Barnes Jonathan. Truth, etc
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260 What is a Connector?
first sentence says. But it is not a two-placed sentential connector: it is an
Aristotelian articulator, an item which connects without unifying.
That seems right for English; and I think it is right for Latin—at any
rate, ‘itaque’ looks like a sentential adverb, and an indexical one at that. In
other words, Victorinus is right, against all his Latin colleagues. What about
the Greeks? Are the items which the Stoics and Apollonius characterize as
inferential connectors not connectors at all but rather adverbs?
Greek certainly has inferential adverbs or adsentences. For example, there
are ‘οὕτως’, ‘ταύτῃ’, ‘διὰ τοῦτο’, … The grammarians say very little about
them. Apollonius cites ‘οὕτως’and‘ταύτῃ’ as adverbs (e.g. adv 123.4,
138.20, 151.23); but he does not comment on their inferential use. His only
remark on ‘διὰ τοῦτο’ occurs when he is arguing that the connectors ‘ὅτι’
and ‘διότι’ are not synonymous:
If you say
διότι it is day, it is light,
‘διὰ τοῦτο’ is missing; but
ὅτι it is day, it is light
is complete.
(conj 242.14–16)¹⁴⁴
That rather implausible contention shows that Apollonius did not take
‘διὰ τοῦτο’ to be a connector; but it sheds no positive light on his opinion.
The Byzantine grammarian, George Choeroboscus, has the following to say:
‘διὰ τοῦτο’ is a demonstrative pronoun, or rather a demonstrative adverbial or an
explanatory connector. It comes from the preposition ‘διά’ and the pronoun ‘τοῦτο’.
(epim in Psalm 58.32–34)¹⁴⁵
Choeroboscus often preserves ancient wisdom; but I should not like to make
anything of this particular passage.
However that may be, it is clear that if in Greek you introduce a sentence
with,say,‘διὰ τοῦτο’ alone, then you have an asyndeton; and if you want to
avoid asyndeton you may do so easily enough by adding a connector—‘δέ’
or ‘οὖν’ or what you will—after the ‘διὰ τοῦτο’. Now the items which the
Greek grammarians class as syllogistic connectors are not like that at all. True,
¹⁴⁴ τὸ γὰρ οὕτω λεγόμενον, διότι ἡμέρα ἐστί, φῶς ἐστι λείπει τῷ διὰ τοῦτο· τὸ δέ ὅτι
ἡμέρα ἐστί, φῶς ἐστιν αὐτοτελές.
¹⁴⁵ διὰ τοῦτο· δεικτικὴ ἀντωνυμία, ἢ μᾶλλον ἐπιρρηματικὴ δεικτική, ἢ σύνδεσμος αἰτιο-
λογικός. γίνεται δὲ ἐκ τῆς διὰ προθέσεως καὶ τῆς τοῦτο ἀντωνυμίας.
Arguments and Sayings 261
you may introduce the conclusion of an argument with ‘διὰ τοῦτο’ and you
may introduce it with ‘ἄρα’; but the latter, unlike the former, does not make
for an asyndeton; and so the latter, unlike the former, will not tolerate a
supplementary ‘δέ’or‘οὖν’.
The facts are a good deal more nuanced than that. And in any event
what I have just said does not prove that the Greek inferential connectors
really are connectors. Nonetheless, there are, on the one hand, pertinent
differences between, say, ‘ἄρα’and‘διὰ τοῦτο’; and there are, on the
other hand, pertinent similarities between, say, ‘ἄρα’and‘ἀλλά’. So is
there anything to say against the thesis that inferential connectors are
connectors?
Look first at the easier case—the case of co-assumptional connectors.
Apollonius’ example was this:
εἰ ἡμέρα ἐστί, φῶς ἐστί· ἡμέρα δέγε ἐστι.
In English, that is roughly:
If it’s day, it’s light; but it’s day.
According to Apollonius, ‘δέγε’ links a saying to a saying. He does not
explicitly say that ‘δέγε’ unifies, and he does not explicitly say that it makes
a single saying from a plurality of sayings. But I have already argued that he
takes connectors in general to unify, and that the result of unifying a plurality
of sayings can only be a saying. If that is so, then Apollonius holds that ‘δέγε’
functions as a two-placed connector, just like ‘εἰ’or‘if’.
The English ‘but’ is not always a sentential connector; but sometimes it is.
After all, ‘and’ often functions as a sentential connector; and where ‘and’ so
functions, it can be usually replaced by ‘but’ without syntactical abuse. For
example, ‘and’ is indubitably a sentential connector in
He is white and he is ugly.
So how cannot ‘but’ be a sentential connector in
I am black but I am comely?
The case passes the embedding test—you can say, for example,
Do you know that she is black but she is comely?
(Well, you can, but you won’t. But you will say, without a qualm
Do you know that she is black but comely?
And come to that, what was actually said was this:
I am black but comely, O ye daughters of Jerusalem.)
Thus if P and Q are sentences, then ‘P but Q’ will at least sometimes be
a sentence.
262 What is a Connector?
Perhaps the co-assumptional use of ‘but’ is different? After all, it will surely
be thought odd, or uncouth, to utter anything like
I know that if it’s day, it’s light, but it’s day.
The thing is a mouthful, and it is likely to be mispunctuated and misunder-
stood. But is it ungrammatical? I cannot see that it is; and so I incline to
think that the English word ‘but’ functions as a sentential connector when it
is used to connect a supplementary premiss to the rest of an argument.
Is the same true of the Greek ‘δέγε’ and of the other items which the
grammarians present as co-assumptional connectors? Is, say,
εἰ ἡμέρα ἐστί, φῶς ἐστί· ἡμέρα δέγε ἐστι
embeddable? Can it be negated, or can you say that you know or believe
that …? The answers to those questions, I take it, are Yes.
It is another question what the truth-conditions for such compound
sentences are. Some will suggest that ‘but’ may be replaced by ‘and’, and
‘δέγε’by‘καί’ (in an appropriate position), without any change in truth-value
or indeed in sense. Apollonius, however, says that ‘δέγε’promisesormeans
something other than ‘δέ’. And in his view, something of the form ‘P, Q
δέγε’ will presumably express a truth if and only if, first both P and Q are
true, and secondly, Q is a supplementary premiss in an argument in which P
is a preceding premiss.
What, finally, of inferential connectors, the connectors which introduce
the conclusion of an argument? Consider:
εἰ ἡμέρα ἐστί, φῶς ἐστί· ἡμέρα δέγε ἐστι· φῶς ἄρα ἐστι.
There, ‘ἄρα’ is—according to the ancient theory—a connector. It connects
the saying, ‘φῶς ἐστι’, to the sayings which precede it: it connects the
sentence which is the conclusion of the argument to the sentences which
are the premisses of the argument. The whole Greek sequence which I have
just cited—two premiss-sentences and a conclusion-sentence—is therefore
supposed to constitute a single saying, the structure of which might be
represented as follows:
(P, Q δέγε)Rἄρα
And that is comparable to, say,
If (P and Q) then R
To be sure, no ancient text actually says as much. But I cannot see what else
an ancient text could have said on the subject.
Nevertheless, if that was what the Greek grammarians wanted to say, surely
they were wrong? After all, translate the Greek into English and you get this:
If it’s light it’s day; but it’s light: therefore it’s day.
Arguments and Sayings 263
That is a connected sequence of sentences—but it is not a single sentence. It
cannot be embedded. For example,
I know that if it’s light it’s day, but it’s light, therefore it’s day
is not an English sentence; and it is not an English sentence in the same way
and for the same reason that
I know that he came, he saw, he conquered
is not an English sentence.
That is the truth aboutEnglish. What about Greek? There isa bronzeanswer
to the question: ‘English ‘therefore’ and Greek ‘ἄρα’ are on all fours; ‘ἄρα’is
not a sentential connector; and the Greek grammarians—like grammarians
all over the world—were wrong about their own language.’ There is a silver
answer: ‘English ‘therefore’ and Greek ‘ἄρα’ are on all fours; each is a connector
but not a sentential connector; and the Greek grammarians were right to
make ‘ἄρα’ a connector which makes sayings from sayings—so long as the
made sayings are not taken to be sentences.’ There is a gold answer: ‘English
‘therefore’ and Greek ‘ἄρα’ are not on all fours; ‘ἄρα’, unlike ‘therefore’, is a
sentential connector; and the Greek grammarians are victorious.’
I am inclined to think that—whatever else may be the case—‘therefore’
is not on all fours with ‘ἄρα’. The sentential adverb or adsentence ‘therefore’
is the exact English for the Greek sentential adverb or adsentence ‘διὰ
τοῦτο’. There is no exact English for ‘ἄρα’. (Translators will continue to use
‘therefore’, and rightly so.) The nearest English is perhaps ‘and therefore’. In
English, ‘and therefore’ functions as a sentential connector. To be sure, it is a
compound connector—it is compounded from a connector and an adverb;
but it is still a connector, and expressions of the form ‘P and therefore Q’
pass the embedding test. Plainly, if ‘and therefore’ is a connector, it is an
inferential connector. A sentence of the form ‘P and therefore Q’ is true if
and only if P is true and, for that very reason, Q is true. One way of putting
such things into Greek is by means of the formula ‘P, Q ἄρα’.
4. Forms of Argument
SPECIES OF SYLLOGISM
Logic, in the good old days, was the art of thinking—more precisely, it
was the art or science of reasoning. Logicians were supposed to consider
arguments—deductions, inferences, syllogisms, what you will—and to sort
them into the good and the bad, the valid and the invalid. There is an
endless number of arguments, and a logician cannot survey each and every
one of them. Nor would he want to. Rather, and like any other scientist,
a logician is interested in the universal, in the general: he is not concerned,
save incidentally, with the individual items which roam his territory but with
types or kinds or classes of individuals.
Towards the end of his Introduction to Logic, Galen announced that
there is also another, third, species of syllogism useful for proofs, which I say come
about in virtue of something relational, while the Aristotelians are obliged to number
them among the predicative syllogisms.
(inst log xvi 1).¹
The announcement was, I suppose, designed to shock, or at least to astound.
Everyone knew that, when you did logic, there were two species of syllogism
to mug up: there were the predicative syllogisms which Aristotle had put on
the market; and there were the hypothetical syllogisms, which were the pride
but not the property of Stoic logic. Who knew about a third species? No
one—until Galen discovered and published it.
The two familiar species are apparently quite different from one another.
The Greek for ‘predicative’ is ‘κατηγορικός’: the word, in this context, is
usually translated by—or transliterated as—‘categorical’, so that histories
of logic discuss categorical syllogisms and categorical syllogistic. The word
‘categorical’ is entrenched; but ‘predicative’ is the right translation: predicative
syllogisms are arguments the validity of which turns on the properties of
¹ ἔστι καὶ ἄλλο τρίτον εἶδος συλλογισμῶν εἰς ἀποδείξεις χρήσιμον, οὓς ἐγὼ μὲν ὀνομάζω
κατὰ τὸ πρός τι γενέσθαι, βιάζονται δ᾿ αὐτοὺς οἱ περὶ τὸν ᾿Αριστοτέλην τοῖς κατηγορικοῖς
συναριθμεῖν.
Species of Syllogism 265
predicative propositions—that is to say, on the properties of one or other of
the four styles of Aristotelian predication. The constituent propositions of a
predicative syllogism—its premisses and its conclusion—are all essentially
such that in them x is predicated in style S of y.
The Greek for ‘hypothetical’ is ‘ὑποθετικός’. Like ‘categorical’, ‘hypo-
thetical’ is a transliteration rather than a translation; and it is potentially
misleading. But in this case there is, I think, no better or less misleading trans-
lation. In any event, ‘hypothetical’ syllogisms are arguments the validity of
which turns on the logical properties of certain compound propositions—and
in particular (in the standard cases), on the logical properties of conditionals
and disjunctions and conjunctions. At least one of the premisses of a
hypothetical syllogism is essentially compounded in one or other of those
ways.
Aristotle claimed that every syllogism and every proof come about through
one of the three figures of his predicative syllogistic; and some Stoics appear
to have made a similar claim on behalf of their hypothetical syllogistic. The
two theories might therefore appear to be rivals; and certainly, if you were
a card-carrying Peripatetic, you were likely to claim—with Alexander of
Aphrodisias—that the predicative syllogisms were the only kosher variety;
and if you were a good Stoic you might hold the converse view.
Nevertheless, it is clear that in later antiquity both theories—or rather,
derivative and simplified versions of both theories—were taught as comple-
mentary parts of the single science of logic. Galen mentions disagreements
over priority, and he urges that
as far as disputes of that sort are concerned, it is no great matter whether you discover
the truth or remain in ignorance. For you need to learn both sorts of syllogisms—that
is what is useful. You may say that one lot is prior, or teach it to be prior, as the
mood takes you—but you must not be ignorant of the others.
(inst log vii 3)²
Most teachers, no doubt, took the Galenic line.
Galen alone offers a third species of syllogism: it was produced as a
necessary supplement to the existing two species and not as a rival to them;
and if Galen once or twice seems to intimate that his relational syllogisms
² ἀλλὰ περὶ μὲν τῶν τοιούτων ἀμφισβητήσεων οὔτε εὑρεῖν οὔτε ἀγνοῆσαι μέγα· χρὴ γὰρ
ἀμφότερα τὰ μέρη γιγνώσκειν τῶν συλλογισμῶν, καὶ τοῦτ᾿ ἔστι τὸ χρήσιμον, ὀνομάζειν
δὲ τοὺς ἑτέρους ἢ διδάσκειν προτέρους ὡς ἑκάστῳ φίλον· οὐ μὴν ἐκείνοις γε ἀγνοεῖσθαι
προσῆκεν.
266 Forms of Argument
are more useful than either predicative or hypothetical syllogisms, he never
states that that is so—and no doubt he did not mean to suggest that it
is so.
Galen discusses his third species at some length in the Introduction to Logic.
But he mentions the matter nowhere else in his vast œuvre; and no one else
in antiquity ever notices relational syllogisms. We only possess a fraction of
ancient writings on logic, and it is likely enough that relational syllogisms
were mentioned in some lost texts. But it is improbable that they were ever
widely known; and it is certain that they never became—as Galen apparently
thought they ought to become—the third part of a tripartite logic.
However that may be, Galen thought that there were three species of
syllogisms, and most ancients thought that there were two. If the word
‘species’istakentechnically,andnotasavariantonsuchinformaltermsas
‘kind’ or ‘sort’, then the ancients will have supposed that the word ‘syllogism’
is a generic term, and they will have divided the genus into two—or in
Galen’s case, three—species, each definable as ‘syllogism of such-and-such
a sort’. Presumably the genus itself is a species of a higher genus, the
genus of argument or inference; for certainly a syllogism is a particular
sort of argument. And presumably the species of syllogisms themselves
divide into subspecies, and so on until we arrive at lowest items or infimae
species.
There is any number of ways of dividing a species of arguments into
subspecies, just as there is any number of ways of dividing any species
into subspecies. And just as some methods of dividing animal species are
appropriate for a zoologist, so some methods of dividing argumentative
species are appropriate for a logician. When Alexander wants to show that the
conclusion of a syllogism cannot be the same as one of its premisses, he says
that ‘we might learn the truth if we went through the species of syllogism’;
he then mentions apodictic syllogisms, dialectical syllogisms, and eristical
syllogisms, and shows that in none of the three cases can the conclusion be
the same as a premiss; and he concludes thus:
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 the syllogism in general.
(in APr 19.1–3)³
³ 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.
Species of Syllogism 267
The terms ‘genus’ and ‘species’, as Alexander’s argument shows, must here
be taken seriously. One way of dividing up the genus of syllogisms is by a
trichotomy—apodictic, dialectical, eristic.
Alexander’s trichotomy distinguishes species of syllogism according to the
character of their premisses—a syllogism is eristic if its premisses falsely
seem to be true, it is dialectical if its premisses are reputed to be true, and
it is apodictic if its premisses actually are true (and also satisfy several other
conditions). That sort of distinction might be deemed to be epistemological
rather than strictly logical. There were also strictly logical divisions. Alexander,
like all other Aristotelians, divided types of predicative syllogisms into what
were called figures, of which there were three. And each figure was divided
into a certain number of moods. The exact number of moods was contested
in antiquity (and after); but all agreed that it was in principle finite and in
fact pretty small—the most exuberant enumerators did not go beyond 24.
Argument, syllogism, predicative syllogism, first figure syllogism, Bar-
bara—such a sequence of terms could then be picked off the Porphyrean tree
or formal division which represents the domain of logic. As far as I know, no
ancient text records such a sequence or describes such a tree. But it is the sort
of thing which a logician of late antiquity would have loved.
There is at least one difficulty with it. No item can belong to two species
unless one of them is subordinate to the other; and hence no item can belong
to more than one lowest species. Those are trivial truths about the classificatory
principles which underlie Porphyrean trees. But might not one and the same
argument belong to two non-subordinate kinds, being (say) both a predicative
and a hypothetical syllogism? And could not one and the same argument
belong to two lowest kinds, being equally or indifferently a predicative syllo-
gism in Cesare (say) or in Camestres? There is no discussion of the question in
any surviving ancient text; but there are a few wisps of evidence on the matter.
The first wisp is found in the list of Chrysippus’ logical writings which
Diogenes Laertius copied down and conserved for us. It contains the following
two adjacent items:
Concerning the fact that the same argument is ordered in several modes (1 book).
On what has been urged against the fact that the same argument has been ordered in
a syllogistic and in a non-syllogistic mode (2 books).
(vii 194)⁴
⁴ Περὶ τοῦ τάττεσθαι τὸν αὐτὸν λόγον ἐν πλείοσι τρόποις· α´.
Πρὸς τὰ ἀντειρημένα τῷ τὸν αὐτὸν λόγον ἐν συλλογιστικῷ καὶ ἀσυλλογίστῳ τετάχθαι
τρόπῳ· β´.
268 Forms of Argument
We know nothing of those essays save what can be deduced from their titles.
Tosaythatanargumentis‘orderedinamode’istantamounttosayingthat
it is of a certain kind or that it has a certain form. The Stoics distinguished
between arguments on the one hand, and what they called ‘τρόποι’ or ‘modes’
on the other. Diogenes Laertius explains the distinction as follows:
An argument … is something consisting of an assumption and a co-assumption and
an inference, as for example:
If it is day, it is light; but it is day: therefore it is light.
… A mode is as it were a shape of an argument, as for example:
If the 1
st
,the2
nd
; but the 1
st
: therefore the 2
nd
.
(vii 76)⁵
I shall have much more to say about modes later on; but it is plain—and it
is enough to be going on with—that a mode is something like an argument-
form or an argument-schema. So the first of the two Chrysippean titles
must have discussed the notion that one and the same argument may have
two or more forms; and the title implies—a little less strongly, it must be
confessed, in the Greek than in my English—that Chrysippus accepted the
notion.
If Chrysippus held that a single argument may have different forms,
then—it might be inferred—he cannot have thought, or at least, cannot
consistently have thought, that arguments divide into genera and species. But
the inference is hasty. As far as the title goes, the different modes which order
a single argument may themselves be arranged as species and genus—one of
the modes may be subordinate to, or an instance of, the other. It may be
helpful to look at the matter in terms of a simple example; and it is convenient
to take a vacuous argument. (The Peripatetics jeered at such things, but the
Stoics had nothing against them.)
So consider the empty argument:
If it is day, it is day; but it is day: therefore it is day.
What modes might that argument be ordered in? What forms might it be
supposed to have? Well, a modern logician will notice that the argument has
at least the following four forms or that it is ordered in at least the following
four modes.
⁵ λόγος δέ ἐστιν, ὡς οἱ περὶ τὸν Κρῖνίν φασι, τὸ συνεστηκὸς ἐκ λήμματος καὶ προσλήψεως
καὶ ἐπιφορᾶς, οἷον ὁ τοιοῦτος· εἰ ἡμέρα ἐστί, φῶς ἐστι· ἡμέρα δέ ἐστι· φῶς ἄρα ἐστί. ...
τρόπος δέ ἐστιν οἱονεὶ σχῆμα λόγου, οἷον ὁ τοιοῦτος· εἰ τὸ πρῶτον, τὸ δεύτερον· ἀλλὰ μὴν
τὸ πρῶτον· τὸ ἄρα δεύτερον.
Species of Syllogism 269
(a) If P, P; P: therefore P
(b) IfP,Q;P:thereforeQ
(c) P; Q: therefore Q
(d) P, Q: therefore R
(For the moment, I represent forms or modes in that standard schematic
manner, and I suppose that the manner is understood or understandable. The
questions of how to represent forms of argument and of how to understand
such schematic formulas will be a major theme of the later parts of this
chapter.)
The first three of those modes are valid, the fourth is not. Every argument
which is ordered by (a) or by (b) or by (c) is thereby a valid argument,
its conclusion following by necessity from its premisses; but an argument
ordered by (d) is not thereby valid. Mode (a) is a special instance of
mode (b). If Chrysippus had looked at the vacuous argument when he
remarked that ‘the same argument can be ordered in several modes’, which
of those four modes would he have had his eye on? If on (b) and (c)—if,
that is to say, he would have adverted to the fact that the argument is
ordered both by (b) and also by (c)—then that would be directly per-
tinent to the question of genera and species of syllogisms; for (b) is not
a special case of (c) nor (c) of (b). But if he would have been think-
ing rather, say, of (a) and (b), then there would be no consequences to
be drawn.
There is still the evidence of the second of the two Chrysippean titles. It
suggests that someone had claimed that one and the same argument may
be ordered in a syllogistic and in a non-syllogistic mode; that someone had
brought objections against the claim; and that Chrysippus then defended
the claim against the objections. What precisely the claim was depends
on the sense which is to be given to the term ‘non-syllogistic’. On one
hypothesis—it is not the only one, and I am not even sure that it is the
most likely one—the word means ‘non-concludent’ or ‘invalid’. In that case,
Chrysippus’ second essay will have defended the thesis that one and the
same argument may have both a valid and an invalid mode. The thesis is
true—as is shown by the fact that the vacuous argument has among its
modes both (a), which is valid, and (d), which is not. Indeed, Chrysippus
might have defended a stronger version of that thesis. It is a trivial truth
that every argument with n premisses exhibits, among other modes, the
mode:
P
1
,P
2
,…,P
n
: therefore Q