The Liar and the Liar Denier – Or “Will the real Liar please stand up and then sit down where she’s told to?”
Here are some thoughts (by a non-logician) on our recent workshop on semantic paradoxes.
Some Turkish philosophers have strange dreams. Erdinc’s are so weird and wonderful that he cannot even tell us about them. And after our two day workshop on paradoxes with Graham Priest and Stephen Read, Ilhan told us about a strange dream in which he met ‘The True’ who appeared as to him as a small talking metallic sphere. And I think it makes some sense to think of sentences as such spheres. So let’s suppose that sentences are small shiny metallic spheres that (a) can be named and (b) tell us what they mean. And let’s think of three such spheres. One of them is called “The Liar” and the second is called “The Liar-Denier” The third is called “The Liar-Affirmer”. The Liar and The Liar-Denier are twin sisters. The Liar-Affirmer is their younger sister. [Perhaps they’re all daughters of ‘The True’ who Ilhan met in his dream.] Anyway, this is what we know about each sphere:
(1) The Liar says: “What the Liar says is False”.
(2) The Liar-Denier says: “What the Liar says is False”.
(3) The Liar-Affirmer says: “What the Liar says is True”
Now, Stephen Read thinks that what The-Liar says is false, because while saying “what the Liar says is false” she says (probably very quietly under her breath, so we may not notice it), “What the Liar says is True” and also says “What the liar says is True and What the Liar says is false” – and Stephen thinks that what a sentence says is false (and not True) if anything the sentence says is false. So Stephen claims that what the liar says is false. Graham Priest thinks that what the Liar says is both true and false.
So what about her sisters? Well, according to Stephen, what the Liar-Denier says is clearly True (and not false), while what the Liar Affirmer says is False (and not True). But what about Graham? Well it seems to me that what he should say is that what the Liar-Denier says is True (and not False) and what the Liar Affirmer says is also True (and not false). But in our workshop he argued that what the Liar-Denier says is both True and False. Now I thought that, according to his account, what the Liar says is both True and False. So what the Liar-Denier says is thus clearly True, but it will only also be false if what the Liar says is both True, False and not-false, and as far as I understand his semantics being both true and false does not entail not being false. So I did not understand why he thought the liar denier was both true and false. I asked him about this and he explained that: “An atomic sentence Pa is true if the denotation of ‘a’ is in the extension of ‘P'; and it is false if ‘a’ is in the anti-extension of ‘P’. By the Liar argument, your (1) is both true and false. The subject of (2) refers to the same subject as the subject of (1), and the predicates of (1) and (2) are the same. Hence (2) is true and false (2)… Note that, in general, though, if Qb is both true and false, both T<Qb> and F<Qb> may be true only. They do not have co-referring subjects and
predicates.” But, given this, I’m wondering what it is, if anything, that makes the Liar and the Liar denier distinct sentences on his approach.
I have two big questions about this little story I’ve told: (1) What is it for two individual sentences to be distinct sentences? And (2) How do sentences get their names?
(1) One might be committed to the principle of the identity of indiscernibles for sentences, and so think that if two sentences say the same thing, they are the same sentence. But if this is right, then we might think that the Liar and the Liar-Denier are really the same sentence with two names. There is only one little metallic sphere which has been given two names. Or perhaps we have one little metallic sphere which is in two places at the same time? [and if this is the case then it looks like we have a problem analogous to Zeno’s problems with motion]. Now if Stephen is right on his account of what the Liar says, then we don’t have this problem as the Liar and the Liar-denier don’t really say the same thing at all, for the Liar does not only say “the Liar is false” but also says “the Liar is True and the Liar is False”. Now one might think that the difference between what the the Liar and the Liar Denier say corresponds to the difference between :
(a) ‘This sentence is false’
(b) “’This sentence is false’ is false”
These do seem to be distinct sentences. But if we are thinking of sentences as small talking metallic spheres and we’re committed to the identity of indiscerbibles for sentences, then, unless we tell a story like Stpehen Read’s it is plausible to think that the Liar and the Liar Denier are really the same sentence. So, here’s a question, how many sentences you do think we have here? One or Two?
(2) Perhaps we can answer this question by thinking a bit more about how sentences get their names. It seems to me that there are two options here: either they are named by someone else, or they name themselves. So one story is would be that the Liar and the Liar Denier were named at birth by their mother, The True. This is an odd story. It sort of commits us to the view that individuals have real names, that are perhaps unknowable by anyone. Perhaps The True names her daughters, but how does she tell them apart when she bumps into them? They say the same thing, and so seem to be indistinguishable. So perhaps they have the names they have, but once they’ve been named there is no way, in principle, of telling them apart. So I think it makes more sense to think that these sentences name themselves. And if this is the case it looks like we have two sentences that are not indiscernible:
(a) one sentence says: “I am called the Liar and what the Liar says is False”
(b) another sentence says: “I am not called the Liar and what the Liar says is False”
Now, what are we to make of what the first sentence says? When she claims that “I am called the Liar” is she saying something true? There is no fact of the matter that makes this claim true. So perhaps we should take it as something more like an order or a stipulation: “Call me the Liar!” And it seems to me that perhaps the proper response to this order is to say “I can’t do that”, because it forces us into a performative contradiction. If someone tells you “Call me Lucas and don’t call me Lucas” what they are saying is not paradoxical, they’re just telling you to do something that you don’t know how to do. It is difficult to see how the liar can succeed in naming herself. But what about the second sentence? Well, this now appears to be a sentence containing a name with an unknown reference, so it looks like we just don’t know its truth value. But what about the following sentence:
(b’) “I am not-called the Liar and what the sentence who claims to be called the liar says is False.”
Given my account of (a) this might look like a way of referring to (a) and so given the fact that (a) is false this sentence is saying something True. Unfortunately, this suggests a way of constructing a stronger version of (a):
(a’) “I am the sentence that claims to be called the Liar, and what the sentence that claims to be called the Liar says is false.”
Now, with this sentence the question is whether the term “the sentence that clams to be called to Liar” should be read as a name or a definite description. If we take it as a name we get the following sentence:
(a’n) “I am called ‘the sentence that claims to be called the Liar’ and what the sentence called ‘the sentence that claims to be called the Liar’ says is false.”
If we read the sentence in this way, then it seems to me that I can give the same sort of analysis as before which is that the whole sentence is false because either (a) the first of the conjuncts is false or (b) because the sentence commits a performative contradiction by trying to do something that it cannot do. Or perhaps (b) is the explanation of (a). But what if we take “the sentence that claims to be called the Liar” as a definite description? Here things are more difficult. But one thing to bear in mind is that “I am the x” will be false if there is more than one x. So if there is more that one sentence that claims to be called the liar then a’ will be false because the first of the conjuncts will be false. So to avoid paradox, The True just has to give birth to at least two daughters who claims to be called the Liar.
(3) Anyway – Perhaps it’s a mistake to think of sentences as small talking metallic spheres. So lets supposed that sentences are strings of words written on bits of paper, and let’s suppose that there are five of them in a hat and we’re told to firstly look at them individually and secondly to take them out one by one and put them in random order – and we produce the following list:
(1) “2+2=5” (F)
(2) “The second Sentence consists of five words” (F)
(3) “The third sentence consists of seven words” (T)
(4) “The fourth sentence is false” (F?)
(5) “The fourth sentence is false” (T?)
The fourth sentence seems to be the Liar –if it is True it is False and if it is False it is True, and the fifth sentence seems to be the Liar-Denier. Now what truth values do these sentences have? Now I think that this is a trick question. If we just randomly look at a sentence while they are still in the hat the only sentence that clearly has a truth value is the first sentence. While they are in the hat none of these sentences are paradoxical. It’s only when we take them out and put them in a certain order that paradox arises. Another strange thing is that when we look at the sentences in the hat we are are unable to distinguish between the liar and the liar denier. If we switch the positions of the 4th and 5th sentence the liar sentence becomes the liar denier and the liar denier becomes the Liar. Which one is which depends upon its order in the list. So perhaps we don’t really have 5 sentences in the hat, but only four – one of which we copy down twice.
Now it seems to me that that the paradox here only arises because we take a set of sentences and try and put them in a certain order. And so my suggestion for blocking the paradox is to revise our theory of order and position. We have assumed that when we are told to randomly take five sentences out of a hat and put them in order this is something we know how to do, and that we will end up with a list with five members. But this is an assumption that we can reject. We have no problem with taking our the first three sentences and putting them in order. But when we take out the next one, something strange happens – our theory of order breaks down. The rules of our game are set up in such a way that when we try and put the sentence “the fourth sentence is false” in the fourth position on our list it ends up somewhere else, and the fourth position is filled by the empty set. We try and put the sentence we’ve just pulled out of the hat just after the third sentence, but instead we put nothing there and the sentence we try and put there ends up somewhere else.
My suggestion for solving this paradox, then, is to claim that what I have been calling the “fourth sentence” is false because there is no fourth sentence. If we think that “The King of France is bald” is false because there is no King of France, then if there is no 4th sentence then “the fourth sentence is false” is false for the same reason. But why would one think that there is no 4th sentence? My thought here is that when we try and put the sentence “The Fourth sentence is false” in the 4th position we necessarily fail. Just as an attempted act of self-naming is not guaranteed to succeed, an attempt to put a sentence in a certain place on a list does not necessarily succeed either. Sometimes, however hard you try, you just can’t get seven people into a taxi, they just don’t fit.
So when we play our game, the list we end up with actually looks more like this:
(1) “2+2=5” (F)
(2) “The second Sentence consists of five words” (F)
(3) “The third sentence consists of seven words” (T)
(5) “The fourth sentence is false” (F)
(?) “The fourth sentence is false” (F)
I’m not even sure that we can say how many things we have on this list. But the fact that we’re told to take a bunch of things and put them in arbitrary order doesn’t mean we can necessarily do it. Where on our list is the sentence marked (?)? I think the best answer is: We just don’t know. [So the suggestion I’m making here is that we can avoid the paradox by questioning the assumption that sets of sentences can necessarily be well ordered.]
In the case of the sentence who claims to be the Liar, we really don’t know who she is. In this paradox we really have no idea where the sentence is on the list. The Liar is not paradoxical; She’s just very naughty. She won’t tell you who she is, and she refuses to sit where you tell her.
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