I’d like to start with a question: could ‘I’d like to talk about sheep’ ever be a self-referential sentence? The answer is: Maybe!

This is a post about paradoxes, normal sentences and self-referential sentences. I mean for it to be short, but it probably won’t be…

Let’s start with a normal sentence. Nothing weird happening yet: ’17 + 13 = 30′. Well, is it true or false? It’s got to be one of them. Let’s see: 17 + 13 = 10 + 10 + 7 + 3 = 20 + 10 = 30. Yes, definitely true. And let’s ask a calculator as well, just to be sure: 17+13=? 30. There is no doubt here.

Another sentence: ’41 x 11 = 431′. Is it true? 41 x 11 = 40 x 11 + 1 x 11 = 440 + 11 = 451. No, it’s false.
Let’s do one more. ‘I am writing this paragraph on the day of the Queen’s Platinum Jubilee.’ Actually, although it’s close and celebrations are still happening, it isn’t the actual day, so that is also false.

Now to make things go wrong, because that’s where it becomes interesting. Just one more sentence: ‘This sentence is true.’

Well, is it true, or false? Hmm. How do we check? Well, it’s true if and only if it’s true, but that’s circular logic. Not helpful. Maybe it’s false? It’s false if and only if it’s false.
OK. None of that is helping… What if we try a proof by contradiction? I think this sentence seems a bit shady, so let’s try to prove it’s false. It might be true, but let’s try for falsehood first. Assume it’s true. If the sentence is true, then the sentence must be saying something correct. Therefore, as it says, it is true. No contradiction there, I’m sorry to say. Oh well. I guess it’s true then.

But wait. Here’s something odd: What if we reverse time, and try to prove it TRUE instead of false?

…I think this sentence deserves to be true, considering how interesting it is. It might be false, but let’s try for truthfulness first. Assume it’s false. If the sentence is false, then the sentence must be saying something incorrect. Therefore, as it does not say, it is false. No contradiction there, I’m sorry to say. Oh well. I guess it’s false then.

Hang on a minute! Both ways came to the same conclusion, which means that ‘This sentence is true’ is simultaneously true… and false? That’s very odd, and also a paradox. Oof.

Here’s something else just as odd: ‘This sentence is FALSE.’

Let’s use the same method. Of course we have to check both sides, but that won’t be hard.

Let’s assume it’s true. That means that it’s false, because that’s what it says. Oh, so then it’s simultaneously true and false. Ok. What about falsehood? Let’s assume it’s false. That means that it’s true, because that’s what it doesn’t say. Oh, so then it’s simultaneously true AND false. O…K…

So if it’s true, then it’s a paradox, and if it’s false, then it’s a paradox. Definitely a paradox…

What if you wanted to BAN paradoxes from corrupting logic? Maybe you find them interesting, like I do, but maybe… Maybe you just want to ban them. I’m going to now play the part of someone trying to get rid of paradoxes: Ahem. ‘

This is an outrage! There are paradoxes everywhere. They must be destroyed, removed from logic. They are foul creatures… We must get rid of them, at all (well, almost all) costs! What if we say ‘Paradoxes don’t exist’? That’s just ignoring the problem, and making our logic even more inconsistent by planting a falsehood in it and calling it true. We certainly shouldn’t do that. What if we change our system of logic, make it… more fortified?’

I’d just like to interrupt myself and say that this is the position that quite a few mathematicians took. This is not a made-up thing, many people have disliked paradoxes; and attempted to remove ‘these foul creatures’ from mathematics. Now, on with the show!

‘How do we do that then? Perhaps we should…Uh… I know! Attack them at the root, where all paradoxes begin. But what is that root? It would be something that all paradoxes have in common. But what do paradoxes have in common? Um, Ah, and Er…

Perhaps we need more paradoxes. So now to find these putrid creatures… Here’s one. ‘1. Sentence 2 is false. 2. Sentence 1 is true.’ And… I can’t think of any more. Huh. Anyway, they all have the word ‘Sentence’ in them, somewhere or another. So that’s it then! But wait… this isn’t a paradox, but it contains the word ‘Sentence’: ”Sentence’ is a word.’ And for that matter, ‘Sentence’ is contained in this sentence, the previous, the previous, not the previous, but the previous. But wait, all those sentences contained ‘Sentence’ in quotes. It was being mentioned, not used, in those sentences. Whereas in the sentence ‘This sentence is false’ then it is being used. It isn’t in quotes. But hang on once again… This sentence uses the word ‘Sentence’, but it isn’t paradoxical. So it isn’t that. Maybe it’s… what the sentences were doing with the word ‘Sentence’. They were using it to describe themselves. So it’s self-description that we’re after.

But wait once again once again… (I just realised, I use elipsises – no I mean, elipsi, no I mean elips…æ? Anyway, I use [elipsis’ pluralisation] far too much.) One, two, three, four, five, six, seven sentences ago, a sentence was self-descriptive: ‘But hang on once again… This sentence uses the word ‘Sentence’, but it isn’t paradoxical.’ It’s talking about itself, but it still isn’t paradoxical. Ugh! This is hard…’

Later…

‘We were on a wild goose chase! It isn’t something that cleanly splits logic into paradoxes and non-paradoxes that we need, it’s something that removes all the paradoxes. Maybe it removes some non-paradoxes too, but just as long as all the paradoxes are gone, then everything’s fine. So without further ado, I propose a new logic system, where self-reference is banned! Wait. This is just ‘Paradoxes don’t exist’ again… We can’t just directly ban them, that makes no sense. There has to be an actual formal constraint. This constraint will be that sentences cannot in any way mention themselves. (That sentence there mentioned itself…) Anyway. All paradoxes… are gone! Wait… What about this one? ‘1. Sentence 2 is true. 2. Sentence 1 is false.’ Neither sentence mentions itself, but the group is paradoxical. Um…’

Even later…

‘Maybe… maybe it isn’t that it can’t reference itself… but that… Hm. What if we had levels? Different levels. Each level would be stacked on top of other levels, and there is one constraint: higher levels cannot be referenced by lower or equal levels. So then sentence 1 of the duo could be on level 1, and sentence 2 would then have to be higher than it, say level 2. But now sentence 1 cannot reference sentence 2! Solved!’

After much celebrating about the new paradox-free logic, a stranger walked in with a question:

Stranger: Hello. I have a question about your logic, or more… a statement I suppose. But it’s framed like a question.
Logician: Go ahead!
Stranger: What about the sentence, ”Is false upon being spoken after its quotation.’ is false upon being spoken after its quotation.’?
Logician: Hm… that’s a strange one. Explain, please.
Stranger: Well, the sentence is saying that ‘Is false upon being spoken after its quotation.’ (in quotes) is false upon being spoken after its quotation. In other words, since ‘Is false upon being spoken after its quotation’ is spoken after its quotation in the actual sentence, then the sentence reduces to, in a strange and coiling way, ‘This utterance is false.’
Logician: Oh! But wait. If it reduces to ‘This sentence is false’, then surely it is subject to my constraint! It is a sentence on level 1, which is referencing a sentence on level 1. That is not allowed.
Stranger: Yes, but it is not referencing itself to begin with. Only its simplification is.
Logician: I see what you mean… So I cannot apply the constraint to the simplification, only the original sentence. Ah. Well, I suppose I shall do that then. ‘Is false upon being spoken after its quotation’ is however, just a sentence fragment, and a level cannot be assigned to it. But when spoken after its quotation, it becomes a sentence which references something in a way that means that the something becomes itself… um… but surely this does not count… no… yes? No? No…
Stranger: Sorry to ruin things for you.
Logician: It’s… okay…

And that was that. The End.

By the way, the sentence ”Is false upon being spoken after its quotation’ is false upon being spoken after its quotation’ is the Quine sentence, named after the logician Quine who originally created the idea. The idea can be materialised in many ways, but it usually just becomes ”X’ X’ where X is a sentence fragment missing a referent, that says that the referent is false when slotted, somehow, into this mold: ”_’ _’. And since replacing the blanks with ‘X’ returns ”X’ X’ – the original sentence – then the sentence is simply saying that it is false.

I hope I didn’t break any copyrights there.

Now for some more self-referencials and other odd sentences. First, ‘This sentence is a paradox’. It seems very much like it should be paradoxical – therefore being true – but is actually not. Probably. Here’s my logic.

First, assume it’s true. If it’s true then that means it is a paradox. This is a contradiction, so it is not true – proof by contradiction says so. Oddly enough, it’s such an easy contradiction that on analysing this sentence, I missed the contradiction and only spotted it the next time I tried to analyse it. I found the sentence so interesting that I decided to write a short blog post on it.

Oops.

Anyway, if it’s not true, then let’s try for falsity. Assume it’s false. This means it is not a paradox, therefore it has one, and only one, correct answer. Assume this answer is truthfulness. If it’s true then that means it is a paradox, just as we said before, so it can’t be true. If it’s not true, then let’s try for falsity. This leads us around a loop until we are back here, at the fork between truthfulness and falsity. But it can’t be true, and if repeatedly going down falsity never leads to a paradox, then it. Must. Be. False. Sentence. Fragment. Another. Sentence. Fragment.

What about ‘This sentence is not a paradox’?

Assume it’s true. If it’s true, then it isn’t a paradox, so it has two possible routes: true and false. If it’s true then this never leads to a paradox, but we have to check falsity before saying that the sentence is not paradoxical – it might be like ‘This sentence is true’. Let’s reverse all the way to the first fork.

Assume it’s false. If it’s false then it is not not a paradox, which means it is a paradox. Contradiction! This is just the same as ‘This sentence is a paradox’ but with truth and falsity reversed.

Hopefully I don’t break any copyrights now…

‘This sentence contians three erorrs.’ Think about it. (It’s good).

Have you thought about it? I’m about to spoil the answer, so make sure you’ve decided whether it is paradoxical or not before reading (if you want to enjoy the puzzle).

How many erorrs does it contian? 2. But it says it contains 3! That’s an erorr. That means it does contian three erorrs, which removes the third erorr! It got the number of erorrs correct! But now it only contians 2 erorrs again. So it contains three.

Is it true? Is it false? Or is it a paradox? It’s a paradox, because if it’s true then it’s immediately false, and if it’s false then it’s immediately true.

Paradoxes, in a way, ‘break out of the system’. There are ‘supposed’ to be two boxes: right and wrong, true and false. But paradoxes don’t fit in either box, because as soon as you assign them one box, you suddenly find them in the other. So you have to put them in a new box: paradoxical. So there is now a new system with three boxes. Can we find a sentence that breaks out of this new system?

It would have to be a sentence that is a paradox if and only if it isn’t. Might ‘If this sentence is paradoxical, then it is false’ do it? Well, let’s see…

Assume it’s true. If it’s true then if it’s paradoxical then it’s false. Is it paradoxical or not? If it is, then that’s immediately a contradiction as that is the definition of a paradox – all answers you can give lead to contradictions. So it isn’t paradoxical, and lo and behold, this is stable. It can stay in the ‘true’ box.

Assume it’s false. If it’s false then it is not the case that if it is paradoxical then it’s false, but this does not mean that if it is not paradoxical then it is not false, as that is not how this works. You can’t distribute the falsity like that. This is also stable.

If there are two stable states then it is a paradox. If the sentence is true then it is false, because that’s what the sentence says and it is a true sentence, and if the sentence is false then it can stay false. Therefore… um… yes I think… it’s false! I think. Probably.

‘Thit sentence is not self-referential because ‘thit’ is not a word.’
In order to understand it then you have to make it false, but it certainly seems true… this is a puzzling one, definitely. Humans can understand ‘thit’ to mean ‘this’ because of context. But thit means that it is self-referential, because humans can understand ‘thit’. But it says that it isn’t self-referential… or does it? The sentence itself is only meaningful if you take ‘thit’ to mean ‘this’ But that yields the self-referential and false sentence ‘This sentence is not self-referential because ‘thit’ is not a word’. ‘Thit’ is enough of a word to have some meaning. So really, ‘Thit sentence is 90% self-referential because ‘thit”s meaning can be obtained from context’.

‘Thit setnense it nos slef|rerefetnila bicose mots of ist worsd hav magnld splelinsg.’
Can you understand it? If you can, then for you it is slef|rerefetnila because you can understand it, making it false. Interestingly, as soon as the sentence becomes too mangled to understand, its ‘hidden meaning’ becomes a true sentence. The hidden meaning goes something like this: ‘This sentence is not self-referential because its many errors cannot be corrected by humans without context’.

‘Ftil cetn3s sis tno refl/selfretfailc cabos ft cntans 2 meanie rerors 2 B Crsedtc wthit cxnte.’
I told you the ‘hidden meaning’. You should be able to get most of its meaning from that, although as the spelling worsens then the meaning changes. This sentence is ‘supposed’ to read ‘This sentence is not self-referential because it contains too many errors to be corrected without context’. So pretty much the same as the hidden meaning. But the thing is, show this sentence to a random person who is interested enough to give it a go, and they will have a very hard time Crsedtcing fts meanie rerors wthit the cxnte that you have.

‘Hits gentsgc ist nst reah_sflettrec reabcs ftoo catsntis weot marrnay areas tero breee cweortectes whaot contacts.’
This is certainly impossible whaot contacts. But the question remains, is it reah_sflettrec? I’m pretty sure that the readers of this post should be able to figore out what I mean, because all the context is here. But show a random person ‘reah_sflettrec’ and there’s no way they could figure out what it was supposed to mean. So is this sentence self-referential? It depends on the contacts.

‘I dope tat ic antrik yoo witeh tish fnail stnecents, bcause sit in’ts itn the smae viena az teh pverisuo sefl-reserventail snetneces wore.’
Think about it for a bit. Can you get the original meaning? I’ve given you context. The original meaning is…
‘I hope that I can trick you with this final sentence, because it isn’t in the same vein as the previous self-referential sentences were.’
Wait, what? It isn’t the extreme of this pattern of sentences? Or is it? In any case, the context gave you the wrong sentence, I bet. Now here’s another.

‘Teh pvresios sntencets aer inetserestgni ansd ale, btui raley wnta tu jsut tlakab otu sheep.’
My question is: can this sentence be made self-referential? The hidden meaning is ‘The previous sentences are interesting and all, but I really want to just talk about sheep.’ If I was to put it in the spot of the previous sentence (whose hidden meaning is arguably self-referential) then would people be fooled? Probably someone would be. Now what about ‘The prvius sentncs ar intrestng and al, but Ireali want to juts tkal abot sheep.’ I don’t know how many people would be fooled by that one. And finally, if I was to replace the last paragraph’s sentence with ‘The previous sentences are interesting and all, but I really want to just talk about sheep.’, would anyone be fooled? I don’t think so. Is there a context you could put that sentence in to make it self-referential? Perhaps you could, but someone else couldn’t. Perhaps all sentences are self-referential with context! Hmm…

Well, I hope you enjoyed this totally short journey through the land of self-referential sentences. Goodbye!

PS: I got quite a lot of the stuff for this post from Douglas R. Hofstadter’s books, ‘Gödel, Escher, Bach’ and ‘Metamagical Themas’. If you enjoyed this post then definitely read them, they’re great. And read his other books, because they’re also great.