Analytics hints at it.

Outside the Analytics Aristotle has little to say about syllogisms. But a couple

of passages in the Physics are traditionally called upon in this context. One of

them occurs in a chapter where Aristotle is discussing the place of necessity

in nature:

Necessity is found in a similar sort of way both in mathematics and in what comes

about by nature: since the straight is such-and-such, necessarily the triangle has angles

equal to two right angles; but not since the latter, the former—rather, if not the

latter, then the straight is not such-and-such. In what comes about for the sake of

something, it is the other way about: if the goal will be, or is, then the earlier item

will be, or is; if not, then—just as in mathematics if the conclusion is not the case

then the ﬁrst principle will not be the case—so here with the goal and that for the

sake of which.

(Phys 200a15–22)¹⁶

At ﬁrst glance, the passage seems wholly irrelevant to the question at issue: after

all, it does not mention syllogisms, and it does not mention matter and form.

Against that, it is to be remarked, ﬁrst, that although Aristotle speciﬁes

mathematical arguments, what he says of mathematical arguments plainly

applies to syllogisms in general; and secondly, that the text explicitly invokes

one of the four Aristotelian causes—the goal or ﬁnal cause—and so tacitly

invokes the others. And so—with a nod to Simplicius’ commentary (which

¹⁶ ἔστι δὲ τὸ ἀναγκαῖον ἔν τε τοῖς μαθήμασι καὶ ἐν τοῖς κατὰ φύσιν γιγνομένοις τρόπον

τινὰ παραπλησίως· ἐπεὶ γὰρ τὸ εὐθὺ τοδί ἐστιν, ἀνάγκη τὸ τρίγωνον δύο ὀρθαῖς ἴσας ἔχειν·

ἀλλ᾿ οὐκ ἐπεὶ τοῦτο, ἐκεῖνο· ἀλλ᾿ εἴ γε τοῦτο μὴ ἔστιν, οὐδὲ τὸ εὐθὺ ἔστιν. ἐν δὲ τοῖς γιγνο-

μένοις ἕνεκά του ἀνάπαλιν, εἰ τὸ τέλος ἔσται ἢ ἔστι, καὶ τὸ ἔμπροσθεν ἔσται ἢ ἔστιν· εἰ δὲ

μή, ὥσπερ ἐκεῖ μὴ ὄντος τοῦ συμπεράσματος ἡ ἀρχὴ οὐκ ἔσται, καὶ ἐνταῦθα τὸ τέλος καὶ τὸ οὗ

ἕνεκα.

Syllogistic Form and Syllogistic Matter 281

it would be fastidious to cite)—it has been imagined that according to

Aristotle the goal or ﬁnal cause of a syllogism is to be found not in its

premisses but in its conclusion. Thence, inasmuch as the ﬁnal cause is

frequently identical with the formal cause, we may infer that the conclusion

is the form of the syllogism—and therefore that the premisses constitute

the matter.

It would be merely fantastical to draw such conclusions from the paragraph

of the Physics which I have quoted were they not in part supported by the

second of the two pertinent pieces of that work, which is part of Aristotle’s

account of the four types of cause. He remarks that

the hypotheses are causes of the conclusion in the sense of that from which.

(Phys 195a18–20)¹⁷

By ‘hypothesis’ here Aristotle presumably means ‘premiss’; and ‘that from

which’ presumably designates the material cause. So Aristotle here applies the

notion of matter to the analysis or description of arguments. He does not

explicitly mention form; but the term ‘matter’ is relational, and its correlative

is ‘form’. So there must be a form for which the premisses of an argument

are matter: what is it? And of what item are the matter and the form matter

and form?

The text seems to say that the premisses of an argument are the material

cause of its conclusion. But, as Alexander saw, that notion hardly makes

sense:

The premisses do not inhere in the conclusion—rather, they are productive of the

conclusion: they inhere in the syllogism as a whole and have the r

ole of matter in it,

while the conclusion has the r

ole of form.

(Simplicius, in Phys 320.7–9)¹⁸

(SimpliciusisquotingfromAlexander’slostcommentaryonthePhysics.)

The premisses not the matter of the conclusion, for the conclusion is not

composed of the premisses; rather, they are the matter of the syllogism,

the conclusion of which is its form. That is Alexander’s view: it is unclear

whether he offered it as a correction or as a charitable interpretation of

Aristotle’s text.

¹⁷ … καὶ αἱ ὑποθέσεις τοῦ συμπεράσματος ὡς τὸ ἐξ οὗ αἴτιά ἐστιν.

¹⁸ αἱ δὲ προτάσεις οὐκ ἐνυπάρχουσι τῷ συμπεράσματι, ἀλλὰ τούτου μὲν ποιητικαὶ μᾶλλόν

εἰσιν, ἐν δὲ τῷ παντὶ συλλογισμῷ ὑπάρχουσι καὶ ὕλης ἔχουσιν ἐν αὐτῷ λόγον, τὸ δὲ

συμπεπερασμένον εἴδους.

282 Forms of Argument

Simplicius did not think that Alexander’s view was a true interpretation of

the text; but he spared no more than a single sentence to denounce Alexander

and to defend Aristotle, thus:

Or perhaps the premisses are in a way in the conclusion and are one with it.

(in Phys 320.9–11)¹⁹

The suggestion is empty. As Alexander indicates, if X is matter of Y, then X

inheres in Y—that is part of the Aristotelian deﬁnition of matter. Well then,

Simplicius supposes, the premisses do ‘in a way’ inhere in the conclusion. But

in what way? Simplicius does not care to tell us—and that is because there is

nothing to tell.

Alexander’s view is scarcely any better. You can say if you like that the

premisses are the matter of the argument and the conclusion its form—and

you can say the opposite, that the conclusion is matter and the premisses

form. Such remarks face a dilemma: either they use the words ‘form’ and

‘matter’ in something like their original Aristotelian way, in which case what

they say is trivially false; or else they use the words in some other way, in

which case they are toying.

Was the view which Alexander criticized and Simplicius defended the

view which Aristotle intended to advance? Perhaps it was; but the text

in the Physics is both isolated and obscure—indeed, a glance at the

immediate context shows that it is not even certain that Aristotle really

means to designate the premisses as material causes. In any event, this

little text scarcely helped to generate, and certainly does not help to

elucidate, the later distinction between material consequence and formal

consequence.

Another passage from Alexander—this time from his extant commentary

on the Topics—is more encouraging:

Aristotle and his followers … lay it down that dialectic is a certain syllogistic method;

and they think that syllogisms do not in the least differ one from another inso-

far as they are syllogisms—their differences are, some of them, according to the

forms of the propositions, some according to the moods and the shapes, some

according to the matter with which they are concerned. The ﬁrst of these dif-

ferences makes some syllogisms probative—or predicative, as we call them—and

others hypothetical. The second makes some perfect and others imperfect, and some

in a ﬁrst shape, some in a second and some in a third … And the third—the

¹⁹ μήποτε δὲ καὶ ἐν τῷ συμπεράσματι τρόπον τινά εἰσιν αἱ προτάσεις καὶ ἕν ἐστιν.

Syllogistic Form and Syllogistic Matter 283

difference according to matter—makes some demonstrative and some dialectical and

some eristical.

(in Top 1.19–2.16)²⁰

One of the ways in which one syllogism may differ from another is in its

matter. You would expect Alexander to say that another way—or rather, that

the other and complementary way—in which one syllogism may differ from

another is in its form. He does not do so.

Instead, he mentions two other types of differentiation: ﬁrst, difference

in ‘the forms of the propositions’, and secondly, difference in ‘the moods

and the shapes’. In effect, then, the matter of a syllogism is contrasted

with three other items—with the shape or ﬁgure of the syllogism, with its

mood, and with the form of its constituent propositions. That is a different

contrast from the one made in the commentary on the Analytics:therethe

ﬁgure or shape of a syllogism was identiﬁed with its form and so made up

one half of the contrast; here the ﬁgure of a syllogism is not—and cannot

be—identiﬁed with its form, and ﬁgure is one of four items and not one

of two. Although Alexander does not mention the form of a syllogism in

this passage, nonetheless—insofar as matter carries form with it—we may

properly ask what he would have taken the form of a syllogism to be. The

question has a ready answer: The three items with which matter is contrasted

together compose the form of a syllogism.

In fact, the three items are not independent of one another. If you specify

the mood of a syllogism—declaring, for example, that it is a syllogism in

Darapti—you thereby determine its ﬁgure. If you specify the ﬁgure of a

syllogism—declaring, for example, that it belongs to the third ﬁgure—you

thereby determine it to be a ‘probative’ or predicative syllogism and hence

you determine the pertinent general form of its constituent propositions. So

the three differences which Alexander announces—and which seem in his

text to be four rather than three—reduce to a couple: the mood of a syllogism

contrasts with its matter. Hence we may ascribe to Alexander the view that

the matter of a syllogism is constituted by its three concrete terms and that

²⁰ ᾿Αριστοτέλης δὲ καὶ οἱ ἀπ᾿ αὐτοῦ ... τίθενται μὲν αὐτὴν μέθοδόν τινα εἶναι συλλογιστικήν,

ἡγούμενοι δὲ τὸν συλλογισμόν, καθ᾿ ὃ συλλογισμός ἐστι, μηδὲν ἄλλον ἄλλου διαφέρειν, εἶναι

δὲ αὐτῶν τὴν διαφορὰν τὴν μὲν κατὰ τὰ εἴδη τῶν προτάσεων, τὴν δὲ κατὰ τοὺς τρόπους

καὶ τὰ σχήματα, τὴν δὲ κατὰ τὴν ὕλην περὶ ἥν εἰσιν, ὧν ἡ μὲν πρώτη διαφορὰ ποιεῖ τῶν

συλλογισμῶν τοὺς μὲν δεικτικούς, οὓς κατηγορικοὺς καλοῦμεν, τοὺς δὲ ὑποθετικούς, ἡ δὲ

δευτέρα καθ᾿ ἣν τοὺς μὲν τελείους τοὺς δὲ ἀτελεῖς, καὶ τοὺς μὲν ἐν πρώτῳ τοὺς δὲ ἐν δευτέρῳ

τοὺς δέ ἐν τρίτῳ σχήματι, ... ἡ δὲ τρίτη ἡ κατὰ τὴν ὕλην τοὺς μὲν ποιεῖ ἀποδεικτικοὺς τοὺς

δὲ διαλεκτικοὺς τοὺς δὲ ἐριστικούς.

284 Forms of Argument

the form of a syllogism is constituted by its mood. Take the syllogism which

might be expressed by saying that since philosophers are both intelligent

and industrious, some intelligent people must be hard workers. What is

its form?—Darapti. What is its matter?—‘philosopher’, ‘intelligent item’,

‘industrious item’.

Not infrequently elsewhere Alexander refers to the terms of a syllogism as

its matter; and a passage in the Ammonian commentary on the Prior Analytics

has been taken to show that Alexander’s teacher Herminus had done so before

him:

Herminus said that the conclusion is not always necessary but only in the case of

certain matter; for if we take animal, man and walking, a necessity is inferred, but if

animal, man and moving a possibility.

([Ammonius], in APr 39.31–35)²¹

Herminus was discussing a crux in Aristotle’s modal logic. According to

Aristotle, the following form is valid:

Necessarily A holds of every B

B holds of every C

Therefore necessarily A holds of every C

According to Herminus, the modal conclusion follows only in the case of

certain matter—that is to say, only in the case of certain triads of terms.

(Herminus’ thesis is logically inept; but the ineptitude is of no concern here.)

So Herminus spoke of the matter of a syllogism, which he identiﬁed with its

triadofterms.

We might reasonably hesitate to infer from this late text that Herminus

had actually used the word ‘matter’ to characterize the terms of a syllogism.

But if the inference is doubtful, the conclusion is independently plausible. For

when Alexander refers to the matter of a syllogism, he gives the impression

that the conceit was a commonplace—and in that case it will surely have

been known to Herminus.

The later commentators, both Greek and Latin, continue the habit.

They will often use ‘πρᾶγμα’ or ‘object’ instead of ‘ὕλη’ or ‘matter’. They

will sometimes contrast the matter of a syllogism with the combination

²¹ ῾Ερμῖνος δ᾿ ἔλεγεν ἀναγκαῖον γίνεσθαι τὸ συμπέρασμα οὐκ ἀεί, ἀλλ᾿ ἐπί τινος ὕλης· εἰ

μὲν γὰρ λάβωμεν ζῷον ἄνθρωπον περιπατοῦν, ἀναγκαῖον συνάγεται· εἰ δὲ ζῷον ἄνθρωπον

κινούμενον, ἐνδεχόμενον.

Syllogistic Form and Syllogistic Matter 285

of the premisses—with their πλοκή or συμπλοκή or complexio.Theywill

sometimes contrast the ‘nature’ of the premisses with their ‘force’ or meaning;

and Latin authors will speak of the vis terminorum. All these are so many

different ways of indicating a single contrast: the contrast between the terms

of a predicative syllogism and the rest of the thing.

That, no doubt, lies at the origin of the distinction between formal and

material consequence—a consequence is material if it depends on the nature

or force of its terms, it is formal otherwise. But three riders need to be

attached to that unoriginal claim. First, the terms ‘formal inference’ and

‘material inference’ are not, so far as I have noticed, found in any ancient text.

Indeed, although the contrast between the formal and the material aspect of

a syllogism is a commonplace in late antiquity, the language of form and

matter is invoked to express the contrast far less often than might have been

expected.

Secondly, the distinction between the two aspects of a syllogism was made

within the context of predicative syllogistic; and it is not evident how it might

be more widely applied. Take the hypothetical syllogism:

If Socrates is a man, he is mortal.

Socrates is a man.

Therefore Socrates is mortal.

The terms, an Aristotelian will surely be inclined to say, are clear enough: they

are ‘man’, ‘mortal’ and ‘Socrates’; and inasmuch as those terms constitute the

matter of the argument, the form is what is left. Any non-Aristotelian will

surely be inclined to take a different view: in that hypothetical syllogism, he

will suggest, it is the two propositions, ‘Socrates is a man’ and ‘Socrates is

mortal’, which constitute the matter; and everything else is the form; for it

is the constituent propositions of a hypothetical syllogism which correspond

in the appropriate way to the constituent terms of a predicative syllogism.

Doubtless that is right—I mean, if you want to talk of matter and form in

connection with hypothetical syllogisms, then that is the best way to go about

it. But in order to explain why it is the best way, you need some general

notion of the formal and material aspects of an argument.

And, thirdly, there are no such notions in the ancient texts: there is no

ancient theory about the difference, in general, between material and formal

inferences. Take the argument:

Socrates is a man.

Therefore Socrates is mortal.

286 Forms of Argument

Ithasasoneofitsforms:

xisaman

Therefore, x is mortal

Every argument of that form is valid. Why not say that the argument

is formally valid? A standard answer to that question, as I have already

indicated, is that the form is not a logical form, and an argument is formally

valid if and only if it is valid in virtue of a logical form. So when is a form a

logical form? There is a standard answer to that question too: a form is logical

if and only if it can be speciﬁed exclusively in terms of logical constants—of

words like ‘all’ and ‘none’ and ‘if’ and ‘or’. And how are logical constants

distinguished from non-logical expressions? That question—which has been

much debated by modern logicians—was not raised, let alone answered, by

the ancient logicians.

Perhaps they were wise. After all, consider the syllogisms which Galen

offers us in On Seed. What are the formal and what the material elements

in them? If the syllogisms are construed predicatively, then their matter will

be constituted by triads of terms and their form will be everything else.

If the syllogisms are construed hypothetically, then their matter will—on

the best account—be constituted by pairs of propositions and their form

will be everything else. So what, really, is the matter and what the form of

those syllogisms? That question is entirely parallel to the question: Are the

syllogisms really predicative or really hypothetical? To which the answer was:

They are really both.

If that is right, then the matter and form of an argument are determined

relative to a way of construing the argument—or rather, relative to a system

of logic within which the argument is construed. No doubt the form of

an argument is ﬁxed by its logical constants; but what counts as a logical

constant is itself ﬁxed by, and in that sense relative to, a logical system.

CIRCUMSCRIPTIONS

Stoic logicians sometimes spoke of the shape of an argument. Peripatetic

logicians sometimes distinguished between the matter and the form of

an argument. And in any event, even if their terminology was ﬂuid and

their theorizing exiguous, the ancient logicians did in fact discuss forms of

argument—after all, what else could they have discussed? If they discussed

Circumscriptions 287

forms, then they needed some way to present forms—some way to indicate

which items they were talking about. How, then, can a form of argument be

speciﬁed?

In various ways. You might, for example, offer a paradigm: ‘A syllogism

in Barbara is an argument like this: …’. You might present a rule of

inference—a permission or an instruction: ‘From this, that and the other

premiss, infer such-and-such a conclusion’. Antiquity took neither of those

paths. Rather, ancient logicians followed two other fashions in specifying

syllogistic forms. One of them I shall call circumscriptive, in honour of

Alexander of Aphrodisias, and the other I shall call schematic, for want of a

better word.

In the Prior Analytics Aristotle introduces the ﬁrst two sorts of argument

which his predicative syllogistic recognizes in the following way:

When three terms are so related to one another that the last is in the middle as in

a whole and the middle either is or is not in the ﬁrst as in a whole, it is necessary

that there is a syllogism of the extremes … For if A of every B and B of every C, it is

necessary that A is predicated of every C. … Similarly, if A of no B and B of every C,

that A will hold of no C.

(APr 25b32–26a2)²²

The ﬁrst sentence compactly describes a couple of syllogistic forms: ‘when

three terms …’—the sentence gives circumscriptions of the forms which we

know as Barbara and Celarent. The second and third sentences comment

on the two forms, and do so schematically. They do not, strictly speaking,

represent the two forms schematically; but schematic representations are, as

it were, implicit in them. Barbara, say, may be presented thus:

AofeveryB

BofeveryC

Therefore A of every C

Both circumscriptive and schematic speciﬁcations were also used in expos-

itions of hypothetical syllogistic. According to Galen,

²² ὅταν οὖν ὅροι τρεῖς οὕτως ἔχωσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλῳ εἶναι

τῷ μέσῳ καὶ τὸν μέσον ἐν ὅλῳ τῷ πρώτῳ ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἄκρων εἶναι

συλλογισμὸν τέλειον. ... εἰ γὰρ τὸ Α κατὰ παντὸς τοῦ Β καὶ τὸ Β κατὰ παντὸς τοῦ Γ, ἀνάγκη

τὸ Α κατὰ παντὸς τοῦ Γ κατηγορεῖσθαι· ... ὁμοίως δὲ καὶ εἰ τὸ μὲν Α κατὰ μηδενὸς τοῦ Β,

τὸ δὲ Β κατὰ παντὸς τοῦ Γ, ὅτι τὸ Α οὐδενὶ τῷ Γ ὑπάρξει.

288 Forms of Argument

the logicians name the shapes of arguments modes—e.g. in the case of the argument

which concludes from a conditional and its antecedent to its consequent (which

Chrysippus calls a ﬁrst unproved), the mode is this:

If the 1

,the2

; but the 1

: therefore the 2

(inst log vi 6)²³

Just as Aristotle presents Barbara and Celarent ﬁrst by way of circumscriptions

and then—in a virtual fashion—schematically, so Galen identiﬁes the

Chrysippean ‘ﬁrst unproved’ by a circumscription and then sets out its shape

or form, which he calls a mode. A formula which represents a mode is a

schematic representation of the syllogistic form which is, or is associated with,

the mode.

A circumscription is a description of an argument form. The word

‘circumscription’, or ‘περιοχή’, comes from Alexander, who writes thus:

You might grasp this from the circumscription of the species of the syllogism.

The circumscription of this sort of syllogism is this: The syllogism which from a

disjunction and one of the disjoined items infers the opposite of the other.

(in Top 11.22–25)²⁴

Alexander uses the word ‘περιοχή’ in the same sense in his commentary on

the Prior Analytics (see 274.20–25). But otherwise the word seems to be

unknown to science. (To be sure, it turns up three times in [Themistius]—see

in APr 81.7; but there [Themistius] is simply copying out parts of Alexander’s

commentary.) It has been conjectured that Alexander’s employment of the

word derives from its use to mean ‘summary’—as in the Summaries of

Menander’s Plays by Sellius, a.k.a. Homer (Suda, s.v. ῞Ομηρος). But in truth,

that latter usage is hardly widespread; and the similarity between a summary

and a circumscription is not overwhelming.

However that may be, circumscriptions specify logical forms by describing

them. Roughly speaking, a circumscription will say something of the form:

‘From this, that and the other premiss there is a syllogism to such-and-

such a conclusion.’ The ancient logicians show some uniformity in their

circumscriptive descriptions, and some scholars think that the Stoics, at least,

²³ … οἷον ἐπὶ μὲν τοῦ ἐκ συνημμένου καὶ τοῦ ἡγουμένου τὸ λῆγον περαίνοντος, ὃν ὁ

Χρύσιππος ὀνομάζει πρῶτον ἀναπόδεικτον, ὁ τοιοῦτος τρόπος ἐστίν· εἰ τὸ πρῶτον, τὸ

δεύτερον· τὸ δὲ πρῶτον· τὸ ἄρα δεύτερον.

²⁴ τοῦτο δὴ λάβοι τις ἂν ἐκ τῆς περιοχῆς τοῦ εἴδους τοῦ συλλογισμοῦ. ἔστι δὲ ἡ περιοχὴ

αὕτη τοῦ τοιούτου συλλογισμοῦ· ὁ ἐκ διαιρετικοῦ καὶ ἑνὸς τῶν ἐν τῇ διαιρέσει τὸ ἀντικείμενον

ἐπιφέρων τοῦ λοιποῦ.

Circumscriptions 289

developed a canonical set of such things. But no ancient text, so far as I know,

has any discussion of the matter.

The syllogistic circumscriptions which we ﬁnd in ancient logical texts are

not always satisfactory. For example, Galen circumscribes Barbara in this way:

In the case of predicative syllogisms, in the ﬁrst ﬁgure there are four unproveds—ﬁrst,

the one which from two universal afﬁrmatives infers a universal afﬁrmative

conclusion …

(inst log viii 3)²⁵

The conclusion of a syllogism in Barbara, according to Galen, will be

universal and afﬁrmative. But which universal afﬁrmative proposition will it

be? We know, from Galen’s general account of the syllogistic ﬁgures, that

it will predicate one of the extreme terms of the other; but Galen does not

specify which of the two extremes will be predicated in the conclusion, and

his circumscription is thus indeterminate between two forms, one of them

Barbara and the other invalid.

Aristotle, it must be added, does not always do better than Galen. Indeed,

in the passage which I cited a moment ago he does worse; for, having

described the conjugations for Barbara and Celarent he simply says that ‘it is

necessary for there to be a syllogism of the extremes’—and that in principle

leaves us with a choice among eight possibilities for each conjugation.

Those criticisms are no doubt pifﬂing. If Aristotle’s circumscription is

inadequate, it does not appear alone—it is complemented by the schematic

comment on the ﬁrst two predicative forms, and that comment is perfectly

precise and determinate. As for Galen, he was writing a textbook which

you might read through with the aid of a teacher or else use as a sort of

aide-m´emoire. In any event, whether or not syllogistic circumscriptions were

always in fact done well, they can in principle be done well. Barbara, for

example, might be circumscribed thus:

From a ﬁrst ﬁgure conjugation of a pair of universal and afﬁrmative

unmodalised propositions, there is a valid inference to a universal and

afﬁrmative unmodalised proposition in which the extreme term which in

the conjugation is predicated of the middle term is predicated of the other

extreme term.

That is a mouthful; but it is digestible.

²⁵ ἐπὶ δὲ τῶν κατηγορικῶν ἐν μὲν τῷ πρῶτῳ σχήματι τέσσαρες εἰσιν ἀναπόδεικτοι, ὅ τε

ἐκ δυοῖν καθόλου καταφατικῶν καθόλου καταφατικὸν ἐπιφέρων συμπέρασμα ...