Barnes Jonathan. Truth, etc

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420 The Science of Logic

uncertainties affect numerous minor points of detail. But quite apart from

those difﬁculties, many of which are insoluble, there is another more general

problem; for in different passages Galen appears to present very different

versions of his metatheorem.

In any event, what relevance has the metatheorem for the question I am

now in principle addressing? Galen does not seem to be talking speciﬁcally of

primary syllogisms or of axiom-like syllogistic forms. Rather, his metatheorem

seems to be formulated for syllogisms in general. Again, he does not seem to

be talking about our knowledge of syllogisms and their validity: rather, he

is interested in the force of a syllogism, and his metatheorem is metalogical

in nature rather than epistemological. Those points are well made, and they

must be met. But for the moment I shall swagger on regardless.

The ﬁrst thing to do is to establish the scope of the metatheorem: it is a

thesis about the force syllogisms—but of which syllogisms? The business is

introduced in the third part of the Introduction, in which Galen describes

his own third species of syllogisms. At the very end of the discussion, Galen

remarks—according to the received text—that

all such syllogisms should be said to belong to the genus of relational arguments and

to the species of arguments constituted in accordance with the power of an axiom, as

Posidonius notes when he says that he calls them concludent in virtue of the power

of an axiom.

(inst log xviii 8)⁶⁶

That remark indicates that the metatheorem applies to the particular kind of

relational syllogisms which Galen has just been discussing and to which he

refers by ‘all such syllogisms’. The syllogisms in question are arguments based

on proportion or analogy. Arguments ‘constituted in accordance with the

power of an axiom’—that is to say, arguments for which the metatheorem

holds—are therefore a sub-species of the ‘third species of syllogisms’. The

metatheorem has a very restricted scope.

But that ﬂies in the face of everything else which Galen has said about his

metatheorem in this last third of the Introduction; and the received text of

xviii 8 is undoubtedly corrupt. It has been proposed that something like the

following should be read:

⁶⁶ τοὺς δὴ τοιούτους ἅπαντας συλλογισμοὺς τῷ γένει μὲν ἐκ τῶν πρός τι ῥητέον, ἐν εἴδει

δὲ κατ᾿ ἀξιώματος δύναμιν συνισταμένους, ὥσπερ καὶ ὁ Ποσειδώνιός φησιν ὀνομάζειν αὐτοὺς

συνακτικοὺς κατὰ δύναμιν ἀξιώματος.

Galen’s Metatheorem 421

all such syllogisms should be said to belong to the genus of relational arguments and

to the species of arguments <based on proportion>, being constituted in accordance

with the power of an axiom …⁶⁷

There might be disagreement about the details of that proposal; but its

general thrust must, I think, be right. And in that case the passage tells us

nothing about the scope of the metatheorem.

Turn, then, from its last to its ﬁrst appearance in the text of the Introduction.

Near the beginning of his discussion of relational syllogisms Galen claims

that

there is a large quantity of such syllogisms, as I said, among the arithmeticians and

the calculators, and what they all have in common is the fact that they have the same

constitution depending on certain axioms.

(inst log xvi 5)⁶⁸

Here the phrase ‘such syllogisms’ refers unequivocally to the relational

inferences which have just been introduced as the third class of syllogism;

and at ﬁrst blush you will suppose that Galen means to pick out a feature

which uniﬁes the class of relational syllogisms and distinguishes them from

the predicatives and the hypotheticals.

That natural supposition appears to be falsiﬁed by a sentence in xvi 10,

where the metatheorem is mentioned for the second time:

Similarly in the case of all the others, the constitution of the probative syllogisms will

be in virtue of the force of an axiom …⁶⁹

By ‘all the others’ does not Galen mean ‘all the syllogisms other than relational

syllogisms’? In that case, the metatheorem has the widest possible scope. But

although the phrase may possibly have that broad reference, the context makes

it far more likely that ‘all the others’ is short for ‘all relational syllogisms other

than those I have thus far described’.

In any event, the metatheorem is certainly applied only to relational

syllogisms on its third occurrence:

⁶⁷ τοὺς δὴ τοιούτους ἅπαντας συλλογισμοὺς τῷ γένει μὲν ἐκ τῶν πρός τι ῥητέον, ἐν εἰδει

δέ κατ᾿ <ἀναλογίαν, κατ’> ἀξιώματος δύναμιν συνισταμένους ...

⁶⁸ πολὺ δὲ πλῆθός ἐστιν, ὡς ἔφην, ἐν ἀριθμητικοῖς τε καὶ λογιστικοῖς τοιούτων συλλογισ-

μῶν ὧν ἁπάντων ἐστὶ κοινὸν ἔκ τινων ἀξιωμάτων τὴν αὐτὴν ἴσχειν σύστασιν ...—The MS

has ‘συστάσεως’, which gives no grammar; and ‘τὴν αὐτήν’ makes dubious sense. But whatever he

wrote, Galen must have meant to say that the constitution of these syllogisms depends upon axioms.

⁶⁹ ὁμοίως δὲ κἀπὶ τῶν ἄλλων ἁπάντων ἡ σύστασις τῶν ἀποδεικτικῶν συλλογισμῶν κατὰ

δύναμιν ἀξιώματος ἔσται ...—The next words in the received text are a conundrum; but they do

notaffectthepointatissue.

422 The Science of Logic

In the same way, syllogisms propounded in accordance with any relation whatever

will have their constitution and the force of the proof warranted by a general axiom.

(inst log xvi 12)⁷⁰

In short, chapter xvi of the Introduction suggests that the metatheorem is a

thesis about relational syllogisms: whatever may underwrite predicative and

hypothetical syllogisms, relational syllogisms are underwritten by axioms.

The beginning of chapter xvii, however, tells strongly—perhaps decis-

ively—in a different direction. The state of the text is lamentable; but

although the following translation is speculative with regard to most of its

details, its overall sense must correspond to what Galen wrote. (That is to

say, either this is more or less what Galen wrote or else we must hold up our

hands in sceptical resignation.) So:

In fact, pretty well all syllogisms have their constitution warranted because of the

universal axioms which are superordinate to them. That is something which I came to

realize later, having stated it neither in the books On Proof nor in On the Number of

Syllogisms—even though in those works I was aware of relational syllogisms and had

discovered their constitution and warrant. That all probative syllogisms are probative

because of the warranty of universal axioms may be learned more clearly if we survey

all arguments of this sort, in whatsoever manner they are propounded.

(inst log xvii 1–2)⁷¹

According to this passage, the metatheorem applies to ‘pretty well all

syllogisms’, or to ‘all probative syllogisms’: do the words mean what they

appear to mean, or are we to suppose a restriction to pretty well all relational

syllogisms and to all probative relational syllogisms?

Two points appear to tell against supposing any such restriction. First, the

textcontinuesasfollows:

Asisthecasewiththisargument:

You say: It is day.

Butyouaretellingthetruth.

⁷⁰ ὡσαύτως δὲ καὶ οἱ καθ᾿ ἡντινοῦν σχέσιν ἐρωτώμενοι συλλογισμοὶ γενικῷ ἀξιώματι

πιστὴν τὴν σύστασιν ἕξουσι καὶ τὴν τῆς ἀποδείξεως δύναμιν.

⁷¹ καὶ σχεδὸν ἅπαντες οἱ συλλογισμοὶ διὰ τὴν τῶν ἐπιτεταγμένων αὐτοῖς καθολικῶν

ἀξιωμάτων πιστὴν ἔχουσι τὴν σύστασιν, ὃ ὕστερόν ποτέ μοι νοηθὲν οὔτε ἐν τοῖς Περὶ

ἀποδείξεως ὑπομνήμασιν οὔτε ἐν τῷ Περὶ τοῦ τῶν συλλογισμῶν ἀριθμοῦ γέγραπται—καίτοι

τοὺς εἰς τὸ πρός τι συλλογισμοὺς ᾔδειμεν καὶ κατ᾿ ἐκείνας τὰς πραγματείας, εὑρηκότες τὸν

τῆς συστάσεως τρόπον αὐτῶν καὶ τῆς πίστεως. ὅτι δὲ πάντες οἱ ἀποδεικτικοὶ συλλογισμοὶ

διὰ τὴν τῶν καθόλου πίστιν ἀξιωμάτων εἰσὶ τοιοῦτοι, μαθεῖν ἔνεστιν ἐναργέστερον ἅπασι

τοῖς ὁπωσοῦν ἠρωτημένοις λόγοις τοιούτοις ἐπιβλέψασι ...

Galen’s Metatheorem 423

Therefore it is day.

That sort of syllogism too is probative inasmuch as the universal axiom under which

it falls is true …

(inst log xvii 3)⁷²

That is surely not a relational argument; but it illustrates the truth of

the metatheorem: therefore the metatheorem is not restricted to relational

arguments.

That may seem a pretty powerful consideration. Yet in fact it has no force

at all. For the argument must have counted, in Galen’s eyes, as a relational

syllogism. True, we should not so class it; true, Galen does not explicitly so

class it. But it is certainly not a predicative syllogism, and it is certainly not

a hypothetical syllogism. There is no fourth species of syllogisms, and no

syllogism fails to belong to a species. Hence it is a relational syllogism. If that

conclusion implies that Galen’s notion of a relational syllogism was pretty

indeterminate, then that is an implication which there are also other grounds

for accepting.

The second point which tells against supposing a restriction on the

metatheorem has more force. It is this. The fact that ‘pretty well all syllogisms

have their constitution warranted because of the universal axioms which are

superordinate to them’ is something which Galen had not discovered when

he wrote, as a young man, his vast work On Proof and its appendix On the

Number of Syllogisms. But ‘in those works I was aware of relational syllogisms

and had discovered their constitution and warrant’. If Galen had discovered

the constitution and warrant of relational syllogisms, then he had surely

discovered that relational syllogisms owe their constitution and warrant to

the universal axioms to which they are subordinated. At any rate, I cannot

see what else Galen might possibly have had in mind. But in that case, the

early works recognized that the metatheorem applies to relational syllogisms

but did not yet recognize that it applies to all syllogisms. Hence when, in the

Introduction, Galen says that the metatheorem applies to all syllogisms, he

means all syllogisms and not all relational syllogisms.

Posidonius had used the formula ‘concludent in virtue of the power of an

axiom’ (inst log xviii 8), so that the idea on which the metatheorem is based

derives from him. But he used the formula—if my reconstruction of the text

⁷² … καθάπερ ἔχει καὶ ὁ τοιόσδε· λέγεις ἡμέρα ἐστίν· ἀλλὰ καὶ ἀληθεύεις· ἡμέρα ἄρα ἐστίν.

ἀποδεικτικός ἐστι καὶ ὁ τοιοῦτος συλλογισμὸς διότι καὶ τὸ καθόλου ἀξίωμα ᾧ ὑποπέπτωκεν

ἀληθές ἐστι ...

424 The Science of Logic

is right—to refer to a subspecies of relational syllogism. When, as a young

man, Galen discovered the constitution and warrant of relational syllogisms,

he generalized the Posidonian notion from the subspecies to the species. Later,

he saw that the metatheorem applied to all—or to pretty well all—syllogisms.

That, I am inclined to suppose, is the story we should read into the text. It

is not a revolutionary story—on the contrary, I have laboriously argued for a

commonplace. But the commonplace needs an argument to support it, and

its support is less robust than it is often taken to be. For it depends wholly

on inferences drawn from a single passage of the Introduction in which the

Greek is quite certainly very corrupt.

At xvii 1 Galen says that the metatheorem holds of ‘pretty well all’ syllo-

gisms, and of ‘all probative syllogisms’. The second formula does not imply

any restriction on the theorem. It is true that here, and indeed throughout the

Introduction, Galen speaks expressly of probative syllogisms—that is to say,

of syllogisms which may serve to present proofs; but in his view an argument

which is not in this sense probative is not a syllogism at all. So ‘syllogism’

and ‘probative syllogism’ are equivalent terms.

What about the qualiﬁcation ‘pretty well all’? Well, that may simply mean

‘all’; for the adverb ‘pretty well [σχεδόν]’ is frequently used as a mark of

modesty or caution rather than as a sign of genuine limitation. But here the

adverb probably does indicate a limitation; for it is echoed a little later when

Galen remarks that

most of the things which men syllogize and prove are said in accordance with the

force of an axiom.

(inst log xvii 7)⁷³

Not all but ‘most of the things’: then what are the exceptions?

There is an indication a little later in the text—or rather, there once was

an indication. At the end of a lengthy discussion of an argument even the

bare bones of which are—once again for textual reasons—wholly uncertain,

the received text makes Galen remark that

‘Dio always speaks the truth’ is assumed in the place of the universal axiom.

(inst log xvii 9)⁷⁴

⁷³ τὰ πλεῖστα γὰρ ὧν οἱ ἄνθρωποι συλλογίζονται καὶ ἀποδεικνύουσι, κατὰ δύναμιν

ἀξιώματος λέγεται.

⁷⁴ τὸ δὲ πάντῃ ἀληθεύειν ∆ίωνα ἐν χώρᾳ τοῦ καθόλου ἀξιώματος εἴληπται.

The Sense of the Metatheorem 425

So in this argument no universal axiom is involved: something else takes its

place and performs its part. Now the proposition that Dio always tells the

truth is certainly not a universal axiom. But it is utterly remote from anything

which might be deemed a universal axiom, and I am unable to imagine how

or why Galen could have thought that it played the rôle of a universal axiom.

Despite the horrible state of the text, it is reasonably plain that, somehow

or other, an explanation of the meaning of the word ‘true’ comes into the

discussion of the argument which Galen is examining. It is therefore tempting

to suppose that it was that explanation, rather than the proposition about

Dio, which Galen took to play the rôle of an axiom. After all, explanations

of meanings, or deﬁnitions, are, in a pretty obvious way, axiom-like; and in

Aristotle’s theory of scientiﬁc proof deﬁnitions feature strongly among the

ﬁrst principles of the sciences.

If a deﬁnition or explanation of ‘true’ plays the rôle of axiom in the

concrete argument which Galen is discussing, then it is very tempting to

suppose that the exceptions to the rule that syllogisms depend on axioms are

precisely those cases in which they depend instead upon deﬁnitions. In that

case, a revised version of the metatheorem would claim that every syllogism

depends either on an axiom or on a deﬁnition.

However that may be, Galen certainly thought that most syllogisms, or

pretty well all syllogisms, depend on the force of an axiom. He cannot, then,

have supposed that no predicative syllogisms and no hypothetical syllogisms

depend on axioms; and despite the textual darkness, it is reasonable to

conclude that the metatheorem includes in its scope both predicative and

hypothetical syllogisms.

THE SENSE OF THE METATHEOREM

So much for the scope of the metatheorem: what of its content? First, it

is certain that, according to Galen, it is axioms which (somehow or other)

support syllogisms. The Greek word ‘ἀξίωμα’ is ambiguous in logical texts:

sometimes it means ‘assertible’ and sometimes ‘ﬁrst principle’. The former

use is Stoic: it originated in the Stoa, and it rarely travelled abroad. Galen does

not much like it—he does not much like Stoic terminology in general—and

it would be surprising had he adopted it here to express a theorem of his own.

In any event, he tells us expressly that

426 The Science of Logic

we should remember the signiﬁcation of the word axiom: we decided in the present

exposition so to name sayings which are self-warranted.

(inst log xvii 7)⁷⁵

In other words, in the metatheorem the term ‘axiom’ is used in the sense of

‘ﬁrst principle’—in more or less the sense in which it was generally used by

the mathematicians, and also (sometimes) by Aristotle.

Axioms are, of course, true. They are also self-warranting (so xvi 6 and 7,

as well as the formal announcement at xvii 7). They are therefore unproved

inasmuch as they do not need proof. It may be supposed—although the

Introduction does not explicitly say so—that they are also primary inasmuch

as they cannot be proved. Again, the axioms to which Galen’s theorem

appeals are universal propositions. (So at xvi 6, and a further eight times;

and once, if the easiest emendation at xvi 12 is right, they are ‘general

[γενικός]’). Whether Galen thought that all axioms were universal is another

matter; but the axioms which ground syllogisms—the syllogistic axioms,

as I shall henceforth call them—undoubtedly are universal truths. Galen

surely took the syllogistic axioms to be necessary truths; but they are

certainly not, or not all, items which we should class as logical truths:

among them, for example, are at least some of the axioms of Euclid’s

Elements.

One passage has been taken to show that syllogistic axioms are, all of them,

conditional propositions (xvi 10). But the text is—yet again—certainly

corrupt, and even the general gist is wholly obscure. Moreover, the illustrative

axioms which Galen introduces into the discussion are not in general expressed

as conditional propositions. Nevertheless, conditionality will come into its

own at a later stage in the discussion.

There is more than one syllogistic axiom: Galen’s use of the plural form

proves that different syllogisms are supported by different axioms; the point

is conﬁrmed by the various examples which Galen provides; and in any event

it is obvious enough. But how many syllogistic axioms are there?

Evidently, there is not one axiom for each concrete syllogism—syllogisms

which share their pertinent logical forms will also share their axiom. But

it is tempting to suppose that syllogistic forms and syllogistic axioms form

monogamous couples: all Stoic ﬁrst unproveds, say, will be supported by

⁷⁵ … μεμνημένων ἡμῶν καὶ αὐτοῦ τοῦ κατὰ τὴν ἀξίωμα φωνὴν σημαινομένου· τὸν γὰρ ἐξ

αὑτοῦ πιστὸν λόγον οὕτως ὀνομάζειν ὑπεθέμεθα κατὰ τὴν προκειμένην διδασκαλίαν.

The Sense of the Metatheorem 427

one and the same axiom, and that axiom will support no syllogisms but ﬁrst

unproveds; there will be an axiom which serves every syllogism in Barbara

and which is strictly faithful to Barbara; and so on.

In the case of relational syllogisms, that supposition may well be trivially

true—for how shall we distinguish one relational form from another unless

by reference to its supporting axiom? But the supposition is not trivial

for predicative and hypothetical syllogisms, where the different forms are

determined in advance of any application of Galen’s metatheorem; and in

fact the supposition is dubious. No doubt there is an axiom which answers to

the ﬁrst Stoic unproved, and an axiom which answers to Aristotle’s Barbara.

But are there also axioms corresponding to derived syllogistic forms? Is there

an axiom corresponding, say, to Baralipton?

Well, if Baralipton is derived from Barbara, will not the axiom which

supports Barbara thereby also support Baralipton? More generally, it must

seem reasonable to suppose that axioms—which are essentially unproved—

correspond to unproved syllogistic forms, and that if anything corresponds

in the same sort of way to a proved form, it will be a theorem rather than

an axiom. So Baralipton, say, will be subordinated to a syllogistic axiom—to

the syllogistic axiom to which Barbara is subordinated; and a derived hypo-

thetical syllogism will be subordinated to the axiom to which—say—the

ﬁrst unproved is subordinated.

It must be allowed that there is nothing in Galen’s text which suggests

that the metatheorem bears, so to speak, directly upon primary syllogisms

and indirectly on derivative syllogisms. But I cannot see how he could have

maintained that every valid syllogistic form had a private axiom to support

it; and that being so, the view which I have just sketched is the obvious one

to embrace.

That meets one of the two points which were raised and not met earlier:

since Galen’s metatheorem is about syllogisms in general and not about

primary syllogisms in particular, how is it pertinent to the present concerns?

The answer is: fundamentally—but never explicitly—the metatheorem

concerns primary syllogisms.

The next question: Is each syllogistic form supported by a single axiom?

Galen’s illustrative examples never appeal to a plurality of axioms; he often

uses the singular in referring to the axiomatic backing of a syllogism (inst

log xvi 10, 11; xvii 3, 7; xviii 6); and he says, introducing a new species of

relational argument that,

428 The Science of Logic

in these cases too, the syllogism will be in accordance with one of the axioms.

(inst log xvi 10)⁷⁶

Such facts suggest that, according to the metatheorem, each syllogistic form

has a single axiom behind it.

On the other hand, if Baralipton does not possess an axiom of its own,

insofar as it depends on Barbara, it might seem plausible to think that

it in fact rests on a pair of axioms. For the standard proof of Baralipton

depends on Barbara and a principle of conversion. If the conversion rule

has the status of an unproved rule of inference, then perhaps it too will

be backed by an axiom; and in that case Baralipton will been underwritten

by two axioms—namely, the axiom which underwrites Barbara and the

axiom which corresponds to the principle of conversion. More generally,

derived forms will depend on a plurality of axioms. Nothing in Galen’s

text strictly excludes that possibility. But nothing in the text remotely

hints at it, and it is doubtless miles away from anything which Galen ever

dreamed of.

The axioms underwrite or support their syllogisms: what sort of support

do they give? I said at the start, with deliberate vagueness, that syllogisms

owe their force to their associated axioms. Galen himself uses a multiplicity

of linguistic turns to describe the support. Sometimes a syllogism is said to

be ‘in accordance with an axiom’ (inst log xvi 10), or ‘in accordance with the

power of an axiom’ (xvii 7; xviii 8). Once he says that certain syllogisms ‘have

the force of proof ’ by an axiom (xvi 12). But most often he expresses himself

either by way of the term ‘constitution [σύστασις]’, or by means of the word

‘warranty [πίστις]’ or a cognate word. The two terms come together at xvi

12, where Galen refers either to ‘the warranted constitution’ of a syllogism or

to ‘the warrant of its constitution’. (The manuscript reading is an impossible

combination of the two.) So too at xvii 2, where he refers to ‘the manner of

their constitution and of their warranty’.

As for ‘πίστις’ and its family, Galen refers sometimes to the warranty

of the superordinate axioms (xvii 1—if the text is correct—and 2); that

is to say, he invokes the fact that axioms are self-warranting (xvii 7). In

a similar vein he notes that in the case of certain arguments the pertin-

ent axiom ‘is both thought of and believed by everyone’ (xviii 6).⁷⁷ But

he also refers to the warranty of syllogisms (xvii 2), and says of certain

⁷⁶ καὶ γὰρ ἐπὶ τούτων ὁ συλλογισμὸς ἔσται κατά τι τῶν ἀξιωμάτων.

⁷⁷ καθολικὸν δὲ καὶ κατὰ τοὺς τοιούτους λόγους ἀξίωμα νοεῖταί τε καὶ πιστεύεται πᾶσι.

The Sense of the Metatheorem 429

syllogisms that ‘their warranty is taken from certain of the universal axioms’

(xviii 1).⁷⁸

The word ‘πίστις’ often means ‘belief ’, and also ‘trust’. But an item

which is πίστος—which is reliable or trustworthy—may be said to have

πίστις (to have reliability or trustworthiness) or to be or provide πίστις for

something else (to warrant it). The metatheorem, insofar as it invokes πίστις,

surely means that syllogisms are reliable, or trustworthy, or warranted, in

virtue of an associated axiom; and that they derive their reliability from the

reliability or trustworthiness of those axioms. In what might the reliability

or trustworthiness of a syllogism consist? Presumably in its validity. The

syllogistic axioms, then, underwrite syllogistic validity. That unadventurous

conclusion is conﬁrmed when Galen remarks that Posidonius called certain

arguments ‘concludent in virtue of the force of an axiom’ (xviii 8).⁷⁹ He

evidently thinks that the expression is apt.

But if a syllogism is trustworthy inasmuch as it is valid, the word

‘πίστις’ does not, of course, mean ‘validity’: it means ‘trustworthiness’ or

‘warrantability’ or something of that sort. The word has an epistemological

sense; and what Galen means to say is that what warrants our accepting a

syllogism, what makes it right for us to trust its validity, is the axiom on

which it rests, or perhaps the fact that it rests on an axiom. That being so,

Galen presumably thinks that we know a syllogism to be valid insofar as we

have grasped the axiom to which it is subordinated.

That meets the second of the two points which were raised earlier: the

metatheorem is metalogical, not epistemological—so how can it be relevant to

the present concerns? The answer is: the metatheorem has an epistemological

element to it.

The σύστασις or constitution to which Galen refers is always the con-

stitution of the syllogism in question. At xvi 5 certain syllogisms ‘have the

same constitution on the basis of certain axioms’ (but the text is uncer-

tain). At xvi 10, ‘the constitution of probative syllogisms’ is in accordance

with the power of an axiom. There is a reference to ‘the constitution of

the reasoning’ at xvi 11. I have already cited xvi 12, and xvii 2. What

does the word ‘constitution’ mean in these passages? Or rather, what is the

meaning of the Greek word ‘σύστασις’, which I have thus far translated by

‘constitution’?

⁷⁸ καὶ τούτων ἡ πίστις ἐκ τῶν καθόλου τινῶν ἀξιωμάτων εἴληπται.

⁷⁹ συνακτικοὺς κατὰ δύναμιν ἀξιώματος.