Barnes Jonathan. Truth, etc

Подождите немного. Документ загружается.

500 When is a Syllogism not a Syllogism?

So either there are syllogisms in which the conclusion repeats a premiss or

else the synthetic theorem must be abandoned or modiﬁed. An apparently

innocuous stipulation has unanticipated effects. Tinker with a system here,

and it is likely to go out of kilter there. To be sure, the unanticipated effect

in the case before us is not a catastrophe; for it is easy enough to modify the

synthetic theorem. But modiﬁcation is one of those activities which goes on

and on—you can never be sure that there isn’t another unexpected difﬁculty

which requires another modiﬁcation. Is it worth the fuss? Why not allow

that some syllogistic forms are useless? The stipulation that a syllogism be

useful, and that it be capable of delivering the probative goods, is merely

that—a stipulation. Nothing is lost by dropping it: you will not ﬁnd yourself

suddenly producing non-probative arguments as if they were proofs.

DUPLICATED ARGUMENTS

Alexander does not discuss repetitive arguments which are predicative in

structure. He thinks that the phrase ‘something other’ in Aristotle’s deﬁnition

was designed to outlaw certain argument-forms which the Stoic logicians

would later patronize. If we take Alexander’s words woodenly, then what he

says is scarcely credible; for there is no reason to believe that Aristotle—or

any of his contemporaries—had dreamed up duplicated arguments, and his

deﬁnition of the syllogism cannot have been intended to outlaw a set of

arguments of which he had never heard. But Alexander is right if we construe

his words more generously; for presumably Aristotle deliberately excluded

any arguments, of whatever form they might be and whencesoever they might

emanate, in which the conclusion repeated one of the premisses.

Aristotle’s deﬁnition of the syllogism excludes repetition. We have no rival

deﬁnition from the Stoics—or from anyone else. Sextus says that, according

to the logicians, an argument is a system of premisses and conclusions; and

he further explains—in a passage I have already quoted—that

of arguments, some are concludent and others non-concludent. An argument is

concludent when the conditional which starts from the conjunction of its premisses

and ends in its conclusion is sound.

(PH ii 137)

But not all concludent arguments are syllogisms, and Sextus offers no account

of how to differentiate syllogistic from non-syllogistic arguments—indeed,

Duplicated Arguments 501

he does not notice a distinction between syllogistic and non-syllogistic

arguments. Diogenes Laertius tells us that, according to the Stoics,

an argument is a system constituted from assumptions and conclusion, and a

syllogism is a syllogistic argument so constituted.

(vii 45)⁶²

But that is perfectly unilluminating.

True, a later passage in Diogenes’ account of Stoic logic seems more

promising.⁶³

Of valid arguments, some share their name with the genus and are called valid,

whereas others are called syllogistic. Syllogistic are those arguments which either are

unproved or else reduce to the unproveds by virtue of one or more of the themata.

(vii 78)⁶⁴

Syllogistic arguments are presumably the same things as syllogisms; so we

may infer that an argument, according to the Stoics, is a syllogism if and only

if either it is one of the unproveds or else it can be proved valid on the basis

of the unproveds and by the aid of the ‘themata’ or rules of reduction.

No doubt that is true—but it is not an explanation of what the word

‘syllogism’ means. It says which arguments are in fact syllogisms: it does not

saywhatitisaboutthemwhichmakesthemsyllogistic.Itisasthoughyou

weretosaythatamanisamaﬁosoifandonlyifheiseitheraheadofoneof

the Five Families or a henchman of one of the heads.

In the light of that meagre evidence, it is difﬁcult to determine how the

Stoic conception of a syllogism differed from the Aristotelian conception (if

indeed it differed at all). Nevertheless, scholars generally suppose that the

Stoic notion differed from the Aristotelian in at least one signiﬁcant particular;

for whereas Aristotle required that the conclusion of a syllogism be other than

its premisses, surely the Stoics did not—on the contrary, they welcomed

repetitive syllogisms. Perhaps they did; but it should be said that even that

orthodox and unexciting claim goes beyond the texts at our disposal. For

although Alexander says that the Stoic logicians studied certain arguments

which are not syllogisms and which therefore—in his view—should not fall

⁶² εἶναι δὲ τὸν λόγον αὐτὸν σύστημα ἐκ λημμάτων καὶ ἐπιφορᾶς· τὸν δὲ συλλογισμὸν

λόγον συλλογιστικὸν ἐκ τούτων.

⁶³ Susanne Bobzien reminded me of the passage.

⁶⁴ τῶν δὲ περαντικῶν λόγων οἱ μὲν ὁμωνύμως τῷ γένει λέγονται περαντικοί· οἱ δὲ

συλλογιστικοί. συλλογιστικοὶ μὲν οὖν εἰσιν οἱ ἤτοι ἀναπόδεικτοι ὄντες ἢ ἀναγόμενοι ἐπὶ τοὺς

ἀναποδείκτους κατά τι τῶν θεμάτων ἤ τινα.

502 When is a Syllogism not a Syllogism?

under the eye of the serious logician, he does not explicitly say that the Stoics

treated those arguments as syllogisms.

However that may be, among the Stoic arguments which sinned by

repetition were the so-called duplicated arguments, of which the stock

example is this:

If it is day, it is day.

But it is day.

Therefore it is day.

No doubt it is perfectly evident that no argument of that form can prove its

conclusion, and hence—on the Alexandrian view—that the form is not a

syllogistic form.

Nonetheless, in the commentary on the Analytics Alexander offers a suite

of seven reasons for thinking that no such argument is a syllogism. The most

interesting of them runs like this:

Again, if it is non-syllogistic to co-assume the consequent of a conditional, and if

in conditionals of this sort the antecedent is the same as the consequent, then the

co-assumption is no more of the antecedent than of the consequent.

(in APr 20.13–16)⁶⁵

Iarguethus:

If it is day, then it is day

It is day

Therefore: it is day

What is my second premiss? Why, it is the consequent of my ﬁrst premiss; so

that my argument is a case of afﬁrming the consequent—it has the general

form:

If P, then Q

Therefore P

Circumscriptively:

From a conditional and its consequent, there is a valid inference to its

antecedent.

And that form is certainly non-syllogistic—after all, it is invalid.

⁶⁵ ἔτι τε εἰ ἡ τοῦ ἑπομένου πρόσληψις ἐν τοῖς συνεχέσιν ἀσυλλόγιστος, ἐν δὲ τοῖς τοιούτοις

συνεχέσι ταὐτὸν τὸ ἡγούμενον τῷ ἑπομένῳ, ἡ πρόσληψις οὐ μᾶλλον τοῦ ἡγουμένου ἢ τοῦ

ἑπομένου γίνεται.

Duplicated Arguments 503

Alexander is right to this extent: the duplicated argument which he cites

does have the invalid form which we associate with the vice of afﬁrming the

consequent. But so what? Alexander invites us to infer that the duplicated

argument is non-syllogistic on the grounds that it has an invalid form. But—as

I have already recalled—every valid argument has numerous forms, among

them invalid forms. Every predicative syllogism has invalid forms—after all,

every predicative syllogism has the form

P, Q : therefore R.

That does not show that no predicative argument is not a syllogism, still less

that none is valid. An argument is valid provided that at least one of the

several forms which it has is a valid form, and it is syllogistic provided that at

least one of its several forms is a syllogistic form.

In that case, Alexander’s duplicated argument is a valid argument. For it

has the following form, among others:

If P, then Q

Therefore Q

That is to say, the argument is a special case of the Stoic ﬁrst unproved. So if

Alexander rejects the argument as non-syllogistic, then presumably he must

reject the Stoic ﬁrst unproved as non-syllogistic.

Well, in fact Alexander does not think that ﬁrst unproved arguments are

syllogisms. Indeed, he does not think that any hypothetical arguments are

syllogisms. When he comments upon Aristotle’s ‘syllogisms based on a hypo-

thesis’, which he identiﬁes with hypothetical arguments, he has this to say:

As to Aristotle’s expression, it is worth noting that he does not say that the hypothet-

icals are syllogisms without qualiﬁcation but he does say that they conclude … and that

they are syllogisms based on a hypothesis (the phrase being taken as a whole). … They

will be simply concludent when the co-assumption is not posited by way of a

syllogism, and hypothetical syllogisms when they have a co-assumption taken by way

of a syllogism. Thus for Aristotle things are the opposite of what the more recent

thinkers claim: hypothetical arguments are concludent but not syllogisms, as we have

already said, whereas predicative arguments are syllogisms.

(in APr 390.9–18)⁶⁶

⁶⁶ ἐκεῖνο δὲ ἄξιον ἐκ τῆς λέξεως ἐπισημήνασθαι ὅτι συλλογισμοὺς μὲν ἁπλῶς οὐ λέγει τοὺς

ὑποθετικοὺς εἶναι, περαίνειν μέντοι αὐτοὺς λέγει, ... καὶ τὸ ὅλον ἐξ ὑποθέσεως συλλογισμούς·

... εἶεν δ᾿ ἂν περαίνοντες μὲν μόνον ὧν ἡ πρόσληψις οὐ διὰ συλλογισμοῦ τίθεται, ἐξ ὑποθέσεως

504 When is a Syllogism not a Syllogism?

‘As we have already said’: earlier Alexander has explained that predicative

arguments

are syllogisms without qualiﬁcation: the hypotheticals are not syllogisms without

qualiﬁcation but hypothetical syllogisms (the phrase being taken as a whole). For

predicative arguments have no need of hypotheses in order to show the point at issue

(that is why they are syllogisms without qualiﬁcation, being sufﬁcient in themselves),

whereas hypothetical arguments show nothing without predicatives.

(in APr 265.19–23)⁶⁷

The phrase ‘hypothetical syllogism’, like the phrase ‘rhetorical syllogism’,

does not designate a kind or species of syllogism: hypothetical syllogisms are

not syllogisms at all. For a hypothetical syllogism cannot prove anything—or

at least, it cannot prove anything unless it is aided by a predicative syllogism.

No ﬁrst unproved is a syllogism. Every ﬁrst unproved is a concludent or

valid argument. A ﬁrst unproved is a hypothetical syllogism if it is embedded

in a longer argument—if its second premiss is the conclusion of a genuine

syllogism. But hypothetical syllogisms are not syllogisms. They are false

noses. Some Stoics—or at any rate, some of the younger logicians—had

claimed that predicative arguments are not syllogisms and that hypothetical

arguments are syllogisms. The truth—Alexander maintains—is the other

way about: predicative arguments are syllogisms and hypothetical arguments

are not syllogisms.

Alexander’s reason for denying that hypothetical syllogisms are syllogisms

will not stand examination; and as for the young Turks who denied that

predicative syllogisms are syllogisms, we do not know how they urged their

case, nor is it easy to imagine any persuasive advocacy on their behalf.

However that may be, insofar as Alexander holds that hypothetical argu-

ments in general are not syllogisms, he has no need to argue that his duplicated

argumentinparticularisnotasyllogism.Nevertheless,hedoesso.Doesthatsig-

nifythat, despite hisofﬁcialline, he issometimes prepared toaccept that certain

δὲ συλλογισμοὶ οἱ τὴν πρόσληψιν ἔχοντες εἰλημμένην διὰ συλλογισμοῦ· ὥστε ἀνάπαλιν κατ᾿

αὐτόν ἢ ὡς οἱ νεώτεροι ἀξιοῦσιν, οἱ ὑποθετικοὶ λόγοι περαντικοὶ μὲν οὐ συλλογισμοὶ δέ,

ὥσπερ φθάνομεν εἰρηκότες, συλλογισμοὶ δ᾿ οἱ κατηγορικοί.

⁶⁷ διὸ καὶ ἁπλῶς ἐκεῖνοι συλλογισμοί, οἱ δ᾿ ἐξ ὑποθέσεως οὐχ ἁπλῶς, ἀλλὰ τὸ ὅλον τοῦτο

ἐξ ὑποθέσεως συλλογισμοί· οἱ μὲν γὰρ κατηγορικοὶ οὐδέν προσδέονται πρὸς τὸ δεῖξαι τὸ

προκείμενον τῶν ὑποθέσεων (διὸ καὶ ἁπλῶς συλλογισμοὶ ἀρκοῦντες αὑτοῖς), οἱ δὲ ὑποθετικοὶ

χωρὶς τούτων οὐδὲν δεικνύουσιν.

Duplicated Arguments 505

hypothetical arguments—for example, non-duplicated ﬁrst unproveds—are

genuine syllogisms? Or should we rather infer that when he argues against his

duplicated argument he intends to show not merely that it is not a syllogism

but that it is not even a hypothetical syllogism—better, that it cannot even

function as a hypothetical syllogism?

The latter option is seductive; and although it must be admitted that there

is no whiff of it in Alexander’s text, it is certainly an option which he might

have taken. For consider the following form of argument:

A holds of every B; B holds of every C; if A holds of every C, then P:

therefore P.

That form can be shown to be valid, thus:

(1) A holds of every B premiss

(2) B holds of every C premiss

(3) If A holds of every C, then P premiss

(4) A holds of every C 1, 2, Barbara

(5) P 3, 4, ﬁrst unproved

An argument of that sort might, according to Alexander, be a proof that

P (or perhaps rather a hypothetical proof that P). And the hypothetical

argument which forms part of it—steps (3) to (5)—may therefore function

as a hypothetical syllogism.

But not every argument of that sort could be a proof that P, or even

a hypothetical proof that P. In particular, an argument in which the ﬁrst

unproved is duplicated cannot be a proof. For example, you couldn’t prove

that A holds of every C by this argument:

(1) A holds of every B premiss

(2) B holds of every C premiss

(3) If A holds of every C, then A holds of every C premiss

(4) A holds of every C 1, 2, Barbara

(5) A holds of every C 3, 4, ﬁrst unproved

More generally, nothing of the form

X; Y; if Z then Z: therefore Z

could ever be a proof that Z, nor even a hypothetical proof that Z.

Why not?—Because one of the premisses of any such argument must be

redundant, and syllogisms may not have redundant premisses. I cite again

Aristotle’s deﬁnition:

506 When is a Syllogism not a Syllogism?

A syllogism is a saying in which, certain items being laid down, something other than

the items laid down comes about by necessity from the fact that they are the case.

(APr 24b18–20)

The conclusion must come about ‘from the fact that they are the case’:

Alexander comments that

by this addition he will exclude arguments which contain any redundant element.

(in APr 22.30–23.1)⁶⁸

And in his commentary on the Topics he explains that

in arguments in which a redundant proposition has been assumed, the conclusion is

not inferred by way of the items laid down inasmuch as even if that proposition is

removed it is still possible to make the same inference.

(in Top 13.29–31)⁶⁹

In other words, if

,…,P

: therefore Q

is a syllogism, then

,…,P

n+1

: therefore Q

is not a syllogism. The notion of redundancy requires some reﬁnement; and

it may be wondered if Alexander’s interpretation of the pertinent clause in

the Aristotelian deﬁnition is not too elaborate; but it will be allowed that, at

bottom, Alexander must be right. Of course, redundancy does not make an

argument invalid. But surely it makes it non-probative.

However that may be, Alexander certainly rejects duplicated arguments

as non-syllogistic: not every argument which, from a conditional assertible

and its antecedent, infers the consequent of the conditional is thereby a

syllogism—not even a hypothetical syllogism, or an argument capable of

functioning as a hypothetical syllogism. But of course, just as it was possible to

keep Barbara while rejecting as non-syllogistic some of its apparent instances,

so too it is with the ﬁrst unproved. The apparent instances of Barbara were not

genuine instances inasmuch as Barbara is a syllogistic form and predicative

syllogisms—according to the revised deﬁnition—have three distinct terms.

⁶⁸ παραιτοῖντο δ᾿ ἂν διὰ τῆς προσθήκης ταύτης καὶ οἱ παρέλκον τι προσκείμενον ἔχοντες.

⁶⁹ ἐν οἷς γὰρ λόγοις πρότασίς τις εἴληπται παρέλκουσα, οὗτοι οὐ διὰ τῶν κειμένων

συνάγουσι τὸ συμπέρασμα, εἴ γε καὶ ἀφαιρεθείσης ἐκείνης ἔτι οἷόν τε τὸ αὐτὸ συνάγεσθαι.

Disjunctive Arguments 507

In the same way, it might be urged that the apparent instances of the ﬁrst

unproved are not genuine instances inasmuch as a ﬁrst unproved is a syllogistic

form—a hypothetical syllogistic form; and such forms—or so we shall expect

a reﬁned deﬁnition to require—must have two distinct assertibles.

As a ﬁrst shot, you might simply stipulate that in a ﬁrst unproved the

antecedent of the conditional premiss be different from its consequent, thus:

Fom a conditional in which the antecedent differs from the consequent,

together with its antecedent, there is an inference to its consequent.

But that is not quite good enough to meet Alexander’s requirements.

For a ﬁrst unproved can only function as a hypothetical syllogism if the

antecedent of its constitutive conditional proposition can be the conclusion

of a syllogism—that is to say, of a predicative syllogism. So we must think of

something along these lines:

From a conditional in which the antecedent both differs from the con-

sequent and has a predicative form, together with the antecedent, there is

an inference to the consequent.

Such a stipulative redeﬁnition will have consequences; and some of them are

likely to be tiresome. No Stoic logician had any particular reason for teasing

out those consequences and avoiding the unwanted implications; and no

Peripatetic logician was likely to bother his head over such matters.

DISJUNCTIVE ARGUMENTS

Alexander’s attitude to duplicated instances—or apparent instances—of the

ﬁrst Stoic unproved is scarcely surprising; and we shall expect him to take a

similar attitude in the case of the other Stoic unproveds, each of which has a

duplicated cousin. To the second unproved there answers

If P, P; not P: therefore not P.

Forthethirdunprovedwemaysetdown:

Not both P and not P; P: therefore P.

The fourth and the ﬁfth unproveds suggest:

Either P or not P; P: therefore P.

and

Either P or not P; not P: therefore not P.

Those four schemata are instances of the last four unproveds in just the way

in which the schema of Alexander’s duplicated argument is an instance of a

508 When is a Syllogism not a Syllogism?

ﬁrst unproved. (And they can be turned into merely specious instances in the

same way as the duplicated form could be turned into a specious instance of

the ﬁrst unproved.)

Alexander says nothing about the ﬁrst two of those four schemata. But

he discusses the last two. Each, he says, is a ‘disjunctive syllogism based

on a contradiction’. The word ‘syllogism’ is not a slip. For, contrary to

every expectation, Alexander holds that disjunctive arguments based on a

contradiction are syllogistical—that is to say, they are, or they may function

as, hypothetical syllogisms. This is what he says in the commentary on the

Analytics:

A disjunctive syllogism based on a contradiction does not infer its conclusion insofar

as it is the same as the re-assumption (or the co-assumption, as the more recent

thinkers say). For if you say

Either it is day or it is not day,

and then co-assume one of the items in the disjunction—either the negative, ‘But it

is not day’, or the afﬁrmative, ‘It is day’—you have as conclusion either ‘Therefore it

is not day’ or ‘Therefore it is day’, which seem to be the same as the co-assumptions,

which were either ‘But it is not day’ or ‘But it is day’. But the conclusion is inferred

not insofar as it is the same as the co-assumption but rather insofar as it is opposite

to the other item in the disjunction. It happens that, in such syllogisms, the opposite

of that item is the same as the co-assumption. But there is a vast difference between

taking the conclusion as being primarily the same as one of the premisses and taking

it as something different which then happens to be the same as it.

(in APr 19.3–15)⁷⁰

Alexander’s illustrative arguments are these:

Either it is day or it is not day. Either it is day or it is not day.

It is day. It is not day.

Therefore it is day. Therefore it is not day.

⁷⁰ ὁ γὰρ ἐξ ἀντιφάσεως διαιρετικὸς συλλογισμὸς οὐχ ὡς ταὐτὸν τῷ μεταλαμβανομένῳ

ἤ, ὡς οἱ νεώτεροί φασι, προσλαμβανομένῳ τὸ συμπέρασμα ἐπιφέρει· ὁ γὰρ λέγων ἤτοι

ἡμέρα ἐστὶν ἢ οὐκ ἔστιν ἡμέρα, εἶτα προσλαμβάνων τὸ ἕτερον τῶν ἐν τῷ διαιρετικῷ, ἢ

τὸ ἀποφατικὸν τὸ ἀλλὰ μὴν οὐκ ἔστιν ἡμέρα ἢ τὸ καταφατικὸν τὸ ἡμέρα ἐστίν, ἔχει μὲν

συναγόμενον ἢ τὸ οὐκ ἄρα ἡμέρα ἐστίν ἢ τὸ ἡμέρα ἄρα ἐστίν, ὃ δοκεῖ ταὐτὸν εἶναι τῷ

προσειλημμένῳ, ἤτοι τῷ ἀλλὰ μὴν οὐκ ἔστιν ἡμέρα ἢ τῷ ἀλλὰ μὴν ἡμέρα ἐστίν· οὐ μὴν ὡς

ταὐτὸν ὂν αὐτῷ ἐπιφέρεται, ἀλλ᾿ ὡς ἀντικείμενον τῷ ἑτέρῳ τῶν ἐν τῷ διαιρετικῷ· συμβαίνει

δὲ τὸ ἐκείνῳ ἀντικείμενον ἐν τοῖς τοιούτοις συλλογισμοῖς ταὐτὸν γίνεσθαι τῇ προσλήψει.

πάμπολυ δὲ διαφέρει τὸ προηγουμένως ταὐτὸν τῶν κειμένων τινὶ λαβεῖν τὸ συμπέρασμα ἢ

λαβεῖν μὲν αὐτὸ ὡς ἄλλο συμπεσεῖν δὲ αὐτῷ τὸ ταὐτὸν αὐτῷ γενέσθαι.

Disjunctive Arguments 509

Surely those arguments could not be used to prove that it is day or that it is

not day? Surely, no argument of the forms

Either P or not-P Either P or not-P

PNot-P

Therefore P Therefore not-P

could ever be used to prove anything? And for precisely the reason that in

each case the conclusion is the same as one of the premisses.

Why in heaven does Alexander want to defend such arguments? And how

on earth can he do so? I shall address those questions as they apply to the

repetitive version of the fourth unproved: the case of the ﬁfth unproved is

entirely parallel and needs no separate discussion.

Alexander does not explain why he defends the repetitive fourth unproved;

but I suppose that he has his mind on a remark in the Posterior Analytics:

That everything is said or denied is assumed by proofs by way of the impossible—and

that not always universally but rather so far as is sufﬁcient, and it is sufﬁcient if it is

limited to the genus. (By to the genus I mean the genus about which the proofs are

being produced.)

(APst 77a22–25)⁷¹

The phrase ‘Everything is said or denied’ is Aristotle’s way of introducing

the Law of Excluded Middle; and he means to claim that propositions of

the form ‘Either P or not P’ will sometimes appear in proofs—in proofs by

reduction to the impossible. In order to prove that P, you hypothesize that

not-P. From the hypothesis you derive an impossibility, thereby establishing

that you cannot accept that not-P. But either P or not-P. Hence P.

I have said earlier that ancient accounts of the workings of reduction to

the impossible are surprisingly unsatisfactory; and the account implicit in the

Posterior Analytics is curious enough. But it is Aristotelian—and presumably it

explains why Alexander feels himself obliged to approve of certain arguments

which, on general grounds, you would expect him to reject. Not that he need

feel obliged to accept the arguments as syllogisms. Indeed, he should not do

so, for they cannot be used on their own to formulate proofs. But he must

accept them as hypothetical syllogisms.

⁷¹ τὸ δ᾿ ἅπαν φάναι ἢ ἀποφάναι ἡ εἰς τὸ ἀδύνατον ἀπόδειξις λαμβάνει, καὶ ταῦτα οὐδ᾿ ἀεὶ

καθόλου, ἀλλ᾿ ὅσον ἱκανόν, ἱκανὸν δ᾿ ἐπὶ τοῦ γένους. λέγω δ᾿ ἐπὶ τοῦ γένους οἷον περὶ ὃ γένος

τὰς ἀποδείξεις φέρει.