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490 When is a Syllogism not a Syllogism?
certain items laid down something other results’—then we might reasonably
have set aside the strict implications of the word ‘other’. But what he actually
says is that the conclusion is ‘something other than the items laid down’.
And the expression ‘than the items laid down’ show that the word ‘other’ was
conscientiously chosen and that Aristotle intended thereby to exclude certain
arguments.
So a syllogism may not conclude to one of its own premisses—it may
not be repetitive, as I shall say. According to Alexander, repetitive arguments
are not syllogisms inasmuch as they have no syllogistic utility. Why not?
We might expect a speedy answer. Repetitive arguments cannot be used to
prove anything, and for that reason they are not syllogisms. They cannot
be used to prove anything because no argument can prove one of its own
premisses. The premisses of a proof, as Aristotle put it, must be prior to
the conclusion. Nothing can be prior to itself. So the conclusion of a proof
cannot be identical with one of its premisses. And quite apart from any
Aristotelian reflections on priority, it must seem blindingly evident that a
repetitive argument is no way to prove anything. The repetitive argument
quoted by Alexander—
If it is day, it is light.
But it is day.
Therefore it is day.
—is not and cannot be a proof that it is day. The useless sort of argument
which that example illustrates is characterized by the following schema:
If P, then Q; P: therefore P
Circumscriptively:
From a conditional and its antecedent, there is a valid inference to its
antecedent.
No argument of that form could serve as a proof.
That would be enough for Galen. In principle, it is enough for Alexander.
But in fact Alexander says rather more:
We shall see that this species of argument is not useful if we run through the species
of arguments and ask in which of them it might be appropriate to infer one of
the items laid down. In probative arguments?—But such arguments try to reveal
what is unclear by way of what is evident and familiar, and what is posterior by
Some Predicative Problems 491
way of what is prior. In dialectical arguments?—But such arguments try to show
something which the interlocutor is unwilling to concede, and to do so by way of
reputable items and items which he concedes, thus bringing him to a contradiction.
In eristic arguments?—But they too are supposed to bring the answerer either to a
contradiction or to an apparent contradiction by way of what he grants. They do
not conclude to what he grants: they conclude by way of what he grants to what he
is unwilling to grant, so that it is clear that they will conclude to something other
that what is granted. Now if a genus exists in its own species, and if the syllogism is
a genus of its species, and if in none of the species is the conclusion the same as a
premiss, then it will not be so either in syllogism in general.
(in APr 18.22–19.3)⁵⁴
All species of syllogism require a certain asymmetry between premisses and
conclusion; and those asymmetries make it impossible for a syllogism to
conclude to one of its premisses.
Alexander’s argument is strange in one respect: surely a repetitive argu-
ment—suitably disguised by a glib sophist—might be just the ticket for a
neat eristic refutation? In eristic argument, anything goes. You would expect
Alexander to have urged that eristic syllogisms are syllogisms only homonym-
ously, or else that the phrase ‘eristic syllogism’, like ‘rhetorical syllogism’,
must be construed as a whole and does not designate a species of syllogism.
SOME PREDICATIVE PROBLEMS
Galen thinks that the Stoic logicians are peculiarly given to the study of
useless forms of argument.
⁵⁴ ὅτι δὲ οὐ χρήσιμον τὸ τοιοῦτον εἶδος μάθοιμεν ἄν εἰ ἐπέλθοιμεν τὰ εἴδη τοῦ συλλο-
γισμοῦ καὶ ἐξετάσαιμεν τίνι αὐτῶν οἰκεῖον τὸ τῶν κειμένων τι ἐπιφέρειν. πότερον γὰρ τῷ
ἀποδεικτικῷ; ἀλλ᾿ οὗτός γε τὸ ἄδηλον πειρᾶται διὰ τῶν φανερῶν καὶ γνωρίμων ἐκκαλύπτειν
καὶ τὸ ὕστερον διὰ τῶν πρώτωῷ. ἀλλὰ τῷ διαλεκτικῷ; ἀλλὰ καὶ οὗτος ὃ μὴ βούλεται
συγχωρεῖν ὁ προσδιαλεγόμενος, τοῦτο πειρᾶται διὰ τῶν ἐνδόξων καὶ ὧν συγχωρεῖ δεικνύναι,
εἰς ἀντίφασιν περιάγων. ἀλλὰ τῷ ἐριστικῷ; ἀλλὰ καὶ τούτῳ πρόκειται καὶ αὐτῷ ἢ εἰς
ἀντίφασιν ἢ εἰς φαινομένην ἀντίφασιν περιαγαγεῖν τὸν ἀποκρινόμενον ἐξ ὧν δίδωσιν· οὐχ ὃ
δίδωσι γοῦν συμπεραίνεται, ἀλλ᾿ ἐξ ὧν δίδωσιν ὃ οὐ βούλεται δοῦναι· δῆλον οὖν ὡς ἄλλο τι
τοῦ δεδομένου ἐποίσει. εἰ δ᾿ ἐστὶ τὸ γένος ἐν τοῖς εἴδεσι τοῖς αὑτοῦ, καὶ ἔστιν ὁ συλλογισμὸς
γένος τῶν αὑτοῦ εἰδῶν, ἐν οὐδενὶ δὲ αὐτῶν ταὐτὸν τῷ εἰλημμένῳ τὸ ἐπιφερόμενον, οὐδ᾿ ἂν
ἐν συλλογισμῷ εἴη ὅλως.
492 When is a Syllogism not a Syllogism?
Against the Stoics, it is necessary to develop a long argument. They are men who are
very well trained in the useless parts of logical theory whereas in the useful parts they
are both entirely untrained and also brought up on bad methods of argument; so
that it is necessary not only to teach them what is useful but also, and first, to wean
them off the bad.
(PHP v 225–226)⁵⁵
Galen exaggerates—Galen always exaggerates. But one of the items on which
he has his mind is the repetitive argument, and he may well have supposed
that such arguments were a Stoic property. Certainly, all the examples
of repetitive arguments which Alexander considers are taken from Stoic
logic; and Alexander does not stop to ask whether they may not have their
counterparts in Aristotle’s predicative syllogistic.
In fact, it is perfectly easy to construct such counterparts. Here is one:
Horses are animals, and animals are animals—so horses are animals.
The argument is repetitive inasmuch as its conclusion is identical with its first
premiss. So the argument is not a syllogism: it cannot show any syllogistic
utility; and in particular it cannot be used to prove that horses are animals.
More generally, no argument of the form:
A holds of every B
B holds of every B
Therefore A holds of every B
can be probative. Such items are exactly on a level with the repetitive Stoic
items to which Alexander refuses the title of syllogism and which he takes
Aristotle’s definition to exclude.
In arguments of that sort there are only two distinct terms, A and B.
Consider, then, another argument which has only two distinct terms in it:
Men can laugh and anything which can laugh is a man—so men are men.
That is not a repetitive argument insofar as its conclusion is not identical with
either of its premisses; but it is an argument which could not constitute an
Aristotelian proof, if only because its premisses are not prior to its conclusion.
⁵⁵ πρὸς μέντοι τοὺς Στωϊκοὺς ἀναγκαῖόν ἐστι μακρὸν ἀνύεσθαι λόγον, ἀνθρώπους ἐν μὲν
τοῖς ἀχρήστοις τῆς λογικῆς θεωρίας ἱκανῶς γεγυμνασμένους, ἐν δὲ τοῖς χρησίμοις ἀγυμ-
ναστοτάτους τε ἅμα καὶ μοχθηραῖς ὁδοῖς ἐπιχειρημάτων ἐντεθραμμένους· ὥστε ἀναγκαῖον
εἶναι μὴ μόνον διδάσκειν αὐτοὺς ὅσον χρηστὸν ἀλλὰ πολὺ πρότερον ἀποστῆσαι τοῦ μοχθηροῦ.
Some Predicative Problems 493
In general, the Aristotelian conditions on proof will not tolerate any argument
of the form:
A holds of every B.
B holds of every A.
Therefore A holds of every A.
Hence—on Alexander’s and Galen’s principles—such an argument form is
not syllogistic.
Unlike the repetitive form, that form is not excluded by the clause in
Aristotle’s definition which demands that the conclusion be ‘something
other’ than the premisses; nor is it excluded by any other clause in the
definition. How then is it to be shut out? Aristotle begins his discussion of
predicative syllogisms by saying that ‘when three terms are so related to one
another…’ (APr 25b32);⁵⁶ and he thinks he can show that ‘every proof
will be through three terms’ (41b36).⁵⁷ He does not say ‘three different
terms’; and his definition of the syllogism does not specify that the terms
of a syllogism be different—indeed, it does not mention terms at all. But
all of Aristotle’s illustrative syllogisms invoke three different concrete terms;
and a reader of the Analytics is likely to suppose, without reflecting on the
matter, that the three terms of any syllogism must be different from one
another—and perhaps he will also surmise that the different Greek letters
which Aristotle uses picture that fact.
Alexander pretty clearly thought that a syllogism must contain three dis-
tinct terms, although so far as I have noticed he nowhere says so explicitly.
Forexample,heonceortwicereferstothemiddletermofasyllogismas‘the
term which is taken twice over’ (e.g. in APr 46.21):⁵⁸ that description picks
out a particular term only on the assumption that every syllogism contains
three distinct terms.
Just as a modern logician might hope that Aristotle did not really mean to
require that the conclusion of a syllogism be different from its premisses, so
he might hope that Aristotle did not really mean to insist that the terms of a
syllogism be distinct from one another. The second hope is less forlorn than
the first. After all, Aristotle never speaks explicitly of three different terms.
Moreover, there is a passage in the Analytics in which—or so it is generally
said—he expressly allows for the repetition of a term:
⁵⁶ ὅταν οὖν ὅροι τρεῖς οὕτως ἔχωσι πρὸς ἀλλήλους ὥστε ...
⁵⁷ πᾶσα ἀπόδειξις ἔσται διὰ τριῶν ὅρων ...
⁵⁸ ὁ δὶς λαμβανόμενος.
494 When is a Syllogism not a Syllogism?
When A holds of the whole B and of C and is predicated of nothing else, and B holds
of every C, then it is necessary for A and B to convert. For since A is said only of B
andofC,andBispredicatedbothofitselfandofC,itisevidentthatBwillbesaid
of everything of which A is said except of A itself.
(APr 68a16–21)⁵⁹
The argument runs thus: Suppose that A holds of every B and of every C,
and of no other term. Suppose, secondly, that B holds of every B and of every
C. Then B holds of every term of which A holds.
The argument explicitly supposes that B holds of every B; it implicitly
supposes—in its last clause—that A holds of every A. And surely Aristotle
must think, quite generally, that any term holds of all of itself? After all, to
think that is to think that every so-and-so is a so-and-so—and who would be
bold enough to deny that every gander is a gander and every goose a goose?
Nonetheless, it is only in this passage that Aristotle plainly admits that all
ganders are ganders; and the passage is a curious one. It is curious in part
because it invokes conversion in an odd sort of way. Normally, to say of two
terms, A and B, that they convert is to say that A holds of every B and B of
every A. Here, Aristotle says that A and B convert inasmuch as A holds of
every B and B holds of all of everything of all of which A holds: that is to
say, A holds of every B and for any X, if A holds of every X, then B holds of
every X. What Aristotle says is true; for—as I have already noted in another
context—B holds of every A if and only if B holds of all of everything of
all of which A holds. Nonetheless, Aristotle’s formulation is unusual—and it
has misled more readers than one.
And there is more. For what Aristotle actually says is that A holds of every
B and B holds of everything of which A holds except of A. And that is worse
than odd—it is incoherent. At any rate, Aristotle appears to be denying that
B holds of every A—and yet the suppositions of the argument ensure that
there is no A of which B fails to hold.
Perhaps a closer analysis will find a more charitable interpretation. But
whatever it means, the passage does not suggest, let alone establish, that
syllogisms may contain fewer than three distinct terms. After all, it does not
produce or discuss any syllogism. It is one thing to allow that every term
holds of all of itself, another to admit such self-predications as components
⁵⁹ ὅταν δὲ τὸ Α ὅλῳ τῷ Β καὶ τῷ Γ ὑπάρχῃ καὶ μηδενὸς ἄλλου κατηγορῆται, ὑπάρχῃ δὲ
καὶ τὸ Β παντὶ τῷ Γ, ἀνάγκη τὸ Α καὶ Β ἀντιστρέφειν· ἐπεὶ γὰρ κατὰ μόνων τῶν Β Γ λέγεται
τὸ Α, κατηγορεῖται δὲ τὸ Β καὶ αὐτὸ αὑτοῦ καὶ τοῦ Γ, φανερὸν ὅτι καθ᾿ ὧν τὸ Α, καὶ τὸ Β
λεχθήσεται πάντων πλὴν αὐτοῦ τοῦ Α.
Some Predicative Problems 495
of syllogisms. No doubt all geese are geese—but perhaps such empty
propositions have no place in real syllogisms?
There are a few other passages in the Organon, and in the commentaries
upon it, which bear upon—or might be taken to bear upon—the question.
But they do not overthrow the strong and general impression that syllogisms
must contain three distinct terms. And I suppose that that impression is a
fair representation, not of Aristotle’s considered view of the matter, but of his
implicit preconceptions about it.
If a syllogism must contain three distinct terms, then the argument
A holds of every B
B holds of every A
Therefore A holds of every A
is not a syllogism. Moreover, if syllogisms must contain three distinct terms,
and if in addition each term in the conclusion must appear in exactly one
of the premisses, then the conclusion of a syllogism must be different from
each of its premisses, so that no repetitive argument is a syllogism. And that
suggests that a utilitarian logician might have done well to rewrite Aristotle’s
definition: replace the clause which forbids repetitiveness in the conclusion
by a clause which forbids repetition among the terms. The latter clause is
necessary to exclude certain non-syllogisms; and if the latter clause is in place,
the former clause is superfluous.
But how, it will be asked, can an Aristotelian consistently reject the two
forms
A holds of every B
B holds of every B
Therefore A holds of every B
and
A holds of every B
B holds of every A
Therefore A holds of every A?
Surely those forms are special cases of Barbara—special cases of the general
form:
A holds of every B
B holds of every C
Therefore A holds of every C?
Rejecting the two special cases involves the rejection of Barbara.
496 When is a Syllogism not a Syllogism?
Not so. If Aristotle’s definition is understood aright—or if it is appro-
priately rewritten—then the two unwanted forms are not special cases of
Barbara; for Barbara is a syllogistic form, and the two special cases are not
syllogistic forms. True, any syllogism of the form:
A holds of every B
B holds of every C
Therefore A holds of every C
isinBarbara;butitdoesnotfollow,anditisnottrue,thatanyargumentof
the form
A holds of every B
B holds of every C
Therefore A holds of every C
is in Barbara. For not every argument of that form is a syllogism.
It might be objected that the revised definition of the syllogism is
unsatisfactory until some account of the identity of terms is supplied: if
we can’t count terms—if we don’t know the criteria for their identity and
diversity—then we shan’t be able to tell a syllogism from a non-syllogism. The
objection is a good one; but the same sort of remark can be made in connection
with the unrevised definition: in order to understand that definition, we must
be competent at counting propositions; for otherwise how can we tell if the
conclusion is or is not other than the premisses? And counting Aristotelian
propositions, it may be suspected, involves counting terms.
Such questions of identity have a modern ring to them; and there is no
general discussion of them in any ancient text which I know of. But they are
addressed in a particular context by Alexander. The context will occupy me a
little later. Here is an anticipatory snippet:
Itisnotbecause‘Itisnotnight’isotherinexpressionthan‘Itisday’thattheyare
not the same as one another. For where items which differ in expression signify the
same item in the same way and primarily, then they make the same proposition,
even if they are different utterances. For ‘He has a dagger’ and ‘He has a poniard’,
if they are said of the same person, are the same proposition, because the same item
is signified primarily by ‘dagger’ and by ‘poniard’. So if ‘It is not night’ meant that
it is day in the same way as ‘It is day’ does, in that case the propositions—the one
saying it is day and the one saying it is not night—would not be different just
because the expressions are different. Rather, it is because the two—‘It is day’ and
‘It is not night’—do not mean the same thing primarily that they are other than
and not the same as one another. For ‘It is day’ primarily posits that it is day and
Some Predicative Problems 497
incidentally rejects that it is night, whereas its not being night primarily rejects night
and incidentally posits day inasmuch as if it is not night then necessarily it is day.
(in Top 12.10–24)⁶⁰
There are minor infelicities there of a sort which makes modern logicians turn
pale and grind their teeth. But the central message is clear. The condition of
identity for propositions is this: Two sentences express the same proposition
if and only if they have the same primary meaning. No doubt we may say
something similar for terms: Two term-expressions express the same term if
and only if they have the same primary meaning.
Alexander’s criterion is clear to the extent that the adverb ‘primarily’ is
clear. In other words, not to a very great extent. Consider the following
case. Definitions are among the principles of any science; and the principles
function as premisses of the probative syllogisms which establish the theorems
of the science. So a sentence such as
Every man is a two-legged rational animal
ought to be able to feature in syllogisms—in probative syllogisms. Hence it
ought to have a different primary meaning from the sentence
Every man is a man,
a sentence which cannot feature in the expression of a syllogism. That is to
say,
Every man is a two-legged rational animal
ought to contain two distinct terms. In that case, ‘man’ and ‘two-legged
rational animal’ are distinct terms, and so ought to have different primary
meanings. But a definiens and its definiendum must surely have the same mean-
ing—how, otherwise, could the former define the latter? Well, of course they
have the same meaning. But perhaps—or so Alexander must and presumably
would claim—they do not have the same primary meaning. Perhaps that
presumed claim is actually a true claim; but in order to assess it, we shall need
⁶⁰ οὐ γὰρ διότι ἕτερον τῇ λέξει τὸ οὐκ ἔστι νύξ τοῦ ἡμέρα ἐστί, διὰ τοῦτο οὐκ ἔστιν αὐτῷ
ταὐτόν· ἐν οἷς γὰρ διαφέροντά τινα κατὰ τὴν λέξιν τοῦ αὐτοῦ τινος ὁμοίως καὶ προηγουμένως
ἐστὶ δηλωτικά, ταῦτα τὴν αὐτὴν πρότασιν ποιεῖ, εἰ καὶ φωνὴν ἔχει διάφορον· τὸ γὰρ οὗτος
ξίφος ἔχει τῷ οὗτος μάχαιραν ἔχει, εἰ ἐπὶ τοῦ αὐτοῦ λέγοιτο, ἡ αὐτὴ πρότασις, ὅτι ὑπὸ τοῦ
ξίφους καὶ τῆς μαχαίρας τὸ αὐτὸ προηγουμένως δηλοῦται. οὕτως οὖν, εἰ καὶ τὸ οὐκ ἔστι νύξ
τὸ ἡμέραν εἶναι ὁμοίως ἐσήμαινεν τῷ ἡμέρα ἐστίν, οὐ διὰ τὸ ἑτέραν εἶναι τὴν λέξιν ἕτεραι ἂν
αἱ προτάσεις ἐγένοντο, ἥ τε λέγουσα ἡμέρα ἐστί καὶ ἡ λέγουσα οὐκ ἔστι νύξ, ἀλλ᾿ ἐπεὶ ταὐτὸν
προηγουμένως ἑκάτερον οὐ σημαίνει, τό τε ἡμέρα ἐστί καὶ οὐκ ἔστι νύξ, διὰ τοῦτο ἕτερα καὶ
οὐ ταὐτὰ ἀλλήλοις. τὸ μὲν γὰρ ἡμὲρα ἐστί προηγουμένως μὲν τίθησι τὸ ἡμέραν εἶναι, κατὰ
συμβεβηκὸς δὲ ἀναιρεῖ τὸ εἶναι νύκτα· τὸ δὲ μὴ εἶναι νύκτα προηγουμένως μὲν ἀναιρεῖ τὴν
νύκτα, κατὰ συμβεβηκὸς δὲ τίθησι τὴν ἡμέραν τῷ μὴ οὔσης νυκτὸς ἐξ ἀνάγκης ἡμέραν εἶναι.
498 When is a Syllogism not a Syllogism?
to be told how to distinguish between the primary meaning of a term and the
meaning of a term. Alexander does not help us towards such a distinction.
Nevertheless, let us suppose that such questions have been answered, and
let us accept the revision of Aristotle’s definition which stipulates that all the
terms in any syllogism be distinct. Does not everything then swim along as
before? Well, stipulations sometimes have unforeseen consequences. Here is
asimpleexample.
In his commentary on the Analytics Alexander states—and ascribes to
Aristotle—a general principle which he calls the synthetic theorem:
Here he delineates for us, rather clearly, the so-called synthetic theorem, which he
himself discovered. Its circumscription is this: When an item is concluded from
certain items, and the item concluded together with an item or items concludes to
an item, then the items which were concludent of it, together with the item or items
with which the last item is concluded, will themselves conclude to that same item.
For, he says, the items which concluded to A and B through which C (say) was
shown themselves conclude to what is concluded by A and B, namely C.
(in APr 274.20–25)⁶¹
Alexander’s circumscription is marginally less opaque and convoluted in his
Greek than it is in my English. This is what he means: ‘When an item is
concluded from certain items’—that is to say, given that
P
1
,P
2
,…,P
n
: therefore Q
is a syllogism; ‘and the item concluded [i.e. Q] together with an item or items
concludes to an item’—that is to say, given also that
Q, R
1
,R
2
,…,R
m
: therefore S
is a syllogism; ‘then the items which were concludent of it [i.e. P
1
,P
2
,…,P
n
,
which concluded to Q], together with the item or items with which the last
item is concluded [i.e. R
1
,R
2
,…,R
m
, which conclude to S], will themselves
conclude to that same item’—that is, to S.
In other words:
If ‘P
1
,P
2
,…,P
n
: therefore Q’ is a syllogism, and ‘Q, R
1
,R
2
,…,R
m
:
therefore S’ is a syllogism, then ‘P
1
,P
2
,…,P
n
,R
1
,R
2
,…,R
m
: therefore
S’ is a syllogism.
⁶¹ ὑπογράφει ἡμῖν φανερώτερον τὸ λεγόμενον συνθετικὸν θεώρημα, οὗ αὐτός ἐστιν
εὑρετής. ἔστι δὲ ἡ περιοχὴ αὐτοῦ τοιαύτη· ὅταν ἔκ τινων συνάγηταί τι, τὸ δὲ συναγόμενον
μετὰ τινὸς ἢ τινῶν συνάγῃ τι, καὶ τὰ συνακτικὰ αὐτοῦ, μεθ᾿ οὗ ἢ μεθ᾿ ὧν συνάγεται ἐκεῖνο,
καὶ αὐτὰ τὸ αὐτὸ συνάξει. τὰ γὰρ τῶν Α Β συνακτικά, δι᾿ ὧν δείκνυται τὸ Γ φέρε εἰπεῖν,
ταῦτα συνάγειν φησὶ καὶ τὸ ὑπὸ τῶν Α Β συναγόμενον, ὃ ἦν τὸ Γ.
Some Predicative Problems 499
Alexander states that the theorem is to be found in the Analytics,andhe
also states that the Stoics took it and divided it up in order to form certain
theorems of their own. Neither of those statements is particularly plausible,
so far as the history of logic is concerned; and from a logical point of view
it is not clear how—or that—you can cut the theorem up and produce the
Stoic theorems. But I shall not chase those nifty hares. For in any event, the
synthetic theorem seems to be a true theorem, and it is surely a powerful
theorem.
Consider, for example, the three-premissed syllogism which I discussed
briefly in another context:
A holds of every B; B holds of every C; C holds of every D: therefore A
holds of every D.
The validity of that syllogism can be proved, as we saw, on the basis of
Barbara. It can equally well be proved by an application of the synthetic
theorem. An instance of the theorem is this:
If ‘A holds of every B, B holds of every C: therefore A holds of every C’ is
a syllogism, and ‘A holds of every C, C holds of every D: therefore A holds
of every D’ is a syllogism, then ‘A holds of every B, B holds of every C, C
holds of every D: therefore A holds of every D’ is a syllogism.
The antecedent of that conditional proposition is true, for it says of a couple
of syllogisms in Barbara that each is a syllogism. Hence the consequent is
true—and the consequent states that the argument in question is a syllogism.
But now consider the following argument, which is a special case of the
three-premissed syllogism which has just been proved valid:
A holds of every B, B holds of every C, C holds of every B: therefore A
holds of every B.
That argument is not a syllogism; for its conclusion is identical with one of
its premisses. And yet it can be proved by an application of the synthetic
theorem. For
A holds of every B, B holds of every C: therefore A holds of every C
and
A holds of every C, C holds of every B: therefore A holds of every B
are two impeccable syllogisms in Barbara.