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290 Forms of Argument

In the case of hypothetical syllogisms, circumscription again presents

no theoretical difﬁculty—although again, ancient circumscriptions are not

always strictly adequate. Sometimes, indeed, it is not clear what are the

criteria of adequacy. According to Galen,

in the case of complete conﬂict, there will be two syllogisms, if we assume in addition

either that one of the items holds or that it does not hold and infer that the other

does not hold (when the ﬁrst holds) and holds (when the ﬁrst does not).

(inst log v3)²⁶

A group of items—of assertibles or propositions—is in complete conﬂict

if and only if exactly one of them must be true. Complete conﬂicts are

appropriately expressed by disjunctive sentences, where the disjunction is to

be understood in a strongly exclusive sense. So the two syllogisms—that

is to say, the two syllogistic forms—which Galen here mentions might be

circumscribed thus:

(1) From an exclusive disjunction and one of its disjuncts, there is a valid

inference to the negation of its other disjunct.

(2) From an exclusive disjunction and the negation of one of its disjuncts,

there is a valid inference to its other disjunct.

Those two forms, according to Galen, exhaust the inferential powers of

disjunctions. Of course, there are many more—inﬁnitely many more—valid

arguments one of the premisses of which is a disjunction. Take, say:

Either he’s alive or he’s dead. He’s alive if and only if he’s breathing. So

either he’s breathing or he’s dead.

But when Galen discusses hypothetical syllogisms, he has his mind on what

were sometimes called mixed hypotheticals or syllogisms which have one

complex and one simple premiss. And on that assumption, his disjunctive

circumscriptions may well appear to be exhaustive.

Or rather, they may appear to be exhaustive on the assumption that

the pertinent disjunctions have exactly two disjuncts. But ancient dis-

junctions—unlike the disjunctions of standard modern logic—were not

necessarily two-placed items; and an ancient logician should want to circum-

scribe disjunctive syllogisms the disjunctive premiss of which may have any

²⁶ κατὰ μὲν οὖν τὴν τελείαν μάχην δύο συστήσονται συλλογισμοὶ προσλαμβανόντων ἡμῶν

ἤτοι γε ὑπάρχειν ἢ μὴ ὑπάρχειν τὸ ἕτερον αὐτῶν, ἐπιφερόντων δὲ θάτερον οὐχ ὑπάρχειν μὲν

ὅταν ὑπάρχῃ τὸ ἕτερον, ὑπάρχειν δὲ ὅταν οὐχ ὑπάρχῃ.

Circumscriptions 291

number of disjuncts. How might that be done? The obvious suggestion goes

like this:

(1*) From an exclusive disjunction and one of its disjuncts, there is a valid

inference to the conjunction of the negations of its other disjuncts (or to

the negation of its other disjunct if there is only one other disjunct).

(2*) From an exclusive disjunction and the negation of one of its disjuncts,

there is a valid inference to the exclusive disjunction of its other disjuncts

(or to its other disjunct if there is only one other disjunct).

But whereas (1) and (2) perhaps exhaust the inferential possibilities for

two-placed disjunctions, it is plain that (1*) and (2*) are not exhaustive for dis-

junctions in general. For example, from a three-membered disjunction and the

negations of two of its disjuncts, there is a valid inference to the third disjunct:

Either 2

is greater than 2 or it is equal to 2 or it is less than 2. But 2

isn’t

equal to 2, nor is it less than 2. So it’s greater than 2.

That inference is not covered by (2*)—nor, of course, by (1*).

So if exhaustivity is the aim, then (2*) must be modiﬁed. The modiﬁcation

might look something like this:

(2**) From an exclusive disjunction and the negations of at least one of its

disjuncts (but not of all of them), there is a valid inference to the exclusive

disjunction of its other disjuncts (or to its other disjunct if there is only

one other disjunct).

Ought Galen to have offered (1*) and (2*) rather than (1) and (2)? Ought he

to have offered (1*) and (2**) rather than (1*) and (2*)? The answer to the

ﬁrst of those two questions is, I am sure, Yes. The answer to the second is less

evident inasmuch as the syllogisms which (2**) circumscribes may be analysed

as or reduced to sets of syllogisms each of which is circumscribed by (2*).

There is another point. The circumscription which I quoted from Alexan-

der’scommentaryontheTopics was this:

The syllogism which from a disjunction and one of the disjoined items infers the

opposite of the other.

(in Top 11.24–25)

That answers to one of Galen’s two syllogisms ‘in the case of complete

conﬂict’. Like Galen, Alexander is thinking of two-placed disjunctions, so

that his circumscription matches (1). But the match is not perfect: where

292 Forms of Argument

Galen speaks of negations, Alexander speak of opposites. There is not merely

a difference there: the two circumscriptions are not equivalent to one another.

For if x is an opposite of y, then y is an opposite of x; but if x is a negation

of y, then y is not a negation of x. ‘So-and-so’ is an opposite, but it is not a

negation, of ‘It is not the case that so-and-so’. Then which circumscription is

right? Which is the better? What are the criteria of judgement here?

Whatever the answers to such questions, it seems plain that the circum-

scriptive way of specifying a syllogistic form is always available: any type

of argument—or so I suppose—can in principle be clearly and distinctly

circumscribed. Later ancient logicians generally preferred the circumscriptive

mode. In Galen’s Introduction, for example, and in Apuleius’ de Interpreta-

tione, predicative syllogisms are presented by way of circumscriptions; and

although Apuleius comments on the Peripatetic and the Stoic uses of schem-

atic formulas, and although Galen suddenly makes use of schemata when

he sketches ecthetic proofs of validity, no schemata appear in their ofﬁcial

presentations of predicative syllogistic.

But circumscriptions have certain practical drawbacks. First, it is easy to

misformulate them—both Galen and Apuleius, as well as Aristotle himself,

illustrate the point. Secondly, even in simple cases circumscriptions can be

heavy; and the more complicated the case, the harder the circumscription is

to digest. Thirdly, ancient logicians were concerned not only to set out valid

syllogistic forms but also to explain or justify their validity: typically, they

tried to prove the validity of one syllogistic form on the basis of another. Such

proofs can be expounded in the circumscriptive mode; but the expositions

are obliged to choose between imprecision and a laborious cumbersomeness.

In short, circumscription leads to circumlocution.

SCHEMATIC REPRESENTATIONS

One of the more intricate pieces of exposition in the Prior Analytics is the

so-called pons asinorum. In his commentary, Alexander ﬁrst sets out Aristotle’s

theory without using any schematic formulas. He then remarks that

Aristotle proves what we have just said by using letters—for the sake of clarity.

(in APr 304.32)²⁷

²⁷ ἐπὶ στοιχείων ἃ προειρήκαμεν δείκνυσι σαφηνείας χάριν.

Schematic Representations 293

And Alexander himself sets out the theory again, this time using schematic

formulas. The result, it will be agreed, is far clearer—that is to say, it is far

more readily understood and assessed; and Alexander remarks that

just as a geometer will construct a diagram for the sake of clarity in his exposition,

so a logician will use letters (in APr 379.28–29)²⁸

In a similar vein, when Sextus embarks on a discussion of a certain complex

syllogism, he says that

this argument is composed from a second and a third unproved—as we can learn

from an analysis, which will be more clear if we set out the exposition in the form of

a mode, thus:

If the 1

and the 2

,the3

; but not the 3

; but the 1

: therefore not the 2

(M viii 235)²⁹

An analysis—roughly speaking, a proof of validity—will be more clear, or

more readily followed, if it is done on modes than if it is done on arguments.

In other words, the substitution of symbols for concrete terms lends clarity to

the enterprise. Here Sextus alludes to the advantages of symbols over concrete

terms, not over circumscriptions. But the underlying point is very similar.

(And it makes not a whit of difference whether you use letters or numerals.)

Schematic representations make for clarity; and they are in at least certain

respects comparable to the geometers’ use of diagrams in their proofs of

universal theorems. (I shall return to the comparison.) In some ways,

schemata are undeniably preferable to circumscriptions. But do they—can

they—do the same job as circumscriptions?

Galen, as we have seen, puts circumscriptions and schemata together in

one of his accounts of certain hypothetical syllogisms. So consider his account

of the second of the ﬁve Stoic unproveds:

In the case which, from a conditional and the opposite of its consequent, infers the

opposite of its antecedent, … the mode is this:

If the 1

,the2

; but not the 2

: therefore not the 1

(inst log vi 6)³⁰

²⁸ ὡς γὰρ ὁ γεωμέτρης ὑπὲρ σαφηνείας τῆς κατὰ τὴν διδασκαλίαν καταγραφὴν ποιεῖταί

τινα ...

²⁹ συνέστηκε γὰρ ὁ τοιοῦτος λόγος ἐκ δευτέρου τε ἀναποδείκτου καὶ τρίτου, καθώς πάρεστι

μαθεῖν ἐκ τῆς ἀναλύσεως, ἥτις σαφεστέρα μᾶλλον γενήσεται ἐπὶ τοῦ τρόπου ποιησαμένων

ἡμῶν τὴν διδασκαλίαν, ἔχοντος οὕτως· εἰ τὸ πρῶτον καὶ τὸ δεύτερον, τὸ τρίτον· οὐχὶ δὲ γε

τὸ τρίτον· ἀλλὰ καὶ τὸ πρῶτον· οὐκ ἄρα τὸ δεύτερον.

³⁰ ἐπὶ δὲ τοῦ ἐκ συνημμένου καὶ τοῦ ἀντικειμένου τῷ εἰς ὃ λήγει τὸ τοῦ ἡγουμένου

ἀντικείμενον ἐπιφέροντος, ... τοιοῦτός ἐστιν· εἰ τὸ πρῶτον, τὸ δεύτερον· οὐχὶ δὲ τὸ δεύτερον·

οὐκ ἄρα τὸ πρῶτον.

294 Forms of Argument

Take this argument: ‘If she’s not in Italy, then she can’t be in Milan. But

she’s certainly in Milan—so she’s in Italy.’ Is that a second unproved? Yes,

according to the circumscription. For ‘She’s in Milan’ is the opposite of the

consequent of the conditional premiss, and ‘She’s in Italy’ is the opposite

of its antecedent. But is the argument a second unproved according to the

mode?

The modern schema which corresponds to the Stoic mode is—or seems

to be—this:

If P, then Q; but not Q: therefore not P.

The concrete argument which I have just rehearsed ﬁts the following modern

schema:

IfnotP,thennotQ;butQ:thereforeP.

There are—to all appearances—two different schemata there: each embraces

some but not all of the arguments which meet Galen’s circumscription. If

Galen’s Stoic mode corresponds to the ﬁrst schema, then it does not match

his circumscription: the mode captures only some of the arguments which

the circumscription encloses.

Or again, this is Galen’s presentation of the third Stoic unproved:

So too in the case of the third (according to Chrysippus), which from a negated

conjunction and one of the items in it gives the opposite of the other item, the mode

is this:

Not at the same time the 1

and the 2

; but the 1

: therefore not the 2

(inst log vi 6)³¹

The circumscription is clear, and so is the mode. But does the mode repeat

the circumscription? Consider the following argument: ‘Well, I’m sure she’s

not in England—after all, she’s in Paris, and she can’t both be in England

and in Paris.’ Is that an example of a third unproved?

According to the circumscription, it undoubtedly is. But according to the

schematic version? A contemporary logician will declare that the argument is

an instance of the schema

Not (both P and Q); but Q: therefore not P.

That is a distinct schema from

Not (both P and Q); but P: therefore not Q,

³¹ … ὥσπερ γε κἀπὶ τοῦ τρίτου κατὰ τοῦτον, ὃς ἐξ ἀποφατικοῦ συμπεπλεγμένου καὶ ἑνὸς

τῶν ἐν αὐτῷ τὸ ἀντικείμενον τοῦ λοιποῦ παρέχει, τοιοῦτος ὁ τρόπος ἐστίν· οὐχ ἅμα τὸ

πρῶτον καὶ τὸ δεύτερον· τὸ δὲ πρῶτον· οὐκ ἄρα τὸ δεύτερον.

Schematic Representations 295

which is the schema corresponding to Galen’s mode. (Let it be added that if

the third unproved is stated in such a way as to allow for conjunctions with

any number of conjuncts, then things get far trickier.)

Has Galen bungled? Perhaps he has miscircumscribed these two unproveds,

or else given the wrong modes? I do not think so; and I think that the

circumscriptions trump the schemata, at least in Galen’s text—that is to say,

if we want circumscriptions and schemata to match, then we must modify

the schemata. But how is that to be done? Indeed, is it to be done at all?

Take a third and more testing example. In Chapter xv of the Introduc-

tion Galen considers syllogisms based on what the ancient logicians called

παραδιεζευγμένα or quasi-disjunctions: a quasi-disjunction is true if and

only if at least one of its quasi-disjuncts is true; in other words, a quasi-

disjunction is what a modern logician would call an inclusive disjunction.

The discussion is lengthy, complicated, and textually corrupt. But the gist

of the matter is plain. Galen describes the following form of hypothetical

syllogism:

From an inclusive disjunction and the opposites of one or more of the

disjuncts (but not of all the disjuncts) there is a valid inference to the

inclusive disjunction of the other disjuncts (or to the other disjunct if there

is only one).

That is a clear and distinct circumscription, even if it is a little lumpy. What

is the mode which corresponds to it?

Well, a modern logician will ﬁrst come up with the familiar schema:

Either P or Q

Not P

Therefore Q

Any instance of that schema is an example of the syllogism which Galen

describes; but there are indeﬁnitely many examples of Galen’s syllogism

which are not instances of the schema. Alongside the familiar schema must

be set, ﬁrst, the equally familiar schema:

Either P or Q

Not Q

Therefore P

And then there will be six schemata for three-placed disjunctions, among

them

296 Forms of Argument

Either P or Q or R

Not Q

Either P or R

and

Either P or Q or R

Not P and not R

Therefore Q

And so on. A whole family of schemata corresponds to the single circum-

scription; and this particular family has an inﬁnite number of members—or

at least, it has an indeterminately vast number of members.

No ancient text raises such issues. But where the ancient logicians appeal to

circumscriptions and to schemata side by side, they tacitly presuppose that, in

principle at least, the two fashions of specifying syllogistic forms pair off. Can

the two fashions be made to pair off? Perhaps we should look a little more

narrowly at schemata; for it might be objected to the preceding argument

that it construes the notion of schematization too priggishly and that it takes

an unnecessarily puritanical view of the relation between a schema and a

concrete argument.

The three hypothetical circumscriptions under scrutiny invoke opposition.

Opposites here are contradictories; that is to say, the term ‘opposite’ is to

be understood in the Stoic fashion. Now according to a text which I have

already cited, the Stoics

say: opposites are items the one of which exceeds the other by a negation. For example

It is day—It is not day.

For the assertible ‘It is not day’ exceeds ‘It is day’ by a negation, namely ‘not’, and is

for that reason opposite to it.

(Sextus, M viii 89)³²

That being so, we should—strictly speaking—talk of an opposite of X rather

than of the opposite of X; for every proposition which is governed by a

negation will have two opposites—both

It’s day

and also

³² φασὶ γὰρ ἀντικείμενά ἐστιν ὧν τὸ ἕτερον τοῦ ἑτέρου ἀποφάσει πλεονάζει, οἷον ἡμέρα

ἔστιν—οὐχ ἡμέρα ἔστιν. τοῦ γὰρ ἡμέρα ἔστιν ἀξιώματος τὸ οὐχ ἡμέρα ἔστιν ἀποφάσει

πλεονάζει τῇ οὐχί, καὶ διὰ τοῦτ᾿ ἀντικείμενόν ἐστιν ἐκείνῳ.

Schematic Representations 297

It’s not not day

are opposite to

It’s not day.

Now where the three circumscriptions refer to opposites, the three modes

or schemata use negation; and that is one of the reasons for the mismatch.

So why not replace opposition by negation in the circumscriptions? (In fact,

as we have seen, Galen uses negation in his circumscription of a couple of

disjunctive syllogisms.) Well, that would be a retrograde move: it would

reduce the ambit of the circumscription for no good theoretical reason.

Better, then, emend the modes by substituting oppositions for negations.

For example, the mode for the second unproved will be not

If the 1

,the2

;butnotthe2

: therefore not the 1

but rather

If the 1

,the2

; opp(the 2

): therefore opp(the 1

)

—where ‘opp(X)’ means ‘an opposite of X’. That schema, unlike its pre-

decessor, seems to coincide with Galen’s circumscription of the second

unproved.

In order to deal with the schema for the third unproved, it is not enough

to replace negation by opposition. But another dodge might be called upon:

why not write the mode something like this:

Notboththe1

and the 2

;butthen

: therefore not the m

where either n = 1andm = 2orn = 2andm = 1

That dodge is suggested by a modern device—the device of subscription.

With the aid of subscription, you might think to represent the syllogistic

form like this:

NotbothP

and P

;butP

: therefore opp(P

)

where either i = 1andj= 2ori= 2andj= 1.

(That is to say, the second premiss of any argument of that form is either

the ﬁrst or the second of the negated conjuncts; and the conclusion is the

opposite of the other conjunct.) Such devices have sometimes been used

by expositors of the Stoic logic. True, they were not used by any ancient

logicians, and the use of subscripts is a thoroughly modern convention. But

then the modern schemata are offered as superior replacements for the old,

not as explanatory interpretations of them.

Even so, it is not easy to get them to do all the work which is required

of them. How, for example, shall we ﬁnd a schema to correspond to a

298 Forms of Argument

multi-placed negated conjunction? Or a schema for Galen’s quasi-disjunctive

syllogism? We might start with this:

Either P

or P

or…orP

where a generous soul will claim to understand the three dots and the

subscripted ‘n’. But what comes next? Something like:

Not Q

and not Q

and … and not Q

where n > m > 1andeachQ

is identical with some P

And the conclusion? Try this:

or R

or…orR

where k = n − m, each R

is identical with some P

and no R

is identical

with any Q

Perhaps that—or some variant upon it—will do the trick.

But, ﬁrst, such a schematic representation loses the advantage which was

claimed for schemata over circumscriptions. For that particular schema is

surely far less clear—far less immediately intelligible—than the correspond-

ing circumscription.

And secondly, is the thing really a schema at all? I shall postpone that

second question for a few pages.

PREDICATIVE SCHEMATA

If problems of mismatch between schemata and circumscriptions arise in the

case of hypothetical syllogisms, do they also haunt predicative syllogistic? In

general, no; but they have been thought to haunt one particular predicative

mood.

The ﬁrst syllogistic form in Aristotle’s third ﬁgure, or Darapti, is described

thus by Galen:

Of the syllogisms in the third ﬁgure, the ﬁrst, from two universal afﬁrmative premisses,

has a particular afﬁrmative conclusion, being reduced—by way of conversion of the

premiss on its minor term—to the third syllogism in the ﬁrst ﬁgure.

(inst log x1)³³

And here is the Aristotelian original:

³³ τῶν δ᾿ ἐν τῷ τρίτῳ σχήματι συλλογισμῶν ὁ μὲν πρῶτος ἐκ δυοῖν προτάσεων καθόλου

καταφατικῶν ἐν μέρει καταφατικὸν ἔχει συμπέρασμα, δι᾿ ἀντιστροφῆς τῆς πρὸς τῷ ἐλάττονι

τῶν ὅρων προτάσεως ἀναγόμενος εἰς τὸν ἐν τῷ πρώτῳ σχήματι τρίτον.

Predicative Schemata 299

There is no perfect syllogism in this ﬁgure either; but there will be a potential

syllogism both when the terms are universal and when they are not universal in

relation to the middle term—when they are universal, when both P and R hold of

every S, that P will hold of necessity of some R. For since the afﬁrmative converts, S

will hold of some R, so that since P holds of every S and S of some R, it is necessary

that P hold of some R—for there is a syllogism in the ﬁrst ﬁgure.

(APr 28a15–22)³⁴

The question is this: does the schematic speciﬁcation in Aristotle match the

circumscriptive speciﬁcation in Galen?

True, Aristotle does not, strictly speaking, offer a schema for Darapti.

Nonetheless, what he does offer is tantamount to the schema:

P holds of every S.

R holds of every S.

Therefore P holds of some R.

Any syllogism which ﬁts that schema will also ﬁt Galen’s circumscription of

Darapti. But is the converse also the case? Consider the following schema:

P holds of every S.

R holds of every S.

Therefore R holds of some P.

Any argument which ﬁts that schema plainly ﬁts Galen’s circumscription

of Darapti. But is it not a different schema from Aristotle’s? And do not

two separate schemata then correspond to one circumscription? If so, then

in predicative syllogistic as in hypothetical syllogistic there is a mismatch

between circumscriptions and schemata, even if the predicative mismatch is

far less impressive.

The second of the two predicative schemata is the schema for the mood

known as Daraptis. Daraptis is not mentioned in the chapter of the Prior

Analytics in which Aristotle deals with the third ﬁgure; but it is implicitly

acknowledged in a later paragraph:

³⁴ τέλειος μὲν οὖν οὐ γίνεται συλλογισμὸς οὐδ᾿ ἐν τούτῳ τῷ σχήματι, δυνατὸς δ᾿ ἔσται

καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων πρὸς τὸ μέσον. καθόλου μὲν οὖν ὄντων, ὅταν

καὶ τὸ Π καὶ τὸ Ρ παντὶ τῷ Σ ὑπάρχῃ, ὅτι τινὶ τῷ Ρ τὸ Π ὑπάρξει ἐξ ἀνάγκης· ἐπεὶ γὰρ

ἀντιστρέφει τὸ κατηγορικόν, ὑπάρξει τὸ Σ τινὶ τῷ Ρ· ὥστ᾿ ἐπεὶ τῷ μὲν Σ παντὶ τὸ Π, τῷ

δὲ Ρ τινὶ τὸ Σ, ἀνάγκη τὸ Π τινὶ τῷ Ρ ὑπάρχειν· γίνεται γὰρ συλλογισμὸς διὰ τοῦ πρώτου

σχήματος.