The Law of ContradictionAuthor(s): Jonathan BarnesSource: The Philosophical Quarterly, Vol. 19, No. 77 (Oct., 1969), pp. 302-309Published by: Blackwell Publishing for The Philosophical QuarterlyStable URL: http://www.jstor.org/stable/2217842 .Accessed: 24/01/2011 19:01

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302

THE LAW OF CONTRADICTION

BY JONATHAN BARNES

Section I presents an argument for a proposition which might reasonably be expressed by the sentence ' The Law of Contradiction is a Law of Thought'. Section II examines a similar argument given by Aristotle in his Meta- physics. Section III sets out some propositions which the argument does not prove.

The symbols ' aB:(P) ' and ' aD:(P) ' abbreviate ' a believes that P' and 'a disbelieves that P '; the rest of the notation is standard, except for the relational constant 'C' and the operator ' ' which are explained as they appear.

I The argument has three premisses, each of which is necessarily true; in

the case of each premiss I shall offer a few considerations designed to bring this out.

The first premiss is: (1) (x) ((xB:(P & Q)) - ((xB:(P)) & (xB:(Q)))). This is a special case of: (1.0) (x) ((xB:(P & P & . . . & Pn))-+ ((xB:(P1) )& (xB:(P2)) &

... & (xB:(Pn) ) ) ). Informally, (1.0) states that anyone who believes a conjunction of proposi- tions believes each of the conjoined propositions. I shall try to support the weaker thesis (1), which is all that the argument requires; but analogous points could plainly be advanced in support of (1.0).

To deny the necessity of (1) is to assert that a man may at one time both (a) believe that both P and Q, and also (b) either not believe that P or not believe that Q or both not believe that P and not believe that Q. And this is absurd: if a man does not believe, say, that Q, how can he possibly believe that both P and Q?

Let us try to imagine circumstances in which someone says that he believes some conjunction and yet at the same time denies that he believes one of the conjuncts. Suppose that a foreign student reads in a digest of English literature the sentence ' Pepys and Evelyn wrote diaries', but does not meet with either of the sentences 'Pepys wrote a diary' and 'Evelyn wrote a diary'; and suppose that this leads him in an examination to assert that he believes that Pepys and Evelyn wrote diaries and to deny that he believes that Evelyn wrote a diary. Could we make anything of this answer ? We might conjecture that the candidate was making a curious joke or attempting to simulate lunacy; but if explanations of this sort were elimin- ated, I think we could only conclude that he did not understand the con-

THE LAW OF CONTRADICTION 303

junctive sentence he wrote and hence could not believe the conjunctive proposition he claimed to believe. (This might happen in more than one

way : he might, for example, be ignorant of the meaning of the connective

'and'; or think that Pepys and Evelyn were collaborators like Beaumont and Fletcher; or imagine that 'wrote diaries' was a relational predicate like 'corresponded with one another '.)

In short, if anyone asserts that he believes that P and Q, and also asserts that he does not believe that P, one at least of his assertions is not true: either he is lying or joking or otherwise unserious, or else he does not under- stand one of the sentences which express the propositions which he claims to believe.

It is worth noting that the converse of (1): (1.1) (x) ( ( (xB:(P) ) & (xB:(Q) ) ) - (xB:(P & Q) ) does not hold-it is possible to believe a number of propositions and yet be ignorant of the operation of conjunction. It is clear too that the dis-

junctive and conditional analogues of (1): (1.2) (x) ( (xB:(P v Q) - (x:() )v (xB:(Q) ) ) ) and:

(1.3) (x) ( (xB:(P - Q) ) ( (xB:(P) ) (xB:() ) ) ) are not valid; and neither are the disjunctive and conditional analogues of

(1.1). The notion of contrary predicates is deployed by Aristotle in the argu-

ment which I shall discuss in Section II; and although the notion is not

necessary to the present argument, it is convenient to introduce it here be- fore setting down premiss (2).

Contrariety has a long, but not unambiguous, history. It is useful to

distinguish between three relations, each of which Aristotle seems to have

thought of as a type of contrariety. In the following definitions RF-the

range of F-is, roughly, the class of objects which can be called " F " with- out commission of a category mistake; ' Ci(F, G) ' reads 'G is contraryi to F'. (D1) C1(F, G) = (x) ( (RF = RG) & D (Gx -- Fx) ) (D2) C2(F, G) = (x) ((RF = RG) & D (Gx - Fx) &

((x s RF) -> D (Fx v Gx))) (:D3) C3(F, G) = (x) ( (RF =RG) & D (Gx -> Fx) & ( (H) ( ( (RH =

RF) & (H # F) & (H # G) ) -> (H is between F and G)))) Contrariety, might be called incompatibility; paradigm contraries1 are red and green, hot and cold. Contrary2 predicates might be called contradictory predicates; typical contraries2 are odd and even, guilty and innocent.

Contrariety3 might be called polar opposition; examples of contraries3 are black and white, bald and hirsute.

These definitions are not very satisfactory: some account is needed of what the range of a predicate is, and in (D3) the metaphor of being between demands explanation. However, I shall not attempt to satisfy these require- ments here since I do not think that anything in Aristotle's argument turns

304 JONATHAN BARNES

on the more questionable features of the definitions. In what follows I shall not be concerned with either contrariety2 or contrariety3, and hence- forth I shall omit the subscripts and use 'contraries' and 'C' to refer to contraries1 and Cl.

The second premiss of the argument is: (2) (x) ((xD:(P ))- ( xB:(P)) ). This is a necessary truth if (but not only if) believing that P and disbelieving that P are contraries. The Oxford Dictionary asserts that disbelief is the state contrary to that of belief; and although this is not strictly true,1 it seems to me clear that belief and disbelief-like praise and blame, love and hate, approbation and disapprobation-do fulfil the defining conditions of the relation C.2 In particular, it seems indisputable that anyone who dis- believes that P must thereby not believe that P: for how could a man both believe that P and also not not believe that P ? Disbelieving is simply one way of not believing, just as loathing a person is simply one way of not liking him. And just as there are other ways of not liking a man-for ex- ample, feeling completely indifferent to him-so there are other ways of not believing a proposition-for example, feeling uncertain about its truth- value. This last point shows that the converse of (2): (2.0) (x) ( (i xB:(P) ) ->(xD:(P))) is not a necessary truth. Indeed, if the range of the variable ' x ' is restricted to, say, stones, (2.0) is necessarily false.

The necessary truth of: (3) (x) ((xB:(-7 P) ) (xD:(P)) ) is evident. The conditional may be strengthened to a biconditional to give: (3.0) (x)((xB:(. P))) (xD:(P))) which might serve as a contextual definition of disbelief. To disbelieve a man's testimony just is to believe that his testimony is false; and in general, disbelieving that P just is believing that not-P.

It might be objected against (3.0) that 'a doesn't believe that P' is

compatible with 'a doesn't believe that not-P ' (a might be unsure whether or not P), so that the sentence:

(S) If a doesn't believe that P, then a believes that not-P does not express a necessarily true proposition. This, of course, can only hold as an objection against (3.0) if' a doesn't believe that P ' is synonymous with 'a disbelieves that P'. But it does appear that 'a doesn't believe that P' at least sometimes means the same as 'a disbelieves that P '-I can

reject a story as well by saying 'I don't believe you ' as by the more ponder- ous 'I disbelieve you '.

I think that this objection only shows that the everyday negative ' doesn't believe that' is ambiguous. We can say both of the atheist and of the

agnostic that he doesn't believe that God exists, but when we say it of the

l(D1) gives no guarantee of uniqueness, and so does not warrant our talking of the contrary of a given predicate.

2They might well fit (D3) on some satisfactory account of the relation of being between; I suspect that Aristotle thought of them as contraries of the third type.

THE LAW OF CONTRADICTION 305

former we mean that it is the case that he believes that God does not exist, and when we say it of the latter we mean that it is not the case that he believes that God does exist. It is plain that if we take 'a doesn't believe that P' in the atheist sense, then 'a doesn't believe that P' is incompatible with ' a doesn't believe that not-P '; and if we take it in the agnostic sense, then ' a doesn't believe that P' and ' a doesn't believe that not-P ' are com- patible, but their compatibility does nothing to show that (3.0) is false.

The argument from these three premisses can best be put in the form of a reductio ad absurdum. Let us assume that: (4) (3x) (xB:(P & m P)) and also that: (5) aB:(P & 7 P). From (1), with '-i P' for 'Q ', and (5) there follows by universal instantia- tion and modus ponendo ponens : (6) (aB:(P) )& (aB:(-i P) ). By suitable substitutions in the theorem: (T) (P & Q) - ( (Q - R) - (P & R) ) and two applications of modus ponens, (6) and the relevant instantiation of (3) yield: (7) (aB:(P)) & (aD:(P)). And (7) and an instantiation of (2), again by substitution in (T) and two applications of modus ponens, give: (8) (aB:(P)) & (aB:(P) ). Quantification over (8) yields: (9) (3x)( (xB:(P) ) & (xB:(P)) ). Since (9) follows from (1), (2), (3) and (5), and its derivation satisfies the standard conditions for existential generalization, (9) follows from (1), (2), (3) and (4). But the contradictory of (9): (10) (x) - ((xB:(P)) & (-xB:(P)) ) is a variant of the theorem: (11) (x) - (Fx & - Fx). Hence (4) is incompatible with (1), (2) and (3); and so its contradictory: (12) i (3x) (xB:(P & -i P)) is a necessary truth.

The argument does not depend on the properties of any particular proposition P; its scope is quite general. Thus its conclusion, (12), can be put informally as follows : no-one can believe the conjunction of any pro- position and its negation. A slightly more dramatic way of expressing this is to say that the Law of Contradiction is a Law of Thought.

II The traditional doctrine of the Laws of Thought can be traced back to

a passage in Aristotle. It should be stressed, however, that Aristotle does not recognise four Laws, but only one; and that he does not put forward psychological points or use the vague phrase 'a Law of Thought', but gives

306 JONATHAN BARNES

a short and rigorous argument for a precisely formulated conclusion. Aris- totle's argument is similar to the one given in Section I; since it is difficult to follow as it stands in the text, and since its ingenuity has not been

appreciated by the commentators, I shall set out the passage in translation and then offer a few explanatory remarks.3

(A) A principle firmest4 of all is one concerning which it is impossible to be mistaken. . . . What this is let us now say: It is impossible for the same thing at the same time both to belong and not to

belong to the same thing and in the same respect (and let us suppose added all the other qualifications which we should add to guard against the dialectical difficulties).5 This, then, is firmest of all the

principles, (107T) for it answers to the definition given above.

(106T) For it is impossible for anyone to believe that the same

thing is and is not, as some think Heraclitus says. For it is not

necessary that a man believes what he says. But if (102T) it is not possible for contraries to belong at the same time to the same

thing (let us suppose the usual qualifications added to this pro- position too), and if (103T) the opinion contrary to an opinion is the opinion of the contradictory, (105T) it is evident that it is im-

possible for the same man at the same time to believe that the same thing is and is not; (104T) for the man who was mistaken about this would have contrary opinions at the same time. (Meta- physics, F 3, 1005bll-12; 18-32)

This passage can be supplemented by an extract from a later chapter of the book:

(B) Since (100T) it is impossible for contradictories to be true at the same time of the same thing, it is evident that (102T) it is not possible for contraries to belong at the same time to the same thing either.

(101T) For of contraries one is no less a privation, and a privation is a denial of being to some determinate subject-matter.6 (ib. 6, 1011b15-20)

These passages demand close reading; it may help if the steps of the

argument are re-arranged somewhat and the proof set out more formally. The proof is conducted, Aristotle says (ib. 4, 1006a4), by means of the

Law of Contradiction; the Law appears in the form of:

(100) (F) (x) -

(Fx & - Fx). From (100) and:

3The numerals inserted in the translation key sentences of the text-hence the adscript ' T '-with formulae in the explanatory remarks. The other adscripts I use are self-explanatory.

4The superlative implies that Aristotle thought only one logical law could be a Law of Thought.

5For these see also : de Interpretatione, 6, 17a35-37; Topics, I 5, 166b37-167a35; and compare Eudemian Ethics, B 3, 1221b4-7. The dialectical difficulties can safely be ignored in a modern presentation of Aristotle's argument.

6The Greek is uncertain here. I have translated the reading of MSS E and J; the text offered by Ab is hardly possible. Most editors, including Ross and Jaeger, follow Alexander's hybrid version of the passage.

THE LAW OF CONTRADICTION 307

(101)7 (F) (G) (x) ( (C(F, G) ) - (Gx -m - Fx))

Aristotle infers (" .7.t ", 1011b15 ; " yap ", 1011b18): (102) (F) (G) (x) ((C(F, G) ) - - (Fx & Gx)). This is the argument of (B). Aristotle seems to find its validity patent; it could readily be set out in an amplified version of any standard treatment of the predicate calculus.

Next let us take: (103) C(B:(P), B:( P)). Aristotle does not argue for (103) in the Metaphysics; some commentators have thought that he tries to prove it in chapter 14 of the de Interpretatione, but the chapter is extremely obscure. However, my comments in Section I on (2) and (3) ought together to be enough to establish the truth of (103). Aristotle's argument might have been clearer had he interpolated: (103a) C(B:(P), D:(P) ).

From (102) there follows: (104) (x) ( (C(B:(P), B:(Q) ) & ( (xB:(Q) ) ) ). The inference is a case of universal instantiation. Next, (103) and (104) yield (" s' ", 1005b26; " yap ", 1005b31) :

(105) (x) - ((xB:(P)) & (xB:( - P))) by substitution of '-i P ' for ' Q ' in (104) and modus ponens.8

Finally, Aristotle passes from (105) to: (106) (x)- (xB:(P & - P) ) which is logically equivalent to (12). It is not clear from the text what Aristotle thought the relation was between (105) and (106); indeed, it is not even clear that he saw any difference between the two propositions. However, (106) is the most natural reading of the Greek of (106T); and at the same time it is hard to suppose that Aristotle thought of (106) as an immediate inference from (103) and (104), so that it seems reasonable to introduce (105) into the argument as a representation of (105T). To get from (105) to (106) Aristotle of course needs something like my premiss (1).

III My account of Aristotle's argument misrepresents the text on at least

two counts; both concern the modal qualification of propositions. First, some of the steps in the argument are prefixed in the Greek text,

but not in my account of it, by modal operators. For example, a stricter

interpretation of (105T) would be: (105M) (x) -i ((xB:(P)) & (xB:( P)) ),

7(101) is not an adequate representation of (101T) on any reading of the Greek there (see note 6). However, I think that (101) must be entailed by whatever proposition Aristotle means to express by (101T). I take my version to be expressing something like:

(101') (F) (G) (x) ((C(F, G)) -> (Gx -> (- Fx & (x RF) ) ) ). Aristotle has almost certainly said this before, at r 2, 1004a10-16; but there too the text is corrupt.

8Strictly, we need an intermediate application of modus ponens to a suitable substi. tution-instance of:

(T') ( (x) (P Fx) - (P --> (x) (Fx ) ).

308 JONATHAN BARNES

The same applies to (100T), (102T) and (106T); but not to (101T), (103T) or (104T). However, this is probably trivial: I suspect that the function of these operators is simply to indicate that the steps in the argument are necessary truths; the indications are admittedly casual and sporadic, but that is in line with the carefree attitude towards modal operators which Aristotle displays in the rest of Metaphysics F and elsewhere. At all events, I shall not pursue the point any further.

The second point is more serious. Aristotle claims to have proved not merely (106) but (107T), or more fully: (107T') It is impossible to be mistaken concerning the principle that it is

impossible for the same thing at the same time both to belong and not to belong to the same thing.

If we read 'be mistaken concerning the principle that' as ' (wrongly) be- lieve that it is not the case that ', this becomes: (107M) (x) - O (xB:(O (3F) (3x) (Fx & - Fx) ) ). Let us simplify this to: (107) (x) - (xB:(O (P & - P) ) ). It is quite clear that (107) is neither equivalent to nor entailed by (106): in general, no-one's believing that actually P is compatible with there being men who believe that possibly P.

Moreover, no argument analogous to Aristotle's or to that of Section I could prove (107), for although the modal analogue of (1): (1M') (x) ( (xB:(O (P & Q) ) ) - ( (xB:(O P) ) & (xB:(O Q) )) ) seems to hold, the requisite modal analogue of (3): (3M') (x) ( (xB:(O -m P) ) - (xD:(O P) ) ) is plainly false.

Aristotle's deduction of (107) is thus invalid. A similar fallacy would be committed if we tried to infer a quantified version of (107): (107Q) (x) - (xB:( (3P) (P & - P) ) ). The fallacy turns on the fact that the quantifier ' (3P) ' in (107Q) falls within the intensional context 'xB: . . .'. If we quantify over ' P ' in (106) to get: (106Q) (P) (x)- (xB:(P & - P)) we can see that in the passage from this to (107Q) the initial quantifier is captured by the epistemic context 'xB: . . .'. An analogous point holds of the move from (106M) to (107) ; and it is possible that Aristotle's willingness to make the move was due to his ignorance of the snares of intensionality.

A further proposition which the argument does not prove is: (106U) (x) (xB:( - (P & - P) )) Nor does it prove the propositions similarly analogous to (107) and (107Q): (107U) (x) (xB:( -i (P & - P) ) ) and: (107QU) (x) (xB:( -7(3P) (P & -7 P) ) ). Moreover, if, as I claimed, (1.1) is invalid, (106) does not even yield: (106&) (x) -7 ((xB:(P) ) & (xB:(-7 P) )).

THE LAW OF CONTRADICTION 309

Finally, there is an interesting proposition which is all but stated by Aristotle at F 3, 1005b13-17, a sentence which I omitted in my translation of passage (A). It is: (108) (x) ( ( (3P) (xB :(P) )-(xB:( (P) (P & P) )) ). Neither (108) nor any modalized version of it follows from (106).

Any of the propositions mentioned in the last three paragraphs might not unreasonably be glossed as the proposition that the Law of Contra- diction is a Law of Thought; and there are certainly many more analogous propositions equally open to such a gloss. Furthermore, there are normative propositions about how all men ought to think and psychological propositions about how all men do think which might also claim the right to the gloss. Some of these propositions may well be true. My claim that the Law of Contradiction is a Law of Thought is meant as nothing more than the claim that (12) or (106) is a necessary truth.

Oriel College, Oxford.