Barnes Jonathan. Truth, etc

Подождите немного. Документ загружается.

220 What is a Connector?

and what little they say on the matter tends to suggest—without ever coming

straight out with it—that all connectors are in principle multi-placed. So

consider this sentence:

If I’m in Germany I drink beer if I’m thirsty.

Why not say that the ﬁrst ‘if ’ is two-placed, attaching ‘I drink beer’ to ‘I’m

in Germany’ whereas the second ‘if ’ is three-placed, attaching ‘I’m thirsty’ to

‘If I’m in Germany I drink beer’?

Finally, something similar may be said for sentences which mix their

connectors. For example, in

We’re in Europe or we’re in Asia if we’re in Turkey

the ‘if ’ connects the saying which follows it to the two which precede it. It

adds a third carriage to a two-carriage train.

The railway-train model has a certain appeal—at any rate, it has the

appeal of naïvet

e. But it faces difﬁculties, some of them syntactical and others

semantic. Consider, ﬁrst, a syntactical point.⁸⁴ It may be introduced by way

of the last illustrative example; for that sentence might be parsed in either of

two ways; and if it is claimed that the ‘if ’ connects ‘we’re in Turkey’ to what

precedes it, then it must be recognized that the connection may have either

of two distinct characters. The two characters and the two parsings can be

crudely indicated as follows:

[We’re in Europe or we’re in Asia] if we’re in Turkey

We’re in Europe or [we’re in Asia if we’re in Turkey]

Any decent theory of connectors will recognize the two parsings, and the

difference between them.

The issue is a general one. True, it does not arise for conjunctive connectors.

In modern logic the conjunctive connective is two-placed, so that a formula

of the form

P&Q&R

is ill-formed: if you want to conjoin three items you must write either

(P & Q) & R

or else

P&(Q&R).

That might lead you to think that in English three-part conjunctions admit,

in principle, three parsings, namely:

IwalkandImoveandIread

⁸⁴ Paolo Crivelli brought the point to my attention.

Multiple Connections 221

[I walk and I move] and I read

I walk and [I move and I read]

But the parsings make no difference: whichever way it is parsed, the con-

junctive sequence says the same thing, expresses the same saying.

Things are different in the case of disjunction. Just like a three-part

conjunction, a three-part disjunction such as

He’s good or he’s bald or he’s ugly

might be parsed in any of three ways, namely:

He’s good or he’s bald or he’s ugly

[He’s good or he’s bald] or he’s ugly

He’s good or [he’s bald or he’s ugly]

And here the different parsings are not indifferent. A disjunctive connector, as

Apollonius puts it, ‘announces the holding of one and the removal of the other

or of the others’ (conj 216.14–16), so that a disjunction is true if and only if

exactly one of its disjuncts is true. That being so, the three-placed disjunction

He’s good or he’s bald or he’s ugly

will say something different from the two-placed disjunction

[He’s good or he’s bald] or he’s ugly.

Suppose that, as a matter of horrid fact, he’s good and he’s bald and he’s

ugly. (That’s why the example is unorthodox by a letter.) Then plainly the

three-part disjunction

He’s good or he’s bald or he’s ugly

is false; for exactly none of its disjuncts is true. But consider the two-placed

disjunction. Since he’s good and he’s bald, the disjunction

He’s good or he’s bald

is false. But in that case, exactly one of the disjuncts in the two-placed

disjunction

[He’s good or he’s bald] or he’s ugly

is true, namely ‘He’s ugly’; so the two-placed disjunction is true.

Such phenomena certainly raise questions for multi-placed connectors; but

I am not sure that they raise any genuine or peculiar difﬁculties. No doubt

the ancient grammarians should somehow have distinguished between the

three-placed and the two-placed readings of

He’s good or he’s bald or he’s ugly.

They did not do so, so far as we can tell. But nothing in their theory prohibits

them from doing so. Should they not at least have told us how to determine

the number of items which a connector connects in a given saying? Well, it

222 What is a Connector?

would have been nice had they pointed out that the question is not to be

answered by merely enumerating the simple sentences which are constituents

of the connected sentence. But that apart, what else is there to say?

‘How many items does this connector here connect?’ Sometimes the answer

is immediately evident—if, say, the saying contains a single connector which

connects two simple sayings. Sometimes the answer is given by some turn of

idiom—for example, there is no doubting the intended parsing of

Sometimes I sits and thinks and sometimes I just sits.

Sometimes the context, or general background knowledge, or simple savvy,

will do the trick. I recently saw on the slate in my local brasserie:

Plat + entr

ee ou dessert:

¤ 16.

I was not troubled by the formal ambiguity. That is all there is to be said, on the

general level. The fact that more cannot be said should not worry a partisan

of multiple connections. For two-placed connectors raise exactly similar

questions, to which—outside the conﬁnes of artiﬁcial languages—there are

no general answers to be had.

So much for the syntactical difﬁculty. Consider now a semantic point. I

said that in the case of the three-part disjunction

IstandorIsitorIlie,

the ﬁrst ‘or’ is two-placed and connects ‘I sit’ to ‘I stand’, whereas the second

‘or’ is three-placed and connects ‘I lie’ to ‘I stand or I sit’. But if that is so, then

the ﬁrst ‘or’ must be taken to be semantically inert—to contribute nothing

to the sense of the sentence. For if it is taken as a two-placed connector and

if it is assigned its standard sense, then either the sentence must be parsed as

[I stand or I sit] or I lie,

or else it is simply ill formed.

Perhaps it is semantically inert? After all, in

Either I stand or I sit,

the word ‘either’ was taken to be a disjunctive connector and yet to be

semantically inert. It is, so to speak, a punctuation mark—or better, an

indication of disjunctive things to come. Why not say something similar

about the ﬁrst ‘or’ in

IstandorIsitorIlie?

Well, perhaps you might develop that line of thought; but there is surely a

better one.

I asked how many places should be assigned to the ﬁrst and to the second

‘or’ in the triple disjunction. If there is an answer it can only be that the

pair of ‘or’s has, as a pair, three places. But it was a bad question. You may

Connection and Uniﬁcation 223

reasonably ask how many disjunctive connectors are present in a sentence, or

how many disjuncts a sentence disjoins; but that is all there is to ask—there is

no further question as to the number of disjuncts each disjunctive connectors

disjoins. In

IstandorIsitorIlie

there are two disjunctive connectors, and the sentence expresses a three-

part disjunction. What more is there to be known about its disjunctive

nature? Nothing.

Well, its disjunctive sense is still to be explained. Earlier, following

Apollonius, I suggested that a disjunctive connector announces that precisely

one of the items which it disjoins is the case. That is a poor suggestion:

if disjunctions are allowed to have two or three or … members, then it

is better to explain what they mean in the way which Sextus adopts,

namely:

A disjunction announces that one of the items in it is sound and the other or the

others false (together with conﬂict)

(PH ii 191)⁸⁵

Similarly, if you want to explain the sense of the Greek disjunctive connector,

then why not say something roughly along these lines:

Asentenceofthesort:‘ἤτοι’ + P

+ ‘ἤ’ + ··· + ‘ἤ’ + P

is true if and

only if precisely one P

is true.

And perhaps, after all, that is what Apollonius means when he says that

the announcement of the disjunctive connectors announces the holding of one and

the removal of the other or of the others.

(conj 216.14–16)

CONNECTION AND UNIFICATION

Connectors connect a plurality of items. But what exactly is a connection? The

two-placed sentential connectives of contemporary logic unify, in the sense

that they take a couple of sentences and make a single sentence. Aristotle says

that connectors unify, and that a connection is a uniﬁcation—or at least, an

accidental uniﬁcation. The ancient grammarians also speak, occasionally, of

⁸⁵ τὸ γὰρ ὑγιὲς διεζευγμένον ἐπαγγέλλεται ἓν τῶν ἐν αὐτῷ ὑγιὲς εἶναι, τὸ δὲ λοιπὸν ἢ τὰ

λοιπὰ ψεῦδος ἢ ψευδῆ μετὰ μάχης.

224 What is a Connector?

uniﬁcation. But the view that connectors unify was contested in antiquity—at

any rate, according to Plutarch,

there are some who think that connectors do not make a unity but that the connected

discourse is an enumeration, as when rulers or days are listed one after the other.

(quaest Plat 1011c)⁸⁶

That anonymous view is not otherwise reported; and it is hard to take seriously

the notion that all connectors simply make for a sequential enumeration.

But there is in principle nothing odd about the notion of a non-unifying

connection.

In the Poetics Aristotle distinguishes between what he calls ‘σύνδεσμοι’

and what he calls ‘ἄρθρα’, or between connectors and articulators. (The

standard translations give ‘conjunctions’ and ‘articles’: ‘articles’ is wildly

misleading—and, as I have already said, Aristotle’s use of ‘ἄρθρον’has

nothing to do with the use of the word in later grammatical texts. As for

‘σύνδεσμος’, ‘conjunction’ is infelicitious in any logical context, and we

should not suppose without some ado that in Aristotle the word has the sense

which it bears in later authors.) The text of the passage is corrupt—and

in particular, Aristotle’s illustrative examples cannot be recovered with any

certainty. But it is plain that articulators serve to articulate a text without

thereby unifying it.

How might that be done? Suppose you are giving a number of short

reasons in favour of a proposition. You might introduce the second and

subsequent reasons with the word ‘again’; and Aristotle might use the word

‘ἔτι’. If I write

X. Again, Y.

IhaveconnectedYtoX,butIhavenotmadeasingleuniﬁedsayingoutof

X and Y. (Why not?—Well, ‘X. Again, Y’ hasn’t got a truth-value: each of

its elements has its own truth-value.—Why not give it a truth-value, saying

that it is true if and only if each of its elements is true?—That is a question

to which I shall half return.) I suggest that ‘again’—in that usage—is an

Aristotelian articulator; and if that is on the right lines, then it is not difﬁcult

to identify further items which articulate without unifying.

The grammarians do not take up Aristotle’s articulators; but some of the

items which they classify as connectors Aristotle might well have classiﬁed

as articulators. So you might surmise that the grammarians’ connectors

⁸⁶ τοὺς δὲ συνδέσμους εἰσὶν οἱ μὴ νομίζοντες ἕν τι ποιεῖν, ἀλλ᾿ ἐξαρίθμησιν εἶναι τὴν

διάλεκτον, ὥσπερ ἀρχόντων ἐφεξῆς ἢ ἡμερῶν καταλεγομένων.

Connection and Uniﬁcation 225

were meant to include both Aristotle’s unifying connectors and also his

non-unifying articulators. And in that case, the grammarians’ connectors will

not always unify.

Evidence in favour of that surmise has been found in Apollonius. Of the

expletive connector ‘δή’ he remarks that

everyone is aware that ‘δή’ is sometimes superﬂuous; but that it also often makes a

transition of sayings is clear from examples such as the following … For we think of

the cessation of one saying and the beginning of another.

(conj 251.19–23)⁸⁷

At least one connector signals the end of one saying and the beginning of

another: so if I write something of the form ‘P—Q δή’, I do not unify a

couple of items—I hold them at arm’s length from one another.

That passage has no fellow in Apollonius’ surviving works; it is part of

an argument, which is often strained, to show that expletive connectors

are genuine connectors; and it concerns the function of ‘δή’whenitis

working together with another connector. So even if the passage unam-

biguously implied that there are non-unifying connectors, we should be

wary of putting much weight on it. But in fact the passage does not

imply that ‘δή’ does not unify. Consider a parallel case. Some people had

wondered how disjunctive connectors could possibly be connectors: surely

they don’t connect things—they disjoin or separate them. Apollonius answers

thus:

The connectors in question are called connectors because they connect phrases and

so possess the characteristic feature of connectors. They are called disjunctive because

of what they indicate—for while being connective of the whole phrase, they disjoin

the objects in it.

(conj 216.2–6)⁸⁸

A disjunctive connector connects (and uniﬁes) syntactically and at the same

time it disjoins semantically. In the same way, Apollonius might have said, the

connector ‘δή’ marks a break semantically and at the same time it constructs

a unity syntactically. He might have said that—and I think he would have

said it.

⁸⁷ ἔτι ὁ δή ὡς μὲν παρέλκει, παντὶ προῦπτον· ὡς δὲ καὶ πολλάκις μετάβασιν λόγου

ποιεῖται, σαφὲς ἐκ τῶν τοιούτων ... νοοῦμεν γὰρ λόγου ἔκλειψιν καὶ ἀρχὴν ἑτέρου.

⁸⁸ οἱ δὴ προκείμενοι σύνδεσμοι εἴρηνται μὲν σύνδεσμοι ἕνεκα τοῦ συνδεῖν τὰς φράσεις,

ὥστε τὸ κοινὸν τῶν συνδέσμων αὐτοὺς ἀναδεδέχθαι· ἕνεκα δὲ τοῦ ἀπ᾿ αὐτῶν δηλουμένου

διαζευκτικοὶ ὠνομάσθησαν. ὅλης γὰρ τῆς φράσεως ὄντες συνδετικοί, τὰ ἐν αὐτῇ πράγματα

διαζευγνύουσιν.

226 What is a Connector?

Apollonius speaks of uniﬁcation in several different linguistic contexts. For

example,

sometimes supervening accidents have produced a sort of uniﬁcation—when ‘τί

ποτε’ suffers from syncope and is uniﬁed in ‘τίπτε σὺ δείδοικας…’.

(adv 149.10–13)⁸⁹

Or again,

‘οἶκον δέ’and‘τὸν οἶκον δέ’ are not the same: ‘I shall go οἶκον δέ’, not ‘I shall

go τὸν οἶκον δέ’. Relying on such arguments, they say that such things have been

uniﬁed into an adverbial derivative.

(adv 181.6–9)⁹⁰

Or again,

It is not plausible, as Trypho says in his On Prepositions, that the prepositions are

uniﬁed with the verbs and yet do not receive any inﬂexions externally inasmuch as,

being prepositions, they ought not to have anything in front of them.

(synt iv 36 [464.9–12])⁹¹

One word is sometimes uniﬁed with another: that is to say—as all the

examples make clear—two words sometimes become a single word.

Apollonius says very little about uniﬁcation in Connectors. But the following

passage which introduces the causal connector ‘γάρ’ or ‘for’ is pertinent.

The particle produces the same construction as ὅτι [or ‘because’], and it has the same

force. But it is different in that … it is not taken at the beginning of the sayings but

in subordination. That is why it displaces into second order the causes which are

placed at the beginning when they are with prepositive connectors, and in that way

the saying is true. Take examples:

Becauseitisday,itislight

If we remove ‘ὅτι’andadd‘γάρ’, the saying becomes false …

(conj 239.9–15)⁹²

⁸⁹ ἔσθ᾿ ὅτε γὰρ τὰ ἐπισυμβαίνοντα πάθη ὡς ἕνωσιν τῶν μορίων ἀπετέλει, ὅτε καὶ τὸ τί

ποτε ὑπὸ συγκοπὴν πεσόντα ἥνωται ἐν τῷ τίπτε σὺ δείδοικας.

⁹⁰ οὐ μὴν ταὐτόν ἐστιν ἐν τῷ οἶκον δέ καὶ τὸν οἶκον δέ· οἶκον δέ γὰρ ἐλεύσομαι, οὐ μὴν

τὸν οἶκον δέ ἐλεύσομαι. τοῖς τοιούτοις λόγοις ἐπανέχοντές φασιν ἡνῶσθαι τὰ τοιαῦτα εἰς

ἐπιρρηματικὴν παραγωγήν.

⁹¹ οὐ γὰρ ἐκεῖνο πιθανόν, καθό φησιν Τρύφων ἐν τῷ περὶ Προθέσεων, ὡς ἡνωμέναι μὲν

εἰσιναἱπροθέσειςμετὰτῶνῥημάτων,οὐμὴντὴνπροσγινομένηνκλίσινἔξωθενἐπιδέχονται,

καθὸ προθέσεις οὖσαι οὐκ ὀφείλουσιν πρὸ αὑτῶν τι ἔχειν.

⁹² τὴν αὐτὴν σύνταξιν ποιεῖ τὸ μόριον τῷ ὅτι, δύναμίν τε τὴν αὐτήν. ἔχει δὲ παραλλαγὰς ...

τὸ ἐν ἀρχῇ μὴ παραλαμβάνεσθαι τῶν λόγων, ἐν ὑποτάξει δέ. διὸ καὶ κατ᾿ ἀρχὴν τιθέμενα τὰ

αἴτια μετὰ τῶν προτακτικῶν συνδέσμων εἰς δευτέραν τάξιν μεθίστησι, καὶ οὕτως ἀληθεύει ὁ

λόγος. ἔστω δὲ ὑποδείγματα· ὅτι ἡμέρα ἐστί, φῶς ἐστιν. εἰ ἀφέλοιμεν τὸν ὅτι καὶ προσθείημεν

τὸν γάρ, ψευδὴς ὁ λόγος γίνεται.

Connection and Uniﬁcation 227

The connectors ‘because’ and ‘for’ have the same force; but ‘because’ is

prepositive whereas ‘for’ is postpositive or subordinative. That is to say,

Because it is day, it is light

is equivalent not to

It is day; for it is light

but to

It is light; for it is day.

Not everything in that passage is uncontestable; and Apollonius’ analysis of

‘for’—or rather, of ‘γάρ’—might be questioned. But it is plain that he takes

it to be a unifying connector, just like ‘because’. In other words, he thinks that

It is light; for it is day.

constitutes a single uniﬁed item. True, he does not here use the word ‘unify’;

but he calls

It is day; for it is light

asaying,andheascribesatruth-valuetoit.

We might have guessed that the word ‘for’ would be counted as an

Aristotelian articulator, or as a connector which links without unifying.

Apollonius takes it to be a uniﬁcatory expression. Was that an eccentric point

ofview?InhiscommentaryontheAnalytics Alexander tries to explain how

a certain argument which is not a genuine syllogism can be pummelled into

syllogistic shape. The argument is this:

A is equal to C

B is equal to C

Therefore A is equal to B

To turn that inference into a syllogism, we need to do two things: ﬁrst, we

must add a universal premiss; and then

we must condense the items which were assumed as two propositions into a single

proposition which has the same force as the two—namely: A and C are equal to the

same thing for they are equal to B.

(in APr 344.17–19)⁹³

Now scholars generally—and understandably—take the single condensed

proposition to be

AandCareequaltothesamething.

⁹³ ... τὰ εἰλημμένα ὡς δύο προτάσεις εἰς μίαν συστείλωμεν πρότασιν ἣ ἴσον ταῖς δύο

δύναται· ἔστι δὲ αὕτη τὸ δὲ Α καὶ Γ τῷ αὐτῷ (τῷ γὰρ Β) ἴσον.

228 What is a Connector?

And having so construed Alexander, they then complain, with justice, that

that proposition is not in fact equivalent to the conjunction of the two

original premisses of the argument. But perhaps the complaint is misplaced,

and perhaps the single proposition is meant to be:

A and C are equal to the same thing, for they are equal to B.

Apollonius would have counted that as a single saying. Perhaps Alexander

did too?

However that may be, there are two further Apollonian passages to be

adduced. Each of them makes a general statement. One comes in Adverbs

and the other in the Syntax.InAdverbs Apollonius says that

connectors never signify anything on their own, but they connect sayings, ordering

them consecutively and thus interconnecting and unifying them.

(adv 133.25–134.1)⁹⁴

That clearly indicates that connectors always unify—indeed, it suggests that

connecting is precisely a matter of uniﬁcation. And in the Syntax:

The connectors which accompany the sayings sometimes unify two or even more

sayings … or again, being absent, make a dissolution of the sayings.

(synt i 10 [11.3–7])⁹⁵

That has been read as suggesting that connectors sometimes unify and

sometimes merely connect. But that is not Apollonius’ meaning. The word

‘sometimes’ is picked up by ‘again’: connectors sometimes, by their presence,

unify, and sometimes, by their absence, dissolve. In other words, connectors

always unify.

Connectors unify—they take a number of items and they produce a single

item. But a single what? There is, of course, no reason to expect any answer

to that question beyond the empty ‘A single expression’; for what item a

connector makes will depend in part on what items it takes. Let ‘and’ take

‘you’ and ‘I’ to make ‘you and I’. What sort of an expression is that? We

might call it a pronominal phrase, or a pronominal complex; and similarly we

might call ‘if and when’ a connective phrase, ‘now if ever’ an adverbial phrase,

and so on. But the ancient grammarians, as I have already remarked, had

no terminology for and no theoretical interest in such complex expressions

⁹⁴ οἱ δὲ σύνδεσμοι οὔποτε κατ᾿ ἰδίαν σημαίνουσί τι, συνδέουσι δέ τοὺς λόγους, ἑξῆς

τάσσοντες καὶ οὕτως ἐπισυνδέοντες καὶ ἑνοῦντες.

⁹⁵ ἀλλὰ κἀν τοῖς λόγοις οἱ παρεπόμενοι σύνδεσμοι ἔσθ᾿ ὅτε ἑνοῦσι δύο λόγους ἢ καὶ πλείους,

... ἢ πάλιν ἀποστάντες διάλυσιν τῶν λόγων ποιοῦνται.

Connection and Uniﬁcation 229

which are longer than a word and shorter than a sentence. On Aristotle’s

account of things, such items are sayings; but had the question been put, it

may be doubted if he—or anyone else—would so have classiﬁed them.

Suppose, in particular, that a connector takes a number of sayings and

makes one item out of them: what will the new item be? Surely, a saying.

For it can hardly be anything smaller than a saying, and ancient grammar

recognizes no unit larger than a saying. Any compound of sayings will be a

compound saying. Then consider the following three examples:

I came, I saw, I overcame.

First I came, then I saw, then I overcame.

I came, and I saw, and I overcame.

The ﬁrst is a connectorless sequence of sentences; the second adds three

adverbs—or perhaps three Aristotelian articulators; and in the third example,

there is the connector ‘and’. The third example is, on (almost) anyone’s

account, a single saying and a single sentence. (It is also, of course, three

sentences and three sayings.) But if it is a single saying, then are not the ﬁrst

two examples also single sayings? For aren’t the three examples just three ways

of saying the same thing? When Caesar said

veni vidi vici

he said something. How many things did he say? Three things, no doubt;

and also, no doubt, one thing. What was that one thing? Well, in uttering

that sequence of expressions Caesar said that he came and saw and overcame.

It is not evident that Apollonius would have resisted that conclusion. He

apparently countenanced certain connectorless connections, as we shall see.

For example, he apparently thought that

It’s day. It’s night

is a disjunction, and hence a single saying. Then why should he not think

that

veni vidi vici

is a conjunction, and hence a single saying? Well, I suspect that he would in

fact have denied that the Caesarian sequence is a conjunction. For if

It’s day. It’s night

is a disjunction, then that is so inasmuch as its two elements are ‘naturally’

disjoined—and the elements of Caesar’s remark are not naturally conjoined.

(There are no natural conjuncts.) In any event, neither Apollonius nor anyone

else will suppose that any sequence of sayings which is not uniﬁed by any

connectors constitutes a conjunction.