Barnes Jonathan. Truth, etc

Подождите немного. Документ загружается.

470 When is a Syllogism not a Syllogism?

Therefore gall and safﬂower are phlegm-like.

That is, or can be construed as, a predicative syllogism in Barbara. It is

not a modal syllogism. None of its component propositions carries a modal

operator. In particular, its conclusion is, as we should expect, the proposition

that

Gall and safﬂower are phlegm-like

and not the problematic proposition that

It is possible that gall and safﬂower are phlegm-like.

Nonetheless, it is certainly meant as an example of a proof which concludes

‘that it possibly holds’.

Galen remarks that ‘it is what is possible or likely or contingent which

holds of the argument’. That sentence is strange, and the text may be corrupt;

but I think that Galen must have meant to say that his syllogism concludes

to a proposition which may—for all that the argument itself shows—hold

contingently rather than necessarily. For when he says that ‘it is what is

possible which holds of the argument’, he cannot mean that the argument

concludes to something of the form ‘It is possible that … ’—for it does not.

Nor can he mean that the argument concludes to a proposition which in

fact holds contingently—for he will proceed to argue that the conclusion in

fact holds of necessity. So—as far as I can see—he can only mean that the

argument does not show that necessarily gall and safﬂower are phlegm-like.

And to be sure, it does not.

Can that really be what Galen means? He distinguishes between proofs

which conclude that of necessity this holds of that and proofs which conclude

that possibly this holds of that. He then singles out a special case of the

latter sort of proof, and says that his gall and safﬂower syllogism is an

example of such a case. So he says of his syllogism, implicitly, that it proves

‘that possibly it holds’; and that strongly suggests that he has in mind a

syllogism the conclusion of which is a problematic proposition. ‘Strongly

suggests’?—Surely it’s more than that: doesn’t Galen state quite explicitly

that the syllogism has a problematic conclusion? Well, I agree that that is the

plain sense of his words. Nonetheless, he can’t in fact have meant what the

plain sense makes him mean; so I infer that he meant something else. What

he meant, if I am right, is, of course, perfectly compatible with al-Farabi’s

report that Galen rejected problematic syllogistic.

A Logical Theorem 471

What, then, is the logical theorem which supposedly enables Galen to

conclude that gall and safﬂower are phlegm-like not contingently but as a

matter of necessity? It is natural to suppose that the theorem must tell us

that when a certain syllogism is valid, then so is another. Galen offers a hint

of the content of the theorem when he claims that the transformation from

possibility to necessity occurs ‘when they follow necessary principles’. So why

not this?

If an argument from a given set of premisses to a given conclusion is

valid, then so too is the argument from the same premisses apodictically

modalized to the same conclusion apodictically modalized.

Or schematically:

If ‘P

,…,P

: therefore Q’ is valid, then ‘Necessarily P

, Necessarily

, … , Necessarily P

: therefore necessarily Q’ is valid.

The suggestion that that is Galen’s logical theorem has two things in its

favour: ﬁrst, it is a genuine theorem—it is true; and secondly, it will do the

work which Galen expects his theorem to do.

For the theorem will transform Galen’s ﬁrst, non-modal argument into

this:

Necessarily gall and safﬂower attract phlegm.

Necessarily what attracts phlegm is phlegm-like.

Therefore necessarily gall and safﬂower are phlegm-like.

In other words, a non-modal syllogism in Barbara is transformed into an

apodictic syllogism in Barbara.

The apodictic syllogism is a proof only if its premisses are true. Galen

says that he has elsewhere proved that attraction is a matter of similarity,

and of similarity in essence. So if gall and safﬂower attract phlegm, then

they do so in virtue of their essence: they attract phlegm essentially—and

therefore necessarily. Again, Galen takes himself to have established the

general thesis that

What attracts X is essentially X-like,

so that, in particular,

What attracts phlegm is essentially phlegm-like.

472 When is a Syllogism not a Syllogism?

And that implies that

What attracts phlegm is necessarily phlegm-like.

And there are the two apodictic premisses for the apodictic Barbara.

Now as a matter of fact I do not think that what Galen had in mind

was that particular logical theorem. But I shall leave Galen in suspense for a

few pages: the logical theorem, whether or not it is Galen’s, has an ancient

history, to which a few paragraphs may be consecrated.

The logical theorem in front of us—which I shall henceforward call the

Modal Theorem—will produce apodictic syllogisms quite generally from

non-modal syllogisms. In particular, it will show, at a single blow, that if

X, Y: therefore Z

is a valid predicative syllogism, then so too is

Necessarily X, Necessarily Y: therefore necessarily Z.

In other words, it will establish the validity of all and only those apodictic

syllogisms which Aristotle acknowledges in the Prior Analytics.

Aristotle in fact spends very little time on apodictic syllogisms—a bare

half-page. He remarks 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.

(APr 29b36–30a2)³⁸

‘Things are … the same’ insofar as

Necessarily X, Necessarily Y: therefore necessarily Z

is a valid predicative syllogism if and only if

X, Y: therefore Z

is a valid predicative syllogism.

True, Aristotle does not say that things are the same: he says that they are

pretty much the same. But the explanation of that qualiﬁcation comes in the

next lines:

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

‘of every’ in the same way. So in most cases the conclusion will be proved necessary

³⁸ 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.

A Logical Theorem 473

by way of conversion, as in the case of holding. But in the middle ﬁgure, when the

universal is afﬁrmative and the particular negative, and in the third ﬁgure, when the

universal is afﬁrmative and the particular negative, the proof will not be the same.

(APr 30a2–9)³⁹

The proofs of apodictic syllogisms will mimic the proofs of their non-modal

cousins—save in the case of Baroco and Bocardo.

For in those two cases, the proofs cannot rely on the principles of

conversion; and Aristotle used a proof by reduction to the impossible. That

is to say, in the case of Baroco, he began his proof like this:

(1) B holds of every A premiss

(2) B does not hold of some C premiss

(3) It is not the case that A does hypothesis

not hold of some C

From those three propositions a contradiction is readily inferred: hence the

negation of the hypothesis, (3), which is the conclusion of Baroco. Aristotle

cannot use the very same proof for apodictic Baroco—for the simple reason

that the hypothesis for apodictic Baroco would have to be not (3) but rather:

(3*) It is not the case that necessarily A does not hold of some C

The upshot is this: the apodictic syllogisms are only ‘pretty much the same’,

not in the sense that they do not quite pair off with non-modal syllogisms,

but rather inasmuch as two of them cannot be proved in the same manner as

the corresponding non-modal syllogisms.

Aristotle does not advance the Modal Theorem in support of his general

thesis, and Alexander does not mention the Theorem. To be sure, the

Theorem does not sufﬁce to prove Aristotle’s general thesis. It will show one

half of the thesis, namely that if

X, Y: therefore Z

is a valid predicative syllogism, then so too is

Necessarily X, Necessarily Y: therefore necessarily Z.

But it does not sufﬁce to show that if

Necessarily X, Necessarily Y: therefore necessarily Z

³⁹ τό τε γὰρ στερητικὸν ὡσαύτως ἀντιστρέφει, καὶ τὸ ἐν ὅλῳ εἶναι καὶ τὸ κατὰ παντὸς

ὁμοίως ἀποδώσομεν. ἐν μὲν οὖν τοῖς ἄλλοις τὸν αὐτὸν τρόπον δειχθήσεται διὰ τῆς ἀντιστρο-

φῆς τὸ συμπέρασμα ἀναγκαῖον ὥσπερ ἐπὶ τοῦ ὑπάρχειν· ἐν δὲ τῷ μέσῳ σχήματι, ὅταν ᾖ τὸ

καθόλου καταφατικὸν τὸ δ᾿ ἐν μέρει στερητικόν, καὶ πάλιν ἐν τῷ τρίτῳ, ὅταν τὸ μὲν καθόλου

κατηγορικὸν τὸ δ᾿ ἐν μέρει στερητικόν, οὐχ ὁμοίως ἔσται ἡ ἀπόδειξις.

474 When is a Syllogism not a Syllogism?

is valid, then so too is

X, Y: therefore Z.

But Aristotle himself does nothing to show the second half of his thesis; and

he would have lost nothing by appealing to the Theorem for the ﬁrst half.

Perhaps Aristotle simply had not thought of the Theorem? Or perhaps he

thought that, despite its apparent evidence, there might in fact be something

dodgy about it? The second of those suggestions vaut le d´etour.

The theorem is closely allied to a thesis which was familiar to ancient

logicians. It is stated thus by Cicero:

If what is anterior in a conditional is necessary, then what is posterior too is necessary.

(fat vii 13)⁴⁰

That is to say,

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

The thesis was not undisputed; for having stated it, Cicero immediately

adds that ‘Chrysippus does not think that this holds in all cases’.⁴¹ And

since anyone who doubts or rejects the thesis will surely doubt or reject the

Theorem, the Theorem must have been found dodgy by at least some ancient

logicians.

Why did Chrysippus reject the thesis? Cicero offers no explanation; but

his remark is generally glossed with the aid of a text in Alexander:

Aristotle proves that it is not possible for the impossible to follow the possible … But

Chrysippus says that nothing prevents the impossible from following the possible.

He says nothing against Aristotle’s proof, but he tries to show that things are not so

by way of certain examples—which are not soundly put together.

(in APr 177.19–27)⁴²

In other words, Chrysippus denied that:

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

One of the two alleged counterexamples which Alexander reports has already

been discussed in a different context. It is this:

If Dio is dead, then this man is dead.

⁴⁰ si … quod primum in conexo est necessarium est, ﬁt etiam quod consequitur necessarium.

⁴¹ quamquam hoc Chrysippo non videtur valere in omnibus.

⁴² ᾿Αριστοτέλης μὲν οὖν ὅτι μὴ οἷόν τέ ἐστι δυνατῷ ἀδύνατον ἕπεσθαι δείκνυσι ...

Χρύσιππος δὲ λέγων μηδὲν κωλύειν καὶ δυνατῷ ἀδύνατον ἕπεσθαι πρὸς μὲν τὴν ὑπ᾿

᾿Αριστοτέλους εἰρημένην δεῖξιν οὐδὲν λέγει, πειρᾶται δὲ διὰ παραδειγμάτων τινῶν οὐχ ὑγιῶς

συγκειμένων δεικνύναι τοῦτο μὴ οὕτως ἔχον.

A Logical Theorem 475

According to Chrysippus, the conditional is true; its antecedent is possible;

and its consequent is impossible.

If that was Chrysippus’ was of dealing with the thesis which Alexander

ascribes to Aristotle, then surely he would have dealt similarly with the thesis

which Cicero reports. For consider the contrapositive of Chrysippus’ example:

If this man is not dead, then Dio is not dead.

Chrysippus must surely hold that the conditional is true, the antecedent

necessary, and the consequent not necessary. If Chrysippus is right, then the

thesis which Cicero reports is false; and if the thesis is false, then the Modal

Theorem which partners it is also false.

Alexander argues at length against Chrysippus; but I shall not follow him.

For I doubt if Dio’s death and the logical behaviour of the demonstrative

phrase ‘this man’ have any relevance to Cicero’s remark. At any rate, in the

Ciceronian context Chrysippus’ maverick views on demonstrative pronouns

are not in the least to the point, and it is likely that Cicero has something

quite different in mind.

The truth or falsity of the thesis which Cicero reports—of the thesis that

If necessarily P, and if P then Q, then necessarily Q

—depends in part on the sense of its conditional component ‘If P then Q’.

The point is trivial. Alexander took it; or rather, he took it in the case of the

parallel thesis about possibility:

Aristotle shows that it cannot be the case that something impossible follows what

is possible but that necessarily what the impossible follows is itself impossible—in

the case of every necessary implication. A necessary implication is not an occasional

implication, but rather one in which the consequent always follows what is taken as

the antecedent. For the conditional

If Alexander exists, Alexander talks

If Alexander exists, he has such-and-such an age

is not true—not even if he is in fact of such-and-such an age when the proposition

is said.

(in APr 175.34–176.6)⁴³

⁴³ … δείκνυσιν ὅτι μὴ οἷόν τε δυνατῷ τι ἀδύνατον ἀκολουθεῖν, ἀλλ᾿ ἀνάγκη ἀδύνατον εἶναι

ᾧ τὸ ἀδύνατον ἀκολουθεῖ, ἐπὶ πάσης ἀναγκαίας ἀκολουθίας. ἔστι δὲ ἀναγκαία ἀκολουθία οὐχ

ἡ πρόσκαιρος, ἀλλὰ ἐν ᾗ ἀεὶ τὸ ἑπόμενον ἕπεσθαι ἔστι τῷ τὸ εἰλημμένον ὡς ἡγούμενον εἶναι.

οὐ γὰρ ἀληθὲς συνημμένον τὸ εἰ ᾿Αλέξανδρος ἔστιν, ᾿Αλέξανδρος διαλέγεται, ἢ εἰ ᾿Αλέξανδρος

ἔστι, τοσῶνδε ἐτῶν ἐστι, καὶ εἰ εἴη ὅτε λέγεται ἡ πρότασις τοσούτων ἐτῶν.—‘ἑπόμενον’is

Mueller’s correction of the received ‘εἰλημμένον’.

476 When is a Syllogism not a Syllogism?

So that

Aristotle shows that it cannot be the case that an impossible follows a possible inas-

much as in a true conditional the consequent must follow the antecedent by necessity.

(in APr 177.19–21)⁴⁴

What Alexander says is correct for the thesis that

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

It is also correct for the thesis that

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

That is to say, the two theses are true when ‘If P, then Q’ is construed

as a necessary implication (assuming, of course, that the modal operators

‘possibly’ and ‘necessarily’ are given a normal interpretation).

There are, and in antiquity there were, other ways of construing conditional

propositions. Philo of Megara, for example, proposed that a conditional is

true if and only if it is not the case that both its antecedent and the negation

of its consequent are true. If the thesis which Cicero reports is construed in

that Philonian fashion, then it is equivalent to this:

If necessarily P, and if it is not the case both that P and that not Q, then

necessarily Q.

And that is false. (To see that it is false, take take any necessary truth for ‘P’

and any contingent truth for ‘Q’. Thus

Necessarily no married man is a bachelor

and also

Aristotle was married.

Hence

It is not the case that both necessarily no married men are bachelors and

also Aristotle wasn’t not married.

But it is false that

Necessarily Aristotle was married.)

The Ciceronian context shows—or at the very least, makes it stupendously

probable—that when Chrysippus denied that the thesis ‘holds in all cases’

he meant that it does not hold for every type of conditional connec-

tion; and—whatever we may make of Dio’s death—on that point he was

quite right.

⁴⁴ ᾿Αριστοτέλης μὲν οὖν ὅτι μὴ οἷόν τέ ἐστι δυνατῷ ἀδύνατον ἕπεσθαι δείκνυσι διὰ

τοῦ δεῖν μὲν ἐν τῷ ἀληθεῖ συνημμένῳ ἐξ ἀνάγκης ἕπεσθαι τὸ λῆγον τῷ ἡγουμένῳ.

A Logical Theorem 477

So the passage in Cicero’s On Fate does not prove that Chrysippus thought

that the Modal Theorem was false, or even dodgy. It shows only that he

thought that it must be correctly understood. And who would have denied

that?

But Dio is still dead; and if the case of this man’s death does not

explain anything in the Ciceronian text, it may nevertheless be pertinent to

Chrysippus’ attitude to the Theorem. For the case was taken by Chrysippus

to show that there are exceptions to the thesis that

If possibly P, and if P then Q, then possibly Q;

and he can hardly have failed to see that the case must show equally that there

are exceptions to the thesis reported by Cicero, namely:

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

So if he also thought that

If Dio is dead, then this man is dead

is a necessary conditional, then he must have rejected the Modal Theorem—or

rather, then he could not coherently have accepted the Modal theorem. Did

he take that conditional to be necessary? Plainly, the sentence

If Dio is dead, then this man is dead

may be used to express contingent truths (and contingent falsehoods)—

suppose that in it ‘this man’ refers to Theo rather than to Dio. But

in Chrysippus’ example, ‘this man’ is, of course, used to refer to Dio; and

hence—or so it is plausible to think, or at least to think that Chrysippus would

have thought—it is not possible for Dio to be dead and this man still alive.

However that may be, and whether or not Chrysippus found—or should

have found—the Modal Theorem dubious, there is not the slightest reason

to ascribe any similar doubts to Aristotle. Alexander asserts that Aristotle has

shown that

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

This is what Aristotle says:

First, let it be said that if, if A is the case then necessarily B is the case, then if A is

possible then necessarily B too will be possible.

(APr 34a5–7)⁴⁵

He then offers an argument for his claim—and he uses the claim to validate

certain modal syllogisms.

⁴⁵ πρῶτον δὲ λεκτέον ὅτι εἰ τοῦ Α ὄντος ἀνάγκη τὸ Β εἶναι, καὶ δυνατοῦ ὄντος τοῦ Α

δυνατὸν ἔσται καὶ τὸ Β ἐξ ἀνάγκης.

478 When is a Syllogism not a Syllogism?

That being so, it is hardly likely that Aristotle—or Alexander—would

have found anything dodgy about the thesis that

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

Moreover, although I do not ﬁnd the thesis in the text of the Analytics,the

allied Modal Theorem is in fact inconspicuously announced there:

IfCofDandDofF,thennecessarilyCofF;andifeachofthetwoispossible,

then the conclusion too is possible. So if you were to make A the premisses and B

the conclusion, it will follow not only that if A is necessary then B too is necessary

but also that if it is possible it is possible.

(APr 34a19–24)⁴⁶

Suppose that

A: therefore B

is a syllogism. Then so too are

Necessarily A: therefore necessarily B

and

Possibly A: therefore possibly B.

That is what Aristotle says: so he recognized the Modal Theorem, and

he might perfectly well have used it in order to establish his apodictic

syllogisms.

It is more than time to return to Galen and to his Simple Drugs.The

Modal Theorem is a good theorem and a powerful theorem. It was stated

by Aristotle; and although he did not make use of it, he could properly have

done so. But what about Galen? How could he have made use of it? The

theorem tells you how to generate an apodictic argument from a non-modal

argument. Yet according to al-Farabi, Galen had no time for any modal

syllogisms—he thought that hyparctic or non-modal syllogisms were all the

syllogisms a scientist needed and hence all the syllogisms that there are.

al-Farabi is reporting what Galen had said in On Proof ;soperhapsGalen

changed his mind between writing On Proof and compiling Simple Drugs?

Hardly: after all, Simple Drugs refers explicitly to the earlier work for a proof

of the logical theorem, and you will not believe that On Proof both rejected

modal syllogistic and also offered a proof of a theorem which infers modal

from non-modal syllogisms.

⁴⁶ εἰ γὰρ τὸ Γ κατὰ τοῦ ∆, τὸ δὲ ∆ κατὰ τοῦ Ζ, καὶ τὸ Γ κατὰ τοῦ Ζ ἐξ ἀνάγκης· καὶ

εἰ δυνατὸν ἑκάτερον, καὶ τὸ συμπέρασμα δυνατόν. ὥσπερ οὖν εἴ τις θείη τὸ μὲν Α τὰς

προτάσεις, τὸ δὲ Β τὸ συμπέρασμα, συμβαίνοι ἂν οὐ μόνον ἀναγκαίου τοῦ Α ὄντος ἅμα καὶ

τὸ Β εἶναι ἀναγκαῖον, ἀλλὰ καὶ δυνατοῦ δυνατόν.

A Logical Theorem 479

Then is al-Farabi simply mistaken? Did Galen reject problematical syl-

logisms but keep his arms round apodictic syllogistic? Well, you may be

attracted by such a view if you are persuaded that Galen’s logical theorem is

the Modal Theorem; but in fact there are pretty good reasons for thinking

that it was not.

First, the Modal Theorem, as I have said, has universal scope: it will

make an apodictic syllogism of any non-apodictic syllogism. Galen’s logical

theorem, on the other hand, is supposed to have a circumscribed domain of

application; for Galen explicitly says that in some cases—and he evidently

means in some cases only—the theorem allows us to move from a possible

conclusion to a necessary conclusion. If the logical theorem is identiﬁed as the

Modal Theorem, then we shall have to suppose that Galen somehow came to

convince himself that its scope was restricted to a certain class of syllogism.

Secondly, I said that we might readily suppose the logical theorem to state

something of the form: ‘When such-and-such a syllogism is valid, then so too

is such-and-such another syllogism.’ (The Modal Theorem has that form.) But

although the supposition is readily made, there is no direct warrant for it in

Galen’s text. Galen does not speak of transforming one syllogism into another:

he speaks of showing that the conclusion of a given proof not only holds but

holds of necessity. There is no explicit mention of any second syllogism, and

perhaps there is no need to insert a second syllogism into the text.

Earlier, I argued that when Galen says that ‘some proofs conclude that … it

possibly holds’, we should forget the plain sense of his words and take him

to mean not that some proofs conclude to propositions of the form ‘Possibly

so-and-so’ but rather that some proofs conclude to ‘So-and-so’, where—for

all we yet know—it may be contingently the case that so-and-so. That is to

say, the proof of possibility is simply this:

Gall and safﬂower attract phlegm.

What attracts phlegm is phlegm-like.

Therefore gall and safﬂower are phlegm-like.

It is a proof of possibility insofar as, for all that has been thus far argued, the

conclusion of the syllogism may be contingently rather than necessarily true.

Now if that is so, then by the same token, a proof which concludes that ‘of

necessity this holds of that’ ought to conclude not to ‘Necessarily so-and-so’

but rather to ‘So-and-so’, where in fact, as we can show, it is necessary that

so-and-so. In other words, the proof of necessity will be exactly the same as

the proof of possibility, namely: