Lies, lies, and more lies: A plea for propositions

J O R D A N H O W A R D S O B E L

L I E S , L I E S , A N D M O R E L I E S : A P L E A

F O R P R O P O S I T I O N S

(Received 20 June, 1991)

ABSTRACT. To resolve putative liar paradoxes it is sufficient to attend to the distinction between liar-sentences and the propositions they would express, and to exercise the option of turning would-be deductions of paradox (of contradictions) into reductios of the existence of those propositions. Defending the coherence of particular resolutions along these lines, leads to recognition of the non-extensionality of some liar- sentences. In particular, it turns out that exchanges of terms for identicals in the open- sentence '-- does not expression a true proposition' are not invariably truth-preserving because they are not invariably proposition-expression preserving. All of this recommends propositions as fruitful subjects of interesting renewed research.

W. V. O. Quine feels that crude forms of the paradox of the Liar such as 'I am lying' and 'This statement is false', call for tinkering to secure them against scoffers.

The problem is to devise a sentence that says of itself that it is false without venturing outside the timeless domain of pure granunar and logic. Here is a solution:

(3) 'Does not yield a truth when appended to its own quotation' does not yield a truth when appended to its own quotation.

The eleven-word quotation, with its quotation marks, is a noun and the subject of (3). The quotation names that eleven-word expression) (3) tells us that if we append those eleven words to the quotation itself, the result will not be true. 2 Carrying out the instruction, we end up with (3) itself. (3) achieves self-denial. [The quality of antinomy is right on the surface here; the statement is true if and only if it is false.] [Quine 1987, pp. 148-913

I. Sentence (3), Quine's Liar , 4 is held to achieve paradox through self- denial. Here is a putative deduction of a contradiction: Sentence (3) is either true or not true. Everything is [pace, Sainsbury, p. 124]. But sentence (3) tells us that it is not true. So it is not true, for if it were true, then it would be not true; and it is not not true, i.e., it is true, for if it were not true, then it would be true. Therefore, being neither true nor not true, sentence (3) is not either true or not true.

PhilosophicalStudies 67: 51--69, 1992. �9 1992 KluwerAcademic Publishers. Printed in the Netherlands.

52 J O R D A N H O W A R D S O B E L

What are we to make of this deduction? Certainly there is something wrong with it, for it yields a contradiction. But what is wrong with it? A bold though not definitive indictment with which it is convenient to begin would charge that since appending those eleven words to their quotation results in a sentence, this result -- that is, the sentence that is Quine's Liar -- tells us nothing, and is neither true nor false. People tell things, and it is what they say, the propositions they affirm, that have truth-values. So sentence (3), being only a sentence, tells us nothing, and, in particular, it does not tell us that it is not true, though in fact, being only a sentence, it is not true.

Such observations challenge the deduction of paradox. They impugn the grounds for the conditional premise according to which, if sentence (3) were not true, then it wouM be true. But possibly saving elaborations are close to hand. For it is unobjectionable that sentences should by metaphorical extension 'say' and 'tell' things that people can use them to say and tell, and that sentences should be true and false derivatively on occasions, according to the truth-values of these things, these propositions, they are used to say. And so perhaps grounds for that conditional premise can be secured by attending to the proposition that sentence (3) would express and be used to affirm. I come back to this possibility in Section III below and maintain that propositional elaborations do not help Quine's Liar itself to paradox. First, however, I consider a propositional surrogate for Qulne's Liar.

II. A promising response to difficulties with Quine's sentential liar is to convert it into one that addresses itself explicitly to propositions (and implicitly, to just one), and that itself purports to be a proposition (implicitly, precisely that one). For this conversion, we might employ the sentence,

(3') 'Does not express a true proposition when appended to its own quotation' does not express a true proposition when appended to its own quotation.

But, following Patrick Grim, I will use the following related sentence of his design:

'Appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition. [Grim, p. 24]

LIES, L IES , A N D M O R E LIES 53

Let that be Grim's Sentence. It is a 'stylistic variant' of sentence (3'): while different in forms and words they are identical in meanings. Here then is what purports to be a propositional liar modelled on Quine's Liar.

(3*) 'Appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition.

And here is our deduction of paradox, modified to stilt: Proposition (3*) s is either true or not true. Everything is. But proposition (3*) tells us that it is not true. That is, its affirmers tell us that it itself, this proposition (3*), is not true: for when they affirm it, what they say is true, if and only if it -- what they say, the proposition that their sentence expresses, this very proposition that they affirm -- is not true. In short, if proposition (3*) were true, then it would not be true; and if it were not true, then it would be true. So, being neither true nor not true, proposition (3*) is not either true or not true.

It can thus seem that nothing is gained by raising propositions, and that their introduction in Section I in connection with preliminary objections to Quine's Liar, far from blocking paradox barely puts it off. So it can seem. But in fact the would-be deduction of paradox just produced proceeds on the basis of a plainly deniable presupposition, and thus makes not a paradox, but a reductio ad absurdurn of that presupposition. This presupposition attaches to the employment in a subject position of 'proposition (3*)'. What is presupposed in this employment, part of what is presupposed, and what is several times implicitly claimed during the course of the deduction, is that there is a

proposition (3*). It is presupposed that the sentence made when the predicate-clause 'appended to its own quotation does not express a proposition' is appended to its own quotation-name expresses a proposition, that is, that there is something that this sentence can be used on uncontrived occasions to say or to affirm. But the would-be deduction of paradox shows that there is not a proposition (3"), that is, that there is not a proposition that by general conventions is expressed by Grim's Sentence,

'Appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition.

5 4 J O R D A N H O W A R D S O B E L

For suppose this sentence on some occasion did, without special contrivance, express a certain proposition P. Then P would be the

proposition expressed without special contrivance and by general conventions by Grim's Sentence on that occasion. So, since, by hypothesis, on that occasion Grim's Sentence is to express a proposition, namely P itself: (i) P would be true, if it was not true (for if P was not true, then Grim's Sentence would express only a proposition that was not true, and, as P would say, would not express a true proposition); and (ii) P would be true, only if it was not true (for otherwise, contrary to what P would say, Grim's Sentence would express a true proposition). So P would NOT be true: for if it were true, then, given the proposition it would be (it would be a proposition that 'said' that Grim's Sentence does not express a true proposition), it would be not true. And yet this very proposition P would BE true: for if it were not true, then, given the proposition it would be, it would be true. 6

The assumption that Grim's Sentence expresses by general conventions and without special contrivance a proposition on some occasion, makes possible the deduction of a contradiction, and so we have a reductio of that assumption, and may conclude that there is not, that there never is, a proposition expressed without special contrivance by Grim's Sentence,

'Appended to its own quotation does not express a true proposit ion' appended to its own quotation does not express a true proposition. 7

That is, we may conclude that,

0. Grim's Sentence does not express a proposition. 8

III. Returning now to Quine's sentential liar itself, I recall that resist- ance in Section I to this sentence's -- to Quine's Liar's itself -- being both true and not true centered on a certain conditional premise, namely, the premise that if sentence (3), Quine's Liar, were not true, then it would be true. It was suggested that, being only a sentence, Quine's Liar is not true, that only propositions have true-values, and that this conditional premise is thus simply false, and so certainly not true. But this, it was allowed, seems high-handed and too simple, for sentences can be allowed to have truth-values derivatively from the

LIES, LIES, AND MORE LIES 55

truth-values of the propositions they express. The possibility left open was that this concession might provide a way to save that conditional premise and to secure paradox for Quine's Liar.

But it does not, for though sentences have truth-values derivatively from the propositions they express, they have truth-values only in this way. A sentence is true (false) on an occasion if and only if it expresses on this occasion a true (false) proposition, or can be used without special contrivance on this occasion to say something that is true (false). When a sentence fails to express a proposition, and cannot be used to say anything at all, then this sentence is not true, or false for that matter. This suffices to confirm the preliminary denial made in Section I of that conditional premise: for by an argument similar to the one used to secure the conclusion, (0), that Grim's Sentence does not express a proposition, the closely related conclusion can be established that,

(0') Quine's Liar does not express a proposition. 9

Briefly, it can be seen that, as with Grim's Sentence, if Quine's Liar (this other simpler sentence) expressed a proposition, it would express one that was true if and only if it was not true. And so it would express one that was both not true and true. Details of the argument in Section 1I above for conclusion (0) concerning Grim's Sentence can be adapted to spell out this argument for the closely related conclusion (0') concerning Quine's Liar. And so the conditional premise opposed in a preliminary way in Section I -- that if it were the case that sentence (3) is not true, then it would be the case that it is true -- can be set down as definitely false. Its antecedent is true, and its consequent is false: since sentence (3), Quine's Liar, does not express a proposition, this sentence does not have a truth-value; so (the antecedent) it is true that this sentence is not true, and (the consequent) it is false that this sentence is true. And with that we may seem done with Quine's Liar, both in its original sentential form, and in its more recent purportedly propositional cast.

IV. But wait. We are not free of these Liars yet. For consider, as Patrick Grim in correspondence has suggested one do, that it is an undeniable consequence of proposition (0) that,


(1) Grim's Sentence does not express a true proposition.

Add to this the fact, confirmed by "close empirical observation," Grim has written, that,

(2) Grim's Sentence is identical with 'appended to its own quotation does not express a true proposition' appended to its own quotation.

And observe that "given simple and standard assumptions regarding identity (1) and (2) would seem to give us" [Grim in correspondence],


That is, propositions we must accept, namely, (1) and (2), seem by the simplest rule of identity, to commit us not only to the existence of a proposition (3*), contrary to conclusion (0) reached above, but even to the truth of this proposition, l~

There is now an appearance of proofs against proofs, and the threat of the miracle of a paradox established if the proof for it is superior. But the threat is hollow, since the first proof against paradox is not only a superior one, but the only bonafide one before us.

The trouble with the second proof, the one that would regain paradox, is that it proceeds by way of an illicit application of 'Leibniz's Law' or The Principle of Substitutivity:

For all expressions a and fi, if (i) ~ = fin expresses a true proposit ion, then, for all sentences S and S', if (ii) S' is like S save for containing an occurrence of fl where S contains an occurrence of a , then if (iii) S expresses a true proposition, (iv) so does S' [cf, Cartwfight 1987b(1971), pp. 136 and 137].

This principle would vouchsafe as always truth-preserving Substitution ofldenticals, the license or rule of inference, given an identity expressed by c~ = ~ and a proposition expressed by S, to infer a proposition expressed by S'.

The Principle of Substitutivity may be contrasted with The Principle of Identity,

ff x -- y, then every property of x is a property of y.

This principle is certainly true: for clearly, if x has a property that y


lacks, that is, if something is true of x that is not true of y, then x and y must be two and not identical [cf., ibid., pp. 136 and 147]. But the Principle of Substitutivity, far from being obvious and certain, is not (as I have presented it, quite without restriction) even true. Though many substitutions that this principle would license are truth-preserving, not a few that it would license are not.

To the number of well-known counter-examples to the (unrestricted) Principle of Substitutivity, can now be added one that is before us. For this counter-example, set out in the manner made precise in Cartwright 1987b (1971), p. 137, I observe first that each of the following three propositions (about sentences)is true:

(i) 'Grim's Sentence is identical with 'appended to its own quotation does not express a true proposition' appended to its own quotation' expresses a true proposition;

(ii) "Appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition' is like 'Grim's Sentence does not express a true proposition' save for containing an occurrence of "appended to its own quotation does not express a true proposition' appended to its own quotation' where 'Grim's Sentence does not express a true proposition' has an occurrence of 'Grim's Sentence'.

(iii) 'Grim's Sentence does not express a true proposition' expresses a true proposition.

To complete the counter-example I note that the following proposition (about a sentence) is false:

(iv) "Appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition' expresses a true proposition.

The subject of proposition (iv) -- Grim's Sentence, i.e., the sentence "appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition' -- does not express a proposition. This has been demonstrated. Afor t ior i

58 JORDAN HOWARD SOBEL

this sentence does not express a true proposition, and (iv) which says that it does express a true proposition is false.

Proposition (iv) and its subject, which is Grim's Sentence, may be instructively compared with the following proposition and sentence:

(v) 'The sentence made of 'appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition' expresses a true proposition.

(4) The sentence made of 'appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition.

Sentence (4) expresses a proposition: used without contrivance it expresses the proposition that another sentence, namely, Grim's Sentence, does not express a true proposition. And this proposition that sentence (4) expresses is true, as has been demonstrated: Grim's Sentence does not express any kind of proposition, true or false. So sentence (4), thanks to the very small way in which it embroiders on Grim's Sentence, can be used to say the true thing about Grirn's Sentence that Grim's Sentence would, were this not impossible, be used to say about itself. Proposition (v) is true, in contrast with proposition (iv) which is false.

Returning to simpler sentences and propositions, and to Grim's argument from (1) and (2) to (3), we have seen that though the proposition,

(1) Grim's Sentence does not express a true proposition,

is true, there/s no proposition.


What makes this disparity remarkable, of course, is the identity of the subject of proposition (1) with what would be the subject of proposition

L I E S , L I E S , A N D M O R E L I E S 59

(3*) were there such a proposition, an identity affirmed by the proposition,

(2) Grim's Sentence is identical with 'appended to its own quotation does not express a true proposition' appended to its own quotationJ

This situation embarrasses applications of Substitution of Identicals, and those "simple and standard assumptions" alluded to by Grim, to (1) and (2).

V. We see that 'Grim's Sentence' inserted into the open sentence, ' - does not express a true proposition' yields a sentence other than Grim's Sentence that can be used to make a true point about Grim's Sentence. And we have seen that this point can be made by inserting into that open sentence a descriptive structural term for Grim's Sentence, namely, 'the sentence made of 'appended to its own quotation does not express a true proposition' appended to its own quotation'. The point can also be made using the quotation-name of Grim's Sentence, which name is ' "appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition' ,)2 And the point about Grim's Sentence can be made with the sentence,

'Appended to itself does not express a true proposit ion' appended to itself is designated by a word or phrase ct such that the sentence,

a does not express a true proposition,

expresses a true proposition. 13

But this point about Grim's Sentence cannot be made by the sentence that results when another descriptive structural name for Grim's Sentence, specifically, "appended to its own quotation does not express a true proposition' appended to its own quotation', is inserted into that open sentence: the result of that insertion, which is Grim's Sentence itself, cannot be used without special contrivance to say anything at all. Exchanges of terms for identicals (even terms that are semantically equivalent) in the context of the open-sentence '-- does not express a true proposition' do not invariably preserve truth, for they do not

60 JORDAN HOWARD SOBEL

invariably preserve even the property of proposition-expression. 14 And so this open sentence joins such venerables as ' -- was so-called because of his size' and 'that - - is greater than 7 is true merely in virtue of the meanings of these words' in the list of 'non-extensional' open-sentences in which exchanges of terms for identicals do not invariably preserve truth. 15 What is distinctive about ' -- does not express a proposition', I stress, is that in its case exchanges of terms for identicals can fail to preserve proposition - - expressionhood, and that it is for this reason that they can fail to preserve truth.

Given the metalinguistic import of completions of '-- does not express a true proposition', that this open-sentence should be non- extensional - - that semantic qualities of its completions should be sensitive not only to what is designated by words inserted, but also to these very words themselves - - is not surprising. What is remarkable (pace, Parsons 1984, p. 14) is not that open-sentences such as this one

should be non-extensional, but that learned discussions of semantic predicates, and of purported semantic paradoxes, even now, after strictures by Skryms' and others, should, as they sometimes do, involve quite unremarked and undefended dependence on these predicates' being extensional. (Consider, for example, Schmidtz 1990.)

Some reflection on the sentence,

(5) 'Appended to its own quotation expresses a true proposition' appended to its own quotation expresses a true proposition.

might convince us that the open sentence ' - - expresses a true proposition' also belongs on that list. For we might well conclude after reflection that sentence (5) quite falls (not on pain of contradiction, but for want of a 'ground' or for want of complete propositional content) to express a proposition, and that for this reason the proposition,

(6) Sentence (5) expresses a true proposition.

is false. Our view would then be that there is no proposition,

(5*) 'Appended to its own quotation expresses a true proposition' appended to its own quotation expresses a true proposition.

at all, and that there is therefore not a false proposition here, and this


notwithstanding the identity of the subject (as distinct from the subject- clause) of the false proposition (6) with what would be the subject of proposition (5*) were there such a proposition. Convinced as we might be that there is no proposition (5*), if we require that substitutions of terms for identicals in extensional open sentences should preserve truth-values generally, and so both truth and falsity, then we would have the open sentence '-- expresses a true proposition' join its negation ' - does not express a true proposition' on the list of non-extensionals.

On a similar note, reflection on the sentence,

(7) 'Appended to itself expresses a proposition' appended to itself expresses a proposition.

might convince us that ' - - expresses a proposition' is non-extensional. For, as with (5), reflection on (7) might convince us that it fails, for

want of a 'ground' or complete propositional content, to express a proposition. If it expresses a proposition, it expresses a curiously vacuous, analytically true one. Similarly, if the sentence,

(8) 'Appended to itself does not express a proposition' appended to itself does not expresses a proposition.

expresses a proposition, it expresses an analytically false one. And if this sentence does not express a proposition, then the open-sentence ' - does not express a proposition' is non-extensional. I leave open these last issues of classification. They turn mainly on whether sentences (5),

(7), and (8) - - given not only that they purport to express propositions about themselves, but what propositions about themselves they purport to express - - are fit or unfit (I think they are unfit) to be used without special contrivance to say these things, or to affirm propositions. (The problem with these sentences, if there is one, is not merely one of purported references back to them by propositions they would express. There is, I think, no problem with the following sentence:

'Appended to itself contains exactly fourteen words' appended to itself contains exactly fourteen words.

This sentence expresses a proposition, indeed a true one, about this very sentence.)

62 J O R D A N H O W A R D SOBEL

C O N C L U S I O N

Quine takes his Liar seriously. He thinks its problems must somehow be accommodated, and that that calls for desperate measures.

The Liar, thus purified, [i.e., Quine's Liar] does to truth what the Heterological did to denotation. The accommodation is similarly desperate: levels again. [Quine 1987, p. 149.] 16

I think that the would-be paradox of Qulne's Liar can be resolved, and that there is already ample accommodation for numerous stylistic variants of this sentence, and indeed for saying precisely what it would be used to say if it could be used to say anything. There is no need for (though there may still be interest in) various desperate and not otherwise mandated measures. The main things, I think, are: to attend to distinctions between words, sayings, and things that can be said; to attend to the primacy of propositions, of things said, as truth-bearers; and to exercise the option of turning would-be deductions of paradoxes into reductio demonstrations of the non-existence of what would be troublesome propositions. If this assessment of the propositional gambit for Liars, all Liars, is correct, it provides a reason for coming to terms with propositions, and getting clear about what they are not, for example sentence tokens or types, or sentential meanings. And, a thing more difficult, it provides a reason for getting clear about what they are, about how propositions may be individuated and distinguished one from another [cf., Cartwright 1987a (1962), p. 51 and passim], about characteristics of sentences that (in certain contexts) without special contrivance express propositions, 17 and of ones that fail to express propositions, and (what is of particular present interest) about why certain truth and untruth predicating sentences fail, barring special contrivance, to express propositions, is But the relevance of propositions to resolutions of Liars is not the only thing that recommends that we come to theoretical terms with them, as propositions are of central importance to any full understanding of thought and talk, and of truth, puzzles and purported paradoxes aside. So the treatment prescribed here, unlike that of levels, calls for nothing new or special for Liars. 19


N O T E S

1 Sentence (3) is in form like,

'Is a predicate-clause' is a predicate-clause.

and,

'Consists of four words ' consists of four words.

Like these sentences, sentence (3) is formed by appending to the quota t ion-name of a predicate clause that very predicate clause. And so whatever problems attend to sentence (3) are not due to its form, since these sentences are quite unproblematic. It is, for example, an obvious t ruth that 'is a predicate-clause' is a predicate-clause. Similarly, it is plainly true that 'consists of four words ' consists of four words. For variety, consider the sentence,

'Is a giraffe' is a giraffe.

It is obviously and unproblematical ly (save in Moorean nightmares - - see Cartwright 1987a(1962)) not true that 'is a giraffe' is a giraffe. For one thing these words do not have a particularly long neck.

Quota t ion-mark conventions: (a) Double quotat ion-marks are used for direct quotations, as in footnote 2 below. (b) Single quotat ion-marks are used to make names of words and punctuat ion marks they enclose, as in sentence (3). Since the result of enclosing a word or words in single quotat ion-marks is a name and so a word, it can be part of what is enclosed by single quotat ion-marks to make a larger name, as in displayed (ii) in Section IV below. Single quotat ion-marks also serve sometimes, as in the last sentence of footnote 3 below, as scare-quotes. (c) In a statement by Richard Cartwright of The Principle of Substitutivity in Section IV below, corners are used in conjunction with meta-linguistic variables to form a stylized abbreviat ion for a descrip- t ion of a form of expression as the result of a simple construct ion by appended concatenation: ' ~ = ~ ' is therein short for ' the expression that consists of a , followed by '= ' , followed by fi'. 2 But then sentence (3) is, prima facie, not a solution to the problem of devising "a sentence that says of itself that it is false" [Quine 1987, p. 148, emphasis added]. 3 Here are somewhat similar passages in earlier works of W. V. D. Quine:

[A complex variant of Epimenides ' paradox is this:]

(3) 'does not produce a true statement when appended to its own quotation' produces a true s tatement when appended to its own quotation.

The above statement is readily seen to say that its own denial is true. [Quine 1964a (1953), p. 134.1

If . . . we are . . . bent on constructing a sentence that does attribute falsity unequivocally to itself, we can do so thus: "Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation' . This sentence specifies a string of nine words and says of this string that if you put it down twice, with quotat ion marks a round the first of the two occurrences, the result is false. But that result is the very sentence that is doing the telling. The sentence is t rue if and only if it is false, and we have our antimony. [Quine 1966, p. 9]

1964a (1953)-passage raises the question, Wha t does Quine mean by 's tatement '? There is evidence that he means a kind of sentence, specifically a 'closed' sentence: consider Quine 1946b (1953, 1937), p. 80, and Quine 1964c (1953), p. 109).


4 Brian Skyrms casts precisely sentence (3) as a "form of Grelling" (Skyrms 1984, p. 120). 5 I use 'proposition (3")' as short for terms such as the following: 'the proposition expressed by Grim's sentence'; ' the proposition that 'appended to its own quotation does not express a true proposition' appended to its own quotation does not express a true proposition'. I practice (see for example, footnote 1 above) the general rule that, for any sentence O, the terms,

the proposition expressed by O,

the proposition that O,

and simply,

that O,

stand for the proposition, if any, that perhaps on a certain occasion made plain by the context is expressed without special contrivance by O. For purposes of the present paper, the convention just explained in connection with 'proposition (3*)' has been added to this general rule for propositional terms. 6 I note that this argument does not use as a premise that every proposition has a truth value and is either true or false. The question, Does every proposition have a truth value, true or false?, is not relevant to this argument, or to any other argument of the present paper. For the record, however, I do endorse the principle that every 'proposition' is either true or false, and note that the adequacy for various theoretical purposes of sets of possible worlds as surrogates for propositions depends on this principle, since every set of possible worlds either includes the' actual world (truth) or does not include it (falsity). 7 In contrast, the following sentence,

'Appended to its own quotation does not express a true proposition' appended to its own quotation expresses a true proposition.

of which Grim is a negation of sorts does express a proposition. Let 'UnGrim' label that sentence. Since, as we have seen, Grim does not (on pain of contradiction) express a proposition, UnGrim says something (not about itself, of course, but about Grim) that is false. It expresses a false proposition, and so of course it expresses a proposition.

And so there is a counter-example to the following prima facie plausible principle for sentences and their negations:

For any sentence O, if O' is a negation of O, then O' expresses a true proposition if and only if O expresses a false proposition.

For the counter-example, let O be UnGrim and O' be Grim: UnGrim expresses a false proposition, but Grim does not express any proposition, let alone a true one. Let the canonical negation of a sentence come from that sentence by prefixing to it 'it is not the case that'. In the just stated principle, 'negation' is used to cover canonical negations, and sentences that come from subject-predicate sentences by replacing predicates by negations of them: for example, UnGrim comes from Grim by putting 'does not express a true proposition' in place of 'expresses a true proposition'.

For a second counter-example that 'goes the other way', let O be the sentence, labelled 'Mirg',

The foUowing clause, prefixed commas and all to its own quotation, expresses a proposition that is not true: 'the following clause, prefixed commas and all to its own quotation, expresses a proposition that is not true:'.

And let O' be the canonical negation of Mirg, labelled 'NoMirg',

L I E S , L IES , A N D M O R E L I E S 6 5

It is not the case that the following clause, prefixed commas and all to its own quotation, expresses a proposition that is not true: 'the following clause, prefixed commas and all to its own quotation, expresses a proposition that is not true:'.

Mirg, the current substitution for O, does not (on pain of contradiction) express a proposition, and so does not express a false proposition. And yet NoMirg, the current substitution for 0 ' , expresses a true proposition: it expresses a proposition that says truly of Mirg that it does not express a false proposition.

The principle under consideration has now been twice impeached, but the following restricted principle has not been challenged.

For any sentence O and negation O' of O, if O and O' express propositions, then O' expresses a true proposition if and only if O expresses a false proposition.

However, I hesitate to say that there are no counter-examples in English to this principle. Consider 'there is resurrection of the dead', which seems to say that all the dead are raised, and 'it is not the case that there is resurrection of the dead', which seems to say that none of the dead are raised. [Cf., Sobel 1977.] 8 This is no t to say that Grim's sentence is meaningless , It is not meaningless. It is not a piece of nonsense. If it were there would be no saying what proposition, if any, it would express on uncontrived occasions. It is by understanding this sentence's meaning (something we accomplish by applying our understandings of its grammatical structure, and of the meanings of its words) that we know what proposition it would express on uncontrived occasions, if it expressed one on such occasions.

Meanings of sentences are not the propositions, if any, that they would express on various uncontrived occasions. (For proof of this see Cartwright 1987a (1962), pp. 4 9 - - 5 1 , corrected on p. 53.) Meanings of sentences are (amongst other things) determinants of the propositions, if any, sentences would express on various uncontrived occasions. Knowing the meaning of a sentence, and, perhaps, knowing certain facts about a particular uncontrived occasion of its use, one can figure out what, if any, proposition it expresses on that occasion. 9 Cf.: "There is a great temptation to say that, in some sense of proposition, the Liar sentence and its ilk simply do not express . . , proposition[s]." (Skyrms 1984, p. 127) 10 R. M. Sainsbury makes use of a simple assumption regarding identity in the course of an argument against the adequacy of a certain theory that would imply,

%2 is not true' is not true.

where it is given that,

L2 = 'L2 is not true'.

Sainsbury contends that:

From these two premises we should be able to infer

L2 is not true.

simply by replacing one name of the sentence by another. [Sainsbury, p. 125.1

Sainsbury allows that this "reliance on the principle of substitutivity of ident ica ls . . , is not uncontroversial," and cites Skyrms as one who would challenge it. Sainsbury thinks "this position is extremely unintuitive" [Sainsbury, p. 139]. J~ Skyrms sketches a particular theory of propositions, and observes that "substitution of coextensive designators can take one from a sentence which expresses a proposition [as construed in this theory] to one which doesn't" (Skyrms 1984, p. 128). ~2 I have inscribed the quotation-name of the quotation-name of Grim's sentence. For


the quotation-name of Grim's sentence, drop outer single-quotes. For Grim's sentence itself, drop also next-to-outer single-quotes. ~3 It is of some interest that the related open-sentence,

- - is designated by a word or phrase a such that the sentence,

a does not express a true proposition,

expresses a true proposition.

is, in its open position, non-extensional. This open-sentence is thus available for use in contexts in which one would, if one could, 'quantify into' the non-extensional context ' - - does not express a true proposition'. 14 Here is a pair of semantically equivalent terms that cannot be exchanged 'saving proposit ion-expressionhood' in this context: ' the sentence made of 'appended to its own quotation does not express a true proposit ion' appended to its own quotation'; and "appended to its own quotation does not express a true proposit ion' appended to its own quotation'. 15 These metalinguistic open sentences are unambiguously non-extensional. In contrast, to cite two other sources of examples of non-extensionality, the modal open-sentence 'it is logically necessary that - - is greater than 7', and the mental attitude open-sentence 'Kaplan believes that - - was a spy', have both extensional and non-extensional interpretations. ~6 Quine's Heterological is that:

No adjective can denote all and only the adjectives that do not denote themselves. [(Quine 1987, p. 147]

Quine writes:

There is no arguing with [this] . . . ; how then are we to come to terms with 'non-self- denoting'? The line taken is polysemy of 'denote': there is denoting, we may say, at this level and that. [ibid.]

For comparison and interest I offer the following Platonic Heterological:

No property is a property of a property if and only if this property is not a property of its elf.

There is no arguing with this. In particular 'being a property that is a property of a property if and only if this property is not a property of itself' does not stand for a property that is a property of a property if and only if this property is not a property of itself. Indeed, since those words would stand for such a property if they stood for anything, they stand for nothing.

A connected point is that because its subject fails to denote anything, the sentence,

Being a property of a property if and only if this property is not a property of itself, is not a property that is a property of a property if and only this property is not a property of itself.

does not say anything - - it does not express a proposit ion true or false. One could, by introducing levels, say parts of what one would, if one could, say with this sentence. A n d one can express without levels, and in one breath, the 'English-image' of all that one would, if one could, say with sentence. One can do that all in one deep breath in the following words:

The words 'being a property of a property if and only if this property is not a property of itself', do not stand for a property that is a property of a property if and only this property is not a property of itself.


These words say something true, something that is an easy consequence of our Platonic Heterological which says that there is no property that is a property of a property etc.. Since there is no such property, no words stand for such a property. 17 It is my practice to count sentences as fit for use without special contrivance to say (affirm, tell) things - - that is, as expressing propositions - - when they are grammatically fit and there are no reasons (for example, contradictions that would be implied, or 'grounding problems') for denying them this status. Theoretical elaboration of this practice, in which I am not alone, would be welcome. is One might guess that a theory of propositions and English could feature very simple rules that made the propositional expressiveness (yea or nay) of truth-functionally compound sentences functions of the propositional expressiveness of their primary sub- sentences. It may seem, for a first simple rule, that a truth-functionally compound sentence expresses a proposition, if all its primary sub-sentences do. For it can seem that for a sentence not to express a proposi t ion is a semantic defect that any truth- functional operation must preserve. (Cf, [Sainsbury, p. 132] on the inheritability of the "semantic defect" of being in a 'gap' with respect to 'true', so that neither 'true' nor 'not true' is applicable.) It may seem, for a second simple rule, that a truth-functionally compound sentence expresses a proposition, only if all its primary sub-sentences do.

Note 7 above provides a counter-example against the second simple rule. Mirg does not express a proposition, though its canonical negation NoMirg does. Against the second simple rule, we have it that though the sentence Mirg*,

It is not the case that prefixed comma and all to its own quotation the clause to be named forthwith expresses a true proposition, 'it is not the case prefixed commas and all to its own quotation the clause to be named forthwith expresses a true proposition, ' .

does not (on pain of contradiction) express a proposition; Mirg* is the canonical negation of the sentence UnMirg*,

Prefixed commas and all to its own quotation the clause to be named forthwith expresses a true proposition, 'it is not the case that prefixed commas and all to its own quotation the clause to be named forthwith expresses a true proposition, ' .

which sentence does express a proposit ion (a false one about the sentence UnMirg*). For a problem with disjunctions under the first simple rule, we have the following

sentences:

(1) The present sentence - - unless 'or' occurs in it, in which case only the part of the present sentence that precedes 'or' - - when disjoined to the left with sentence (2) makes a sentence that expresses a proposit ion that is not true.

(2) The present sentence - - unless 'or' occurs in it, in which case only the part of the present sentence that follows 'or' - - when disjoined to the right with sentence (1) expresses a proposit ion that is not true.

%' and 'R' abbreviate sentences (1) and (2) respectively. Both (1) and (2) express propositions. Each expresses a proposit ion that entails that the sentence 'L or R' (wherein 'or' is weak or inclusive) expresses a proposit ion that is not true. The propositions expressed by (1) and (2) are false. The sentence 'L or R' does not, on pain of contradiction, express a proposition, so it does not express one that is not true. For proof, assume that 'L or R' does express a proposition. This proposit ion is not true, for if the disjunction 'L or R' expressed a proposit ion that was not not true, that is, if it expressed a proposit ion that was true, then at least one of its disjuncts would express a true proposition. But (recall what the proposit ions expressed by (1) mad (2) say) this


true proposition would entail that 'L or R' expressed a proposition that was not true. And the proposition expressed by 'L or R' is true, for if 'L or R' expressed a proposition that was not true, then each of its disjuncts would express a proposition that was true (recall what propositions expressed by the disjuncts of 'L or R' say about it), and so their disjunction 'L or R' would express a proposition that was true. Problems for the first simple rule as it regards truth-functional conjunctions, conditionals, biconditionals, etc., can be similarly concocted.

Sentences (1) and (2) are, Quine might say, not " secure . . . against scoffers" [Quine 1987, p. 148] who would object that in each the description 'the present sentence' and the names '(2)' and '(3)' want cashing, and the sentences they stand for named by explicit quotation. It can be confirmed that "when we . . . do so, we get nowhere" [p. 148]. That is, it can be confirmed that when we do so the results are spelled out sentences that express propositions that are still about (1) and (2) disjoined now explicitly named, and not, as an argument like that of the previous paragraph would require, about these spelled out sentences themselves disjoined. This suggests that perhaps a restricted version of the first simple rule, a version confined to disjunctions and other non-unary truth-functional compounds whose primary sub-sentences do not venture "outside the timeless domain of pure grammar and logic" [p. 148], is safe from the kind of problem made by 'L or R' for the first simple rule unrestricted.

It may be noted that the last two paragraphs have been about non-unary truth- functional compounds and the first simple rule. No reasons have been given why the second simply rule does not hold quite without restriction for non-unary truth-functional compounds. Also, even if neither of these rules holds for sentences, it may be that related rules hold for open-sentences, and that, for example, if completions of an open-sentence can, depending on terms employed to complete them, fail to express propositions, then the same holds for this open-sentence's negation: it may be that ' -- expresses a proposition' is non-extensional if and only if ' -- does not express a proposition'. 19 I am grateful to Ron Clark, Patrick Grim, and to Willa Freeman-Sobel, William Seager, and Arnold Silverberg for comments and discussion. And I thank Anil Gupta for a reference he makes to a paper of Yehoshua Bar-Hillel's that I confess was not known to me when I wrote the present paper.

Commenting on "the role of propositions in the diagnosis of the Liar," Gupta writes (in a review of The Liar, J. Barwise and J. Etchemendy) that:

A number of philosophers have put forward the view that pathological sentences do not express propositions. Some have even seen in this idea a complete resolution of the paradox. [Gupta 1989, p. 708]

He cites Bar-Hillel to this last point, quoting from the summary of a paper first published in 1966. [Let me interject here a very recent statement of the first point by James Cargile, also in a review of The Liar: "In my opinion, the most fundamental fact about the Liar Paradox is that there could not be such a proposition." Cargile 1990, p. 757.] "Here is a somewhat more extensive quote from that summary:

IT]he philosopher's argument for his claim that natural languages contain semantic paradoxes . . . can be met . . . by making, among others, a clear distinction between sentences and statements, realizing that it is to statements that, primarily, truth-values have to be assigned and that the supposedly paradoxical situations can be shown to evaporate by realizing that in those situations no statement had been made at ai1 . . . [Bar-Hillel 1970, p. 285]

Consulting Bar-Hillel's paper I find that several of my arguments elaborate on a theme he sets (without elaboration):

[T]he assumption that a statement was made at all [by the utterance on an otherwise

LIES , LIES , A N D M O R E L I E S 69

blank blackboard of the sentence 'The sentence written on this blackboard is false'] leads to a contradiction . . . . [p. 283]

[I interject Cargile's recent statement of this point: "The hypothesis that there is such a one leads to a contradiction . . . . " p. 757] And the penultimate sentence of my paper is in spirit kin to Bar-Hillel's:

Even if there were no threat of paradoxes, the distinctions between sentence and state- m e n t . . , would have to be made anyway in order to get along best w i th . . , facts of life. [pp. 283--41

R E F E R E N C E S

Bar-Hillel, Y. 'Do Natural Languages Contain Paradoxes,' reprinted in Aspects' of Language, Jerusalem, 1970. (Th~s essay was first published in Studium Generale, vol. 19, 1966, 391--7.)

Cargile, J. 1990, 'The Liar, An Essay in Truth and Circularity, J. Barwise and J. Etchemendy', No~s, vol. 24, 757--73.

Cartwright, R. 1987a, 'Propositions', reprinted with addenda in Richard Cartwright, Philosophical Essays, Cambridge, Mass., 1987. (This essay was first published, without addenda, in Analytical Philosophy, ed. R. J. Butler, Oxford, 1962.)

Cartwright, R. 1987b. 'Identity and Substitutivity', reprinted in Richard Cartwright, Philosophical Essays, Cambridge, Mass. 1987. (This essay was first published in Identity and Individuation, ed. Milton K. Munitz, New York, 1971 .)

Grim, P. The Incomplete Universe: Totality, Knowledge and Truth, Cambridge, Mass., forthcoming.

Gupta, A. 1989. 'Barwise and John Etchemmendy's The Liar,' Philosophy of Science, vol. 56,687--709.

Parsons, C. 1984. 'The Liar Paradox', in Recent Essays on Truth and the Liar Paradox, ed. by R. L. Martin, Oxford. (This essays first appeared in the Journal of Philosophi- calLogic, vol. 3, 1974, 381--412).

Quine W. V. D. 1964a. 'Notes on the Theory of Reference', in From a Logical Point of View: Second Edition Revised (first edition 1953).

Quine, W. V. O. 1964b. 'New Foundations for Mathematical Logic', reprinted with supplementary remarks in From a Logical Point of View: Second Edition Revised (first edition 1953). (This essay first appeared in American Mathematical Monthly, 1937.

Quine, W. V. O. 1964c. 'Lo~c and the Reification of Universals', in From a Logical Point of View: Second Edition Revised (first edition 1953).

Quine, W. V. O. 1966. Ways of Paradox, and Other Essays, New York. Quine, W. V. O. 1987. Quiddities: An Intermittently Philosophical Dictionary, Cam-

bridge, Mass. Sainsbury, R. M. 1988. Paradoxes, Cambridge. Schmidtz, D. 1990. 'Charles Parsons on the Liar Paradox', Erkenntnis, vol. 32, 419--

22. Skyrms, B. 1984. 'Intentional Aspects of Semantical Self-Reference', in Recent Essays

on Truth and the Liar Paradox, ed. by R. L. Martin, Oxford. Sobel, J. H. 1977, 'The Resurrection of the Dead: An Exercise in Critical Analysis',

Teaching Philosophy, vol. 2.

Scarborough College

University o f Toronto

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Lies, lies, and more lies: A plea for propositions