Barnes Jonathan. Truth, etc

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430 The Science of Logic

There are several possibilities, of which the most popular, I think, is

‘validity’. The popularity is comprehensible—indeed, ‘validity’ is the very

word we should most like to hear. But although the sense of ‘validity’ is

frequently claimed for the word, and although there are one or two late texts

in which the sense is at least plausible, I am inclined to doubt its existence.

In any event, it is certainly not a common sense, and it is not a sense found

elsewhere in Galen’s writings.

The common sense of the word, as my translation has already intimated, is

‘constitution’: the σύστασις of a syllogism is its construction or constitution,

the way in which it is put together from premisses and conclusion. And

that this is the sense which the word bears in our texts is shown by two

passages. First, ‘the constitution of the reasoning’ at xvi 11 refers to the

fact that an argument is propounded as a predicative syllogism rather than

as a hypothetical syllogism: it is not the validity of the syllogism but its

form or constitution which is in question. Secondly, at xviii 8 Galen refers

to arguments ‘constituted in accordance with the force of an axiom’.⁸⁰

The participle, ‘constituted’ belongs to the verb of which ‘σύστασις’isa

nominalization; and, again, it is not validity but constitution which is in

question.

If that is right, then syllogisms owe two things to their superordinate

axioms: their trustworthiness, and their constitution. It is not difﬁcult to see

how an argument might owe its trustworthiness to an axiom: the argument

is reliable inasmuch as the axiom is true. But how on earth might a syllogism

be said to owe its constitution to an axiom?

AXIOMS AS PREMISSES

An answer to that question is suggested by the way in which Galen presents

his ﬁrst example of a relational syllogism. The sense of the passage is clear,

and the text—for once—is well preserved:

Given that there is this universal axiom, which has its warranty from itself—namely,

items equal to the same item are also equal to one another—it is possible to syllogize

and prove in the way in which Euclid produced the proof in his ﬁrst theorem where

he shows that the sides of the triangle are equal; for since items equal to the same item

⁸⁰ κατ᾿ ἀξιώματος δύναμιν συνισταμένους.

Axioms as Premisses 431

are also equal to one another, and it has been shown that the ﬁrst and the second are

equal to the third, the ﬁrst will be equal to the second.

(inst log xvi 6)⁸¹

Galen alludes to the proof of the ﬁrst theorem in the ﬁrst book of Euclid’s

Elements—or rather, to a part of it:

Since the point A is the centre of the circle CDB, AC is equal to AB. Again, since

the point B is the centre of the circle CAE, BC is equal to BA. But it has been shown

that CA is equal to AB. Each of CA and CB is therefore equal to AB. But items

equal to the same item are also equal to one another. Therefore CA is equal to CB.

Therefore the three—CA, AB, CB—are equal to one another.

(i i)⁸²

According to Galen, ‘he shows that the sides of the triangle are equal’: CA,

CB and AB are the sides of the triangle ABC. Again, what Galen calls ‘the

ﬁrst’ side is CA, ‘the second’ is CB, and ‘the third’ is AB. And ‘it has been

shown’—in the ﬁrst two sentences I have just cited from the Elements—that

CA is equal to AB and that CB is equal to AC. Galen’s account of Euclid’s

procedure is impeccable.

In Euclid, the axiom is used as a premiss in the argument. There is nothing

untoward about that: the axiom is the ﬁrst of Euclid’s ‘common notions’;

and the common notions, like the other bits and pieces which Euclid sets out

before he begins his proofs, are there precisely in order to serve as premisses

in proofs. Galen’s account of the argument does not state clearly that the

axiom is employed as a premiss; but he can hardly have imagined that Euclid

employed it in any other capacity, and he must have intended to employ it in

the same way himself. In other words, the argument which Galen is inviting

us to consider is this:

Any two items equal to the same item are equal to one another.

CA is equal to AB.

⁸¹ ὄντος γὰρ ἀξιώματος τοῦδε καθόλου τὴν πίστιν ἔχοντος ἐξ ἑαυτοῦ, τὰ τῷ αὐτῷ ἴσα καὶ

ἀλλήλοις ἐστὶν ἴσα, συλλογίζεσθαί τε καὶ ἀποδεικνύναι ἔστιν ὥσπερ Εὐκλείδης ἐν τῷ πρώτῳ

θεωρήματι τὴν ἀπόδειξιν ἐποιήσατο τὰς τοῦ τριγώνου πλευρὰς ἴσας δεικνύων· ἐπεὶ γὰρ τὰ

τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἴσα ἐστίν, δέδεικται δὲ τὸ πρῶτόν τε καὶ τὸ δεύτερον τῷ τρίτῳ

ἴσον, ἑκατέρῳ αὐτῶν ἴσον ἂν εἴη οὕτω τὸ πρῶτον.

⁸² καὶ ἐπεὶ τὸ Α σημεῖον κέντρον ἐστὶ τοῦ Γ∆Β κύκλου, ἴση ἐστὶν ἡ ΑΓ τῇ ΑΒ· πάλιν, ἐπεὶ

τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου, ἴση ἐστὶν ἡ ΒΓ τῇ ΒΑ. ἐδείχθη δὲ καὶ ἡ ΓΑ τῇ

ΑΒ ἴση· ἑκατέρα ἄρα τῶν ΓΑ, ΓΒ τῇ ΑΒ ἐστὶν ἴση. τὰ δέ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν

ἴσα· καὶ ἡ ΓΑ ἄρα τῇ ΓΒ ἐστὶν ἴση· αἱ τρεῖς ἄρα αἱ ΓΑ, ΑΒ, ΒΓ ἴσαι ἀλλήλαις εἰσίν.

432 The Science of Logic

CB is equal to AB.

Therefore CA is equal to CB.

He took that to be a relational syllogism; and he thought that it depended

for its trustworthiness and for its constitution on the axiom which appears as

its ﬁrst premiss.

The syllogism thus owes its constitution to the axiom in a straightforward

way: the axiom is an essential part or premiss of the syllogism, the other

premisses referring to special cases which fall under it. The generalization to

relational syllogisms is obvious: every relational syllogism will have, among its

premisses, a universal axiom, its other premisses introducing special instances

which fall under its general rule.

That such was indeed Galen’s view of the matter is conﬁrmed, it seems,

by other passages in the Introduction. Once, for example, Galen remarks of a

particular relational syllogism that

the constitution of the reasoning will be more forceful with predicative proposi-

tions—there being premissed here too, of course, a universal axiom …

(inst log xvi 11)⁸³

The axiom is expressly said to be premissed, or put forward as a premiss of

the argument. (But the passive participle ‘being premissed’ is a conjectural

emendation of the corrupt manuscript text.) Again, the very ﬁrst example

in the Introduction—which is offered as an illustration of what a probative

syllogism should look like—reads thus:

This proof is constituted from three items: ﬁrst, the ﬁrst one I stated, which was

Dio is equal to Theo;

second, the one after it:

Philo is equal to Dio;

and third, in addition to them,

items equal to the same item are also equal to one another.

From these it will be concluded that Theo is equal to Philo.

(inst log i3)⁸⁴

The text is found on a page of the manuscript the right-hand half of which

has been torn off, and only half the words which I have translated actually

⁸³ κατηγορικαῖς δὲ προτάσεσι βιαιότερον ἔσται ἡ σύστασις τοῦ λογισμοῦ, προτεινομένου

δηλονότι καθόλου κἀνταῦθά τινος ἀξιώματος ...

⁸⁴ καὶ τοίνυν ἡ ἀπό[δειξις ἥδ᾿ ἐκ τριῶν συνίστα]ται, πρώτου μὲν τοῦ πρώτου ῥηθέντος,

ὅπερ ἦν ἴσος ἐστὶ [∆ίωνι Θέων, δευτέρου δὲ τοῦ μετ᾿ α]ὐτό ∆ίωνι Φίλων ἐστὶν ἴσος καὶ τρίτου

πρὸς τούτοις τοῦ τὰ [τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴ]σα· περανθήσεται δὲ ἐξ αὐτῶν ἴσος

Axioms are not Premisses 433

answer to any legible Greek. But the reconstruction is certainly correct in its

general lines; and it is the general lines which count here: they show that

Galen took his axioms to function as premisses.

That view of the function of syllogistic axioms has the great merit of being

simple and intelligible; nor is it evidently foolish—after all, it answers to

Euclid’s actual practice. Moreover, you might conjecture a plausible origin

for it. For the role which, on this view, Galen’s metatheorem gives to the

syllogistic axioms can hardly fail to recall the role which—according to

Aristotle’s Topics —the ‘commonplaces’ or τόποι hold in dialectical argu-

ments. A commonplace, according to the most plausible view of the matter,

is an abstract universal truth. Dialectical syllogisms owe their constitution

to the commonplaces inasmuch as the commonplaces form their leading

premisses, the other premisses adverting to special cases which fall under

them. Galen knew his Aristotelian logic—and so too did that Aristotelizing

Stoic, Posidonius. Galen’s metatheorem is in effect a generalization to all

syllogisms of the Aristotelian notion that dialectical syllogisms are consti-

tuted on the basis of commonplaces—a generalization which replaces the

commonplaces by the syllogistic axioms. Perhaps Galen was inspired by the

Topics? Perhaps Posidonius had been so inspired before him?

AXIOMS ARE NOT PREMISSES

Thus the interpretation which makes premisses of the syllogistic axioms is

ﬁrmly based on the text of the Introduction; it ﬁts with actual geometrical

practice; and it can claim dialectical precedent. Nonetheless, it cannot be

right; or at least, it is confronted by two serious difﬁculties.

The ﬁrst difﬁculty is this. It is conceivable that every relational syllogism

must have an axiom among its premisses—or at any rate, it is conceivable

that Galen thought that every relational syllogism must have an axiom among

its premisses. The argument

CA is equal to AB

CB is equal to AB

Therefore CA is equal to CB

εἶναι Θέων [Φίλωνι.—The letters inside brackets are an imaginative reconstruction of what was

once on the missing half of the page.

434 The Science of Logic

is no doubt valid. But its validity depends on the sense of the term ‘equal’—it

depends (perhaps) on its matter rather than on its form. So perhaps it is not

a genuine syllogism—I mean, perhaps Galen would not have taken it to be

a genuine syllogism. On the other hand, the argument

Any two items equal to the same item are equal to one another

CA is equal to AB

CB is equal to AB

Therefore CA is equal to CB

surely is a syllogism—at any rate, its validity does not depend on the sense

of the word ‘equal’ or on any other aspect of its matter.

The argument without the axiom among its premisses is a relational

argument, but it is not a relational syllogism. The argument with the axiom

is a relational syllogism. And so it is in all cases: a relational argument with

no axiom among its premisses is never a syllogism; all relational syllogisms

have an axiom among their premisses. Now I do not think that in fact Galen

ever entertained such an idea; but I allow that he could or might have done.

(Whatever that means.)

Nonetheless, the idea can hardly be extended to all syllogisms. For

Galen surely cannot have imagined, and quite certainly never suggests, that

hypothetical syllogisms and predicative syllogisms must all always have axioms

among their premisses.

True, some Stoics claimed that predicative syllogisms are not really

syllogisms; but they did not ﬁnd predicative syllogisms wanting because their

premisses contained no axioms. True, Alexander—in some moods—urged

that hypothetical syllogisms are not really syllogisms; but he did not claim

that they fail to meet the grade because they lack an axiom among their

premisses.

Actually there is a passage in the commentary on the Analytics which has

been thought to show that he did take that line; and the passage is worth an

airing. Alexander is discussing this text:

Again, if, being a man, he must be an animal, and being an animal a substance,

then being a man, he must be a substance. But that is not yet a syllogism—for the

premissesarenotaswehavesaid.

(APr 47a28–31)⁸⁵

⁸⁵ πάλιν εἰ ἀνθρώπου ὄντος ἀνάγκη ζῷον εἶναι καὶ ζῴου οὐσίαν, ἀνθρώπου ὄντος ἀνάγκη

οὐσίαν εἶναι· ἀλλ᾿ οὔπω συλλελόγισται· οὐ γὰρ ἔχουσιν αἱ προτάσεις ὡς εἵπομεν.

Axioms are not Premisses 435

The illustrative argument might be construed in more ways than one; but

Alexander read it as a ‘wholly hypothetical’ inference, thus:

If he is a man he is an animal.

If he is an animal he is a substance.

Therefore if he is a man he is a substance.

That is a valid argument, according to Aristotle; but it is not a syllogism—for

‘the premisses are not as we have said’. What does Aristotle mean?

Alexander offers two interpretations of Aristotle’s Delphic phrase. First, he

says this:

The premisses are not as we said they must be if there is to be a syllogism—i.e. the

premisses must be universal, either both of them or at least one. And in this example

neither is taken universally. But since the universal which has been omitted and the

laying down of which would produce a syllogism is true, what seems to follow the

items laid down seems to be true. The universal premiss is:

Everything which follows a certain item also follows whatever that item follows.

… For when the universal which we have just mentioned is assumed, and it is

co-assumed that being a man he is an animal and being an animal a substance, it is

concluded syllogistically that … being a man he is a substance.

(in APr 347.20–348.2)⁸⁶

It is far from clear what syllogism—what predicative syllogism—Alexander

thinks he has constructed; but it is plain, at least, that he thinks he can

turn the wholly hypothetical inference into a syllogism by adding a universal

premiss—and the added universal premiss might perhaps be taken for an

axiom.

But Alexander then offers a second interpretation of his Aristotelian

text:

The premisses are not as we have said because they are not taken deictically or

universally. There will be a syllogism if they are taken in this way:

Every man is an animal

Every animal is a substance

⁸⁶ οὐ γὰρ ἔχουσιν αἱ προτάσεις ὡς εἴρηται δεῖν ἔχειν εἰ μέλλοι συλλογισμὸς ἔσεσθαι·

ἔστι δὲ τοῦτο τὸ δεῖν ἢ ἀμφοτέρας εἶναι τὰς προτάσεις καθόλου ἢ πάντως γε τὴν ἑτέραν.

ἐνταῦθα δὲ οὐδεμία εἴληπται καθόλου. τῷ μέντοι τὴν καθόλου ἀληθῆ εἶναι τὴν παρειαμένην

ἧς τεθείσης συλλογισμὸς ἔσται, ἀληθὲς δοκεῖ τὸ τοῖς κειμένοις ἕπεσθαι δοκοῦν. ἔστι δὲ ἡ

καθόλου πρότασις πᾶν τὸ ἑπόμενόν τινι ἕπεται καὶ ᾧ ἐκεῖνο ἕπεται. ... ληφθέντος γὰρ τοῦ

καθόλου οὗ προειρήκαμεν καὶ προσληφθέντος αὐτῷ τοῦ ἀνθρώπου δὲ ὄντος ζῷόν ἐστι καὶ

ζῴου οὐσία, συνάγεται συλλογιστικῶς τὸ καὶ ... ἀνθρώπου ὄντος οὐσίαν εἶναι.

436 The Science of Logic

… Again, it is possible to criticize

being a man he must be an animal and an animal a substance

as not concluding syllogistically on the grounds that the premisses—‘A man is an

animal’ and ‘An animal is a substance’—are indeﬁnite.

(in APr 348.15–22)⁸⁷

On this second interpretation of Aristotle’s text, the non-syllogistic argument

does not need to be reinforced with a new premiss—let alone with a new

axiomatic premiss: what it needs is some quantiﬁers.

In short, Alexander neither says nor implies that every hypothetical

inference needs to be reinforced by an additional axiomatic premiss in order

to become a syllogism. He holds—what is a perfectly distinct thesis—that

every syllogism, predicative or hypothetical, must have at least one universal

premiss.

There is a second and more fundamental difﬁculty with the interpretation

of Galen’s metatheorem which has it place an axiom among the premisses of

every syllogism. If a syllogistic axiom functions as a premiss in a syllogism, then

how can it also be responsible for the trustworthiness of the syllogism in which

it appears? To be sure, if the syllogistic axiom is an essential premiss in a

syllogism, then if it is removed what is left will not be a syllogism; and in that

sense you might say that the presence of the axiom among the premisses was

responsible for the warranty of the syllogism. But you might say exactly the

same of any non-redundant premiss of any argument: if it is removed, then

the result is not a valid argument—so that, in that sense, the warranty of any

argument depends on each and every one of its premisses.

Moreover, if you were asked to account for the trustworthiness of the

Euclidean argument to which Galen alludes, you would surely not appeal to

the axiom which is found among its premisses—nor yet to the fact that an

axiom is found among its premisses. It is true that the argument has an axiom

among its premisses. But it is not that feature of it which underwrites its

validity. Set the Euclidean argument alongside the following silly argument:

Any two items unequal to the same item are unequal to one another.

CA is unequal to AB.

⁸⁷ οὐ γὰρ ἔχουσιν αἱ προτάσεις ὡς εἴπομεν ὅτι μὴ δεικτικῶς μηδὲ καθόλου ἐλήφθησαν.

ἔσται γὰρ συλλογισμὸς ἂν οὕτω ληφθῶσι· πᾶς ἄνθρωπος ζῷον, πᾶν ζῷον οὐσία. ...

δυνατὸν πάλιν τὸ εἰ ἀνθρώπου ὄντος ἀνάγκη ζῷον εἶναι καὶ ζῴου οὐσίαν αἰτιᾶσθαι ὡς μὴ

συλλογιστικῶς συνάγον ὅτι ἀδιόριστοι αἱ προτάσεις, ἥ τε ὁ ἄνθρωπος ζῷον καὶ ἡ τὸ ζῷον

οὐσία.

Axioms are not Premisses 437

CB is unequal to AB.

Therefore CA is unequal to CB.

The silly argument and the Euclidean argument are both valid, and they are

valid in virtue of a valid form which they share and which might be set out

schematically thus:

For any x, y, z: if xRz and yRz, then xRy

aRc

bRc

Therefore aRb

Every argument of that form is valid. Some arguments of that form—Euclid’s

argument, for example—may have an axiom as their ﬁrst premiss. Others may

have a truth which is not axiomatic. Others—like the silly argument—may

have a staring falsity. All are valid, and valid for the same reason. All are

trustworthy, and trustworthy for the same reason. The fact that, in Euclid’s

case, the ﬁrst premiss is an axiom has absolutely nothing to do with the

question.

There are several possible reactions to those difﬁculties. One of them—and

not the least plausible—is despair: perhaps Galen’s metatheorem is no more

than a muddle, just as his ‘third species of syllogism’ is no more than a

phantom? But if we decline to despair, then the least implausible—and the

most interesting—lines to follow are those which suppose that, despite the

appearances of the text, the syllogistic axioms are not meant to function as

premisses in their arguments.

And the supposition is not entirely baseless. Once or twice in the Intro-

duction Galen quite certainly presents syllogistic axioms as premisses of their

syllogisms. But by far the greater number of his illustrative syllogisms do not

count axioms among their stated premisses. You may always say that in those

cases Galen leaves us to supply the missing axiomatic premiss for ourselves.

But perhaps Galen did not err by omission in those cases where he left out

an axiom: perhaps he erred by commission where he put the axiom among

the premisses.

In any event, if the axioms are not premisses of the syllogisms whose

constitution and trustworthiness they guarantee, then how do they function?

One suggestion tries to have things both ways: the axioms do indeed function

as premisses (as several passages show); but the syllogisms in which they

function as premisses are not the syllogisms whose trustworthiness and

constitution they underwrite. The Euclidean syllogism contains an axiom

438 The Science of Logic

among its premisses. The axiom, by its position there, underwrites another

syllogism, namely:

CA is equal to AB

CB is equal to AB

Therefore CA is equal to CB

That argument has no axiom among its premisses; but it is valid in virtue of

the force of an axiom—namely, of the axiom which appears in the Euclidean

syllogism.

The interpretation may be generalized as follows. Every syllogism is

underwritten by an axiom insofar as the result of adding the axiom to its

premisses produces another syllogism. In other words, if

,…P

: therefore Q

is a syllogism, then there is an axiom A such that

A, P

,…P

: therefore Q

is a syllogism; and the former syllogism is valid in virtue of the rôle of the

axiom in the latter syllogism.

That suggestion has the merit of ingenuity—and several drawbacks.

Amongthemoreevidentdrawbacksisthefactthatitintroducesaninﬁnite

regression. For if Galen’s metatheorem claims that every syllogism is under-

written by an axiom, then the syllogism which employs the underwriting

axiom as a premiss will itself be underwritten by an axiom; that second

axiom will be a premiss in the syllogism which underwrites it; and so on

ad inﬁnitum. There are various ways in which such a regression could be

blocked, or rendered harmless; but, on the whole, it seems best to decide that

the underwriting axioms are not premisses at all—neither of the syllogism

which they underwrite nor yet (in virtue of their ofﬁce) of any other syllogism.

And after all, why should we consider such exotic suggestions? The simple

suggestion is the best: a syllogistic axiom underwrites or supports a syllogism

insofar as the syllogism is trustworthy because the axiom is true. Or (if you

like reading letters), for any syllogism S, there is a proposition P such that it

is an axiom that P and S is trustworthy because P.

CORRESPONDING CONDITIONALS

That simple suggestion in turn has its drawbacks. Most evidently, if that is

what Galen means to say, then how do the syllogistic axioms secure not only

the trustworthiness but also the constitution of their syllogisms?

Corresponding Conditionals 439

Perhaps it will help to ask how Galen came to hit upon, or to convince

himself of the truth of, his metatheorem. All he says on the matter in the

Introduction is that the theorem will become clear to us if we consider

all sorts of syllogisms. So perhaps we should not expect to ﬁnd a general

theoretical argument in favour of the metatheorem: we might rather hope

that a survey of various syllogisms will somehow indicate the truth of the

metatheorem. Such an indication might be discovered if reﬂection on any

and every syllogism did indeed enable us to put a ﬁnger on the appropriate

axioms; and it is hard to imagine how we might ﬁnd ourselves capable of

doing that unless we were able to discover a recipe—or perhaps several

recipes—for cooking up the syllogistic axioms appropriate to any given

syllogism.

Are there such recipes? In his account of Stoic logic, Diogenes Laertius

notices that

of arguments, some are inconclusive and others conclusive. Inconclusive are those in

which the opposite of the conclusion does not conﬂict with the conjunction made

from the premisses.

(vii 77)⁸⁸

Diogenes does not here explain what conclusive arguments are; but the

deﬁnition is readily reconstructed: an argument is conclusive when the

opposite of its conclusion conﬂicts with the conjunction of its premisses.

Now according to an account of conditional propositions which is gener-

ally, and correctly, associated with Chrysippus,

a conditional is sound when the opposite of its consequent conﬂicts with its

antecedent.

(Sextus, PH ii 111)⁸⁹

The deﬁnition of conclusiveness implicit in Diogenes is therefore equivalent

to the thesis that an argument is conclusive when the Chrysippean conditional

formed from the conjunction of its premisses as antecedent and its conclusion

as consequent is true. And that thesis is found a few pages later in the Outlines

of Pyrrhonism:

⁸⁸ τῶν δὲ λόγων οἱ μέν εἰσιν ἀπέραντοι, οἱ δὲ περαντικοί. ἀπέραντοι μὲν ὧν τὸ ἀντικείμενον

τῆς ἐπιφορᾶς οὐ μάχεται τῇ διὰ τῶν λημμάτων συμπλοκῇ.

⁸⁹ οἱ δὲ τὴν συνάρτησιν εἰσάγοντες ὑγιὲς εἶναί φασι συνημμένον ὅταν τὸ ἀντικείμενον τῷ

ἐν αὐτῷ λήγοντι μάχηται τῷ ἐν αὐτῷ ἡγουμένῳ.