I

The Greek philosopher Protagoras, who wrote a treatise on truth, made a contract with a pupil [Euathlus] to whom he was teaching the art of rhetoric for fighting legal cases. The contract was half the fee straight away, and the other half of it if – but only if – the pupil won his first case. The pupil engaged in no case. So after a while Protagoras sued him for the remaining half fee. ‘If I win’, argued the pupil, ‘then by the decision of the court I don't have to pay; if I lose, then by our contract I don't have to pay’. Protagoras argued in parallel. How the court decided isn't on record. I haven't any doubt the whole thing was fixed up between them in advance.Footnote ¹

Protagoras was one of the tribe called Sophists – the first European inventors of logical puzzles. He didn't believe there was any such thing as false opinion – anything anyone thinks is true, it's like perception, it's how things appear to him. The Sophists invented an argument that there can't be false thinking: ‘He who thinks what is false thinks what is not; but what is not isn't anything; so he who thinks what is false isn't thinking anything, but if he isn't thinking anything, he isn't thinking’. Plato tried to solve the problem by saying that one who thinks what is not is thinking something different from what is, and so, after all, is thinking. This doesn't work unless, explaining ‘not so’ as ‘different from so’ I also insist that ‘different from so’ implies ‘not so’. Otherwise, things might be both so and different from so.

The question – rather in the form ‘How is it that a false proposition makes sense?’ – came to interest the Cambridge philosophers Russell, Moore and Wittgenstein in the first quarter of the 20^th century. But other waters had rolled under the bridge in the interval.

II

St Augustine had produced an argument that truth must be eternal: for if it had a beginning, then before that it was true that there was no truth; and if it will have an end, then after that it will be true that there is no truth.Footnote ² Both are absurd. You could construct a parallel argument for the eternity of falsehood. So far as I know, St Augustine didn't consider the point.

The truth and falsehood of propositions of kinds that may be true or false – that for such propositions truth and falsehood are twins – was perceived in the 11^th century by St Anselm. It comes in his De Veritate,Footnote ³ a dialogue between a teacher and a pupil.Footnote ⁴ What is truth in a true proposition, a true ‘enuntiation’?Footnote ⁵ Is it the thing enuntiated? No, says the pupil, the thing it enuntiates is not its truth but the cause of its truth. – Is it the utterance itself, its signification, or anything belonging to the definition of the proposition? No, says the pupil, for if it were any of those things, the proposition would always be true; for all those are the same whether what it enuntiates is the case or is not the case. – Well, what is it? I can't think of anything to say, says the pupil, except that when it signifies the being so of what is so, there's truth in it and it's true.

The teacher then asks him what affirmation is for – what it has been created for – and he says it is for signifying the being so of what is so. This, then, the teacher says, is what affirmation ought to be doing; and when it is signifying what it ought, then it is signifying rightly. We, the readers, should note that this rightness of an action of signifying, performed by an action of affirming that something is so, is explained with the same words as those offered by the pupil as ‘all he can say’ when asked for his positive account of the truth of a proposition. – The teacher briefly adds that the same holds when the proposition signifies the not being so of what is not so.

Now the pupil, who is no stooge, asks: ‘What am I to reply, if someone says that even when an utterance signifies as being so what is not so, it signifies what it ought? For it's been equally given it to signify as being so both what is and what is not so. For if it hadn't been given it to signify as being so even what is not so, it wouldn't signify that being so. Then even when what it signifies is not so, it is signifying what it ought. But then the utterance is right and true even when it enuntiates, as being the case, what is not the case.’

This point the teacher does not deny. It is indeed, he says, not usual to call a proposition true when it signifies as being the case what is not the case – all the same it does have rightness and truth, because it is doing what it ought – i.e. signifying what's been given it to signify. But when it signifies the being so of what is so, there are two ways in which it does what it ought: first, it does its job of signifying what it does signify; and second it signifies what it was created for …. It ‘more ought to do what it was given its signification for than what it was not given it for’. For why was it only given it to signify the being so of something when it isn't so or the not-being so when it is so? Precisely because it couldn't be given it to signify something's being the case solely when it is the case, or not being the case solely when it isn't the case. So there's one kind of rightness which consists in its signifying what it does – and that is a constant property – and another kind which consists in its signifying what it was given its signification for. This rightness is variable and depends on one's use of it. If I say ‘It's daylight’ so that it signifies what actually is the case, I am using the signification of ‘it's daylight’ rightly … But when I use the same utterance to signify its being so when it is not so, I am not using it rightly: it was not made for that.

III

Note that here we have two things: one, a teleology of the kind of proposition in question, and here there is an argument – a proposition (if it's the kind to be true or false) is for being true because the other possibility for it is ancillary. The second is what affirmation was created for – namely the right use of the proposition according to what it itself is for.

St Anselm in this brief stretch of dialogue shows, first of all, that by a proposition he understands a sentence, vocally uttered or written or signed with the fingers in deaf and dumb language, etc. He doesn't mean what we nowadays call an ‘abstract proposition’. He does regard propositions as liable to have multiple occurrences. In consequence he regards propositions that may be true or false as changing from true to false or false to true as things change. In this he contrasts with almost all modern logicians, who regard it as a different proposition if it changes from being true to being false – the time at which it is uttered is a perhaps unmentioned part of it, or of its sense. And so, ‘It's daylight’ said at different times would be as many different propositions.

Anselm speaks separately of affirmation and denial of, or in, propositions. This was quite traditional, but from his saying ‘The argument will work for denial too’ we can assume that he would accept the blanket term ‘assertion’ – the denial of a proposition being the assertion of its negation. Assertion presumably has a personal subject, so we can see that Anselm says both that the proposition signifies things' being thus and so, and that the person using the proposition does so. However, he seems not to notice that a proposition can occur as a subordinate clause, an if clause, for example, and then there is nothing wrong with its being false. We might correct him, then, by saying that a proposition, true or false, performs the task of signifying what it does, and the person who asserts it also uses it to signify what it does, but there is a further duty, on the part of one asserting, of signifying as being the case only what is the case. He can use the proposition so, because if it is the complete thing that is said, that is properly what it is for. It is of course possible that Anselm would only call something a proposition if it was a complete one – i.e. one that is not part of another.

The most signal thing that he does in this stretch of dialogue is to note a certain parity of truth and falsehood in connection with the sort of proposition he is talking about. ‘It's been given to the proposition equally to signify something's being so when it is so, and when it is not so’ and ‘it couldn't be given to it to signify something's being so only when it is so’.

IV

With this observation we find ourselves in the company of Frege and Wittgenstein. Frege thought that so long as a sentence was well-formed and did not contain any proper names that did not stand for anything, the sentence was itself a complex name either of truth or of falsehood. He did not restrict his consideration to propositions that might be either true or false, but neither did he exclude such propositions from it. His conception of ‘the true’ and ‘the false’, or truth and falsehood, as objects, is as wild as it seems, and has one noteworthy objection to it: why should one aim at believing, judging, asserting, only the true? As Michael Dummett has observed, this is rather like describing a game in which there is winning and losing without there being anything to show that the object of a player is to win.Footnote ⁶ Frege misses the teleological points that are so important in Anselm.

Wittgenstein does not think that sentences are complex names at all and explicitly accused Frege of error in this. But he is with Frege, and with Anselm, in thinking that the sense of a proposition is independent of the facts. With this reference to ‘facts’ he shows that he is speaking of propositions which logically may be true or may be false: the sense, then, is independent of what Frege taught us to call the truth value of a proposition, i.e. of whether the proposition is true or false.

Hardly anyone would deny that the sense of a proposition, one that may be true or false, is independent of its truth value. Wittgenstein's observation in fact is: ‘If we don't remember that the sense is independent of the facts, it is easy for us to think of truth and falsehood as equally justified relations between signs and what they signify’ [Tractatus. 4.061].Footnote ⁷ So evidently he denies that truth and falsehood are equally justified relations, and he warns against thinking that, given a proposition p and its negation, ~p, ‘p’, say, signifies in the true way what ‘~p’ signifies in the false way. We are reminded of Anselm saying: it couldn't be given to a proposition to signify that something is so, only when it is, i.e. without its being able to signify that that thing is so, when it isn't. Why couldn't it? Because the thing it is to signify is the same in the two cases, both when it is so and when it isn't. So mightn't one think that it signifies, through that signifying of the same in the two cases, what actually is the case, but sometimes in a peculiar way called ‘the false way’? Anselm makes no such suggestion, and Wittgenstein warns against it. But Wittgenstein goes on to imagine something further: Could we communicate with false propositions, as hitherto with true ones – so long as it was understood that they were ‘meant wrong’, i.e. meant as false? He replies ‘No! For if we use ‘p’ to say that not p and things are as we mean they are, the ‘p’ in the new way of taking it will be true, not false’ [Tractatus. 4.062].

Here he has put his finger on two nerves in the matter. The first is, that you could have a code in which every sentence was to be understood as the negation of itself in non-code use. That this is possible is very striking. It would not be possible for example to have a code in which every adjective was to be interpreted as meaning ‘pink’. You'd be severely restricted in what adjectives of the non-code language you used. But the negation code is a possibility. It has even been used – Joan of Arc used it in communicating with her generals, and signalled that she was doing so by having a little cross put at the top of the page. This brings us to the second nerve. Which of the two opposite senses, that of a proposition and that of its contradictory, is actually being asserted is what fixes that aspect of the sense. And this would not be possible if it were not for the parity between truth and the corresponding (contradicting) falsehood which Anselm's pupil maintains and which Anselm accepts.

V

The dialogue in fact emphasises and insists on the parity. It could not be given to the proposition to say what it says is the case only when it is the case. Anselm indeed does not seem to notice that this means you could use the negation-code. (He is a very succinct writer who might have noticed this without remarking on it.) At any rate if either he or Wittgenstein had been witnesses at Joan of Arc's trial, where she was accused of blasphemously signalling lies with a cross, they could have declared ‘These were not lies – or if they were, not for the reason you think’.

Anselm, the teacher, explains the truth of propositions by a teleological account of – as I will now say – assertion. Assertion is for signifying as being so what is so; so asserting what is not so is a wrong use of assertion, and truth is explained as rightness of assertion precisely as assertion. This could be the explanation of Wittgenstein's warning against regarding truth and falsehood as equally justified relations between signs and what they signify. There is no doubt the German says that, though English translators have funked so translating it.Footnote ⁸ On the other hand, it is notorious that Wittgenstein criticised the ‘assertion sign’ used by both Frege and Russell. It is logically meaningless, i.e. it lacks logical meaning, he says, it merely shows that the authors believe the propositions they mark with it.

He also remarked that affirmation can't give propositions their sense – what is affirmed must already have a sense; and the same holds of denial. So we are left wondering – is the falsehood of a proposition something bad about it – as Anselm seems to think; for, using his word for a proposition, he talks about the two different ways of being true, one, having the sense that it has, and the other, doing what it's for. Also, we remember that the pupil said that the truth of a true proposition couldn't be the thing enuntiated: that must be the cause of its truth. Well, is there a thing enuntiated when the proposition is false? At one point the pupil seems to imply that there is: ‘But if it is right and true by signifying what it ought, as you have shown, an utterance is true even when it enuntiates as being so what isn't so’. Protagoras might say: no enuntiatum, so no enuntiatio.

Well, we have seen that Wittgenstein in fact concedes to assertion a role in determining sense. He does this in his reason for denying that we could communicate with false propositions as hitherto with true ones, so long as it's understood that they're meant to be false. We couldn't, because if we are using ‘p’ to say that not p, and what we mean actually is the case, ‘p’ is true, not false, in the new way of taking it. So assertion, i.e. what you are asserting, does determine which is which of the two opposite senses, those of ‘p’ and ‘~p’. ‘p’ and ‘~p’, he goes on to say, have opposite senses, but there corresponds to these one and the same reality. What reality? we want to know. The answer must be, that being the case which one of them says is the case. Which it is is not given us in their senses. And which sense each has is a matter of what we mean to assert in using them.

This account, however, still seems to leave open the possibility of saying that one of them is related ‘in the true way’ to just what the other is related to ‘in the false way’. Now if that is supposed to characterise their senses, it can't be right – precisely because their senses are independent of the fact, that is: are independent of which either of them is, true or false. If it is not supposed to characterise their senses, it is a harmless way of speaking.

VI

I have remarked that Anselm is at variance with the majority of modern logicians in treating the sort of proposition we are talking about as variable in truth-value. Russell wrote in severe criticism of a logician who took this line, giving as an example ‘Mrs Brown is not at home’. ‘Here’, Russell says, ‘it is plain that what is variable primarily is the meaning of the form of words. What is expressed by the form of words at any given instant is not itself variable, but at another instant something else, itself equally invariable, is expressed by the same form of words.’Footnote ⁹ He makes a comparison with ‘He is a barrister’, which expresses a truth in some contexts and a falsehood in others. ‘Thus the variability involved is primarily in the meaning of a form of words.’ That ‘thus’ is very significant. What Russell says is all right about ‘He is a barrister’ if the point is that the he will likely be a different person on different occasions of the use of that sentence. In order to make ‘Mrs Brown is not at home’ analogous, Russell says ‘we know the time at which this is said, and therefore we know what is meant’. Well, we might not ‘know the time’; and if one says ‘it means: Mrs Brown is not at home now’, one has to say, if one wants to follow Russell, that ‘now’ means a time, and a different time each different time it is said. But ‘now’ is no more the name of a time than ‘today’ is a date. We can conclude that Russell's explanations are not real, and the real thought lies precisely in that ‘Thus’. A certain sentence ‘expresses a truth in some contexts and a falsehood in others. Thus the variability involved is primarily in the meaning of the form of words.’ The whole reason lies in the sentence's expressing a truth in some contexts and a falsehood in others. But that is the very thing that raises the question: must it be a different proposition really, because of that? Is it like a proper name's being a different name, though equiform, if it names a different thing?

Anselm, together with Wittgenstein who has further elaborations on the theme, seems satisfactory except for the unanswered question I have mentioned: is there an ‘enuntiated thing’ when the proposition is false? He made no objection when the pupil said that the ‘thing enuntiated’ was the cause of the truth of the proposition. But neither did he object to the expression ‘what it enuntiates’ used by the pupil of a supposedly false proposition. Again, this may only be an instance of Anselm's undistractability when his eye is on a target. The significance – the sense – of the proposition is the same whether it is true or false. What about ‘what is enuntiated’? Will it too be the same when the proposition is false as when it is true? Is enuntiation the same as signification?

This question should elicit from us the last bit, the keystone of the arch representing the relations of truth, sense and assertion. There is no ‘thing enuntiated’ by a false proposition. A true proposition tells one something if one believes it. A false proposition believed still tells its believer nothing. A person may tell one a falsehood; but, just as we say that a proposition as well as a person says such-and-such, so we may also say that a proposition believed tells its believer something. – But only if it is true. For then it reflects the being so of what is so. But the analogue of this, for a false proposition, would be that it reflects the being so of what is not so. And there is no such thing as either. A paradox, on the other hand, does not say anything. There is no solution to the question whom the court should find right in Protagoras' suit.

Where the Sophists were right is reached in my present formulation: the false proposition, while it does say something, does not, being believed, tell its believers anything. So: he who thinks what is false thinks what is not; he thinks something which tells him nothing; but that does not mean he thinks nothing, i.e. does not think anything.

VII

I will end by speaking again of the paradoxes, and in particular of the one that a librarian in Oxford gave to Russell.Footnote ¹⁰ He knocked on his door, handed him a piece of paper and went away. On the piece of paper was written ‘What's on the other side is true’; Russell turned it over and found ‘What's on the other side is false’. Let's call the first one (the one with ‘true’ in it) a, and the other b. If a is true b is true – for that's what a says. But if b is true a is false – for that's what b says. So if a is true a is false. But if a is false b is false, for a says b is true. But if b is false a is true, for b says that a is false. So each of them if true is false and if false is true. I conclude that neither says anything. But, you may say, your argument was partly from what they did say. True, it was; but I would now insert ‘if a says anything’ after ‘that's what a says’, and similarly for ‘that's what b says’. Note that if b had been something else, like ‘2 × 2 = 4’, there'd be no question, a would then say something. One has to argue it out. Things can turn out not to say anything because of certain extraneous facts. However, the paradoxical character is a tip-off. I applaud Russell's desire to exclude paradoxical propositions, but not his desire to have a general principle by which he does it. The matter needs arguing in the particular case. Indeed, in the case of that piece of paper, the paradoxical character, which of course was what interested Russell, is in fact a distraction. Suppose that ‘What's on the other side is true’ had been written on both sides. It would quickly be evident that nothing was said – one can't find out what is being said to be true. (Similarly for ‘What's on the other side is false’, written on both sides.) And that should be enough to damn the paradoxical case of the same thing. But each paradox (or kind of paradox) needs excluding ad hoc, on its merits. Russell perhaps does this, but he has a sort of formula. He gives a list of words: ‘truth’, ‘falsehood’, ‘function’, ‘property’, ‘class’, ‘relation’, ‘cardinal’, ‘ordinal’, ‘name’, ‘definition’, and says: ‘Any such word, if its typical ambiguity is overlooked, will apparently generate a totality containing members defined in terms of itself and will thus give rise to vicious circle paradoxes.’Footnote ¹¹

That sounds as if he meant that a totality containing members defined in terms of itself was always illegitimate and a possible source of paradoxes. But perhaps not so: for he says: ‘The appearance of contradiction is always due to the presence of words embodying a concealed typical ambiguity and the solution of the apparent contradiction lies in bringing the concealed ambiguity to light.’Footnote ¹²

So it is not just the totality containing members defined in terms of itself that is ‘illegitimate’, but the fact that the totality is generated by means of a word ambiguous in logical type.

Russell is therefore not refuted by the perfect legitimacy of my making a list of all the lists kept in my office, which will include itself if it is kept in my office. Or even a list of all the lists kept in my office which do not include themselves – so long as I don't keep it in my office. But what about a list of all the lists in the world that don't include themselves? That seems to generate a characteristic Russellian paradox – but what is the ambiguity of logical type involved in constructing this description?