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270 Forms of Argument

That mode is invalid. So every argument—and therefore every valid argu-

ment—exhibits at least one invalid form.

Who put forward the claim which had been attacked and which Chrysippus

then defended? It is tempting to guess that it was Chrysippus himself who

did so—that the second of the two titles refers to an essay in self-defence;

and it is very tempting to guess that the claim which Chrysippus defended

had been put forward in the ﬁrst of the two adjacent essays on Diogenes’ list,

and that it constituted the chief theme of that lost essay. In that case, the

ﬁrst essay did not discuss, let alone defend, the thesis that one and the same

syllogism may have different and non-subordinate valid forms.

All that is hopelessly iffy. But it is enough for my present ends: the two

titles do not demonstrate that Chrysippus thought that an argument could

have two different and non-subordinate valid forms. Perhaps he did think so;

but the titles do not prove that he did.

The other wisps of evidence to which I alluded come from Galen. Here,

ﬁrst, is a passage from the Introduction to Logic:

The syllogism will be propounded hypothetically as follows:

If Socrates is a son of Sophroniscus, then Sophroniscus is father of Socrates.

But Socrates is a son of Sophroniscus.

Therefore Sophroniscus is father of Socrates.

But the construction of the reasoning will be more forceful with predicative

propositions …

(inst log xvi 11)⁶

I have there translated an emended text, and the emendations are anything

but certain. But unless the transmitted Greek is unfathomably corrupt, one

thing is plain: Galen supposes that one and the same argument may be

put forward either hypothetically or predicatively. He presupposes, in other

words, that an argument may belong to two distinct and non-subordinate

kinds.

The passage in the Introduction is not isolated. In his work On Seed Galen

argues against those scientists who deny that females produce seed. After a

string of refutatory arguments, he remarks:

⁶ ὑποθετικῶς μὲν οὕτως ὁ συλλογισμὸς ἐρωτηθήσεται· εἰ Σωκράτης υἱός ἐστι Σωφρονίσ-

κου, Σωφρονίσκος πατήρ ἐστι Σωκράτους· ἀλλὰ μὴν ὁ Σωκράτης υἱός ἐστι Σωφρονίσκου·

Σωφρονίσκος ἄρα πατήρ ἐστι Σωκράτους. κατηγορικαῖς δὲ προτάσεσι βιαιότερον ἔσται ἡ

σύστασις τοῦ λογισμοῦ ...

Species of Syllogism 271

Those considerations are enough to refute their opinion. And refutation apart, it is

possible to produce a direct proof, syllogizing in two ways, both hypothetically and

predicatively.

(sem iv 609)⁷

Galen then offers some arguments, which he duly characterizes as predicative

or as hypothetical syllogisms. There are difﬁculties in interpreting the passage;

but here too one thing is plain: Galen supposes himself to be offering a single

probative argument—an argument which is now done in predicative guise

and now in hypothetical.

In another of his technical treatises, on simple drugs, Galen spends a few

pages on the pharmacological claim that olive-oil is astringent. Those who

have advanced the claim, he alleges, have made a logical error.

For it is agreed that everything astringent is rough and that olive-oil is rough.

But from these suppositions it does not follow that olive-oil is astringent, whether

we make the propositions predicative or hypothetical; for nothing follows from two

universal afﬁrmatives in the second ﬁgure, and the conditional is not true of necessity.

In the predicative syllogism the two premisses will be these:

Everything astringent is rough,

Every olive-oil is rough

—and from agreement on them nothing follows. As for the hypothetical premiss,

which the Chrysippeans call a connected assertible, we cannot ﬁnd one which is

true.

(simp med temp xi 499)⁸

Here too—and more clearly—Galen seems to suppose that one and the

same argument (admittedly a faulty argument) may be either predicative or

hypothetical.

How can that be? Well, ‘these suppositions’, or the materials which consti-

tute the starting-points of the faulty argument, may be made into predicative

propositions and they may be made into hypothetical propositions. Thus

⁷ ταυτὶ μὲν ἱκανὰ τὴν δόξαν ἐλέγχειν αὐτῶν. ἔστι δὲ καὶ χωρὶς ἐλέγχου τὴν ἀπόδειξιν ἐξ

εὐθείας ποιεῖσθαι, διττῶς συλλογιζομένοις ὑποθετικῶς τε καὶ κατηγορικῶς.

⁸ ἅπαν μὲν γὰρ ὡμολόγηται τὸ δάκνον εἶναι κερχνῶδες. ὡμολόγηται δὲ καὶ τοὔλαιον ὑπάρ-

χειν κερχνῶδες. ἀλλ᾿ ἐκ τῶν ὑποκειμένων τούτων οὐ περαίνεται δακνῶδες εἶναι τοὔλαιον,

οὔτε κατηγορικὰς οὔτε ὑποθετικὰς ἡμῶν ποιησάντων τὰς προτάσεις, τῷ μήτ᾿ ἐκ δύο καθόλου

καταφατικῶν ἐν δευτέρῳ σχήματι περαίνεσθαί τι μήτε τὸ συνημμένον ἐξ ἀνάγκης ἀληθεύεσ-

θαι. γενήσονται δ᾿ ἐν μὲν τῷ κατηγορικῷ συλλογισμῷ δύο προτάσεις αἵδε· πᾶν τὸ δάκνον κερ-

χνῶδες, πᾶν ἔλαιον κερχνῶδες· ἐξ ὧν ὁμολογηθέντων οὐδὲν περανθήσεται. τὴν δ᾿ ὑποθετικὴν

πρότασιν, ἣν οἱ περὶ τὸν Χρύσιππον ἀξίωμα συνημμένον ὀνομάζουσιν, οὐκ ἔχομεν ἀληθῆ

λαβεῖν.

272 Forms of Argument

we agree that olive-oil is rough; and we might formulate the agreement by

saying, for example,

Everything which is olive-oil is rough.

We might equally well say this:

If anything is olive-oil, then it is rough.

The ﬁrst of those sentences ﬂaunts a predicative form or a subject–predicate

structure: it quite overtly expresses an Aristotelian A-predication, it straight-

forwardly says of every so-and-so that it is such-and-such. The second

sentence just as shamelessly displays a hypothetical form: it is a sort of

conditional, it says that if thus-and-so, then so-and-thus. That being so, are

the ‘suppositions’ on which the bad inferences build really predicative or

really hypothetical? Evidently, they are really both—and of course they are

really both regardless of the ways in which we choose to express them.

Since a supposition may be both hypothetical and predicative, it is easy to

see how a syllogism may be both hypothetical and predicative.

It may be objected that the two sentences with which I expressed the

thought about olive-oil do not in fact express the very same thought—or at

any rate, that there are different thoughts there to be expressed. For from an

A-predication which says ‘rough’ universally and afﬁrmatively of ‘olive-oil’,

it follows—in Aristotle’s logic—that

Some olive-oil is rough.

But from the hypothetical proposition which declares something to be

rough if it is olive-oil, no such consequence follows. The predicative pro-

position has, as they say, ‘existential import’, the hypothetical does not.

Hence the sentence which wears predicativity on its sleeve expresses one

proposition, and the sentence which sports hypotheticality expresses another;

and if I try to say neutrally that olive-oil is rough—perhaps by way of the

sentence ‘Olive-oil is rough’—then what I say is either ambiguous or in-

determinate.

Perhaps that is correct. But if so, it constitutes an objection to Galen, not

to an interpretation of Galen. For Galen plainly supposes that one and the

same proposition is expressed by the two sentences.

It may be objected, secondly, that no ancient text states in so many words

that one and the same argument may constitute both a predicative and a

hypothetical syllogism. On the contrary, most ancient logical texts give the

clear impression—without, I think, ever making a direct statement—that

an argument may be either predicative or hypothetical and cannot be both at

once. As for Galen, the two passages which I have quoted are the only passages

Species of Syllogism 273

which suggest that an argument may be both predicative and hypothetical;

and the Introduction to Logic will give its readers a strong contrary impression.

All of that is quite true. Nonetheless, it is reasonable to conclude from the

wisps of evidence that at least one ancient logician was at least sometimes

aware that every argument is not, so to speak, conﬁned to a single syllogistic

form: an argument may, in principle, have two—or more—distinct and

non-subordinate valid forms.

In that case, syllogisms cannot be arranged under species and genera. You

may talk of kinds or sorts of arguments. You may not talk of species of

arguments—unless, of course, you use the word ‘species’ in a relaxed sense.

If logicians occasionally talk about kinds or sorts or even species of

arguments, they also—and perhaps more often—talk about forms of argu-

ment. Now the forms of argument in which a logician is interested can be

arranged into a genuine Porphyrean tree, from which we may pick—for

example—the sequence: syllogism, predicative syllogism, ﬁrst ﬁgure syllo-

gism, Barbara. That is not the same sequence as before, though it is expressed

in the same words. For in it Barbara is no longer an inﬁma species:itis

an individual—an individual syllogistic form. No such individual belongs

to more than one lowest species, and the classiﬁcatory scheme is saved. As

for arguments and syllogisms—concrete, individual arguments and concrete,

individual syllogisms—they are not forms, and so they do not appear on the

tree at all. Rather, they have or possess or show forms. And an item may have

a plurality of forms without thereby prejudicing the Porphyrean structure of

the tree of forms.

If you want to talk seriously about species of syllogism, then, you had better

be thinking of species of syllogistic forms, not of species of concrete syllogisms.

Strictly speaking, a syllogism—according to the ancient deﬁnitions—is an

argument: it has premisses which are either true or false, and a conclusion

which either follows or fails to follow from the premisses. Barbara is not a

syllogism: Barbara has no premisses and no conclusion—rather, instances or

cases of Barbara have premisses and conclusions. Barbara is a syllogistic mood

or a syllogistic form. That is perfectly clear, and it was perfectly clear to the old

logicians. Nonetheless, the old logicians, Galen among them, will frequently

talk of, say, ‘the ﬁrst syllogism in the ﬁrst ﬁgure’ and thereby designate the

syllogistic form Barbara; and in general, they use the word ‘syllogism’ often

enough to refer to syllogistic forms and not to concrete arguments. Indeed,

I suspect that their most common use of the word ‘syllogism’ fails to accord

with their formal deﬁnition of the word ‘syllogism’. That will vex pedants;

274 Forms of Argument

but it is generally harmless enough—and I myself shall make no serious effort

to say ‘syllogistic form’ rather than ‘syllogism’ when it is a syllogistic form I

mean to talk about.

FORMAL LOGIC

The preceding remarks will, I fear, have excited no one who is not a devotee of

Porphyrean taxonomy. But they have at least served to introduce the notion

of form; and perhaps they have also served to suggest that all logic is—in a

pretty straightforward sense—formal logic. For logic is not concerned with

this or that individual concrete inference, except insofar as the individual

inference is an instance of some particular form; nor yet is logic concerned

with this or that general type of inference, except insofar as the general type

of inference is determined by a general form.

The claim that all logic is formal might be resisted, on several grounds.

One bad ground for resistance is worth mentioning inasmuch as it involves a

vulgar misconception—a misconception which is, I think, common among

non-logicians. It will be afﬁrmed, and truly, that not all logic is symbolic logic:

after all, no ancient logicians showed any interest in artiﬁcial symbolizations,

which hardly obtruded themselves before the nineteenth century. And if not

all logic is symbolic, then—it may be inferred—not all logic is formal. But

the inference is invalid. For although the terms ‘formal’ and ‘symbolic’ are

sometimes confounded, they have two quite different signiﬁcations. Symbolic

logic studies forms of inferences, so that it is a kind of formal logic. But what

makes it symbolic is not that feature but rather the fact that the inferences

which it studies are expressed with the help of artiﬁcial symbols: just as

symbolic arithmetic uses the symbols ‘2’, and ‘+’ rather than the words

‘two’ and ‘plus’, so symbolic logic uses the symbols ‘⊃’and‘∀’ rather than

the words ‘if’ and ‘every’. Symbols have their advantages—sometimes they

have overwhelming advantages. But logic—formal logic—is not obliged to

use them.

A more serious objection to the claim that all logic is formal rests on the

counterclaim that there are non-formal inferences. The mediaeval logicians

recognized such things, which they called ‘material consequences’. They

contrasted material with formal consequences—and that contrast is the

immediate origin of our modern use of the term ‘formal’ in connection with

logic. Consider, for example, the inference:

Formal Logic 275

Socrates runs.

Therefore, Socrates moves.

That is surely a valid deductive inference—it is impossible that its premiss

should be true and its conclusion not true, its conclusion follows necessarily

from its premiss. But—so it was urged—it is a material and not a formal

consequence. Hence not all logic is formal logic.

A short answer to that objection asserts roundly that material consequences

are, despite their mediaeval name, formally valid. The argument

Socrates runs.

Therefore, Socrates moves.

has several forms in common with the rather similar argument

Plato runs.

Therefore, Plato moves.

One of the common forms might be described by saying that, from a premiss

which says of some item that it runs, there is an inference to the conclusion

that that same item moves; or you might say that the form is this:

xruns.

Therefore, x moves.

That is a form of argument, or a mode; the form or mode is valid; and the

two material consequences are valid inasmuch as they are instances of that

form or mode.

That blunt reply is impeccable. But it does not end the discussion. The

two material consequences, it will be allowed, are indeed valid in virtue of

a certain shared form, and for that reason they may, if you like, be called

formally valid. But there are forms and forms: the form in virtue of which the

material consequences are valid is not a logical form; and anyone who denies

that all logic is formal logic means to deny that all valid inferences are valid in

virtue of a logical form. No doubt—trivially—all valid inferences are valid

in virtue of some form or other; but some inferences are valid in virtue of a

logical form, others in virtue of a non-logical form. Any inference of the form

xruns.

Therefore x moves.

is valid. But that form is not a logical form. Therefore logic is not exclusively

formal.

276 Forms of Argument

That conclusion is intelligible insofar as the distinction between logical

and non-logical form is intelligible—and it may be said at once that that

distinction is anything but self-evident. Nevertheless, grant that there is a

distinction between logical and non-logical forms, and grant that there are

valid arguments which are valid in virtue of non-logical forms. Even so, it does

not follow that logic is not exclusively formal. For perhaps logic does not—or

even cannot—deal with non-formal inferences. That, in point of fact, was

the view taken, implicitly, by ancient logicians. Material consequences are

not, and cannot be, objects of scientiﬁc study.

They cannot be objects of scientiﬁc study because they are, as the Stoics put

it, ‘non-methodically concludent’; for whatever exactly that expression may

mean, it is clear that, according to the Stoics, non-methodically concludent

arguments are not possible objects of methodical study—and hence not pos-

sible objects of a science. The Peripatetics, we know, disagreed with the Stoics:

according to them, the arguments which the Stoics labelled unmethodical

were, or at any rate could be remodelled as, predicative syllogisms. Galen

disagreed with the Peripatetic remodelling. He also disagreed with the Stoics.

He held that the allegedly unmethodical arguments, or at any rate some of

them, are in fact syllogisms of his third species. There are sharp differences of

opinion there; but they are grounded on an underlying consensus: the Stoics,

the Peripatetics, and Galen may have disagreed on the question of whether

this or that particular argument was or was not formally valid; but they all

agreed that logic studies formally valid arguments.

What I have just said is true. But it is also—as Aristotle would have

put it—unclear, or unilluminating. It is unilluminating for two reasons.

First, the notion of form which I have been bandying about needs some

explanation—and so, afortiori, does the notion of logical form. Secondly,

although the ancients agreed that logic is formal logic, they did not formulate

their agreement in those terms—indeed, they did not formulate it in any

terms—and the expression ‘formal logic’ has no synonym in any ancient text.

SHAPES OF ARGUMENT

According to Diogenes Laertius, a Stoic mode is ‘as it were a shape of an

argument’ (vii 76). In the same vein, Sextus refers to ‘the modes and as it

were shapes in which arguments are propounded’ (M viii 227),⁹ and Galen

⁹ τρόποι δὲ αὐτῶν καὶ ὡσπερεὶ σχήματα ἐν οἷς ἠρώτηνται ...

Shapes of Argument 277

observes that ‘the logicians name the shapes of arguments modes’ (inst log

vi 6).¹⁰ The word ‘shape’ or ‘σχῆμα’ is in fact sometimes used in more or

less that sense in our texts. Sextus, for example, will say that an argument ‘is

valid inasmuch as it is propounded in a sound shape’ (M viii 413);¹¹ and he

reports a theory which distinguished four ways in which an argument might

go wrong, one of which is

on account of being propounded in an unsound shape, when the shape of the

argument is not concludent.

(PH ii 147)¹²

In that last text, it is difﬁcult not to translate ‘σχῆμα’by‘form’.

And after all, the word ‘σχῆμα’ or ‘shape’ was often used as a synonym,

or near synonym, for ‘μορφή’; ‘μορφή’ was often used as a synonym or near

synonym for ‘εἶδος’; and ‘εἶδος’ is the Greek for ‘form’. But in the texts to

which I have alluded and which are of Stoic origin or Stoic inspiration, the

word ‘shape’, although it may in fact be co-extensive, or nearly co-extensive,

with ‘form’, is never contrasted with ‘matter’. If shapes are forms, the formal

is not set against the material.

In any event, it is no surprise to ﬁnd that the distinction between formal

and material consequence, insofar as it has any ancient ancestors, belongs to

the Aristotelian and not to the Stoic clan. After all, what is it but yet another

application of that Jack-of-all-trades distinction, the distinction between form

and matter, εἶδος and ὕλη?AsPorphyryputit,

objects are constituted of matter and form or else have their constitution analogous

to matter and form.

(Isag 11.12–13)¹³

Pretty well anything is an object, so that matter and form are virtually

ubiquitous. Certainly, arguments are objects. So arguments consist of matter

and form, or at least of something analogous to matter and something

analogous to form. But what is their matter, or quasi-matter, and what their

form, or quasi-form?

In the Analytics Aristotle does not speak of matter and form in connection

with syllogisms. But he does frequently talk of the shape of a syllogism, and

¹⁰ ὄνομάζουσι δὲ τρόπον οἱ διαλεκτικοὶ τὰ τῶν λόγων σχήματα.

¹¹ συνάγει μὲν διὰ τὸ ἐν ὑγιεῖ ἠρωτῆσθαι σχήματι.

¹² παρὰ δὲ τὸ ἐν μοχθηρῷ ἠρωτῆσθαι σχήματι, ὅταν μὴ ᾖ τὸ σχῆμα τοῦ λόγου συνακτικόν.

¹³ τῶν γὰρ πραγμάτων ἐξ ὕλης καὶ εἴδους συνεστώτων ἢ ἀνάλογόν γε ὕλῃ καὶ εἴδει τὴν

σύστασιν ἐχόντων ...

278 Forms of Argument

his commentators found matter and form behind his use of the word ‘shape’.

According to Alexander of Aphrodisias, Aristotle tells us in the Prior Analytics

what a syllogism is, and what it is composed from, and how many syllogistic shapes

there are, and what are the differences among them—for the shapes are like a sort

of common mould: by ﬁtting matter into them, you can mould the same form in

different matters; for just as, in the case of identical moulds, the difference is made

not by the form and shape of what is ﬁtted into them but by its matter, so too is it

with the syllogistic shapes.

(in APr 6.15–21)¹⁴

Syllogisms have form and matter; and their forms are here identiﬁed with,

or at least determined by, their shapes. Now the word ‘σχῆμα’, which I

have thus far translated as ‘shape’, is standardly rendered by ‘ﬁgure’ when it

appears in the context of Aristotelian syllogistic. So according to Alexander,

the form of a syllogism is what we customarily call its ﬁgure.

The ﬁgure of a syllogism, or its shape or form, is like a mould. A mould

ﬁxes the form of the jelly you pour into it, and when one moulded jelly differs

from another, that is in virtue of its matter. So too, the ﬁgure of a syllogism

determines its form, and when one ﬁgured syllogism differs from another,

that is in virtue of its matter.

The Aristotelian notion of ﬁgure is perfectly well determined. The ﬁgure

of a syllogism is ﬁxed by the ‘conjugation’ of propositions which constitute

its premisses. A conjugation is a pair of predicative propositions which

have exactly one term in common. Since the common term must, in each

proposition, be either subject or predicate, there are three possible types of

conjugation: either the common term is once predicate and once subject,

or else it is twice predicate, or else it is twice subject. Those three types

of conjugation determine the three syllogistic shapes or ﬁgures: a syllogism

belongs to the ﬁrst ﬁgure if and only if the common term in its conjugation

is once predicate and once subject; and so on.

No doubt the word ‘σχῆμα’—whetherwegiveitas‘shape’oras

‘ﬁgure’—has had its sense somewhat stretched; but the stretching was done

by Aristotle and not by his commentators, and it is inoffensive inasmuch as

¹⁴ … τί ἐστι συλλογισμός, καὶ ἐκ τίνων σύγκειται, καὶ πόσα σχήματά ἐστι συλλογιστικά,

καὶ τίνες αὐτῶν διαφοραί· τύπῳ γάρ τινι κοινῷ τὰ σχήματα ἔοικεν, ἐν οἷς ἔστιν ἐναρμόσαντα

ὕλην εἶδός τι ἀναμάξασθαι ταὐτὸν ἐπὶ ταῖς διαφόροις ὕλαις· ὡς γὰρ ἐπὶ τῶν τύπων τῶν

αὐτῶν ἡ διαφορὰ οὐ κατὰ τὸ εἶδος γίνεται καὶ τὸ σχῆμα τοῖς ἐναρμοζομένοις ἀλλὰ κατὰ τὴν

ὕλην, οὕτω δὴ καὶ ἐπὶ τῶν σχημάτων τῶν συλλογιστικῶν.

Shapes of Argument 279

the stretched sense is deﬁned. It is easy to see how Alexander glossed ‘ﬁgure’

by ‘form’—for the two words, as I have said, are near synonyms; and it is

easy to understand how he then helped himself to the term ‘matter’—for

where, in Peripatetic philosophy, form intrudes, can matter be far behind?

True, the words ‘matter’ and ‘form’ have now lost all contact with their

origins: the matter of something is no longer its constituent stuff, the form

no longer the outward shape or ﬁgure. But even in Aristotle’s own writings,

the couple of matter and form is often found wandering far from home; and

so long as the two terms are given some clear sense in their new surroundings,

no serious harm is done. (Nonetheless, some unserious harm is done; for

fastidious readers will be repelled and rapid readers may be misled. And in

the credit column there is a large zero.)

However that may be, matter and form occasionally turn up in that way

in the later logical tradition. Here, for example, is Ammonius:

We say that in every syllogism there is something analogous to matter and something

analogous to form—it is the objects themselves by way of which the syllogism is

composed which are analogous to matter, and the shapes which are analogous to

form—for some are in a ﬁrst shape, others in a second, others in a third.

(in APr 4.8–12)¹⁵

Ammonius is in one respect slightly more cautious than Alexander: he speaks

not of matter and form but of something analogous to matter and something

analogous to form. After Porphyry, he supposes that only material objects have

matter and form in the strict sense of the words: non-material objects—such

as syllogisms—have quasi-matter and quasi-form. But he is fundamentally at

one with Alexander: it is the shape of a syllogism—that is to say, the ﬁgure

to which it belongs—which is, or determines, its form or quasi-form.

That, as I have said, is a clear notion, and it is a notion rooted in

Aristotle’s own terminology. But it is not the notion which grounds the

distinction between formal and material consequence and which underlies

the idea of formal logic. Formal consequences are formally valid, or valid

in virtue of their form; but no syllogism is valid in virtue of its ﬁgure. An

argument is valid in virtue of a given feature only if every argument which

shares that feature is valid. Hence a syllogism would be valid in virtue of

its ﬁgure only if every syllogism which shares that ﬁgure is valid. But that

¹⁵ λέγομεν ὅτι ἐν παντὶ συλλογισμῷ τὸ μέν τί ἐστιν ἀνάλογον ὕλῃ τὸ δὲ εἴδει. ὕλῃ μὲν οὖν

ἀναλογεῖ τὰ πράγματα αὐτὰ δι᾿ ὧν ὁ συλλογισμὸς πλέκεται, εἴδει δὲ τὰ σχήματα· οἱ μὲν γὰρ

ἐν πρώτῳ σχήματί εἰσιν, οἱ δὲ ἐν δευτέρῳ, οἱ δὲ ἐν τρίτῳ.