Barnes Jonathan. Truth, etc

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320 Forms of Argument

The pristine expression of a negated conjunction uses the form ‘It is not the

case that both P and Q’. The formula ‘It is false that both P and Q’ is a

degenerate version. That is why Diogenes Laertius’ example is subsyllogistic.

The pristine expression of a conditional assertible uses the connector ‘if’.

To use ‘follow’ is degenerate. That is why Alexander’s example is subsyllo-

gistic.

And—to return to the matter in hand—that is why the Eurostar argument

is not a ﬁrst unproved. It is not a ﬁrst unproved because it is not a syllogism:

rather, it is a subsyllogistic argument.

So we cannot, after all, say on behalf of the Stoics that an argument is a

ﬁrst unproved if and only if it might be expressed by a sequence of sentences

which corresponds to the sequence of matrixes:

If the 1

,the2

;the1

: therefore the 2

For that suggestion turns subsyllogisms into syllogisms. Rather, we must say

that an argument is a ﬁrst unproved if and only if it is in fact expressed by a

sequence of sentences which is derivable from an expression of the mode of

the ﬁrst unproved.

In that case, either the circumscription of the ﬁrst unproved catches numer-

ous arguments which are not ﬁrst unproveds but subsyllogistic counterparts

of ﬁrst unproveds; or else the Stoics must identify conditional assertibles

as those which are expressed—not, which might be expressed—by way of

sentences of a certain speciﬁed type. Neither of those options is appetizing.

Moreover, it is hard to see how either of them—or anything at all like

them—is coherent with the rest of Stoic logic.

What, after all, is the relation between Diogenes Laertius’ subsyllogistic

argument:

Itisfalsethatitisdayanditisnight

It is day

Therefore it is not night

and its syllogistic cousin:

It is not the case that it is day and it is night

It is day

Therefore it is night?

There ought to be two distinct arguments there; for a syllogism can hardly

be the same argument as a subsyllogism. If the arguments are distinct, then

Subsyllogistic Arguments 321

they must be distinct in their ﬁrst premisses. But the ﬁrst premiss of the

subsyllogism is expressed by a sentence which, according to the theory before

us, is a degenerate synonym of the sentence which expresses the ﬁrst premiss

of the syllogism. But in that case, there is one argument in front of us, not

two. How could two synonymous sets of expressions express two different

arguments?

Well, the Stoics apparently held that there were two arguments there, one

of them a syllogism and the other subsyllogistic. Now according to the Stoics,

an argument is a sequence or system of assertibles, so that this argument is

the same as that argument if and only if it is the same system of assertibles.

But in that case, if the Stoics are to ﬁnd two arguments where Alexander

found but one, they must hold that the sentences

Itisfalsethatitisdayanditisnight

and

It is not the case that it is day and it is night

express different assertibles. But the Stoics also hold that the meaning of an

expression is what you can say by uttering it, so that the meaning of a complete

indicative sentence will be the complete sayable—the assertible—which it

expresses. Hence if the two sentences in question express different assertibles,

they have different meanings—and the argument in Diogenes Laertius is

not, after all, subsyllogistic.

In general, suppose that—as before—we have two sequences of sentences

,…,P

: therefore Q

and

,…,R

: therefore S

such that each R

is either the same as or synonymous with the corresponding

and S is either the same as or synonymous with Q. In that case, the two

sequences express exactly the same assertibles. Hence one sequence expresses

a syllogism if and only if the other sequence expresses a syllogism. Hence it

cannot be the case that one sequence expresses a syllogistic argument and the

other a subsyllogistic argument.

If that is right, there are no such things as subsyllogistic arguments: there

are—you might say—subsyllogistic ways of expressing arguments and there

are syllogistic ways of expressing arguments; but if a subsyllogistic expression

and a syllogistic expression express the same argument, then that argument

is a syllogism. The Eurostar argument is a syllogism: it is—it ought to be

counted as—a Stoic ﬁrst unproved.

322 Forms of Argument

STOIC NUMERALS

Schematic characterizations of syllogistic forms are such familiar items

that we scarcely stop to ask how we understand them. I said a little

earlier that the signs and symbols which appear in matrixes are syn-

tactically determinate but semantically inert, that they are place-holders

or place-markers which do not themselves mean anything. No doubt that

is true of the items in a thoroughly modern matrix. But is it also true

of the signs and symbols which the ancient logicians employed? And

how, in any case, did the ancient logicians themselves understand their

symbols?

I begin with a passage in Apuleius’ On Interpretation. The text contrasts

Peripatetic and Stoic schemata:

Thus in the Peripatetic fashion, by the use of letters, the ﬁrst unproved … is this:

A of every B, and B of every C: therefore A of every C.

… The Stoics use numerals instead of letters, for example:

If the ﬁrst the second; but the ﬁrst: therefore the second.

(int xiii [212.4–12])⁴⁹

It is not merely that the Stoic symbols have the syntax of sentences and

represent assertibles whereas the Peripatetic symbols stand in for terms: in

addition, the Stoic schemata use numerals whereas the Peripatetic schemata

use letters. If in an Aristotelian text you ﬁnd a sentence like ‘τὸ Α πάντι τῷ

Βὑπάρχει’, you should take the Greek capitals to indicate the ﬁrst two letters

of the Greek alphabet. If in a Stoic text you ﬁnd something like ‘εἰ τὸ Α,

τὸ Β’, you should take the Greek capitals to be the ﬁrst two members of the

Greek ‘alphanumeric’ system—more particularly, you should construe them

as ordinal numerals. (One standard way of saying or writing ‘The ﬁrst, the

second, the third, …’ in Greek was: ‘The alpha, the beta, the gamma, …’)

Although there is rather little positive evidence in support of Apuleius’

statement that the Stoics used numerals, there is no evidence against it, and

no one doubts it.

English translators of the Analytics invariably give ‘A’, ‘B’, ‘C’, … rather

than ‘alpha’, ‘beta’, ‘gamma’, … ; and perhaps Aristotle wrote ‘τὸ Α’ and the

⁴⁹ ut etiam Peripateticorum more per litteras … sit primus indemonstrabilis: A de omni B, et B de

omni C, igitur A de omni C. … Stoici porro pro litteris numeros usurpant, ut: si primum secundum,

atqui primum, secundum igitur.

Stoic Numerals 323

like rather than ‘τὸ ἄλφα’ and the like. But he and his followers surely said

‘τὸ ἄλφα’ and the like. Perhaps the Stoic logicians wrote ‘τὸ Α’ and the like

rather than ‘τὸ πρῶτον’ and the like. (Galen explicitly recommends that we

write numerals out in full—otherwise the scribes are bound to corrupt them.

But he implies that most Greeks preferred to save a little ink and time; and

later copyists usually opted for the abbreviated forms.) However that may be,

the Stoics certainly said ‘τὸ πρῶτον’ and the like, and we should translate

‘the ﬁrst’ (or ‘the 1

’) and the like.

So the mode which corresponds to the ﬁrst Chrysippean unproved should

be expressed not by, say,

If P, Q; P: therefore Q

but rather by

If the 1

,the2

;butthe1

: therefore the 2

No doubt that is true—but isn’t it a truth of no signiﬁcance? The Stoics

might just as well have expressed the mode by using letters—after all,

schematic letters and schematic numerals are alike in having no sense, and so

they cannot differ from one another in signiﬁcance. Then why not represent

the mode by

If P, Q; P: therefore Q?

Matrixes of that sort will not mislead or alienate a modern reader who has

a smattering of modern logic; and they have exactly the same sense as the

matrixes which employ ordinal numerals.

That is correct. But it is worth pausing to ask why the Stoics opted for

ordinal numerals. We know that they sometimes employed a sort of hybrid

between argument and mode, which they called a λογότροπος or ‘argumode’.

This is how Diogenes Laertius describes the item:

An argumode is what is compounded from both [sc from an argument and a mode],

for example:

If Plato lives, Plato breathes; but the 1

: therefore the 2

Argumodes were introduced so that, when the components of an argument were

rather long, you did not have to state the co-assumption, which was long, and also

the conclusion—rather, you could continue brieﬂy:

But the 1

: therefore the 2

(vii 77)⁵⁰

⁵⁰ λογότροπος δέ ἐστι τὸ ἐξ ἀμφοτέρων σύνθετον, οἷον εἰ ζῇ Πλάτων, ἀναπνεῖ Πλάτων·

ἀλλὰ μὴν τὸ πρῶτον· τὸ ἄρα δεύτερον. παρεισήχθη δὲ ὁ λογότροπος ὑπὲρ τοῦ ἐν ταῖς

μακροτέραις συντάξεσι τῶν λόγων μηκέτι τὴν πρόσληψιν μακρὰν οὖσαν καὶ τὴν ἐπιφορὰν

λέγειν, ἀλλὰ συντόμως ἐπενεγκεῖν· τὸ δὲ πρῶτον· τὸ ἄρα δεύτερον.

324 Forms of Argument

Argumodes are not mentioned by name outside this text (for the entry in the

Suda,s.v.τρόπον, was taken from Diogenes); but the things themselves are

found frequently enough elsewhere. Here is an example from Sextus:

You can deal with what we have just said brieﬂy by propounding the argument as

follows:

If what is apparent is apparent to everyone and signs are not apparent to everyone,

then signs are not apparent.

But the 1

Therefore the 2

(M viii 242)⁵¹

Sextus’ example makes the abbreviatory function of argumodes evident. There

are also exemplary argumodes in reports of Stoic inferences; and some are

found in the rare fragments of Stoic logical texts. The Logical Investigations

of Chrysippus has this:

If there are plural predicates, then there are plurals of plurals ad inﬁnitum.But

certainly not that. So not the 1

(PHerc307, ii 21–26)⁵²

Perhaps that is only a quasi-argumode insofar as Chrysippus says ‘not that’

rather than ‘not the 2

’? Still, there is no reason to doubt that the Stoics

habitually used argumodes, and that they did so primarily for abbreviatory

purposes.

So although the things are called argumodes rather than arguments, and

are said to be compounded from an argument and a mode, that is slightly

misleading. For argumodes are in fact arguments—arguments expressed in

an abbreviated form. Sextus is right when, at M viii 242, he introduces his

item as an argument.

How are the ordinal numerals which appear in an argumode to be

understood? I guess that ‘the 1

’ is a referring expression; that it is short for

‘the ﬁrst assertible’ or ‘τὸ πρῶτον ἀξίωμα’;andthatitreferstotheﬁrst

assertible in the current context—that is to say, to the assertible which, in

Diogenes’ example, forms the antecedent of the conditional premiss of the

argument. In that case, the argumodes are not only abbreviatory but also

⁵¹ ἐνέσται δὲ καὶ βραχέως τὰ προειρημένα περιλαβόντας τοιουτουσί τινας προτείνειν

λόγους. εἰ τὰ φαινόμενα πᾶσι φαίνεται, τὰ δὲ σημεῖα οὐ πᾶσι φαίνεται, οὐκ ἔστι τὰ φαινόμενα

σημεῖα· ἀλλὰ μὴν τὸ πρῶτον· τὸ ἄρα δεύτερον.

⁵² εἰ πληθυντικά ἐστιν κατηγορήματα, καὶ πληθυντικῶν πληθυντικά ἐστι μέχρι εἰς ἄπειρον.

οὐ πάνυ δὲ τοῦτο. οὐδ᾿ ἄρα τὸ πρῶτον.

Stoic Numerals 325

brachylogical: ‘the 1

’ must be taken to stand for something like ‘the ﬁrst

assertible is the case’.

In an argumode, the ordinal numerals must have a determinate reference.

Asequencesuchas:

If the 1

,the2

But Plato lives.

Therefore Plato breathes.

is not an argumode; for the ordinals there have no reference. (To be sure, you

might readily invent a convention which determined a reference for them.)

That seems clear and unremarkable; but in fact it raises a problem for

the Sextan argumode which I have just cited. The argumode was evidently

intended to abbreviate a valid syllogism, namely:

If what is apparent is apparent to everyone and signs are not apparent to

everyone, then signs are not apparent.

But what is apparent is apparent to everyone and signs are not apparent to

everyone.

Therefore signs are not apparent.

In other words, in Sextus’ argumode the formula ‘the 1

’ refers to ‘what is

apparent is apparent to everyone and signs are not apparent to everyone’,

and ‘the 2

’ refers to ‘signs are not apparent’. But then ‘the 1

’ and ‘the

’ do not refer to the ﬁrst and the second assertible expressed in the

argument: the ﬁrst assertible to be expressed in the argument is in fact ‘what

is apparent is apparent to everyone’ and the second is ‘signs are not appar-

ent to everyone’. So doesn’t the argumode abbreviate the following invalid

argument?

If what is apparent is apparent to everyone and signs are not apparent to

everyone, then signs are not apparent.

But what is apparent is apparent to everyone.

Therefore signs are not apparent to everyone.

Well, of course that’s not the argument which Sextus means to propose; of

course no reader has ever imagined that he did intend to propose it; and of

course a little common savvy is enough to determine which argument an

argumode is meant to present.

Nonetheless, there is a theoretical problem. The best solution—perhaps

the only solution—is to stipulate the following convention: The reference

326 Forms of Argument

of ordinal numerals in argumodes is always to simple assertibles. Thus ‘the

’ will refer to the simple assertible which is ﬁrst expressed in the pertinent

patch of argumentative discourse; ‘the 2

’ will pick out the second simple

assertible; and so on. According to that convention, Sextus’ argumode in fact

sets down the invalid argument. What he should have written is this:

If what is apparent is apparent to everyone and signs are not apparent to

everyone, then signs are not apparent.

But the 1

and the 2

Therefore the 3

However that may be, it is plausible to suppose that the Stoic use of ordinal

numerals to articulate their modes derives from their use of ordinal numerals

in argumodes rather than vice versa. For it is in argumodes that the numerals

have their ordinary use and sense. From the argument

If Plato lives, Plato breathes; but Plato lives: therefore Plato breathes

we engender the argumode:

If Plato lives, Plato breathes; but the 1

: therefore the 2

We then bleach the argumode of its concrete content, in the obvious way,

and arrive at

If the 1

,the2

;butthe1

: therefore the 2

And there is a mode of the ﬁrst unproved.

But the transference from argumode to mode is not without its problems.

First, consider the following argument, which I take from Galen’s Intro-

duction:

If food is distributed through the body, either it is pushed or it is sucked

or it is conducted or it moves by itself.

But food is distributed through the body.

Thereforeeitheritispushedoritissuckedoritisconductedoritmoves

by itself.

That, you will say, is a ﬁrst unproved; and Galen agrees. And its mode, you

will therefore say, is this:

If the 1

,the2

But the 1

Therefore the 2

But there Galen disagrees. For this is his comment on the structure of the

syllogism:

Stoic Numerals 327

In one of the modes we have the force of the ﬁrst of the hypothetical syllo-

gisms, namely:

If the 1

, either the 2

or the 3

or the 4

or the 5

Then a co-assumption:

But the 1

Therefore either the 2

or the 3

or the 4

or the 5

(inst log xv 8)⁵³

It is plain why Galen says what he says. Moreover, it is a natural enough thing

to say if you regard modes as argumodes bleached of their concrete content,

and if in an argumode the ordinal numerals refer to the simple constituents

of the complex premiss on which they depend for their interpretation.

Nonetheless, what Galen says has some curious consequences.

First, there will be no such thing as the mode of the ﬁrst unproved. Rather,

there will be an inﬁnite number of such modes; for either the antecedent of

the conditional premiss, or its consequent, or both items, may be complex

assertibles—and assertibles of any degree of complexity. Secondly, it will be

natural to hold that a valid argument is not automatically valid in virtue of

having the mode which it has. Any argument which matches the mode which

Galen sets out at inst log xv 8 is, of course, valid; but its validity derives—or

so, I imagine, we shall be inclined to agree—not from its matching that

particular mode but rather from the fact that it is a ﬁrst unproved—from the

fact that it ﬁts the standard circumscription of the ﬁrst unproved.

Does Galen’s text here reﬂect Stoic thinking? Did the Stoics allow, in

practice or in principle, a multiplicity of modes for their ﬁrst unproved?

Galen does not say that he is following the Stoics; and it would be a

gross error to imagine that everything which Galen says about hypothetical

syllogisms is at bottom Stoic. Nonetheless, Galen is not alone in admitting

multiplicity: there are comparable examples in other authors. Thus Sextus,

having announced that the structure of a certain argument will be clearer if

we conduct its analysis in terms of modes, says this:

So there are two unproveds. One of them is this:

If the 1

and the 2

,the3

But not the 3

⁵³ καθ᾿ ἕτερον γὰρ τῶν τρόπων ἡ τοῦ πρώτου τῶν ὑποθετικῶν συλλογισμῶν δύναμίς

ἐστιν, οὖσα τοιαύτη· εἰ τὸ πρῶτον, ἤτοι τὸ δεύτερον ἢ τὸ τρίτον ἢ τὸ τέταρτον ἢ τὸ πέμπτον.

εἶτα πρόσληψις· ἀλλὰ μὴν τὸ πρῶτον· ἤτοι ἄρα τὸ δεύτερον ἢ τὸ τρίτον ἢ τὸ τέταρτον ἢ τὸ

πέμπτον.

328 Forms of Argument

Therefore not the 1

and the 2

That is a second unproved. The other is a third unproved, thus:

Not the 1

and the 2

But the 1

Therefore not the 2

That, then, is the analysis in terms of modes.

(M viii 236–237)⁵⁴

Themodeofthethirdunprovedisorthodox.Themodeofthesecond

is comparable to Galen’s mode of the ﬁrst unproved; and it has the same

implicit consequences.

It cannot be a coincidence that both Galen and Sextus deal with modes

in this way—and the way is encouraged by the existence of argumodes.

So I suppose that the Stoics too had sometimes spoken in the same vein.

A few pages before the passage I have just quoted, Sextus adverts to a

distinction—apparently made by the Stoic logicians—between simple and

non-simple unproveds. (See M viii 228–229.) The mode of the simple ﬁrst

unproved is presumably this:

If the 1

,the2

;butthe1

: therefore the 2

Sextus’ own example at M viii 236 will presumably be a mode of a complex

ﬁrst unproved. There are inﬁnitely many other complex modes of the ﬁrst

unproved.

If the Stoics spoke along those lines, then they had two rather different

views on the nature of modes. On the one view, the ordinal numerals in

the expression of a mode may be replaced by sentences which express any

assertibles whatsoever, simple or complex. According to that view, there is

one mode of the ﬁrst unproved, namely:

If the 1

, then the 2

;butthe1

: therefore the 2

On the other view, the ordinal numerals in the expression of a mode must be

replaced by sentences which express simple assertibles. According to that view

there will be not one but an inﬁnite number of modes of the ﬁrst unproved.

The former view is preferred—implicitly—by modern scholars; but there

are at any rate traces of the latter view in the ancient texts. There are, so far

as I know, no traces of any discussion of the two views—nor even of any

recognition of the fact that there are two views to discuss.

⁵⁴ ὥστε δύο εἶναι ἀναποδείκτους, ἕνα μὲν τοιοῦτον· εἰ τὸ πρῶτον καὶ τὸ δεύτερον,

τὸ τρίτον· οὐχὶ δέ γε τὸ τρίτον· οὐκ ἄρα τὸ πρῶτον καὶ τὸ δεύτερον, ὅς ἐστι δεύτερος

ἀναπόδεικτος· ἕτερον δὲ τρίτον, τὸν οὕτως ἔχοντα· οὐχὶ τὸ πρῶτον καὶ τὸ δεύτερον· ἀλλὰ

μὴν τὸ πρῶτον· οὐκ ἄρα τὸ δεύτερον. ἐπὶ μὲν οὖν τοῦ τρόπου ἡ ἀνάλυσίς ἐστι τοιαύτη.

Stoic Numerals 329

The transfer of the numerals from argumodes to modes raises another

question—or rather, it has another and more serious consequence; for

in the course of the transfer the ordinal numerals undergo an essential

transmogriﬁcation. If you take Diogenes’ illustrative argumode and replace

its ﬁrst premiss by

If the 1

,the2

then you reach the mode of the ﬁrst unproved, namely:

If the 1

,the2

;the1

: therefore the 2

What do the numerals mean now? They are no longer used to refer; for there

is nothing to which they could refer: in ‘If the 1

,the2

’ neither ordinal has

a determinate referent. More precisely, in the expression of the argumode the

sense of ‘the 1

’ determines its referent; but in the expression of the mode

the sense of ‘the ﬁrst’ determines no referent.

That being so, you might as well give the mode by:

If the 2

,the1

;the2

: therefore the 1

Or, come to that, by:

If the 234

,the19

; the 234

: therefore the 19

There is absolutely no difference in sense among those three schemata.

In modern propositional logic, the schema for modus ponens is likely to be

given in something like this way:

P ⊃ Q; P: Q.

If instead I offer

Q ⊃ P; Q: P.

S ⊃ R; S: R

you may judge me eccentric—but my schemata are impeccable. I may try

something even more eccentric, and choose to represent modus ponens by

X ⊃ Y; X: Z

or by

1 ⊃ 2; 1: 2

or by

♣⊃♥;♣ : ♥.

All those schemata are impeccable, provided that the syntax of the symbols

has been appropriately speciﬁed. It is just the same for the Stoics: instead of

ordinal numerals they might have used letters—or anything else.

In other words, and whatever their history may have been, the Stoic

ordinal numerals, as they were employed in expressions of the modes, have

no ordinal and no numerical function. To be sure, there may still have been