Barnes Jonathan. Truth, etc

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440 The Science of Logic

Of arguments, some are concludent and some inconcludent. An argument is

concludent when the conditional which begins with the conjunction of the premisses

of the argument and ends with its conclusion is sound.

(PH ii 137)⁹⁰

Scholars generally ascribe that deﬁnition of concludence to the Stoics. True,

Sextus does not say, in so many words, that it is Stoic. But in Against the

Mathematicians the same deﬁnition recurs:

There is a concludent argument when, if we conjoin the assumptions and make a

conditional which begins with the conjunction of the premisses and ends with the

conclusion, this conditional is found to be true.

(M viii 417)⁹¹

That is proposed as something which ‘they say’ (415); and it forms part of

a long exposition which is ﬁrmly and explicitly ascribed to the Stoics (396,

399, 400, 406, 407, 408, 425).

Thus according to the Stoics (this from Diogenes Laertius), an argument

,…,P

: therefore Q

is inconclusive or invalid if and only if there is no conﬂict between ‘not-Q’

and ‘P

and P

and … and P

’. Hence (a proper inference from this same

passage in Diogenes) an argument is conclusive or valid if and only if there is

conﬂict between ‘not-Q’ and ‘P

and P

and … and P

’. Now a Chrysippean

analysis of conditional propositions has it that ‘If P, then Q’ is true if and

only if there is a conﬂict between ‘not-Q’ and ‘P’. Hence (it is eminently

reasonable to infer) the Stoics held that an argument

,…,P

: therefore Q

is valid if and only if the corresponding conditional proposition

If P

and P

and … and P

,thenQ

is true. In other words—to put it shortly—an argument is valid if and only

if its corresponding conditional is true. And Sextus conﬁrms that the thesis

was indeed Stoic.

But Sextus exploits the thesis for his own ends on more occasions than

one, and it is plain that he took it to be not a peculiarly Stoic idiosyncrasy

⁹⁰ τῶν δὲ λόγων οἱ μέν εἰσι συνακτικοὶ οἱ δὲ ἀσύνακτοι· συνακτικοὶ μὲν ὅταν τὸ συνημμένον

τὸ ἀρχόμενον μὲν ἀπὸ τοῦ διὰ τῶν τοῦ λόγου λημμάτων συμπεπλεγμένου, λῆγον δὲ εἰς τὴν

ἐπιφορὰν αὐτοῦ, ὑγιὲς ᾖ.

⁹¹ οὐκοῦν ὁ μὲν συνακτικὸς τότε ἐστὶν ὅταν συμπλεξάντων ἡμῶν τὰ λήμματα καὶ

συνημμένον ποιησάντων τὸ ἀρχόμενον μὲν ἀπὸ τῆς διὰ τῶν λημμάτων συμπλοκῆς λῆγον δ᾿

εἰς τὸ συμπέρασμα, εὑρίσκηται τοῦτο τὸ συνημμένον ἀληθές.

Corresponding Conditionals 441

but rather a commonplace of logic. Thus at PH ii 249 it is invoked as a thesis

belonging to ‘the dialecticians’—that is to say, to logicians in general; and at

PH ii 113 and 145 it is invoked as a general test of validity—a test, so Sextus

implies, accepted by all the ‘dogmatic’ philosophers.

Now there is at least a superﬁcial similarity between the test and Galen’s

metatheorem: according to the metatheorem, a syllogism is trustworthy if

and only if it is underwritten by an axiom; according to the test, an argument

is valid if and only if its corresponding conditional is true.

But that is no more than a formal similarity; and we cannot identify the

syllogistic axiom of a syllogism with its corresponding conditional. It is not

merely that Sextus speaks of arguments in general and not of syllogisms

in particular, that he does not hint—let alone state—that the conditional

corresponding to an argument is an axiom, and that he says nothing in

the context about the justiﬁcation or underwriting of arguments. There are

also positive reasons which forbid the identiﬁcation. For whereas Galen’s

syllogistic axioms are universal, the conditional which corresponds to an

argument is never universal: any universal proposition has the form ‘For any

x, such-and-such’; no conditional proposition has that form. And even if

a corresponding conditional is somehow universalized, it will not thereby

become an axiom—there are any number of universal truths which are not

axiomatic.

Nonetheless, if there is any recipe for ﬁnding the axiom appropriate to

a given syllogism, that recipe will surely start by considering corresponding

conditionals—after all, what else could it possibly look to? Consider, then,

the stock Stoic example of a hypothetical syllogism:

If it is day, it is light.

It is day.

Therefore it is light.

The corresponding conditional is this:

If if it is day it is light, and also it is day, then it is light.

Now that is not a universal proposition; nor does it look much like an axiom.

But there are indeﬁnitely many propositions which resemble it formally

insofar as they are all instances of the schema:

If if P then Q, and also P, then Q.

That schema may serve as the material for constructing a universal proposi-

tion, thus:

For any P and Q, if if P then Q, and also P, then Q.

442 The Science of Logic

And if you want to avoid schematic letters, then try:

For any pair of propositions, if if the one holds then the other does, and

in fact the one holds, then the other does.

That is certainly universal, and I suppose that it is axiomatic. Is it not the

universal axiom which—according to Galen—underwrites the stock Stoic

syllogism?

Something similar is readily done for a standard predicative syllogism in

Barbara. The stock syllogism

All men are animals.

All animals are substances.

Therefore all men are substances.

has the following corresponding conditional:

If all men are animals and all animals are substances, then all men are

substances.

That is neither universal nor an axiom. But it shares a form with countless

other conditionals, namely the form:

If all As are B and all Bs are C, then all As are C.

And that schema is material for a universal truth, namely:

For any X, Y, and Z, if every X is Y and every Y is Z, then every X is Z.

You may write that without schematic letters if you prefer.

Consider, ﬁnally, a relational argument, say:

CA is equal to AB.

CB is equal to AB.

Therefore CA is equal to CB.

The corresponding conditional is:

If CA is equal to AB and CB is equal to AB, then CA is equal to CB.

That is not universal, and it is not an axiom; but it admits generalization.

Thus:

For any x, y and z, if x is equal to z and y is equal to z, then x is equal to y.

Or:

For any triad of items, if two are each equal to the third, then they are

equal to one another.

Or:

Items equal to the same item are equal to one another.

Corresponding Conditionals 443

That is certainly universal, and it might be taken for an axiom. And in fact, as

we have seen, Galen did take it for an axiom—after all, it is the ﬁrst common

notion of Euclid’s Elements.

The ﬁrst part of the recipe for producing the syllogistic axiom for a given

syllogism is this: construct the conditional corresponding to the syllogism

and generalize it. So syllogistic axioms will not be conditional propositions,

in the strict sense of that phrase; but they will be derived from conditional

propositions and will have a conditional connection at their core. The

sentence

For any triad of items, if two are each equal to the third, then they are

equal to one another

is not conditional; for it does not have the form

If … , then …

But it is a universalized conditional—and it may be remarked that ancient

logicians sometimes did classify such items as conditionals.

However that may be, the ﬁrst part of the recipe will sometimes be enough

to produce the appropriate axiom. But it will not always produce an axiom:

it will produce one only when the syllogism on which it operates is valid

in virtue of a primary form. If the form is derived, or proved, then the

proposition produced by the recipe will be a theorem and not an axiom. So

the recipe needs an optional second part, which goes like this: if the syllogism

is not primary but derived, then its supporting axiom (or its supporting

axioms) is (or are) to be found by ﬁrst proving it and then applying the recipe

to the items on which the proof depends.

In that way, I suppose, an axiom, or a group of axioms, can be found for

each syllogism. For a universal truth can always be found; and a universal

truth which corresponds in the way I have outlined to a primary syllogism will

be an axiom. And if the recipe fails to work, or sometimes fails to work, then

I cannot see any other general way of generating syllogistic axioms. If Galen

thought that he had a way of showing that all syllogisms are underwritten by

axioms, then he must have followed, at least approximately and for some of

itslength,thelinewhichIhavejusttraced.

There is rather more which could and should be said about the recipe.

In addition, something needs to be added to cater for those cases in which,

according to Galen, the work of a syllogistic axiom is done by an item of a

different sort. (Perhaps in such cases the recipe will in fact come up with an

item which might be counted a deﬁnition?) Again, it would be worth asking

444 The Science of Logic

what is the relation between the syllogistic axioms produced by the recipe

and the dictum de omni et nullo (and the hypothetical dictum de si et aut).

DO SYLLOGISTIC AXIOMS WORK?

Do syllogistic axioms, so discovered, really determine the constitution and

guarantee the trustworthiness of their syllogisms? Well, you might say that

the axiom

For any P and Q, if if P then Q, and also P, then Q,

determines the structure of the syllogism

If it is day, it is light

It is day

Therefore it is light

insofar as it speciﬁes, implicitly, an argument with a conditional proposition

as one premiss, the antecedent of the conditional as another, and the

consequent of the conditional as the conclusion: the structure of the syllogism

corresponds—trivially enough—to the structure of the axiom. In general,

if the axiom answering to a given syllogism is a generalization of the

corresponding conditional of that syllogism, then the constitution of the

syllogism will be determined by the structure of the axiom—you will be able

to read off the constitution from the structure.

That will be so for any primary syllogism. But for derived syllogisms, the

case is different: the axioms which underwrite a derived syllogism will not, in

general, determine its constitution. That follows from the fact that the same

axioms may underwrite two different syllogistic forms. For example, Cesare

and Camestres are two derived syllogisms, two different syllogistic forms;

but they are underwritten—according to the present hypothesis—by the

same set of axioms. Each is proved from Celarent, by the application of the

principle of E-conversion. Each will therefore be underwritten by the axiom

corresponding to Celarent and the axiom corresponding to E-conversion;

and those axioms determine neither the constitution of Cesare nor the

constitution of Camestres.

Perhaps, after all, it is better to give a restricted scope to Galen’s

metatheorem? Not that it should be conﬁned to relational syllogisms, but

that it should be limited in application to all and only primary syllogisms.

Perhaps it should; but Galen never says so, nor even hints as much. Earlier,

Do Syllogistic Axioms Work? 445

I suggested that the metatheorem applies directly to primary syllogisms and

indirectly to derivative syllogisms. That comes to the same thing as placing

a restriction on the scope of the metatheorem; and it is perhaps a preferable

way of making the point. But—as I admitted—there is no hint of that either

in Galen’s text.

So much for the constitutional side of the metatheorem. What of trust-

worthiness? In the case of primary syllogisms, you might say that the axiom

guarantees the validity of the syllogism insofar as the syllogism is valid

precisely because the axiom holds. (And here derived syllogisms do not, in

principle, cause any special trouble: they are valid insofar as their underlying

axioms are true.) You might say that—but why should I accept it if you do?

Suppose that A is the axiom answering to a primary syllogism S: then of

course A is true if and only if S is valid. But the metatheorem supposes that

there is also an asymmetry between A and S; for it supposes that S is valid

insofar as A is true (and it is not the case that A is true insofar as S is valid).

Where does the asymmetry come from?

Or rather, where do the asymmetries come from?—For there is a second

asymmetrical thesis in the wings. The Peripatetic dictum was supposed

to underwrite predicative syllogisms—or rather, perfect predicative syllo-

gisms—in two ways: ﬁrst, it explains their validity; and secondly, grasp of

the dictum givesusagraspoftheirvalidity:

Barbara is valid because the dictum is true.

IknowthatBarbaraisvalidinasmuchasIgraspthetruthofthedictum.

Galen—so I have suggested—implicitly supposes that a syllogistic axiom

will underwrite its syllogism in the same double fashion:

The Euclidean syllogism is valid inasmuch as this axiom is true.

I grasp the validity of the Euclidean syllogism inasmuch as I recognize the

truth of this axiom.

So whence come those fearful asymmetries?

No doubt they depend on the fact that syllogistic axioms—like any

other axioms—possess a very special character: they are self-evident, or self-

warranting. In real money, that means that axioms—or at least syllogistic

axioms—are warranted by the meaning of their constituent terms, or by the

sense of any sentence which expresses them. For example, if you grasp the

sense of the syllogistic axiom

For any P and Q, if if P then Q, and also P, then Q,

then you recognize its truth.

446 The Science of Logic

Now that is certainly a Galenic thought. For Galen more than once states

that the axioms of any science must be anchored in the essences of the

items to which they allude, and he usually adds that the essence must be

determined—or at least approached—from a consideration of the sense of

the terms used to express it.

But Galenic or not, the thought seems to achieve nothing at all. Earlier,

I suggested that the hypothetical dictum de si underwrote the ﬁrst Stoic

unproved in the sense that anyone who grasped the dictum must thereby

recognize the validity of the syllogism. Now I have in effect urged that the

sense of Galen’s metatheorem is this: anyone who grasps the dictum will

thereby recognize the truth of the syllogistic axiom which backs up the ﬁrst

unproved. But why proceed to the syllogism by way of the axiom? Why not

go directly from the dictum to the syllogism?

Again, Aristotle’s dictum de omni is not an axiom but a deﬁnition; and

it supposedly underwrites Barbara without the intervention of any axiom.

The deﬁnition conveyed by the dictum is allegedly such that anyone who

understands Barbara thereby recognizes its validity. Galen’s metatheorem

offers us an axiom to back up Barbara. The axiom, I have suggested, is

presumably something like this:

For any triad of terms, if one holds of all of another and the other holds of

all of the third, then the one holds of all of the third.

No doubt that is a truth, and perhaps it is an axiom. But insofar as it is true, its

truth is underwritten by the dictum de omni. Perhaps the dictum underwrites

the axiom which underwrites Barbara? Perhaps the dictum underwrites both

the axiom and Barbara independently? In any case, it is the dictum which

counts, not the axiom.

I have urged that the dictum cannot in fact do the work which is required

of it, and it might be inferred that on that count the axiom must be superior.

But that is not so. Rather, if the dictum does not work, then the proposition

which was just put forward as the appropriate axiom for Barbara is not an

axiom at all. For if you may master the dictum without thereby grasping the

validity of Barbara, then you may understand the supposed axiom without

thereby recognizing its truth.

Something similar may be said for the ﬁrst Stoic unproved. Anyone who

knows what a conditional is thereby knows that the form

If P, Q; P: therefore Q

Do Syllogistic Axioms Work? 447

is valid. Perhaps he also thereby knows that the conditional proposition

For any P and Q, if if P then Q and P, then Q

is true; and perhaps he thereby grasps the truth of an axiom. But so what?

The appeal to an axiom seems to be otiose: a dictum—a deﬁnition—will

attach itself directly to a syllogism, and there is no need to employ an axiom

as go-between.

Perhaps there is a little more to be said on Galen’s behalf? I have guessed

that in his view a deﬁnition may sometimes play the role of a syllogistic

axiom; and I have supposed that Galen will have accepted that the syllogistic

axioms themselves depend upon deﬁnitions. But there may still be a point in

introducing axioms rather than simply citing deﬁnitions. For the axioms may

be taken to specify that part or aspect of the deﬁnition which is pertinent to

the syllogistic form.

Consider again the fourth of the Stoic unproveds, which makes use of an

exclusive disjunction; and take the syllogism:

Either it is day or it is night

But it is day

Therefore it is not night

The syllogistic axiom which lies behind that argument is—according to the

recipe—this:

For any P and Q, if either P or Q and P then not Q.

Now the nature of disjunction, or the sense of ‘or’, determines both the

truth of the axiom and the validity of the fourth unproved. But there is

nonetheless point in appealing to the axiom. For although the deﬁnition of

‘or’ determines that any fourth unproved is a valid syllogism, it is not, so to

speak, the whole of the deﬁnition which is pertinent to the validity of the

syllogism. Rather, it is that part or aspect of the sense which is presented in

the axiom.

6. When is a Syllogism not a Syllogism?

PLOTINUS ON LOGIC

In the third essay of the ﬁrst Ennead, Plotinus asks

what art or method or practice leads us up to the place to which we must travel?

(enn i iii 1 [1–2])¹

The piece is the twentieth of the twenty-one items which Plotinus had written

before Porphyry joined his circle in 263 (vit Plot iv 61–62). It generally circu-

lated under the title ‘On Dialectic’ (iv 18–19). It is short; and it is light—for

the question which it addresses is neither intricate nor difﬁcult to resolve.

Our destination—the place to which we must travel—is not a secret: we

must ascend to the Idea of the Good. The journey has two stages: ﬁrst, we

must scramble from the sensible world to the intelligible world; and then we

must climb up through the ranges of the intelligible world. Musicians, lovers

and philosophers are all, in their own ways, equipped for undertaking the

journey; but they need to prepare for it, and to prepare for it in different

fashions. As for the philosopher,

to accustom him to thinking and believing about the incorporeal he must be taught

mathematics, which he will accept readily inasmuch as he is a lover of learning; and,

being naturally virtuous, he must be led to perfect the virtues; and after mathematics

he must be given the arguments of dialectic and in sum he must be made a

dialectician.

(enn i iii 3 [5–10])²

Excelsior, excelsior: the climb to the summit is done on the oxygen of

mathematics—and of dialectic.

¹ τίς τέχνη ἢ μέθοδος ἢ ἐπιτήδευσις ἡμᾶς οἷ δεῖ πορευθἣναι ἀνάγει;

² τὰ μὲν δὴ μαθήματα δοτέον πρὸς συνεθισμὸν κατανοήσεως καὶ πίστεως ἀσωμάτου—καὶ

γὰρ ῥᾴδιον δέξεται φιλομαθὴς ὤν—καὶ φύσει ἐνάρετον πρὸς τελείωσιν ἀρετῶν ἀκτέον καὶ

μετὰ τὰ μαθήματα λόγους διαλεκτικῆς δοτέον καὶ ὅλως διαλεκτικὸν ποιητέον.

Plotinus on Logic 449

That being so, Plotinus’ next question is plain: ‘What is dialectic?’; and

when he asks it, he adds that dialectic ‘must be taught to the others too’ (i

iii 4 [1–2])³ —that is to say, the musician and the lover, and not only the

philosopher, must climb on dialectic. The addition is inelegant, inasmuch as

the preceding argument gave a strong impression that dialectic was reserved

for philosophers; but it is not surprising—for according to Plato,

it is the dialectical method alone which travels in this way, taking up the hypotheses,

to the ﬁrst principle itself.

(Rep 533c)⁴

If that is so, then everyone who is to ascend to the Idea of the Good must

take dialectic.

The opening paragraphs of Plotinus’ essay cling closely to Plato and to

Platonic texts. Plotinus’ answer to his question ‘What is dialectic?’ also clings

closely to Plato. Indeed, it is a cento of allusions and paraphrases, which serve

to show that dialectic

is the disposition which is capable of stating, by reason, about each item, what it is

and in what it differs from other items and what it has in common with them.

(enn i iii 4 [2–4])⁵

And dialectic is capable of stating such things about each item ‘by using

Plato’s division’ (4 [12]).⁶ With that compass in his hands, the traveller will

ﬁnd his way safely through the errors of the sensible world and discover the

right tracks to follow upwards in the world of Forms.

Plotinus’ dialectician, then, is not exactly a logician—or at least, he is

not a general logician. Rather, he is a deﬁner and a classiﬁer—an abstract

arboriculturalist of the sort who would later educate himself in Porphyry’s

woods and groves. Plotinus does not dissemble the fact. On the contrary, he

insists that dialectic

assigns to another art the so-called logical discipline which concerns propositions and

syllogisms (just as it consigns knowledge of writing to another art)—some of those

³ τίς δέ ἡ διαλεκτική, ἣν δεῖ καὶ τοῖς προτέροις παραδιδόναι;

⁴ οὐκοῦν, ἦν δ᾿ ἐγώ, ἡ διαλεκτικὴ μέθοδος μόνη ταύτῃ πορεύεται, τὰς ὑποθέσεις ἀναιροῦσα,

ἐπ᾿ αὐτὴν τὴν ἀρχήν;

⁵ ἔστι μὲν δὴ ἡ λόγῳ περὶ ἑκάστου δυναμένη ἕξις εἰπεῖν τί τε ἕκαστον καὶ τί ἄλλων

διαφέρει καὶ τίς ἡ κοινότης.

⁶ … τῇ διαιρέσει τῇ Πλάτωνος χρωμένη.