Barnes Jonathan. Truth, etc

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300 Forms of Argument

Since some syllogisms are universal and some are particular, all the universal syllogisms

always syllogize a plurality of items, while of the particulars, the afﬁrmative syllogize

a plurality but the negative their conclusion only. For the other propositions convert,

but the negative does not convert; and the conclusion predicates something of some-

thing. So the other syllogisms syllogize a plurality of items—e.g. if A has been shown

to hold of every B or of some B, then it is also necessary that B hold of some A, and if

of no B, then that B hold of no A—and that is different from the previous conclusion.

(APr 53a3–12)³⁵

Aristotle did not enumerate the new syllogisms which can thereby be added

to his system. But if we start from the fourteen forms which are recognized

in chapters 4–6, then it seems that there should be a further nine. For the

conclusions of Barbara, Celarent, Darii, Cesare, Camestres, Darapti, Disamis

and Datisi all convert; and in addition, the conclusion of the new syllogism

produced from Barbara has itself a convertible conclusion.

The principle by which Aristotle generates the new syllogisms is a true

principle. In general, if

A, B, … , W: therefore X

is a valid argument, and if

X: therefore Y

is valid, then

A, B, … , W: therefore Y

is valid. And the principle does indeed add some new moods to Aristotle’s

system, moods which chapters 4–6 overlooked. For in those chapters, once

Aristotle had indicated that a conjugation produced a conclusion, he never

stopped to ask whether it also produced a second.

When it is applied to the three relevant moods of the ﬁrst ﬁgure, the

principle each time generates a new mood. Barbara is

A holds of every B

B holds of every C

Therefore A holds of every C.

Its conclusion entails, by conversion:

C holds of some A.

³⁵ ἐπεὶ δ᾿ οἱ μὲν καθόλου τῶν συλλογισμῶν εἰσὶν οἱ δὲ κατὰ μέρος, οἱ μὲν καθόλου

πάντες αἰεὶ πλείω συλλογίζονται, τῶν δ᾿ ἐν μέρει οἱ μὲν κατηγορικοὶ πλείω, οἱ δ᾿ ἀποφατικοὶ

τὸ συμπέρασμα μόνον. αἱ μὲν γὰρ ἄλλαι προτάσεις ἀντιστρέφουσιν, ἡ δὲ στερητικὴ οὐκ

ἀντιστρέφει. τὸ δὲ συμπέρασμα τὶ κατά τινός ἐστιν, ὥσθ᾿ οἱ μὲν ἄλλοι συλλογισμοὶ πλείω

συλλογίζονται, οἷον εἰ τὸ Α δέδεικται παντὶ τῷ Β ἢ τινί, καὶ τὸ Β τινὶ τῷ Α ἀναγκαῖον

ὑπάρχειν, καὶ εἰ μηδενὶ τῷ Β τὸ Α, οὐδὲ τὸ Β οὐδενὶ τῷ Α· τοῦτο δ᾿ ἕτερον τοῦ ἔμπροσθεν.

Predicative Schemata 301

So Barbara validates Baralipton:

A holds of every B

B holds of every C

Therefore C holds of some A.

Baralipton is evidently an additional mood—if only because no existing ﬁrst

ﬁgure syllogism derives an I-predication from a couple of A-predications. It

is easy to verify that the moods generated from Celarent and Darii (and also

from Baralipton) are also new.

Things are different in the second ﬁgure. Cesare, for example, yields

Cesares. Cesare is

A holds of no B

A holds of every C

Therefore B holds of no C.

Conversion of the conclusion yields Cesares, or:

A holds of no B

A holds of every C

Therefore C holds of no B

Now Cesares is not Cesare—but it is not a new mood either. For it is

Camestres, the second mood of the second ﬁgure. As Galen puts it,

in the second and third ﬁgures … one new syllogism is produced by conversion of

the conclusion in the case of the ﬁrst syllogism of the third ﬁgure—and in its case

alone. For the ﬁrst two syllogisms in the second ﬁgure convert into one another by

their conclusions, and so do the third and fourth in the third ﬁgure.

(inst log xi 7)³⁶

By conversion of the conclusion, Cesare generates Camestres, and vice

versa; and Disamis generates Dabitis, and vice versa. But conversion of the

conclusion does produce one new mood outside the ﬁrst ﬁgure; for Darapti

generates Daraptis, and Galen takes Daraptis to be a new mood.

The thesis that Daraptis is different from Darapti was disputed in antiquity.

Here is Apuleius on the subject:

In the third ﬁgure, the ﬁrst mood is the one which infers a particular afﬁrmative from

universal afﬁrmatives, either directly or reﬂexively; for example:

³⁶ κατὰ δὲ τὸ δεύτερον σχῆμα καὶ τρίτον ... ἐκ ... τῆς τοῦ συμπεράσματος ἀντιστροφῆς ἐν

τῷ τρίτῳ σχήματι κατὰ τὸν πρῶτον συλλογισμὸν γίγνεται μόνον· οἱ μὲν γὰρ ἐν τῷ δευτέρῳ

σχήματι πρῶτοι δύο πρὸς ἀλλήλους ἀντιστρέφουσι τῷ συμπεράσματι, ἐν δὲ τῷ τρίτῳ δύο

τρίτος τε καὶ τέταρτος.—Here ‘πρῶτον’isBoche

nski’s correction of ‘τρίτον’.

302 Forms of Argument

Everything just is honest

Everything just is good

Therefore something honest is good

—or

Therefore something good is honest

It makes no difference from which proposition you take the subject part since it

makes no difference which you express ﬁrst. Hence Theophrastus was wrong to think

that, on this account, this is not one mood but two.

(int xi [207.16–24])³⁷

Apuleius thus urges that Daraptis and Darapti are one mood, not two;

and he reports that Theophrastus—like Galen after him—had taken the

opposite view.

Theophrastus seems to have had the majority on his side. For according to

Boethius,

the third ﬁgure has six moods according to Aristotle; but some add a further

mood—among them Porphyry, who was following his predecessors.

(in syll cat 813c)³⁸

The predecessors of Porphyry are Theophrastus and Eudemus, whom

Boethius has just named; and the further mood is later identiﬁed as Daraptis

(819b —where Boethius confesses that he is not himself sure whether the

third ﬁgure contains six or seven moods).

Why did Theophrastus and his followers take Daraptis to be distinct from

Darapti? A passage in Alexander has been cited in evidence:

There are six syllogisms in this ﬁgure. The ﬁrst of them in order is the one which, from

two universal afﬁrmatives, infers a particular afﬁrmative conclusion by conversion of

the minor proposition. It can also come about if the major proposition is converted,

but in that case the conclusion too will have to be converted. That is why some

added this syllogism as different from its predecessor, and say that there are seven

syllogisms in this ﬁgure.

(in APr 95.25–30)³⁹

³⁷ in tertia formula primus modus est qui conducit ex dedicativis universalibus dedicativum

particulare tam directim quam reﬂexim, ut: omne iustum honestum, omne iustum bonum, quoddam

igitur honestum bonum; vel sic: quoddam igitur bonum honestum. quippe non interest ex utra propositione

facias particulam subiectivam quoniam non interest utram prius enunties. ideo non recte arbitratus est

Theophrastus propter hoc non unum modum hunc sed duos esse.

³⁸ … tertia vero auctore Aristotele sex [sc modos habet]; addunt etiam alii unum, sicut ipse Porphyrius,

superiores scilicet sequens.

³⁹ ἓξ δὲ ὄντων συλλογισμῶν ἐν τούτῳ τῷ σχήματι πρῶτος μὲν ἂν αὐτῶν εἴη τῇ τάξει

ὁ ἐκ δύο καθόλου καταφατικῶν ἐπὶ μέρους καταφατικὸν συνάγων κατὰ ἀντιστροφὴν τῆς

Predicative Schemata 303

There are two proofs of Darapti—two proofs, that is, which work by way

of conversion—and that fact led some people to think that there were two

syllogisms.

Alexander does not mention Theophrastus by name; and it has been denied

that he is thinking of Daraptis. After all, when he reports that ‘some added

this syllogism’, the only syllogism in the ofﬁng to which he can readily be

taken to refer is Darapti itself. So the anonymous men who added a seventh

mood did not add Daraptis: they admitted Darapti twice over—or rather,

they admitted two versions of Darapti, one corresponding to each of the two

proofs.

Against that it may be said, ﬁrst, that it would be strange if Alexander

had said nothing about Daraptis while mentioning another—and otherwise

quite unknown—way of augmenting the third ﬁgure; and secondly, that the

argument which, on this interpretation, Alexander ascribes to his anonymous

men is absurd: it is evident that the existence of several proofs of one item is

not an indication that there are several items under proof; and anyone who

denied that evidence could hardly fail to see that he would then have on his

hands not only two versions of Darapti but two or more versions of every

second and third ﬁgure syllogism.

So it is plausible to think that Daraptis—and Theophrastus—come into

the Alexandrian picture. But how? Consider the two proofs of Darapti at

which Alexander hints:

(A)

(1) P holds of every S premiss

(2) R holds of every S premiss

(3) S holds of some R 2, conversion

(4) P holds of some R 1, 3, Darii

(B)

(1*) P holds of every S premiss

(2*) R holds of every S premiss

(3*) S holds of some P 1*, conversion

(4*) R holds of some P 2*, 3*, Darii

(5*) P holds of some R 4*, conversion

ἐλάττονος προτάσεως. δύναται δὲ καὶ τῆς μείζονος ἀντιστραφείσης γενέσθαι, ἀλλὰ δεήσει

καὶ τὸ συμπέρασμα ἀντιστρέφειν· διὸ καὶ τοῦτόν τινες τὸν συλλογισμὸν προστιθέντες ὡς

ἄλλον τοῦ πρὸ αὐτοῦ ἑπτά φασιν τοὺς ἐν τούτῳ τῷ σχήματι συλλογισμούς.

304 Forms of Argument

Each proof is a proof of Darapti—a proof based on Darii and a rule

of conversion. But if proof (B) is curtailed at step (4*), then that is a

proof of Daraptis. The fact that there are two proofs of Darapti persuaded

Theophrastus and others to add a seventh mood to the third ﬁgure not

because they thought that two proofs of Darapti made two moods out

of Darapti, but because the second of the two proofs of Darapti brought

Daraptis to their attention.

But if that explains how Theophrastus hit upon Daraptis, it does not

explain why he took Daraptis to be distinct from Darapti. No ancient text

offers any explanation of that sort; but there is an argument in Theophrastus’

favour which at ﬁrst blush is fetching. Apuleius notes that from the pair of

premisses

Everything just is honest

Everything just is good,

you may conclude either that

Something honest is good

or else that

Something good is honest.

There, then, are two syllogisms with identical premisses but distinct conclu-

sions. So surely they must have different syllogistic forms? If ‘A, B: therefore

X’ and ‘A, B: therefore Y’ are two different valid arguments, then surely they

must be valid in virtue of different forms? Hence Daraptis and Darapti are

different.

One reply to that argument which an Apuleian might well consider denies

that the conclusions of the two arguments are different. No doubt the

sentences

Something honest is good

and

Something good is honest

are different; but they are two different ways of saying the same thing.

Such an idea has a toe-hold in the ancient texts. I have already noted

that according to the Greek grammarians, some connectors indicate an order

and others do not. The connector ‘and’ or ‘καί ’(or‘et’), when it makes a

conjunctive sentence, signals no order. That is to say, there is no semantic

difference between ‘P and Q’ and ‘Q and P’. If that is so, then how can there

be a difference between

Something is A and B

Predicative Schemata 305

and

Something is B and A?

And if, as modern logic generally supposes,

Something A is B

means the same as

Something is A and B,

then how can there be any difference between

Something A is B

and

Something B is A?

Darapti and Daraptis have the same conclusion—they simply express their

conclusions in different ways.

That idea has seduced some logicians, Frege among them. But it is not

evidently correct; for it is not evidently correct that

Something A is B

means the same as

Something is A and B.

And however that may be, the idea surely did not attract Aristotle. For, ﬁrst,

Aristotle takes pains to prove that I-predications convert or that from

A holds of some B

you may infer

B holds of some A;

and that is enough to show that he did not take the two sentences to say the

same thing. Secondly, he observes explicitly that ‘A holds of no B’ is not the

same as ‘B holds of no A’ (APr 53a11–12); and the context indicates that he

thinks the same goes for ‘A holds of some B’ and ‘B holds of some A’.

So the conclusion of Apuleius’ Daraptis is different from the conclusion of

his Darapti, at least in Aristotelian eyes. Must we not then infer that the two

arguments have different forms? No; for the principle according to which if a

couple of arguments share premisses and differ in their conclusions then they

must be valid in virtue of different forms—that principle may be alluring

but it is false. The following circumscription determines a single form of

argument:

From a conjunction there is a valid inference to any one of its conjuncts.

It follows that for every n-placed conjunction, there will be n syllogisms

which have the same form, the same premiss, and different conclusions. Or

again, take the form speciﬁed by the following schema:

P: therefore either P or Q

306 Forms of Argument

(where the disjunction is inclusive). Indeﬁnitely many arguments have that

form, share a premiss, and have different conclusions. Hence the general reas-

on which suggested a distinction between Darapti and Daraptis is a bad reason.

Is there any good reason on the opposite side, any good reason to identify

Darapti and Daraptis? Apuleius states that ‘it makes no difference from which

proposition you take the subject part’; that is to say, it makes no difference

whether the conclusion is ‘P holds of some R’ (where ‘R’ is taken from ‘R

holds of every S’) or rather ‘R holds of some P’ (where ‘P’ is taken from ‘P

holds of every S’). That is so, Apuleius argues, because ‘it makes no difference

which you take ﬁrst’; that is to say, there is no difference between

P holds of every S, R holds of every S

and

R holds of every S, P holds of every S.

Apuleius’ premiss is true—but it is hard to see how it supports his conclusion.

In any event, there is a better argument on his side. We have, on the one

hand, Darapti:

P holds of every S

R holds of every S

Therefore P holds of some R

and on the other hand Daraptis:

P holds of every S

R holds of every S

Therefore R holds of some P

It might seem that there is a difference—the conclusions of the two forms

arrange their terms in different ways. But that difference is an illusion.

Consider the form which I baptize Daraptix:

R holds of every S

P holds of every S

Therefore P holds of some R.

Daraptix is the same syllogism as Daraptis—its schema is merely a notational

variant on the schema for Daraptis. It is also the same as Darapti; for—as

Apuleius says—the order in which the premisses are inscribed or uttered is

perfectly irrelevant to the argument. If Daraptix is the same as Daraptis and

also the same as Darapti, then Daraptis and Darapti are one and the same.

A more general consideration is also available. What are the identity

conditions for schemata? In what circumstances is schema S the same schema

Schemata and Matrixes 307

as schema S*? I shall restrict the question to predicative schemata—for

otherwise various irrelevant obstacles would have to be surmounted; and I

say that a predicative schema S characterizes the same syllogistic form as

a predicative schema S* if and only if every argument which instances S

instances S* and vice versa. Since any argument which instances the schema

for Darapti will also instance the schema for Daraptis, and vice versa, Darapti

and Daraptis are one and the same syllogistic form.

The case of Daraptis does not, after all, show that, in predicative syllogistic,

schematic representations sometimes fail to pair off with circumscriptions.

SCHEMATA AND MATRIXES

So far I have spoken of schematic representations as though they and their

uses needed no explanation. But that is not so.

LookagainatanexamplewhichIusedearlier:‘Ifshe’snotinItaly,then

she can’t be in Milan. But she’s certainly in Milan—so she’s in Italy.’ In

so arguing, have I thereby done a second Stoic unproved? According to the

circumscription, Yes. According to the schematic representation—or so I

urged—No. For although the argument ﬁts the schema

If not P, not Q; Q: therefore P

it does not ﬁt the different schema

If P, Q; not Q: therefore not P;

and it is that second schema which corresponds to the mode of the second

unproved. I then suggested that we might modify the schema, replacing ‘not’

with ‘opp’ and so matching mode to circumscription.

But before accepting the modiﬁcation, it is worth asking why the argument

ﬁts the one unmodiﬁed schema and not the other.

A schematic representation of a syllogism is a sequence of schematic

representations of propositions or assertibles. What exactly is a schematic

representation of a proposition or an assertible? One way of understanding

such schemata—a way which is both simple and common—takes them to

be sets of what I shall call matrixes. Take any sentence. Replace it or any of

its syntactically coherent parts by a symbol or by symbols—by letters, say, or

numerals—and what you get is a sentential matrix. So from the sentence

Aristotle is acute and Plato is profound

you might produce replace the name ‘Aristotle’ by the letter ‘a’ and thereby

produce the matrix:

a is acute and Plato is profound.

308 Forms of Argument

Or you might replace each of the two verbal phrases by letters and produce

this:

F(Aristotle) and G(Plato)

(Shouldn’t that rather be something like

Aristotle(F) and Plato(G)?

Well, there are certain standard conventions which govern the order of

symbols in standard matrixes.)

So a matrix—or rather, a sentential matrix—is a sequence of symbols and

words, or a sequence of symbols, which can be derived from a sentence by an

appropriate replacement of words by symbols; or, equivalently, a matrix is a

sequence which can be turned into a sentence by an appropriate replacement

of symbols by words.

Appropriateness is syntactical appropriateness. Of the symbols which occur

in matrixes, the letters ‘a ’and‘b’ are syntactically names; ‘F’ and ‘G’ are

syntactically verbs (or rather, one-placed predicates); ‘P’, ‘Q’, ‘R’, … are

syntactically sentences. But although the symbols have a syntax, they have

no sense. The symbol ‘a’ names nothing, the symbol ‘F’ can’t be used

to predicate anything, the symbol ‘P’ expresses no thought. The choice of

symbols is, of course, entirely arbitrary: you might use signs like ‘♣’, ‘♦’,

‘♥’, … ; you might use numerals; you might use anything. But letters have

certain practical advantages.

Although the symbols have no sense, they are not merely syntactical devices.

For the repetition of the same symbol in a matrix, or in a connected sequence

of matrixes, has a signiﬁcance. Suppose you start from a sentence and produce

a matrix: if the same expression occurs more than once in the sentence, then

it may—but need not—be replaced each time by the same symbol; but

different expressions must be replaced by different symbols. The sentence

Aristotle was acute and Aristotle was profound

will therefore produce, say,

Fa and Ga

and also

Fa and Gb;

but it will not produce

Fa and Fb

nor yet

Fa and Fa.

Conversely, suppose that you start from a matrix and generate a sentence:

different symbols may—but need not—be replaced by different expressions;

Schemata and Matrixes 309

any symbol which occurs more than once must be replaced by the same

expression each time. From

If P, then Q

you may generate

If today’s Thursday, then this is Paris,

and also

If this is Paris, then this is Paris.

The latter, but not the former, of those two ventures may also be generated

from the matrix

If P, then P.

Thesequenceofformulas

If P, then Q; it is not the case that Q: therefore it is not the case that P

is a sequence of sentential matrixes, the letters of which have the syntax

of sentences. Any appropriate replacement of the letters by sentences will

produce a sequence of sentences which express an argument. So the sequence

of matrixes is a schematic representation of a form of argument.

Return now to the Stoics. In the formula standardly used to express the

mode of the Stoic ﬁrst unproved, namely:

If the 1

,the2

;butthe1

: therefore the 2

the symbols are the numerals—more precisely, they are the items of the form

‘the n

’. They are syntactically sentences; and each of the three elements in

the formula is a sentential matrix. So the standard expression of the mode is

a schematic representation of the ﬁrst unproved.

On the other hand, the revised version of the formula for the mode of the

second unproved, namely

If the 1

,the2

; opp(the 1

): therefore opp(the 2

)

is not a sequence of sentential matrixes. For ‘opp(the 2

)’ is not a senten-

tial matrix: you cannot generate it by starting from a sentence; nor—equiv-

alently—can you get a sentence by replacing ‘the 1

’ in ‘opp(the 1

)’ by

a sentence. Hence although that formula does, of course, represent the

second unproved, and although it uses symbols to do so, it is not—on

the simple and common understanding of the matter—a schematic rep-

resentation of the second unproved. For it is not a set of matrixes.

Take, next, a semi-revised formula for the mode of the third unproved—a

formula which denies itself ‘opp’ but allows subscription, namely:

NotbothP

and P

;butP

: therefore not P

That is a set of matrixes; and it is, as such, a schematic representation of

a form of argument. But it is not a schematic representation of the third