14
Self-Reference and Meaning in Ordinary Language

Theaetetus. Now listen to me attentively, Socrates, for what I shall put before you is not a little tricky.

Socrates. I promise to do my best, Theaetetus, so long as you spare me the details of your achievements in the theory of numbers, and speak in a language which I, an ordinary man, can understand.

Th.	The very next question which I am going to ask you is an extraordinary one, although expressed in perfectly ordinary language.
S.	There is no need to warn me: I am all ears.
Th.	What did I say between your last two interruptions, Socrates?
S.	You said: ‘The very next question which I am going to ask you is an extraordinary one, although expressed in perfectly ordinary language.’
Th.	And did you understand what I was saying?
S.	I did, of course. Your warning referred to a question which you intended to ask me.
Th.	And what was this question of mine to which my warning referred? Can you repeat it?
S.	Your question? Let me see … Oh, yes, your question was: ‘What did I say between your last two interruptions, Socrates?’
Th.	I see you have kept your promise, Socrates: you did attend to what I was saying. But did you understand this question of mine which you have just quoted?
S.	I think I can prove that I understood your question at once. For did I not reply correctly when you first put it to me?
Th.	That is so. But do you agree that it was an extraordinary question?
S.	No. Admittedly, it was not very politely put, Theaetetus, but this, I am afraid, is nothing out of the ordinary. No, I can’t see anything extraordinary in it.
Th.	I am sorry if I was rude, Socrates; believe me, I only wanted to be brief, which was of some importance at that stage of our discussion. But I find it interesting that you think my question an ordinary one (apart from its rudeness); for some philosophers might say that it is an impossible question—at any rate one which it is impossible to understand properly, since it can have no meaning.
S.	Why should your question have no meaning?
Th.	Because indirectly it referred to itself.
S.	I do not see this. As far as I can see, your question only referred to the warning you gave me, just before you asked it.
Th.	And what did my warning refer to?
S.	Now I see what you mean. Your warning referred to your question, and your question to your warning.
Th.	But you say that you understood both, my warning and my question?
S.	I had to trouble at all in understanding what you said.
Th.	This seems to prove that two things a person says may be perfectly meaningful in spite of the fact that they are indirectly self-referring—that the first refers to the second and the second to the first.
S.	It does seem to prove it.
Th.	And don’t you think that this is extraordinary?
S.	To me it does not appear extraordinary. It seems obvious. I do not see why you should bother to draw my attention to such a truism.
Th.	Because it has been denied, at least implicitly, by many philosophers.
S.	Has it? You surprise me.
Th.	I mean the philosophers who say that a paradox such as the Liar (the Megaric version of the Epimenides) cannot arise because a meaningful and properly constructed statement cannot refer to itself.
S.	I know the Epimenides and the Liar who says, ‘What I am now saying is untrue’ (and nothing else); and I find the solution you just mentioned attractive.
Th.	But it cannot solve the paradox if you admit that indirect self-reference is admissible. For, as Russell and Jourdain and Langford have shown (and Buridan before them), the Liar or the Epimenides can be formulated by using indirect self-reference instead of direct self-reference.
S.	Please give me this formulation at once.
Th.	The next assertion I am going to make is a true one.
S.	Don’t you always speak the truth?
Th.	The last assertion I made was untrue.
S.	So you wish to withdraw it? All right, you may begin again.
Th.	You don’t seem to realize what my two assertions taken together amounted to.
S.	Oh, now I see the implications of what you were saying. You are quite right. It is old Epimenides all over again.
Th.	I have used indirect self-reference instead of direct self-reference; that is the only difference. And this example establishes, I believe, that such paradoxes as the Epimenides cannot be solved by dwelling on the impossibility of self-referring assertions. For even if direct self-reference were impossible, or meaningless, indirect self-reference is certainly quite a common thing. I may, for example, make the following comment: I am confidently looking forward to a clever and appropriate remark from you, Socrates.
S.	This expression of your confidence, Theaetetus, is highly flattering.
Th.	This shows how easily it may occur that a comment is a comment upon another one, which in its turn is a comment upon the first. But once we see that we cannot solve the paradoxes in this way, we shall also see that even direct self-reference may be perfectly in order. In fact, many examples of non-paradoxical although directly self-referring assertions have been known for a long time; both of self-referring statements of a more or less empirical character and of self-referring statements whose truth or falsity can be established by logical reasoning.
S.	Could you produce an example of a self-referring assertion which is empirically true?
Th.	… … … … ….
S.	I could not hear what you were saying, Theaetetus. Please repeat it a little louder. My hearing is no longer what it used to be.
Th.	I said: ‘I am now speaking so softly that dear old Socrates cannot make out what I am saying.’
S.	I like this example; and I cannot deny that, when you were speaking so softly, you were speaking truthfully. Nor can I deny the empirical character of this truth; for had my ears been younger, it would have turned out an untruth.
Th.	The truth of my next assertion will be even logically demonstrable, for example by a reductio ad absurdum, a method most beloved of Euclid the Geometrician.
S.	I do not know him; you don’t mean the man from Megara, I presume. But I think I know what you mean by a reductio. Will you now state your theorem?
Th.	What I am now saying is meaningful.
S.	If you don’t mind I shall try to prove your theorem myself. For the purpose of a reductio I begin with the assumption that your last utterance was meaningless. This, however, turns out to contradict your utterance, and thus to entail the falsity of your utterance. But if an utterance is false, then it must clearly be meaningful. Thus my assumption is absurd; which proves your theorem.
Th.	You have got it, Socrates. You have proved my theorem, as you insist on calling it. But some philosophers may not believe you. They will say that my theorem (or the demonstrably false anti-theorem ‘What I am now saying is meaningless’) was paradoxical, and that, since it is paradoxical, you can ‘prove’ whatever you like about it—its truth as well as its falsity.
S.	As I showed, the assumption of the truth of your antitheorem ‘What I am now saying is meaningless’ leads to an absurdity. Let them show, by a similar argument, that the assumption of its falsity (or of the truth of your theorem) leads to an absurdity also. When they succeed in this, then they may claim its paradoxical character or, if you like, its meaninglessness, and the meaninglessness of your theorem also.
Th.	I agree, Socrates; moreover, I am perfectly satisfied that they will not succeed—at least as long as by ‘a meaningless utterance’ they mean something like an expression which is formulated in a manner which violates the rules of grammar or, in other words, a badly constructed expression.
S.	I am glad that you feel so sure, Theaetetus; but are you not just a little too sure of our case?
Th.	If you don’t mind, I’ll postpone the answer to that question for a minute or two. My reason is that I should like first to draw your attention to the fact that even if somebody did show that my theorem, or else my antitheorem, was paradoxical, he would not thereby have succeeded in showing that it is to be described as ‘meaningless’, in the best and most appropriate sense of the word. For in order to succeed he would have to show that, if we assume the truth of my theorem (or the falsity of my antitheorem ‘What I am now saying is meaningless’), an absurdity follows. But I should be inclined to argue that such a derivation cannot be attempted by anybody who does not understand the meaning of my theorem and my antitheorem. And I should also be inclined to argue that, if the meaning of an utterance can be understood, then the utterance has a meaning; and again, that, if it has any implications (that is to say, if anything follows from it), it must also have a meaning. This view, at least, seems to be in accordance with ordinary usage, don’t you think so?
S.	I do.
Th.	Of course, I do not wish to say that there may not be other ways of using the word ‘meaningful’; for example, one of my fellow-mathematicians has suggested that we call an assertion ‘meaningful’ only if we possess a valid proof of it. But this would have the consequence that we could not know of a conjecture such as Goldbach’s—‘Every even number (except 2) is the sum of two primes’—whether it is at all meaningful, before we have found a valid proof of it; moreover, even the discovery of a counter example would not disprove the conjecture but only confirm its lack of meaning.
S.	I think this would be both a strange way and an awkward way of using the word ‘meaningful’.
Th.	Other people have been a little more liberal. They suggested that we should call an assertion ‘meaningful’ if, and only if, there is a method which can either prove it or disprove it. This would make a conjecture such as Goldbach’s meaningful the moment we have found a counter example (or a method of constructing one). But as long as we have not found a method of proving or disproving it, we cannot know whether or not it is meaningful.
S.	It does not seem right to me to denounce all conjectures or hypotheses as ‘meaningless’ or ‘nonsensical’ simply because we don’t know yet how to prove them or disprove them.
Th.	Others again have suggested calling an assertion ‘meaningful’ only if we know how to find out whether it is true or false; a suggestion which amounts more or less to the same.
S.	It does look to me very similar to your previous suggestion.
Th.	If, however, we mean by ‘a meaningful assertion or question’ something like an expression which is understandable by anybody knowing the language, because it is formed in accordance with the grammatical rules for the formation of statements or questions in that language, then, I believe, we can give a correct answer to my next question which again will be a self-referring one.
S.	Let me see whether I can answer it.
Th.	Is the question I am now asking you meaningful or meaningless?
S.	It is meaningful, and demonstrably so. For assume my answer to be false and the answer, ‘It is meaningless’, to be true. Then a true answer to your question can be given. But a question to which an answer can be given (and a true answer at that) must be meaningful. Therefore your question was meaningful, quod erat demonstrandum.
Th.	I wonder where you picked up all this Latin, Socrates. Still, I can find no flaw in your demonstration; it is, after all, only a version of your proof of what you call my ‘theorem’.
S.	I think you have disposed of the suggestion that self-referring assertions are always meaningless. But I am sad at this admission, for it seemed such a straightforward way of getting rid of the paradoxes.
Th.	You need not be sad: there simply was no way out in this direction.
S.	Why not?
Th.	Some people seem to think that there is a way of solving the paradoxes by dividing our utterances or expressions into meaningful statements which, in turn, can be either true or false, and utterances which are meaningless or nonsensical or not properly constructed (or ‘pseudostatements’, or ‘indefinite propositions’ as some philosophers preferred to call them), and which can be neither true nor false. If they could only show that a paradoxical utterance falls into the third of these three exclusive and exhaustive classes—true, false, and meaningless—then, they believe, the paradox in question would have found its solution.
S.	Precisely. This was the way I had in mind, though I was not so clear about it; and I found it attractive.
Th.	But these people don’t ask themselves whether it is at all possible to solve a paradox such as that of the liar on the basis of a classification into these three classes, even if we could prove that it belongs to this third class of meaningless utterances.
S.	I don’t follow you. Assume they have succeeded in finding a proof which establishes that an utterance of the form ‘U is false’ is meaningless, whenever ‘U’ is a name of this very utterance ‘U is false’. Why should this not solve the paradox?
Th.	It would not. It would only shift it. For under the assumption that U is itself the utterance ‘U is false’, I can disprove the hypothesis that U is meaningless with the help of precisely this threefold classification of utterances.
S.	If you are right, then a proof of the hypothesis that U is meaningless would indeed only establish a new statement which can be proved as well as disproved, and therefore a new paradox. But how can you disprove the hypothesis that U is meaningless?
Th.	Again by a reductio. Quite generally, we can read offfrom our classi-fication two rules. (i) From the truth of ‘X is meaningless’ we can derive the falsity of ‘X is true’ and also (what interests us here), the falsity of ‘X is false’. (ii) From the falsity of any utterance Y, we can conclude that Y is meaningful. According to these rules, we find that from the truth of our hypothesis, ‘U is meaningless’, we can derive by (i) the falsity of ‘U is false’; concluding by (ii) that ‘U is false’ is meaningful. But since ‘U is false’ is nothing but U itself, we have shown (by (ii) again) that U is meaningful; which concludes the reductio. (Incidentally, since the truth of our hypothesis entails the falsity of ‘U is false’, it also entails our original paradox.)
S.	This is a surprising result: a Liar who comes back by the window, just when you think you have got rid of him by the door. Is there no way whatever of eliminating these paradoxes?
Th.	There is a very simple way, Socrates.
S.	What is it?
Th.	Just avoid them, as nearly everybody does, and don’t worry about them.
S.	But is this sufficient? Is this safe?
Th.	For ordinary language and for ordinary purposes it seems suf-ficient and quite safe. At any rate, you can do nothing else in ordinary language, since paradoxes can be constructed in it, and are understandable, as we have seen.
S.	But could we not legislate, say, that any kind of self-reference, whether direct or indirect, should be avoided, and thereby purify our language of paradoxes?
Th.	We might try to do this (although it might lead to new difficul-ties). But a language for which we legislate in this way is no longer our ordinary language; artificial rules make an artificial language. Has not our discussion shown that at least indirect self-reference is quite an ordinary thing?
S.	But for mathematics, say, a somewhat artificial language would be appropriate, would it not?
Th.	It would; and for the construction of a language with artificial rules which, if it is properly done, might be called a ‘formalized language’, we shall take hints from the fact that paradoxes (which we wish to avoid) can occur in ordinary language.
S.	And you would legislate for your formalized language, I suppose, that all self-reference must be strictly excluded, would you not?
Th.	No. We can avoid paradoxes without using such drastic measures.
S.	Do you call them drastic?
Th.	They are drastic because they would exclude some very interesting uses of self-reference, especially Gödel’s method of constructing self-referring statements, a method which has most important applications in my own field of interest, the theory of numbers. They are drastic, moreover, because we have learned from Tarski that in any consistent language—let us call it ‘L’—the predicates ‘true in L’ and ‘false in L’ cannot occur (as opposed to ‘meaningful in L’, and ‘meaningless in L’ which may occur), and that without predicates such as these, paradoxes of the type of the Epimenides, or of Grelling’s paradox of the heterological adjectives, cannot be formulated. This hint turns out to be sufficient for the construction of formalized languages in which these paradoxes are avoided.
S.	Who are all these mathematicians? Theodorus never mentioned their names.
Th.	Barbarians, Socrates. But they are very able. Gödel’s ‘method of arithmetization’, as it is called, is especially interesting in the context of our present discussion.
S.	Another self-reference, and quite an ordinary one. I am getting a bit hyper-sensitive to these things.
Th.	Gödel’s method, one might say, is to translate certain non-arithmetical assertions into arithmetical ones; they are turned into an arithmetical code, as it were; and among the assertions which can be so coded there happens to be also the one which you have jokingly described as my theorem. To be a little more exact, the assertion which can be turned into Gödel’s arithmetical code is the self-referring statement, ‘This expression is a well-formed formula’; here ‘well-formed formula’ replaces, of course, the word ‘meaningful’. I felt, you will remember, a little too sure for your liking that my theorem cannot be disproved. My reason was, simply, that when turned into the Gödelian code, my theorem becomes a theorem of arithmetic. It is demonstrable, and its negation is refutable. Now if anybody were to succeed, by a valid argument (perhaps by one similar to your own proof) in disproving my theorem—for example, by deriving an absurdity from the assumption that the negation of my theorem is false—then this argument could be used to show the same of the corresponding arithmetical theorem; and since this would at once provide us with a method of proving ‘0 = 1’, I feel that I have good reasons for believing that my theorem cannot be disproved.
S.	Could you explain Gödel’s method of coding without getting involved in technicalities?
Th.	There is no need to do this since it has been done before—I do not mean before now, the supposed dramatic date of this little dialogue of ours (which is about 400 B.C.), but I mean before our dialogue will ever be concocted by its author, which won’t take place before another 2,350 years have elapsed.
S.	I am shocked, Theaetetus, by these latest self-references of yours. You talk as if we were actors reciting the lines of a play. This is a trick which, I am afraid, some playwrights think witty, but hardly their victims; anyway, I don’t. But even worse than any such self-referring joke is this preposterous, nay, this nonsensical chronology of yours. Seriously, I must draw a line somewhere, Theaetetus, and I am drawing it here.
Th.	Come, Socrates, who cares about chronology? Ideas are timeless.
S.	Beware of metaphysics, Theaetetus!

Note

   First published in Mind, N.S. 63, 1954. (See also my Open Society, vol. ii, note 7 to ch. 24.)