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390 The Science of Logic

It is true that Alexander here retains the modal element and also the notion of

taking; but his version of the dictum de nullo, which I shall cite in an instant,

is enough to show that he did not mean them seriously.

If that is the ﬁrst part of the traditional dictum,namelythedictum de omni,

what is the second part or the dictum de nullo? Aristotle is allusive:

And in the same way for of none.

Had Aristotle formulated the dictum explicitly—had he written out what the

adverb ‘in the same way’ promises—he must presumably have said something

very like this:

We say that it is predicated of none when you can’t take anything of which the other

will be said.³⁸

After four pedantries, that formulation would be replaced by:

AispredicatedofnoBifandonlyifthereisn’tanyBofwhichAis

predicated.

And that, I suppose, is the dictum de nullo—it is how Alexander understood it:

When you can’t take any of the subject of which the predicate will be said, then ‘of

no’ is truly said. For example, neigher of no man—for there is no man of whom

neigher is predicated.

(in APr 25.18–21)³⁹

So too Philoponus:

For example, we say that animal is predicated of every man—and there is not a man

of whom animal will not be said. Similarly for ‘of no’: we say that man holds of no

stone—and there is not a stone of which man will be said.

(in APr 39.11–15)⁴⁰

What is the status of the dicta, or of the two parts of the dictum?I

mean, when Aristotle says ‘we say’, does he intend to report the ordinary

usage of us Greeks, or to indicate the special usage of us logicians or of us

Aristotelians, or to stipulate a special usage for the purposes of the Analytics?

³⁸ λέγομεν δὲ τὸ κατὰ μηδενὸς κατηγορεῖσθαι ὅταν μηδὲν ᾖ λαβεῖν καθ᾿ οὗ θάτερον

λεχθήσεται.

³⁹ ὅταν γὰρ μηδὲν ᾖ λαβεῖν τοῦ ὑποκειμένου καθ᾿ οὗ τὸ κατηγορούμενον ῥηθήσεται, τότε

ἐστὶ καὶ τὸ κατὰ μηδενὸς ἀληθῶς· οἷον τὸ χρεμετιστικὸν κατ᾿ οὐδενὸς ἀνθρώπου—οὐδεὶς

γάρ ἐστιν ἄνθρωπος καθ᾿ οὗ τὸ χρεμετιστικὸν κατηγορεῖται.

⁴⁰ οἷον λέγομεν τὸ ζῷον κατὰ παντὸς ἀνθρώπου κατηγορεῖσθαι· καὶ οὐκ ἔστι τις ἄνθρωπος

καθ᾿ οὗ τὸ ζῷον οὐ λεχθήσεται. καὶ τὸ κατὰ μηδενὸς ὁμοίως· λέγομεν γὰρ τὸν ἄνθρωπον

κατὰ μηδενὸς λίθου ὑπάρχειν· καὶ οὐκ ἔστι τις λίθος καθ᾿ οὗ ὁ ἄνθρωπος λεχθήσεται.

The dictum de omni et nullo 391

Aristotle’s words do not suggest a stipulation (but that signiﬁes little). It

seems unlikely that there was, or that Aristotle thought that there was, a

special logical or Aristotelian sense of words such as ‘every’ and ‘no’. It is

best to suppose that the dictum purports to explain what we—that is to say,

we Greeks—normally mean when we say ‘of every’ and ‘of no’ and the like;

that is, it purports to gloss the normal use, or perhaps a normal use, of those

quantifying expressions.

Did Aristotle understand the normal use aright? That is to say, is the dictum

true? Well, a die-hard Aristotelian ought to have doubts, and for a familiar

reason. Suppose that there are no items at all of a certain sort—that there

is not a single phoenix, for example. And take any predicate you like—say

‘red-feathered’. Now since there are no phoenixes at all, it follows that there

is no phoenix of which ‘red-feathered’ is not predicated. And so (according to

Aristotle’s deﬁnition), every phoenix is red-feathered. Again, since there are

no phoenixes, there is no phoenix of which ‘red-feathered’ is predicated. And

so (according to Aristotle’s deﬁnition), no phoenix is red-feathered. Thus

every phoenix is red-feathered and no phoenix is red-feathered. But surely

that is impossible? Well, impossible or not, it is uncontroversially impossible

in the context of Aristotle’s own logic. For

Every phoenix is red-feathered

and

No phoenix is red-feathered

are contraries, and contraries cannot be true together.

That consideration is part of a familiar difﬁculty, or a familiar group of

difﬁculties. The sign under which the difﬁculty is sold is ‘Existential Import’;

for it is most notoriously manifested in the so-called law of subalternation

which afﬁrms that if A holds of every B then A holds of some B. For example,

if every phoenix is red-feathered, then some phoenix is red-feathered. But if

some phoenix is red-feathered, then there is at least one phoenix. The law of

subalternation thus requires

Every phoenix is red-feathered

to be false. The dictum de omni makes it true.

If the difﬁculty is familiar, so are the various ways of circumventing it. I

shall not rehearse them here; but one or another of them must be adopted by

anyone who wishes both to hold onto the dictum andalsotoacceptthemain

theses of Aristotle’s syllogistic.

Someone who is not a die-hard Aristotelian is likely to ﬁnd the dictum

puzzling for a quite different reason: far from being false, or even disputable,

392 The Science of Logic

the dictum will appear to be trivial, tautologous, vacuous. Look again at the

dictum de nullo:

AispredicatedofnoBifandonlyifthereisn’tanyBofwhichAis

predicated.

That is hardly an elucidation of ‘of no’: it merely shufﬂes the words around.

Aristotle’s formulation of the dictum—and in particular, his failure to

formulate the de nullo part—hides the vacuity. But surely it is there?

Well, perhaps the dictum is a triﬂe. But if it is trivial, then it has at least

one virtue—it is true.

PUTTING THE DICTUM TO WORK

In any event, the question which will exercise me here is another one. It is

this: What use is the dictum?whatworkcanitdo?

Two passages in chapters 4–6 of the ﬁrst book of the Prior Analytics invoke

the dictum, and there are a further couple of passages in the chapters on

modal syllogistic. The two passages in chapters 4–6 in effect contain four

appeals to the dictum, as follows:

[1] If A of every B and B of every C, it is necessary that A be predicated of every

C—for we have earlier said what we mean by ‘of every’. [2] Similarly, if A of no B

and B of every C, A will hold of no C.

(APr 25b37–40)⁴¹

[3] Let A hold of every B and B of some C. Then if being predicated of every is what

we said at the beginning, it is necessary for A to hold of some C. [4] And if A holds

of no B and B of some C, it is necessary for A not to hold of some C—for we have

also deﬁned what we mean by ‘of no’. Hence there will be a perfect syllogism.

(APr 26a23–28)⁴²

Item [1] is the conclusion of Aristotle’s account of the ﬁrst mood of the

ﬁrst ﬁgure—of Barbara. The particle ‘for [γάρ]’ suggests that the account is

⁴¹ εἰ γὰρ τὸ Α κατὰ παντὸς τοῦ Β καὶ τὸ Β κατὰ παντὸς τοῦ Γ, ἀνάγκη τὸ Α κατὰ παντὸς

τοῦ Γ κατηγορεῖσθαι· πρότερον γὰρ εἴρηται πῶς τὸ κατὰ παντὸς λέγομεν. 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.

⁴² ὑπαρχέτω γὰρ τὸ μὲν Α παντὶ τῷ Β, τὸ δὲ Β τινὶ τῷ Γ. οὐκοῦν εἰ ἔστι παντὸς

κατηγορεῖσθαι τὸ ἐν ἀρχῇ λεχθέν, ἀνάγκη τὸ Α τινὶ τῷ Γ ὑπάρχειν. καὶ εἰ τὸ μὲν Α μηδενὶ

τῷ Β ὑπάρχει, τὸ δὲ Β τινὶ τῷ Γ, ἀνάγκη τὸ Α τινὶ τῷ Γ μὴ ὑπάρχειν· ὥρισται γὰρ καὶ τὸ

κατὰ μηδενὸς πῶς λέγομεν· ὥστε ἔσται συλλογισμὸς τέλειος.

Putting the dictum to Work 393

justiﬁed or explained by reference to what ‘we have earlier said’—and what

we have earlier said is, of course, the de omni part of the dictum.Item[2]

does for Celarent what item [1] does for Barbara; and the introductory

adverb ‘similarly’ suggests that just as the de omni part of the dictum can be

invoked in support of Barbara, so the de nullo part can be invoked in support

of Celarent. Item [3] is Aristotle’s exposition of Darii: again the dictum is

invoked, and again with a justiﬁcatory or explanatory function. The same is

true of item [4], where it is a question of Ferio—and where it is the de nullo

part of the dictum which is called upon.

The four perfect syllogisms of the ﬁrst ﬁgure are thus explained, or justiﬁed,

or—as I shall say—underwritten by the dictum de omni et nullo. But what

exactlydoesthedictum underwrite? In the case of Ferio, Aristotle remarks: ‘so

thattherewillbeaperfectsyllogism’.Itmightbeinferredthatthedictum is

supposed to explain why Ferio is perfect rather than imperfect; and so, more

generally, to explain perfection or to justify an ascription of perfection. In

other words, the dictum does not explain why perfect syllogisms are valid—it

explains why they are perfect (or in what their perfection consists).

That argument is unpersuasive. It is not, so far as I can see, clear that the

dictum is invoked to explain perfection rather than to explain validity; but

let that pass. Suppose that the dictum explains perfection. That is to say, the

dictum answers the question ‘Why are Barbara and the others evidently valid?’

Now you might perhaps manage to construe that question so that it was

about evidence and not about validity. (‘Yes, I see that Barbara is valid—but

tell me why it is evidently valid.’) But any such construal would be bizarre;

and the question is best taken at its face value—it asks why Barbara and the

rest are evidently valid.

So the state of affairs with perfect syllogisms is this: A perfect syllogism is

valid; its validity is evident without appeal to any external items, and without

proof; its validity is evident inasmuch as it is underwritten by the dictum de

omni et nullo. That was Alexander’s view of the matter:

All the syllogisms thus far described [i.e. the ﬁrst ﬁgure syllogisms] are perfect; for,

using nothing but ‘of every’ or ‘of no’, which has already been established, they have

an evident validity.

(in APr 61.3–5)⁴³

And that was Alexander’s view of Aristotle’s view of the matter:

⁴³ πάντες δὲ οἱ προειρημένοι τέλειοι· πάντες γὰρ τῷ κατὰ παντὸς μόνῳ ἢ κατὰ μηδενὸς

προσχρώμενοι, ὅ ἐστι κείμενον, φανερὰν τὴν συναγωγὴν ἔχουσιν.

394 The Science of Logic

He deﬁned a perfect syllogism as one which needs nothing else apart from the items

assumed in order for its necessity to be apparent: he says that all the syllogisms

he has shown in this ﬁgure are perfect since all are perfected by way of the items

assumed at the beginning and already established, and they need nothing else. The

items assumed at the beginning by way of which the necessity of these syllogisms is

apparent are ‘of every’, which is equivalent to ‘in as in a whole’, and ‘of no’ and ‘in

no’.

(in APr 69.14–20)⁴⁴

Alexander has an eccentric interpretation of the phrase ‘the items assumed’,

which surely refers to the premisses of the perfect syllogisms rather than to

the dictum;butthatgaffeapart,heissurelyright.

That conclusion may be supported by reference to the heterodox view

of Boethus, who held that the canonical syllogisms—all fourteen of them,

or perhaps twelve of them—are perfect. Themistius reports Boethus’ view

about Darapti:

He says that … in the ﬁrst mood of the third ﬁgure (in which the two extremes are

said of all the middle term), when two things are said of one thing, then it is evident

that each is in the other as a part. As for his formula ‘in every’, we must discover how

he uses it …

(ad Max 192)

The report clearly suggests, although it does not state, that the perfection

of Darapti depended, in Boethus’ book, on the dictum de omni. And the

suggestion is conﬁrmed by a more general remark of Themistius’:

Since Boethus now turns to the third ﬁgure, let us remark on how he comes to commit

sophisms concerning the number of syllogisms—sophisms which are consequences

of the fact that he has explained the formula ‘in every’ in a way different from the

one which Aristotle indicates in the Analytics.

(ad Max 192)

The matter stands thus: Aristotle ﬁnds four perfect syllogisms, and their

perfection depends on their relation to the dictum de omni et nullo. Boethus

ﬁnds fourteen—or perhaps twelve—perfect syllogisms, and their perfection

⁴⁴ ὡρίσατο τὸν τέλειον συλλογισμὸν τὸν μηδενὸς ἄλλου προσδεόμενον παρὰ τὰ εἰλημμένα

πρὸς τὸ φανῆναι τὸ ἀναγκαῖον. φησὶ δὲ πάντας τοὺς ἐν τούτῳ τῷ σχήματι δεδειγμένους

συλλογισμοὺς τελείους εἶναι, ἐπειδὴ πάντες ἐπιτελοῦνται διὰ τῶν ἐξ ἀρχῆς εἰλημμένων τε

καὶ κειμένων καὶ οὐδενὸς ἄλλου προσδέονται. ἔστι δὲ τὰ ἐξ ἀρχῆς λαμβανόμενα δι᾿ ὧν τὸ ἐν

αὐτοῖς ἀναγκαῖον φανερόν ἐστιν τό τε κατὰ παντὸς ἴσον ὂν τῷ ἐν ὅλῳ καὶ τὸ κατὰ μηδενὸς

καὶ ἐν μηδενί.

Putting the dictum to Work 395

depends on the dictum. The difference between Aristotle and Boethus is to

be explained—if we may trust Themistius—by the fact that Boethus gave

a heterodox version of the dictum. Unfortunately we don’t know what that

version was.

So far I have mentioned the use of the dictum in connection with non-

modal syllogistic. But it is also invoked twice in Aristotle’s account of modal

syllogisms—or at least, it appears to be invoked in one passage and it is often

taken to be invoked in a second. Of apodictic syllogisms, or syllogisms each

component proposition of which says that something holds necessarily of

something, Aristotle says that

in the case of necessities things are pretty much the same as in the non-modal cases:

when the terms are placed in the same way, there will be or not be a syllogism both

in holding and in holding of necessity (or not holding)—except that they will differ

inasmuch as holding of necessity (or not holding) is added to the terms. For the

privative converts in the same way, and we shall explain ‘as in a whole’ and ‘of every’

in the same way.

(APr 29b36–30a3)⁴⁵

An apodictic syllogism is valid if and only if its non-modal counterpart

is valid, and the validity of apodictic syllogisms is established in the same

way as the validity of their counterparts (with a couple of exceptions). The

reference to the dictum in the passage evidently applies to the four perfect

apodictic syllogisms: they, like their non-modal partners, have their validity

underwritten by the dictum de omni et nullo.SoAlexander:

‘of every’ and ‘of no’ are taken in the same way in the case of the necessary as they are

in the case of the non-modal, and it is by way of them that the ﬁrst ﬁgure syllogisms

are shown.

(in APr 120.13–15)⁴⁶

The second modal invocation of the dictum is more controversial. It occurs

where Aristotle discusses ﬁrst ﬁgure problematical syllogisms, or syllogisms

all of whose component propositions say that something possibly holds of

something. Here are problematic Barbara and Celarent:

⁴⁵ ἐπὶ μὲν οὖν τῶν ἀναγκαίων σχεδὸν ὁμοίως ἔχει καὶ ἐπὶ τῶν ὑπαρχόντων· ὡσαύτως

γὰρ τιθεμένων τῶν ὅρων ἔν τε τῷ ὑπάρχειν καὶ τῷ ἐξ ἀνάγκης ὑπάρχειν ἢ μὴ ὑπάρχειν

ἔσται τε καὶ οὐκ ἔσται συλλογισμός, πλὴν διοίσει τῷ προσκεῖσθαι τοῖς ὅροις τὸ ἐξ ἀνάγκης

ὑπάρχειν ἢ μὴ ὑπάρχειν. τό τε γὰρ στερητικὸν ὡσαύτως ἀντιστρέφει, καὶ τὸ ἐν ὅλῳ εἶναι

καὶ τὸ κατὰ παντὸς ὁμοίως ἀποδώσομεν.

⁴⁶ … τό τε κατὰ παντὸς καὶ τὸ κατὰ μηδενὸς ὁμοίως καὶ ἐπὶ τοῦ ἀναγκαίου λαμβάνεται

ὡς καὶ ἐπὶ τοῦ ὑπάρχοντος, δι᾿ οὗ οἱ ἐν τῷ πρώτῳ σχήματι δείκνυνται συλλογισμοί.

396 The Science of Logic

Now when A can hold of every B and B of every C, there will be a perfect syllogism

to the conclusion that A can hold of every C. This is evident from the deﬁnition; for

we described ‘can hold of every’ in that way. Similarly if A can hold of no B and B of

every C—that A can hold of no C. For for A not to be possibly predicated of what B

is possibly predicated of was precisely for there to be no exceptions among the items

possibly under B.

(APr 32b38–33a5)⁴⁷

Then Darii and Ferio:

If one of the premisses is assumed as a universal and the other as a particular, then

if the universal is on the major term there will be a perfect syllogism. For if A can

hold of every B and B of some C, A can hold of some C—this is evident from the

deﬁnition of ‘can’. Again, if A can hold of no B and B can hold of some C, then it is

necessary that A can not hold of some C. The proof is the same.

(APr 33a21–27)⁴⁸

There are four perfect problematical syllogisms, and each is supported by

reference—explicit or implicit—to a deﬁnition.

Alexander takes the references to be to the dictum de omni et nullo.Or

rather, he thinks that that is plainly so in the cases of the universal syllogisms.

Thus for Barbara:

‘This is evident from the deﬁnition’ indicates that it is clear from the deﬁnition of ‘of

every’ that in the present conjugation a universal afﬁrmative problematic conclusion

is inferred.

(in APr 167.14–17)⁴⁹

And similarly ﬁve lines later for Celarent. As for Darii and Ferio, Alexander

offers two interpretations, between which he does not choose:

⁴⁷ ὅταν οὖν τὸ Α παντὶ τῷ Β ἐνδέχηται καὶ τὸ Β παντὶ τῷ Γ, συλλογισμὸς ἔσται τέλειος

ὅτι τὸ Α παντὶ τῷ Γ ἐνδέχεται ὑπάρχειν. τοῦτο δὲ φανερὸν ἐκ τοῦ ὁρισμοῦ· τὸ γὰρ ἐνδέχεσθαι

παντὶ ὑπάρχειν οὕτως ἐλέγομεν. ὁμοίως δὲ καὶ εἰ τὸ μὲν Α ἐνδέχεται μηδενὶ τῷ Β, τὸ δὲ

Β παντὶ τῷ Γ, ὅτι τὸ Α ἐνδέχεται μηδενὶ τῷ Γ· τὸ γὰρ καθ᾿ οὗ τὸ Β ἐνδέχεται, τὸ Α μὴ

ἐνδέχεσθαι, τοῦτ᾿ ἦν τὸ μηδέν ἀπολείπειν τῶν ὑπὸ τὸ Β ἐνδεχομένων.

⁴⁸ ἐὰν δ᾿ ἡ μὲν καθόλου τῶν προτάσεων ἡ δ᾿ ἐν μέρει ληφθῇ, πρὸς μὲν τὸ μεῖζον ἄκρον

κειμένης τῆς καθόλου συλλογισμὸς ἔσται τέλειος. εἰ γὰρ τὸ Α παντὶ τῷ Β ἐνδέχεται τὸ δὲ Β

τινὶ τῷ Γ, τὸ Α τινὶ τῷ Γ ἐνδέχεται· τοῦτο δὲ φανερὸν ἐκ τοῦ ὁρισμοῦ τοῦ ἐνδέχεσθαι. πάλιν

εἰ τὸ Α ἐνδέχεται μηδενὶ τῷ Β, τὸ δὲ Β τινὶ τῷ Γ ἐνδέχεται ὑπάρχειν, ἀνάγκη τὸ Α ἐνδέχεσθαί

τινι τῶν Γ μὴ ὑπάρχειν. ἀπόδειξις δ᾿ ἡ αὐτή.

⁴⁹ τὸ δὲ τοῦτο δὲ φανερὸν ἐκ τοῦ ὁρισμοῦ δηλωτικόν ἐστιν ὅτι τὸ συνάγεσθαι ἐν τῇ

προκειμένῃ συζυγίᾳ καθόλου ἐνδεχόμενον καταφατικὸν συμπέρασμα δῆλόν ἐστιν ἐκ τοῦ

ὁρισμοῦ τοῦ κατὰ παντός.

Putting the dictum to Work 397

He says that this too is evident from the deﬁnition of ‘can’, either meaning ‘can hold

of every’ (since ‘of every’ was that of which you can’t take anything of which the

other will not be said) … , or else he means from the deﬁnition of the possible itself.

(in APr 169.23–30)⁵⁰

There is much to be said for the ﬁrst of those two interpretations. For if it

is accepted, then Aristotle will appeal to the dictum de omni et nullo in the

case of all the syllogisms which he recognizes to be perfect, and only in their

cases. A syllogism is perfect if and only if it is underwritten by the dictum.

That is the line which Philoponus, for example, follows—apparently

without any anxiety. But modern scholarship has not sided with him. In the

case of problematical Barbara and Celarent, it is urged, Aristotle must be

invoking his explanation of ‘possible’ when he refers to his account of ‘what

can hold of every’, and not to his deﬁnition of ‘of every’ and ‘of no’. As for

problematical Darii and Ferio, there Aristotle clearly invokes his deﬁnition

of ‘possible’; but he cannot be invoking the deﬁnition in order to justify the

perfection of the two syllogisms—for in fact neither syllogism is perfect (and

the word ‘perfect’ must be cut from the received text).

The reasons for denying perfection to Darii and Ferio are frail. But there is a

curiosity about Aristotle’s presentation of the two particular syllogisms which

has suggested that he cannot be appealing to the dictum in their cases. For of

Feriohesaysthat‘theproofisthesame’;andalthoughperfectsyllogismscan,

of course, be proved, an appeal to the dictum is not at all the same thing as a

proof—so that if we take the word ‘proof ’ seriously, we should not think that

Aristotle is here appealing to the dictum. That is a fair-seeming argument. But

it should not seduce. For on any account of the passage, Aristotle supports

Darii and Ferio by appeal to a deﬁnition. Such an appeal does not constitute

a proof. Hence we should not take the word ‘proof ’ seriously.

However that may be, is the validity of the perfect problematical syllogisms

grounded on the dictum or rather on a deﬁnition of ‘can’? The textual

evidence tells decisively in neither direction, so that we might think to rely

on logical or philosophical considerations—in which direction is the better

sense found?

Well, what exactly are we looking for? The validity of perfect syllogisms, I

have said, is underwritten by the dictum. But what exactly does that mean? It

⁵⁰ καὶ τοῦτο φανερόν φησιν εἶναι ἐκ τοῦ ὁρισμοῦ τοῦ ἐνδέχεσθαι ἤτοι λέγων τοῦ παντὶ

ἐνδέχεσθαι (ἐπεὶ τὸ παντὶ ἦν οὗ μηδὲν ἦν λαβεῖν καθ᾿ οὗ θάτερον οὐ ῥηθήσεται), ... ἢ τοῦ

ὁρισμοῦ αὐτοῦ τοῦ ἐνδεχομένου λέγει.

398 The Science of Logic

seems relatively easy to give the beginnings of an answer to that question: the

dictum underwrites the perfect syllogisms insofar as it articulates the sense of

the quantifying words ‘every’ and ‘no’, or insofar as it explains what universal

propositions (afﬁrmative and negative) are; and the perfect syllogisms are

valid in virtue of the sense of those quantifying words, or in virtue of the

nature of those universal propositions.

No doubt the perfect syllogisms are valid in virtue of what might be

called their quantiﬁcational structure. But then all predicative syllogisms are

valid in virtue of their quantiﬁcational structure—that is what makes them

predicative syllogisms. If that is all the dictum does in the way of underwriting,

then it underwrites Darapti and Cesare as much as it underwrites Darii and

Celarent. But that is not what Aristotle is on about: if an appeal to the dictum

is to do the work Aristotle demands of it, then there must be a special relation

between it and the perfect syllogisms, a relation which it does not have with

any imperfect syllogism. What might that relation be? It may be helpful to

turn brieﬂy from predicative to hypothetical syllogistic.

THE DICTUM DE SI ET AUT

The mode—or one of the modes—of the ﬁrst Chrysippean unproved

syllogism was this:

If the 1

,the2

;the1

: therefore the 2

The complex premiss of a ﬁrst unproved is a conditional proposition, and

of non-simple assertibles, the conditional—as Chrysippus says in the Dialectics and

Diogenes in the Art of Dialectic —is one which is composed with the conditioning

connector ‘if ’. This connector announces that the second follows the ﬁrst, e.g.

If it is day, it is light.

(Diogenes Laertius, vii 71)⁵¹

On the basis of that text, we might invent the dictum de si:

One item holds if another does if and only if the one follows the other.

Or perhaps:

If P, then Q if and only if given that P it follows that Q.

The dictum is uncontroversially correct.

⁵¹ τῶν δ᾿ οὐχ ἁπλῶν ἀξιωμάτων συνημμένον μέν ἐστιν, ὡς ὁ Χρύσιππος ἐν ταῖς

∆ιαλεκτικαῖς φησι καὶ ∆ιογένης ἐν τῇ ∆ιαλεκτικῇ τέχνῃ, τὸ συνεστὸς διὰ τοῦ εἰ συναπτικοῦ

συνδέσμου. ἐπαγγέλλεται δ᾿ ὁ σύνδεσμος οὗτος ἀκολουθεῖν τὸ δεύτερον τῷ πρώτῳ, οἷον εἰ

ἡμέρα ἐστί, φῶς ἐστι.

The dictum de si et aut 399

It is true that there were different views about conditionals in antiquity,

and at least four different analyses of the truth conditions for conditional

propositions. Sextus presents the different views as rivals, and he duly uses the

existence of the dispute to argue that ‘true conditionals will be found to be

inapprehensible’ (PH ii 110).⁵² But Sextus’ argument for inapprehensibility

has no value; and although there is some evidence that the correct analysis

of conditional propositions was indeed a matter of dispute in Hellenistic

antiquity, it is plain that, in principle, each of the analyses might be correct,

or useful, in certain contexts; and there is some reason to hold that Stoic

authors would sometimes prefer one analysis and sometimes another.

In any event, even Sextus allows that all the analyses have something

in common; for all the parties to the dispute—as Sextus characterizes

them—agree on the crucial point:

All the logicians say that, in general, a conditional is sound when its consequent

follows its antecedent—it is about when and how it follows that they dispute with

one another and produce conﬂicting criteria of following.

(M viii 112)⁵³

The dictum de si, in other words, is a commonplace of ancient logic. How

could it not be so?

The Stoics, so far as we know, did not appeal to the dictum de si in the way

in which Aristotle appealed to the dictum de omni. But they could have done.

For the dictum de si stands to hypothetical syllogisms as the dictum de omni

stands to predicative syllogisms. The dictum de si underwrites hypothetical

syllogisms to the extent that they turn about conditional assertibles. No doubt

syllogisms which turn about conditionals are valid insofar as the dictum is

true or insofar as the dictum explains what a conditional assertible is. But in

that way, just as Aristotle’s dictum de omni will underwrite Darii and Darapti

alike, so the dictum de si will underwrite both the ﬁrst Chrysippean unproved

and also, say, the following syllogistic form:

If P, then if Q then R; Q: therefore if P, then R.

If the notional dictum de si is to do the work we notionally demand of it,

then it must have a special relation with the ﬁrst unproved which it does not

have with that second syllogism.

⁵² τὸ ὑγιὲς συνημμένον ἀκατάληπτον εὑρεθήσεται.

⁵³ κοινῶς μὲν γάρ φασιν ἅπαντες οἱ διαλεκτικοὶ ὑγιὲς εἶναι συνημμένον ὅταν ἀκολουθῇ τῷ

ἐν αὐτῷ ἡγουμένῳ τὸ ἐν αὐτῷ λῆγον· περὶ δὲ τοῦ πότε ἀκολουθεῖ καὶ πῶς στασιάζουσι πρὸς

ἀλλήλους καὶ μαχόμενα τῆς ἀκολουθίας ἐκτίθενται κριτήρια.