The Principle of Bivalence in de Interpretatione - F. Ademollo

8/11/2019 The Principle of Bivalence in de Interpretatione - F. Ademollo

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THE PRINCIPLE OF BIVALENCE INDE INTERPRETATIONE4

Francesco Ademollo

[Oxford Studies in Ancient Philosophy38 (2010), 97113; unedited final version.]

In chapter 4 ofDe interpretationeAristotle introduces the notion of a !"#$%, saying,1and draws a

crucial distinction between declarative2and non-declarative sayings (17

a24):

&'$()*+,-.%/0$1'2%[sc.!"#$%],&!!34*5+.&!67898,*:;89/8


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declarative sayings are either true or false, i.e. that an individual saying is declarative if and only if

either it is true or it is false. If this is so, then it follows, among other things, that Aristotle is

committing himself to a general principle of bivalence like the one set forth at Categories4. 2a710

(cf. 10, 13a37b3, b2735):

Every affirmation seems to be either true or false [A')()


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correct. So the question is: how is Aristotles statement of Bivalence in chapter 4 to be reconciled

with his denial of (unrestricted) Bivalence in chapter 9?

There are various possible ways of solving this puzzle. We might supposeas some have

actually donethat the two chapters were composed at different times, that Aristotle changed his

mind about bivalence in the meantime, and that when he added chapter 9 he failed to make chapter

4 harmonize with it.6This sort of hypothesis strikes me as both risky and unattractive, but it cannot

be ruled out. Alternatively, we may suppose that in chapter 4 Aristotle is speaking somewhat

carelessly and that his careless generalization is going to be corrected in chapter 9. This possibility

too is rather unpalatable, although it has to be said that Aristotle is actually careless on this score in

ch. 1, 16a918, where he expresses himself as if he were assuming that any complete sentence (as

opposed to names and verbs by themselves) is true or false.

There is also a third possibility, which has been advocated by Richard Gaskin and Paolo

Crivelli.7Perhaps we should resist the temptation to think that in chapter 4 Aristotle is claiming that

all and onlydeclarative sayings are either true or false and rather take him to mean just that only

declarative sayings are either true or falsewhich does not entail that allare. In other words, an

individual saying is declarativeifit is either true or falsenot if and only ifit is either true or false.8

This interpretation is, in a way, obviously superior to the previous two. For it saves the treatises

consistency without accusing Aristotle of inaccuracy or resorting to hazardous hypotheses about its

composition. But can Aristotles words really mean what this interpretation takes them to mean?

And can this make good philosophical sense as a characterization of the declarative saying? I

incline to believe that the answer to both questions is Yes.

5For a different view see Whitaker, ch. 9, who maintains that the principle discussed in chapter 9, and ultimately

rejected by Aristotle as regards future contingents, is not Bivalence but rather what he dubs the Rule of Contradictory

Pairs (= RCP), according to which of two contradictory sentences one is true and the other is false. Whitaker fails to

address the compelling textual argument advanced by Ackrill, 1334, to show that Bivalence, not RCP, is the principle

under discussion; moreover, I find his interpretation unsatisfactory in various philosophical respects which I will not

dwell upon here.

6D. Frede,Aristoteles und die Seeschlacht(Gttingen, 1970), 81; The Sea-Battle Reconsidered: A Defence of the

Traditional Interpretation, Oxford Studies in Ancient Philosophy, 3 (1985), 3187 at 81.

7R. Gaskin, The Sea Battle and the Master Argument(Berlin and New York, 1995), 180; Crivelli, 867.

8Cf. J.-B. Gourinat, Lhistoire du principe de bivalence slon "ukasiewicz, in R. Pouivet and M. Rebuschi (eds.),

La philosophie en Pologne, 19181939(Paris, 2006), 3766 at 51. This construal is also hinted at by Jonathan Barnes,

who personally confirmed to me that he endorses it. He writes (Truth, 3): you might doubt that Aristotle intendsthereby to define the notion of assertion; and in any event it is not evident that the definition means that every

assertion is either true or false.


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To start with the former question, notice that Aristotle says the declarative saying is the one 4*5

+.&!67898,*:;89/8?@8, with the dative.9Now, in the Aristotelian corpus

there is at least a couple of passages where the phrase ='>?@8, 4* is used with reference to a

situation in which a feature is possessed by some but not all of the members of a certain class. Here

is Categories7, 6b1519:

Contrariety too holds in relatives [R'>?@8,/0 -)S 4*)*+,"+6% 4* +$P%'?"% +,], e.g. virtue is

contrary to vice (and each of them is relative), and knowledge to ignorance. But there is not a

contrary to every relative [$1'2?@8,4*)*+T$*]: for there is no contrary to

the double or the triple or any of such items.10

Some, but not all, relatives have a contrary; and Aristotle comments on this by claiming that

contrariety holds in relatives. Likewise, some but not all declarative sayings are either true or

false; so inInt. 4 Aristotle may well be commenting on this as he claims that the declarative saying

is that in which being true or being false holds. On this interpretation, the fact that onlydeclarative

sayings are true or false is not expressed by the phrase ='>?@8,4*, but rather by Aristotles use of a

definite description: not every saying is declarative, but only that in whichbeing true or being false

holds.

There is another very similar passage at Cat.8, 10b1217:

Contrariety too holds with regard to quality [R'>?@8,/0 -)S 4*)*+,"+6% -)+O+.'$,"*]. E.g.

justice is contrary to injustice, whiteness to blackness, and so on, and also things said to be

qualified in virtue of them but this sort of thing does not hold for all cases [ $1-4'S'>*+U*/0+.

+$,$V+$*]; for there is no contrary to red or yellow or such colours, though they are qualified items.

Here the relevant turn of phrase is not ='>?@8,4*, but ='>?@8,-)+>+ accusative; but it comes

much to the same thing.

A third passage from the Categoriesis 12, 14a

35b

1:

Thirdly, a thing is called prior in respect of some order, as with sciences and speeches. For the

prior and posterior in order hold in the demonstrative sciences [N* +8 #O? +)P%&'$/8,-+,-)P%

9PaceWeidemann, who translates Int.17

a24 as ein Behauptungssatz aber ist nicht jedes, sondern nur eines, dem

es zukommt, wahr oder falsch zu sein, and W. Cavini, Principia contradictionis. Sui principi aristotelici della

contraddizione ( 13) [Principia],Antiqvorum Philosophia, 1 (2007), 12369 at 126, who translates ma non ogni

dichiarativo, se non quello cui appartienelessere vero o falso (my italics throughout).10

Here and in the next quotations from the Categories and the De interpretatione I have modified Ackrills

translation.


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4',?@8,4*alone is not enough to make

the latter point, and Aristotle needs to add the adverb only ("*$*).

You will notice that all the instances of ='>?@8,4*which I have been citing are followed by a

dative plural, whereas in Int. 4 Aristotle uses it with the singular. But I doubt that this is a

significant problem. Indeed, the singular is due to the fact that inInt.4, as in the parallel passages I

have cited, Aristotle is talking about features which hold in kinds; in our particular case, he is

concerning himself with the issue of whether or not the features of being true and being false hold

in various kinds of saying, including the declarative one. And when Aristotle says that a non-

essential feature is in a kind, or belongs to it in some way or other, he need not thereby imply that

the feature belongs to every instance of the kind. Thus at Cat.5, 2b

12 Aristotle says that colour is

in body[+.@?M)4*


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(literally Human being is white) are not equivalent to universal sentences (Every human being is

white) but rather toparticularsentences (Some human beings are white).

These things being so, the first question I asked above is to be answered in the affirmative:

Aristotles words in Int.4 are, to say the least, linguistically compatible with the Gaskin/Crivelli

construal. Let us now turn to the second question. At first blush it seems reasonable to assume that

Aristotle ought to be specifying a feature which, besides belonging onlyto declarative sayings, also

belongs to every declarative saying. Surely, therefore, it is pretty unhelpful to characterize the

declarative saying as we are supposing that Aristotle does?

Not necessarily. It seems to me that Aristotle canbelieve that only but not all declarative sayings

are true or false and yet characterize declarative sayings in terms of their being either true or false, if

he holds that those which are either true or false constitute, as it were, the standardorprimarycase

of declarative saying and that it is somehow by reference to them that also the others, which are

neither true nor false, have to be conceived of. Crivelli, 7, says something which you can regard as

one particular version of this suggestion: declarative sayings coincide with truth-evaluable

sentences, i.e. with the sentences with regard to which the question Is it true or false? can be

reasonably asked this question cannot be reasonably asked with regard to certain sentences (e.g.

prayers). In the case of some sentences with regard to which the question Is it true or false? can

be reasonably asked, the correct answer is Neither.Perhaps this suggestion can be refined a bit further. Faced with the task of identifying a feature

which belongs to all and only declarative sayings, both to those which are either true or false and to

those which are neither true nor false, we should not content ourselves with claiming that both sorts

of saying are truth-evaluable, i.e. that both are such that it can be reasonably asked about them

whether they are true or false; we should also want to spell out why that question is a reasonable

one.

To start with, we can rephrase the point in terms of Hodgess test for declarative sentences,13

thus: a saying is declarative if, and only if, substituting it14for P both in

(5) Is it true thatP?

and in

(6) Is it false thatP?

13

W. Hodges,Logic, 2nd edn. (London, 2001), 56.14

Or rather, if sayings are utterances, substituting a token-inscription of the same type. For the view that Aristotelian

declarative sayings are utterances see Crivelli, 726 (although I do not agree with all of his arguments).


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yields a grammatical result. Sayings like questions, commands and prayers obviously fail this test.

By contrast, sayings like (1), (2), (3) and (4) all pass the test, and hence are declarative. But while

sayings like (1), (2) and (3) are such that in their case the answer to either (5) or (6), though not to

both, is Yes, sayings like (4)declarative sayings about future contingentsare special in that in

their case the answer to both(5) and(6) is No.15

This is a way of making it clearer in what sense it is reasonable to ask with regard to

declarative sayings whether they are true or false. But thereby we have not really brought out why

this is so. To get closer to a real explanation we could try something along the following lines:

although declarative sayings about future contingents are neither true nor false, they are as it were

fit for being true or false. This means that, if the world were different than it isi.e. if the future

were completely determined, as it actually is not, and hence there were no future contingent

events, then such sayings would be either true or false. Indeed, Aristotle himself, on at least one

possible interpretation of his views in Int.9, believes that such sayings will become either true or

false in the course of time, i.e. as soon as the events taking place or failing to take place is

determined. E.g., an utterance of (4) which lacks a truth-value now will become either true or false

by tomorrow.16And even now, when the future is not yet determined and (4) still lacks a truth-

value, declarative sayings about future contingents are already taken to be either true or false by the

determinist, who regards the future as already fixed and hence takes (4) to be either true or false

without thereby displaying any sort of linguistic incompetence.

By contrast, nothing similar holds of sayings like questions and prayers. These would still be

neither true nor false in a determinist world; the course of time will not bring them a whit closer to

acquiring a truth-value; and no one who understood them correctly could take them to be either true

or false. In other words, while the fact that declarative sayings about future contingents are neither

true nor false depends on the way the world is (the future is at least partly open), the fact that

sayings like questions and prayers are neither true nor false depends on something about those

sayings themselves.

15I am assuming that declarative sayings about future contingents are the only kind of declarative sayings which are

neither true nor false. I do so for the sake of simplicity and because they are the only kind recognized in the De

interpretatione. Some interpreters believe that at SE25, 180a34

b7 Aristotle addresses the paradox of the Liar and that

his solution consists in rejecting Bivalence for such paradoxical sentences as I am speaking falsely; but it is less than

clear that this is so. See P. Fait,Aristotele: Le confutazioni sofistiche. Organon VI(Rome and Bari, 2007), 20811.

16

The key text is the famous 19

a

369: Necessarily, one of the two members of the contradictory pair is true orfalsenot, however, this one or that one, but however it chances, and one of the two is more true, but not already[Q/6]

true or false. For discussion see Crivelli, 21626.


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So, if Aristotle has an intuition along these linesif he thinks that (i) any declarative saying is

such that you can ask whether it is true or false, and indeed (i) any declarative saying is, in some

sense, fit for being true or false, even though (ii) only some (most) of them are actually true or

false, then it is understandable that inInt.4 he characterizes the declarative saying as the sort of

saying in which being true or being false hold, i.e. (on my construal) some (most) of whose

instances are either true or false.

We may now want to push our analysis one step further and ask what, according to Aristotle,

makesdeclarative sayings (both those which are either true or false and those which are neither true

nor false) fit, and other sorts of sayings unfit, for being true or false. There are at least two possible

ways of trying to answer this question.

One attempt could go as follows. There is some reason to ascribe to Aristotle the view that all

declarative sayings have this in common as against other kinds of saying, that they are capable of

expressing beliefs. For Aristotles phrase !"#$%&'$()*+,-"%is certainly meant to suggest that the

kind of saying at issue &'$()T*8,, declares or reveals, somethingsomething which is

presumably a belief. For Aristotle holds that linguistic expressions are signs of affections of the

soul (Int. 1) and, in particular, declarative sayings are signs of /"Y),, beliefs (Int. 14, 24b13);

moreover, the verb &'$()T*U has a fairly well-established use in the turn of phrase #*[6*

/"Y)*&'$()T*8,*, to express a view / belief.17Thus we could suppose that, on Aristotles view,

declarative sayings are all fit for being true or false because they can all express beliefs. But this

answer would not be completely satisfactory. For it would be based on something (i.e. declarative

sayings being capable of expressing beliefs) which seems to play a rather marginal role in

Aristotles reflections. Moreover, it would raise further difficult questions: why are beliefs fit for

being true or false in the first place? Would Aristotle answer that this is a primitive fact which

admits of no further explanation?

So I will leave this first attempt aside and turn to a different one, which requires that we move on

to chapter 5. There, among other things, Aristotle draws two distinctions: one between (A)

declarative sayings that are single and (B) declarative sayings that are many; another, narrower

one between (A1) declarative sayings that are single because they indicate one single thing and

(A2) declarative sayings that are single in virtue of a connective. Moreover, from the very

beginning of the chapter (17a89) he makes it sufficiently clear that affirmation (-)+>()


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denial (&'"()()


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is close to his definitions of premiss ('?"+)*


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chapter 1, 16a1213: falsehood and truth have to do with combination and division.

23As far as

declarative sayings about the past or the present are concerned, there is a real combination or

division to which they correspond or fail to correspond, hence they are either true or false. As far as

declarative sayings about future contingent events are concerned, instead, there is no fact of the

matter, no real combination or division for them to match or mismatch; hence they are fit for being

true or false while not being actually (or perhaps yet) true or false.

So far so good. Now, it is interesting to notice that the sort of strategy which I am ascribing to

Aristotle inInt.4 finds a partial parallel in chapter 5 of the Categories. There, among other things,

Aristotle has to find out the proprium of substance. Here is his well-known proposal (4 a1021):

It seems most proper to substance that it is something which, while being numerically one and the

same, is capable of receiving contraries [e>!,


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substance, but not of every substance ("*n0*='>?@8,+X$1?@8,, 78. 5; cf. Simplicius, In Cat. 113. 1315, 2031 Kalbfleisch).25 As

Ackrill, 89, puts it, Aristotle is not speaking of the possibility of mans being both dark and pale

(of there being both dark men and pale men), but of the possibility of one and the same individual

mans being at one time dark and at another time pale. Then what about secondary substances? We

can follow Ackrill in thinking that their proprium will be that they are the genera and species of

individuals which, while remaining numerically one and the same, are capable of receiving

contraries. Primary and secondary substances receive an analogous sort of characterization at the

beginning of the same chapter, 2a1119:

A substancethat which is called a substance most strictly, primarily, and most of allis that

which is neither said of a subject nor in a subject, e.g. the individual human being or the individual

horse. Secondary substances are called the species in which the things primarily called substances

areboth these species and their genera.

Here all that we are given as a general characteristic of substance is a disjunctive feature: X is a

substance if and only if eitherXis neither said of a subject nor in a subject (primary substance) orX

is a genus or species of something that is neither said of a subject nor in a subject (secondary

substance).26Likewise, all that 4a1021 allows us to identify as a general proprium of substance is a

disjunctive feature:Xis a substance if and only if eitherXis an individual which, while remaining

one and the same, is capable of receiving contraries orX is a genus or species of such an

individual.27

But there the case of secondary substances is not explicitly taken into account.

So at the end ofCat.5 Aristotle puts forward as a proprium of substance something which in fact

is only a proprium of an especially relevant class of substances, namely primary ones, leaving it to

us to expand that into something which will hold of secondary substances as well. Likewise, on the

25 In fact Simplicius, 114. 23115. 10 Kalbfleisch, argues that being capable of receiving opposites fails to hold

even of everyprimarysubstance, because it fails to hold of the heavenly bodies like the sun.

Here I will not discuss Aristotles view that being capable of receiving opposites holds onlyof substances. In Cat.5.

4a21

b19 Aristotle himself confronts the apparent counterexample constituted by saying (!"#$%) and belief (/"Y)),

which are capable of receiving truth and falsity. Barnes, Truth, 39 suggests other counterexamples.

26It is up to us to go beyond this merely disjunctive analysis and identify some metaphysically significant common

ground between primary and secondary substances, as is convincingly done by M. Kohl, Substancehood and

Subjecthood in Aristotles Categories,Phronesis, 53 (2008), 15279.

27

Philoponus solution is somewhat different: it seems to be especially proper to the category of substance that theindividuals subordinate to it are capable of receiving contraries in turn (79. 23 Busse). Here the proprium is

formulated as something that holds, not of every substance, but rather of Substance as a highest genus.


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present interpretation, inInt.4 Aristotle puts forward as a characteristic of the declarative saying the

fact that some (indeed most) declarative sayings have a certain feature, leaving it to us to explain

the unity between such sayings and those (which we can regard as special or deviant cases) which

lack that feature while being declarative nonetheless.

A characteristic of the declarative saying. What sortof characteristic? We have already found

evidence that inInt.4 Aristotle does not think he is specifying a definition of the declarative saying;

and perhaps it is no accident that he does not claim he is specifying a proprium either. Still, the

parallel with Cat.5 shows that Aristotle mightregard being either true or false as a proprium of the

declarative saying.

This sort of viewthe view that something which holds only of kind K, but not of every K,

might count as a proprium of Kis not advanced in Aristotles discussion of the proprium in the

Topics. Indeed, it is incompatible with Aristotles assumption, made out in Top. 1. 5, 102a1819

and elsewhere, that the proprium of K must be co-extensive with K (&*+,-)+6#$?8P


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Aristotle immediately goes on to illustrate this by a standard example of co-extensive proprium

(like mortal animal capable of receiving knowledge, in the case of the human being, b356); but

the definition literally means just this: Ais a proprium in its own right ofB=dffor everyX, ifXis

notB, thenXis notA; that is to say, Ais a proprium in its own right ofB=dfeveryA isB; that is

to say, Ais a proprium in its own right ofB=dfonlyBs areA.

And thus the view Aristotle never states came to be stated by the ancient commentators.

Simplicius,In Cat.113. 27 Kalbfleisch, in the course of arguing that the proprium of Cat. 5 holds

only of substance but not of every substance, claims that in the fifth book of the Topicshe defined

the proprium thus, presumably referring to 128b345.31Before him Porphyry, Isagoge12. 1314

Busse, claims that there are four sorts of proprium, or senses of the term proprium, and identifies

this as the first one:

what is an accident only of a certain species, even if not of it all [q"*p+,*S8f/8,


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Bods, R.,Aristote: [Catgories](Paris, 2001).

Cavini, W., Principia contradictionis. Sui principi aristotelici della contraddizione ( 13)

[Principia],Antiqvorum Philosophia, 1 (2007), 12369.

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(Dordrecht, 1986), 4980.

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Frede, D.,Aristoteles und die Seeschlacht(Gttingen, 1970).

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79.

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truth and falsity(Amsterdam and London, 1973).

Owen, G.E.L., Aristotle on the Snares of Ontology. In R. Bambrough (ed.), New Essays on Plato

and Aristotle (London, 1965), 6995; repr. in G.E.L. Owen, Logic, Science and Dialectic.

Collected papers in Greek philosophy(London, 1986), 25978.

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2000).

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The Principle of Bivalence in de Interpretatione - F. Ademollo