Barnes Jonathan. Truth, etc

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350 Forms of Argument

receives no easy answer, and so should we not say that ‘AB’ designates no

individual line in particular, but rather an arbitrary individual line?

In modern logic, the letters ‘a’, ‘b’, ‘c’, … are often used—as I used ‘a’a

moment ago—as singular terms or terms which designate individuals. But

they are usually taken to be a rather special sort of singular term: they are

not genuine proper names but ‘arbitrary names’. Some logicians urge that

arbitrary names are names of arbitrary objects: ‘a’ does not designate Agatha or

Arthur—it designates another sort of item altogether. Perhaps Euclid’s ‘AB’

is like that? I hope not. For there are no arbitrary objects for arbitrary names

to designate—no more than there are variable objects for variable names to

name. In particular, there are no arbitrary lines for the name ‘AB’ to designate.

If arbitrary names are names, or singular designating expressions, then they

are names or designations of actual objects, of ordinary and determinate

individuals. And the question, ‘Which ordinary and determinate line does

‘AB’ designate?’, will not go away.

If there are no arbitrary objects to designate, then why do our modern

logicians use letters—‘a’, ‘b’, ‘c’, … ? Why don’t they use ordinary proper

names or designating expressions? If ‘a’ is a name, then it is a name for some

ordinary item. So why not use an ordinary name for the ordinary item—the

name ‘Aristotle’, say? Well, why not? Pretend that, throughout your logic

handbook, the letter ‘a’ is merely an abbreviation for ‘Aristotle’, ‘b’for

‘Boethius’, ‘c’ for ‘Cicero’: what difference will that make? The answer is that

it will make no difference at all; that is to say, an inference will be valid after

your unorthodox interpretation of the letters if and only if it was valid before.

But surely—it will be said—there is some reason—beyond any reason

which the advantages of brevity might bring—to make use of these letters.

Even if, in principle, we might just as well use ordinary names in their stead,

at least we must choose those ordinary names arbitrarily or at random. After

all, if your individual is chosen with malice aforethought, then you can hardly

be warranted in inferring from a singular proposition about it to a universal

proposition about everything. The letters serve to indicate, or to remind, that

the choice of individual is strictly arbitrary—and in that case, we may as well

continue to call the things arbitrary names (on the understanding that that

expression means not ‘name of an arbitrary object’ but ‘arbitrarily chosen

name of an object’ or ‘name of an arbitrarily chosen object’).

But arbitrariness of choice is quite beside the logical point. You may pick

your names out of a hat if you want to; but you may equally well choose your

own particular favourites. I quite deliberately ﬁxed on ‘Aristotle’, ‘Boethius’

Geometrical Letters 351

and ‘Cicero’ a few moments ago: I didn’t draw them from a hat or open

my telephone book at random. But the fact that they were carefully selected

rather than chosen by chance had no bearing whatsoever on the use to which

they were put. If the designated objects are in fact selected at random, that

does not in itself help the case at all.

Suppose that you pick a name at random out of a hat, and notice that

in fact it is a name of Socrates. What a bit of luck, you think—and you

go on to prove that your randomly selected individual must have a nasty

inferiority complex—after all, he is snub-nosed. If you then make a universal

generalization (‘Everyone has an inferiority complex’), your generalization will

be fallacious; and all the randomness in the world will not protect you from

error. If, on the other hand, you pick Socrates after lengthy deliberation—and

perhaps precisely because he is snub-nosed and you want to discriminate in

favour of the nasally disadvantaged in logic if not in life—then nothing at all

will go wrong—provided, of course, that in your proofs you do not invoke

any feature which Socrates has and some of his mates lack. For what matters

insofar as the universal generalization is concerned is not how the individual

objects are chosen but how they are employed: whether the objects are called

up at random or elected after the most punctilious examination, things will

go well if and only if no appeal is made to any of their peculiarities or to any

feature which they do not have in common with every other pertinent object.

So Euclid, as Proclus saw, designates individual objects, draws individual

conclusions, and then universalizes. He draws a double conclusion: ﬁrst, a

singular proposition, ‘F(a)’; and then a universal proposition ‘F(everything)’.

The second inference is legitimate not because a is an arbitrary individual,

nor because a has been selected at random—it is legitimate insofar as the

proof that F(a) does not depend on any fact about a which is not a fact about

any other individual.

In Euclid i i, the expression ‘AB’ designates an individual ﬁnite straight

line—a real straight line, not some ghostly or arbitrary item. If it is to

function at all, it must designate some genuine line or other; and the fact that

it designates a particular and determinate line does not in the least embarrass

the generalization which will ﬁnally be made. Nevertheless, and again, which

line does the expression designate? Why, the one which Euclid then proceeds

to draw—on the sand or on the blackboard. No doubt; but which line is

that? ‘Surely it is the line which you see in front of you: as Proclus says,

Euclid puts the given item in front of our eyes.’ But that cannot be right; for

you can’t see geometrical lines, and the traces which you see in the sand or

352 Forms of Argument

on the blackboard are not geometrical lines—they are pictures or portraits of

geometrical lines. Ceci n’est pas une ligne. (If that sounds implausible, think

of stereometrical diagrams: a drawing of a sphere cannot itself be a sphere.

What goes for three-dimensional spheres goes for two-dimensional ﬁgures

and one-dimensional lines.)

Then which line does Euclid ﬁrst portray in i i? ‘The line which he is

constructing—after all, that is pretty well what he says.’ But Euclid can no

more construct or make lines than an arithmetician can construct or make

numbers; and even if you think that he can make lines, by joining a couple

of points together (whatever that operation might be), nevertheless he surely

cannot make points—and it is points which are the primary objects of his

naming. It is true that an Aristotelian philosopher might suggest that points

exist potentially until they are actualized by the geometer: the geometer

creates his points by thinking them—that is to say, he raises them from

potentiality to actuality. But that Aristotelian notion is too bizarre to tarry

over. ‘Is there a mid-point on the line AB?’—‘Not yet—wait till I start

thinking about it …’. Is that not absurd?

So which line is Euclid naming? Perhaps we can’t tell? Perhaps he didn’t

know himself? After all, what matters is that he was designating some line or

other—which one, as Proclus insists, does not in the least bit matter. Yet if

we don’t know which line he is talking about, how can we understand what

he is saying? And if he himself didn’t know which line he was talking about,

why suppose that he was talking about any line at all?

Think of the party trick in which someone invites you to choose a number,

multiply it by 5, subtract 3, square the result, … He then tells you that you’ve

got 999. Amazing. Suppose he conducts the game like this: ‘Let’s all think

of a number, any number we like, between 1 and 1,000. Pick your number

carefully. I’ve picked one of my own—I won’t tell you what it is, but I’ll call

it ‘N’ for convenience. You can call yours ‘M’, or ‘K’, or ‘Ned Kelly’. Now

let’s multiply our numbers by 5 …’. The thing drags on. And at the end,

he says: ‘Finally, let’s see what number we’ve each got: mine is 999.’ And

so, of course, is everyone else’s. In effect, he has calculated that f(N) = 999;

and you have calculated that f(M) = 999, or that f(K) = 999, or that f(Ned

Kelly) = 999. And we may thence infer that for any number n between 1

and 1,000, f(n) = 999.

Now imagine that, a few years later, we ask our ex M.C. which number

he had in mind himself, which number ‘N’ in fact designated. ‘It doesn’t

in the least matter’, he replies.—‘I know it doesn’t matter: I ask out of idle

Geometrical Letters 353

curiosity.’—‘I’ve quite forgotten.’—‘Oh dear: then how can you still be sure

that your answer was 999?’ That last question is misconceived. It is necessary

that ‘N’ designates some number; for otherwise sentences of the form ‘f(N)

= n’ would have no sense and no truth-value. It is unnecessary to know—or

to specify—what item ‘N’ designates in order to know the truth-value of

‘f(N) = n’. I may know that that sentence is true without understanding

what it means. (Salvation, Cardinal Newman thought, depends upon such

things.)

So the letters in the game must designate something. But there is no need

to know what they designate. Yet is it not quite absurd? Well, consider the

following proof:

= a.b hypothesis

Therefore a

/a = b

Therefore a

= b

Therefore if a

= a.b,thena

= b

Therefore for any x and y, if x

= x.y, then x

= y

That is an ordinary hypothetical proof, which—in the manner of Euc-

lid—ﬁrst shows something about particulars and then transfers to the

universal. What is the status of the hypothesis? The answer to that question

depends, in part, on how we construe the letters ‘a’and‘b’. Presumably

each letter must designate a number; but if they designate numbers, which

numbers do they designate? Perhaps ‘a’designates7and‘b’ designates 43?

But who would ever hypothesize that 7

= 7×43? Perhaps ‘a’designates2

and ‘b’ designates 4? That is better—but it is a cheat; for the hypothesis has

been formed in such a way as to guarantee its truth.

That line of thought—which could perhaps be put more persuasively—is

misguided. The hypothesis must have a truth-value. To be sure, a hypothesis

is not an assertion: in the Stoic jargon, a hypothetical is not the same as

an assertible. But just as an oath is not an assertion and yet what I swear

is an assertible (for I swear that such-and-such, and it can be asserted that

such-and-such), in the same way a hypothesis is not an assertion and yet

what I hypothesize is an assertible (for I hypothesize that such-and-such

and it can be asserted that such-and-such). Hence what I hypothesize has a

truth-value. Hence any designating expression which the hypothesis contains

must designate some item or other. And if we suppose that, in the case before

us, the items in question are the natural numbers, then ‘a’and‘b’must

designate some natural numbers or other.

354 Forms of Argument

What numbers they designate does not in the least matter. Let ‘a’designate

7and‘b’ 43, if you like. On that understanding of the hypothesis—as on

any other—the argument is impeccable.

Perhaps that works for numbers. But does it also work for geometrical

items—and hence for the geometrical use of letters which, according to the

present conjecture, may explain Aristotle’s use of letters in his syllogistic?

Numbers and geometrical objects differ in several fundamental ways—and

there appears to be at least one pertinent difference between them. ‘Think

of a number.’—‘O.K.’—‘Which number are you thinking of?’ If there is

no answer to that question, then you are not thinking of a number. If

you are thinking of a number, then there is a number of which you are

thinking. ‘Draw a straight line.’—‘O.K.’—‘Which line have you drawn?’.

The question is bizarre; for it seems to have no intelligible answer. (It is no

use saying ‘That line’, pointing to the diagram; for the question was precisely

‘Which line is that?’) And certainly you do not need to be able to answer the

question in order for it to be true that you have drawn a line.

However that may be, no ancient philosopher would have been embarrassed

by the question: Which line? Proclus thought that Euclid makes reference to

individual geometrical points and lines and ﬁgures, and that Euclid’s diagrams

are portraits of such entities. We do not perceive the entities; but we can grasp

them by thought—by the sort of thought which grasps individual intelligible

objects in the way in which perception grasps individual perceptible objects.

An orthodox Aristotelian, too, would have taken Euclid to be designating real

points and lines and ﬁgures—namely, the points and lines and ﬁgures which

his marks in the sand roughly trace out. For even if his diagrams are not

themselves geometrical items but representations of geometrical items, the

items represented are physically close to their representations. Geometrical

items, after all, are nothing but spatial items, so that a drawing of a circle, say,

or of a triangle may in principle follow—roughly, of course—the outline of

the circle or the triangle which it portrays.

LOGICAL LETTERS

Should we then suppose that the Greek letters in Aristotle’s syllogistic, like the

Greek letters in Euclid’s geometry, typically have a particular and determinate

meaning? More precisely, should we suppose that, just as in natural deduction

Logical Letters 355

the sign ‘a’ may be thought to designate, say, Aristotle, and just as in

Euclidean geometry the sign ‘AB’ is supposed to designate that individual

line, so too and in the same way the letters which Aristotle uses in expounding

his syllogistic—the letters which he uses in the schematic presentation of

syllogisms—in fact represent or specify particular predicative terms? That

Aristotle’s ‘A’ is the predicate ‘animal’, his ‘B’ is ‘buffalo’, and so on?

The hypothesis perhaps deserves a rather more pedantic formulation. The

question is: How are we to construe the letter ‘A’ in such formulas as ‘the A’

or ‘the item on which the A is’. The hypothesis is that ‘the A’ is the name of

an expression, and in particular of a predicate—a concrete and determinate

predicateofcourse(sincetherearenoothers).Theformula

The item on which A is holds of …

is exactly on a par with

The item of which ‘aardvark’ is true holds of …

Just as ‘the item of which ‘aardvark’ is true’ designates the aardvark, so

‘the item on which the A is’ designates whatever it is that the A is true of.

That is the hypothesis. But so expressed it is a mouthful; and there is no

harm in expressing it sloppily: ‘Aristotle’s logical letters, in their central and

characteristic use, are concrete and determinate predicate expressions.’

The hypothesis has at least one advantage: it makes for a uniﬁed inter-

pretation of the letters in the Analytics. Certainly, those letters sometimes

stand in for determinate predicates, sometimes function as concrete terms.

So let them always stand in for concrete predicates—even if, most often,

there are no concrete predicates for which they stand in. ‘What did Aristotle

mean by ‘the A’ here?’—‘Who knows?’—‘But then what does ‘the A’ mean

here?’—‘Whatever you like it to mean.’ In the early pages of his Begriffsschrift

Frege makes use of Greek capital letters. They represent judgeable contents,

or (roughly speaking) assertibles. When a capital Greek letter ﬁrst appears,

Frege tells us what it means—it is a shortened form of a particular sentence.

Later, he does not say what the letters mean; and he remarks in a footnote that

wherehedoesn’tglossaletterwemaygiveitwhateversenseweplease.The

Greek letters must always have a sense. For if they did not have a sense—in

particular, if they did not have a judgeable content—then the formulas in

which they appear would be ill-formed. But what sense they have does not

in the least matter.

The paragraph from Boethius’ essay on Predicative Syllogisms which I cited a

little while ago suggests such a Fregean interpretation of the syllogistical letters;

356 Forms of Argument

for Boethius says that when we write ‘A’ and ‘B’ we should understand those

letters to mean ‘animal’ and ‘man’. A passage in Philoponus’ commentary on

the Prior Analytics (it is the continuation of a paragraph which I have already

quoted) may perhaps carry the same suggestion:

A universal thesis is refuted by even a single example, as I have already said; but

it is established either by going through all the particulars, which is inﬁnite and

impossible, or by the warrant of a universal rule. That is what Aristotle does now by

means of letters, giving to each of us, as I said, full liberty in their use so that we may

supply terms of whatever matter we wish instead of the letters. ‘Now if A holds of

no B, B does not hold of any A’. If we take winged instead of ‘A’ and man instead

of ‘B’, we shall ﬁnd that just as winged holds of no man, so conversely no man holds

of winged.

(in APr 47.1–10)⁶⁹

Aristotle proves conversions by means of universal rules. The universal rules

rely upon the use of letters. The letters supply the desired universality insofar

as ‘we have full liberty in their use so that we may supply terms of whatever

matter we wish’: we may take Aristotle’s ‘A’ to mean ‘animal’ or ‘artichoke’

or ‘armillary sphere’—for although the letter ‘A’ must mean something or

other, and hence must mean something or other in particular, what it means

is quite up to us. And what guarantees the universality of Aristotle’s proofs is

precisely the fact that the sense of the letters is up to us, that we may saddle

them with any of an inﬁnity of senses.

Or is that what Philoponus means to say? A different and more down-

to-earth interpretation construes him as saying that Aristotle proves the

conversions universally, that he does so by using letters, and that since the

conversions are universally proved, you may apply them to any particular

case you like simply by replacing Aristotle’s letters with the concrete terms

of your choice. In that case, Philoponus gives no account at all of the letters

themselves.

The text is, I think, indeterminate between those two interpretations. But I

suspect that the latter and less exciting interpretation is the more probable. As

⁶⁹ τὸν μὲν γὰρ καθόλου λόγον ἐλέγχει μὲν καὶ ἓν παράδειγμα, ὡς ἤδη εἴρηται, κατασκευάζει

δὲ ἢ ἡ διὰ πάντων τῶν κατὰ μέρος διέξοδος, ὅπερ ἐστὶν ἄπειρον καὶ ἀδύνατον, ἢ ἡ διὰ

καθολικοῦ κανόνος πίστις· ὅπερ ποιεῖ νῦν διὰ τῶν στοιχείων διδοὺς ἑκάστῳ, ὥσπερ εἴρηται,

ἐπ᾿ ἐξουσίας χρῆσθαι καὶ ὑποβάλλειν ἀντὶ τῶν στοιχείων οἵας ἂν βούληται ὕλης ὅρους. εἰ οὖν

μηδενὶ τῶν Β τὸ Α ὑπάρξει, οὐδὲ τῶν Α οὐδενὶ ὑπάρξει τὸ Β· εἰ γὰρ ἀντὶ μὲν τοῦ Α λάβωμεν

πτηνόν, ἀντὶ δὲ τοῦ Β ἄνθρωπον, εὑρήσομεν ὅτι ὥσπερ τὸ πτηνὸν οὐδενὶ ἀνθρώπῳ ὑπάρχει,

οὕτως ἔμπαλιν οὐδεὶς ἄνθρωπος πτηνῷ ὑπάρχει.

Logical Letters 357

for Boethius, I have already described that passage as muddled, and muddled

it surely is.

Nonetheless, it is taken from Aristotle. For Boethius’ mind is evidently on

the following sentences, which come at the end of Aristotle’s presentation of

the non-modal conversions:

If A does not hold of some B, it is not necessary that B too does not hold of some A.

For example, if B is animal and A man—for man does not hold of every animal, but

animal holds of every man.

(APr 25a22–26)⁷⁰

‘Suppose that A does not hold of some B.’—‘What on earth am I being

invited to suppose?’—‘Well, take it that—for example—‘A’ means ‘man’

and ‘B’ ‘animal’.’ Or again: ‘Suppose that that on which A is does not hold of

some of that on which B is.’—‘But what is A on and what is B on?’—‘Oh,

it doesn’t matter—let’s say that A is on man and B on animal.’

That account of the syllogistic letters works rather well in many of those

Aristotelian texts where the letters can scarcely function as variables and yet

are not presented as stand-ins for given predicate expressions. And it also

works rather well in those passages which seem to invite an interpretation in

terms of variables.

Take, again, the ﬁrst appearance of the syllogistic letters in the Prior

Analytics:

When three terms are so related to one another that the last is in the middle as in

a whole and the middle either is or is not in the ﬁrst as in a whole, it is necessary

that there is a syllogism of the extremes … For if A of every B and B of every C, it is

necessary that A is predicated of every C. … Similarly, if A of no B and B of every C,

that A will hold of no C.

(APr 25b32–26a2)

I cited the passage in an earlier context, claiming that the last couple of

sentences implicitly suggest schematic representations of the two syllogistic

forms, Barbara and Celarent, for which the ﬁrst sentence provides the circum-

scriptions. And so they do. But Aristotle does not simply lay the schematic

sentences alongside the circumscriptions: they are linked to the circumscrip-

tions by the word ‘for’. In that case, they ought somehow to explain or justify

⁷⁰ εἰ δέ γε τὸ Α τινὶ τῷ Β μὴ ὑπάρχει, οὐκ ἀνάγκη καὶ τὸ Β τινὶ τῷ Α μὴ ὑπάρχειν, οἷον

εἰ τὸ μὲν Β ἐστὶ ζῷον, τὸ δὲ Α ἄνθρωπος· ἄνθρωπος μὲν γὰρ οὐ παντὶ ζῴῳ, ζῷον δὲ παντὶ

ἀνθρώπῳ ὑπάρχει.

358 Forms of Argument

or otherwise illuminatingly comment upon what Aristotle has just said. But

what exactly is the nature of the comment?

If the letters are variables, then the schematic sentences set out or explain

in a clearer or more determinate fashion what the circumscriptions have

imperfectly indicated. Thus interpreted, the text paraphrases as follows: ‘If

you’ve got a ﬁrst ﬁgure conjugation with two universal afﬁrmatives, then

there is always a conclusion to be inferred—I mean that, for any triad of

terms A, B, C, ﬁrst, if A holds of every B and B of every C, then A must hold

of every C; and secondly, if A holds of no B and B of every C, then A must

hold of no C.’ The circumscription does not say what syllogisms are there for

the picking: the schematization ﬁlls out the gap.

That, it must be allowed, is a reasonable way to read the text. After all

the circumscriptions are in fact inadequate to specify a pair of syllogistic

forms, and the schematic sentences do in fact succeed in making the

necessary speciﬁcations. Nonetheless, there is something odd about the

interpretation—quite apart from the general considerations which tell against

ﬁnding variables in Aristotle. For it invites us to imagine that Aristotle ﬁrst

gave circumscriptions of Barbara and Celarent, that he then saw that they

were inadequate, and that he decided to make up for their inadequacies by

using schematic letters. Why didn’t he simply repair the circumscription?

After all, that is readily done. Surely he must have had something else in

mind when he wheeled on his Greek letters?

Suppose, then, that we take the letters to be predicates: just as the ‘a’

of natural deduction names Anaximander or Anaximenes or Zeno or what

you will, so the ‘A’ of Aristotle’s syllogistic means the same as ‘aardvark’

or ‘albatross’ or ‘zebu’ or what you will. The schematic sentences introduce

concrete terms, and so hint at—without formulating—concrete syllogisms.

Aristotle ﬁrst describes (imperfectly) the two syllogistic forms Barbara and

Celarent; and he then says something about a couple of triads of concrete

predicates. He appeals to the concrete predicates in order to justify a universal

claim. In other words, Aristotle, like Euclid, takes concrete terms and then

transfers to the universal.

That seems to me to be the best way to take this particular passage from the

Prior Analytics. (How Aristotle thinks that the universal validity of Barbara

and Celarent are thereby established is another matter—and a matter for the

next chapter.) And it seems to me to work quite well throughout most of the

Analytics.

Logical Letters 359

‘Very pretty, Mr Barnes, but you may not call it Aristotle.’ I am not at

all sure that it is very pretty. But in any event, pretty or plain, I do not call

it Aristotle. After all, it is an attempt to answer a question which Aristotle

did not raise—or at any rate, a question which is not raised in the Analytics.

The question is: What on earth can Aristotle’s syllogistic letters mean (if they

mean anything at all)? It is, I think, a good question. But it is not an essential

question—that is to say, Aristotle could get on with his syllogistic without

worrying about it. After all, you can do very well in logic without worrying

much about what your sentences mean. Indeed, you can do pretty well in

logic even if your sentences mean nothing at all.