0. Introduction
Recently, I had a conversation with my friends Matt Dillahunty and Ozy about philosophy. At about the 1:25:00 mark (the link above should be timestamped), we started talking about how there may be considerations which lead philosophers to rationally question the basic ‘laws of logic’, such as the law of non-contradiction (for all p: ~(p & ~p)) and the law of excluded middle (for all p: (p ∨ ~p)). I brought up the liar paradox, as an example of this sort of thing. Matt objected that it is actually an instance of a ‘gappy’ sentence, which is neither true nor false. At the time, I knew there was a phenomena called ‘revenge’ which poses big problems for this strategy, but annoyingly I couldn’t bring the details to the bit of my brain that makes my mouth work. Here I want to right that wrong.
- The Liar
The liar sentence is of the following form:
a) This sentence is false.
The issue with a) is that it leads to a contradiction.
We only have to assume what seems like a very natural assumption about how the word ‘true’ works to get there. This is that if p is true, then p. We can think of this principle like this; if I say that it is daytime, and if what I say is true, then it is daytime. Alternatively; if I say a declarative sentence, and if it is true, then what it says correctly describes the thing that the sentence is about. This seems to be at the very core of idea of truth.
A corresponding idea is there for ‘false’ as well; if p is false, then it is not the case that p. If I say that it is daytime, and if what I say is false, then it is not daytime. If I make a declarative sentence, and it is false, then it incorrectly describes the thing that the sentence is about.
So let’s apply these principles to a):
If a) is true, then a) correctly describes what it is about. But a) is about itself, and it says about itself that it is false. So if it is true, then it correctly describes itself as false. So if it is true, it is false. And that is a contradiction.
So maybe a) is false. And if a) is false, then it incorrectly describes itself; yet what it says about itself is that it is false. If its self description is incorrect, then it isn’t false; and the only other option is that it is true. So if it is false, then it is true. Contradiction again.
So if it is true, it’s false; but if it is false, it’s true. Either way you go, you run into a contradiction. This is the paradox.
2. Gaps.
Yet, maybe there is a solution here. Matt certainly proposed a solution here. His idea was that a) is neither true nor false. So, let’s run through the options and see how it works.
a) says about itself that it is false. And we are now saying that it has no truth-value at all. Well, it certainly doesn’t correctly describe itself, because it says that it is false, and it is ex hypothesi neither true nor false. If something is neither true nor false, then it is not false. So it’s own self-description fails. This seems to leave no reason to consider it true. It says about itself that it is false, but we cannot derive that it is true. So far, no contradiction.
But, it says about itself that it is false, and this is incorrect (because being neither true nor false, it is not false). And it’s hard to see why this wouldn’t count as a case of a falsity. After all, it says that it is false, yet (ex hypothesi) it isn’t (because it’s gappy). It is certainly not true that it is false; it’s own self-description fails. But does this mean that it is false? Well, only if ‘not true’ means false. And, on this assumption, where we have some sentences which are ‘gappy’ (i.e. neither true nor false), there is a difference between being not-true and being false. If we listed all the not-true sentences, it would include all the ones which had no truth-value, and all the ones which were false. Thus, being not-true does not entail being false. Thus, we seem to have got out of the trap.
It is neither true nor false, and when it says about itself that it is false we can consider it’s incorrect self-description to be a case of being not-true, rather than false.
Strictly speaking, this does work as a consistent (i.e. contradiction-free) way to think about a).
So far, so good. However, things are not over. There is a second round.
3. Revenge
Consider the ‘strengthened liar’ sentence:
b) This sentence is not true.
We have, on our assumption of ‘gappyness’, three options. Either b) is true, or it is false, or it is neither true nor false. Let’s take them one at a time:
If b) is true, then it correctly describes itself. Yet it says about itself that it is not true. So if it is true, it is not true. This is a contradiction.
If b) is false, then what it says about itself is incorrect. Yet, if it is false, then it does come under the category of not-true, which is what it says about itself. So if it is false, then what it says about itself is correct, and so it is true. And we have another contradiction.
The only other option is the one we used for a), which is that it is neither true nor false. Yet, if b) is neither true nor false, then it is in the not-true category as well (because anything which is neither true nor false is not true). But, as it says about itself that it is not true, it would seem like it has correctly described itself. If it has correctly described itself, then it is not in the not-true category, but in the true category. So if b) is neither true nor false, then it is true! This is, again, a contradiction.
So, while the gappy proposal got rid of one liar sentence (i.e. a)), it fails to help us with another one (i.e. b)). As a strategy, gappiness won a battle, but it loses the war.
4. Conclusion
The problem that the liar paradox presents is subtle, and still an open question in philosophy and logic. It may be that a solution to the generalised problem exists which involves adopting a logic which has truth-value gaps. That may be the case for all I know. But it seems clear that simply adopting truth-value gaps does not solve the underlying phenomenon. It merely pushes the problem to somewhere else. Even if a) can be got around by postulating truth-value gaps, b) cannot be. The liar paradox has had its revenge.
As the philosopher Tyler Burge put it:
“Any approach that suppresses the liar-like reasoning in one guise or terminology only to have it emerge in another must be seen as not casting its net wide enough to capture the protean phenomenon of semantical paradox.” (Tyler Burge, Semantical Paradoxes, p. 173, (1979))
UseOfReason 
Why not just define such self referencing utterances as gibberish?
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Hi. So there are good reasons for thinking the the liar sentence is not gibberish. I think it is clear that the issue cannot be that self-reference always leads to gibberish. Consider the following:
1) This sentence is a sentence.
1 does not seem to be gibberish. In fact, it is true. 1 is a sentence. It refers to itself, but that does not make it gibberish. Or the following:
2) This sentence is a fried egg.
It is obvious that 2 is false, in contrast to 1. But it is not gibberish either. It makes sense, but what it says is not the case. So this shows that self-referential sentences seem to be truth-apt (they can take truth values).
It might be thought that it is not just the self-reference which makes the sentence gibberish, but the self-reference combined with the attribution of a truth value; i.e. the problem is that the sentence says about itself what its truth value is. The problem with this line is that the following seems fairly straightforward:
3) This sentence is true.
The sentence can be given a truth-value without running into contradiction (seemingly unlike the liar sentence). If 3 is true, then it correctly describes itself. And it says that it is true. So that’s cool. If 3 is false, then it incorrectly describes itself. So it is incorrect when it says that it is true. And that’s cool as well. No contradiction either way.
You might have an issue with 3, which is that there seems to be nothing which makes it true rather than false. Sure, picking a truth-value for it won’t lead to a contradiction, but it seems unsupported somehow. We might say that it’s truth-value is underdetermined. That might make you sympathetic to the idea that it lacks a truth value.
But the liar sentence (‘this sentence is false’), is, if anything, overdetermined with respect to its truth-value. It seems more like something which is both true and false, rather than something which is neither true nor false.
And if we step back for a moment, we can see that English clearly has an operation which consists of predicating truth of sentences. You might say ‘what he said is true’, or point to a sentence when asked to pick out the true one, or say ‘it is true that ‘the best things in life are free”, etc. We clearly ascribe truth to sentences all the time. We also understand self-referential sentences, like ‘this is a sentence’. All the liar paradox needs to get off the ground is these two principles. If you want to say that the result of applying them in this case results in something meaningless, when the results of applying both of them together in 3 does not, then this seems ad hoc. You can say anything if you don’t have to justify it at all. The hard part is finding any principled reason for making that move. I don’t see one. I see principled reasons for thinking that the liar sentence should be taken to be not-gibberish.
Hope that helps.
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Hi Alex. Thanks for such a substantial reply.
Initially I might be inclined to put all 3 of those sentences in the gibberish category.
However, they are all different to the liar sentence in that they do not lead directly to a paradox (by design obviously).
So one route could be to treat self-referencing-paradoxes like we treat one-divided-by-zero. You can write it and you can speak it, but we don’t assign a meaning to it. It is undefined by definition (that sounds weird).
In that case your 3 statements would not be gibberish but the LP would be. Really though I think that 3) is gibberish in the same sense that the LP is.
Alternately, on the view that sentences are the truth bearers, the “truth” is in the sentence not out there in the world somewhere. The cat sat on the mat may be true but the cat is not true nor is the mat. It is the sentence that may be true.
So it’s not that they are self referencing but that they are exclusively self referencing that makes them gibberish. They don’t go anywhere. They don’t reach outside themselves.
In that case
1) may be trivially true.
2) would simply be false.
3) would be gibberish.
Regarding 1). Perhaps such a pure tautology could also be considered gibberish.
Lastly, perhaps the English language is only coherent in a given domain. Like general relativity. So we just say, yeah we need a different approach here. Meanwhile of course we can just ignore such utterances as gibberish. Which is what we do in any case.
I have to stop thinking now for a while.
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“So one route could be to treat self-referencing-paradoxes like we treat one-divided-by-zero. You can write it and you can speak it, but we don’t assign a meaning to it.” Well, I think that in Peano Arithmetic, x/y is just shorthand for there is a t such that x * t = y. When x = 1 and y = 0, then there is no such t. So it can never be true. So this is a principled reason for 1/0 not having a value; it follows from the logic of arithmetic. It isn’t an ad hoc decision to just say that it has no value. If you could show that a similar thing happens for the liar paradox, then great. But just placing a ‘no value’ sticker on it for no reason, and pointing to the value of 0/1 as support seems misguided to me.
“it’s not that they are self referencing but that they are exclusively self referencing that makes them gibberish. They don’t go anywhere. They don’t reach outside themselves.” Well, I did say that 3 seems unsupported somehow. There is nothing other than itself which could possibly make it true, and that is weird. But I’m not following why this makes it ‘gibberish’ (which is equivalent to being meaningless or non-sensical, right?). I seem to have no problem following the meaning of the sentence. The problem, if any, is with truth, but not meaning. That seems quite clear to me.
“Perhaps such a pure tautology could also be considered gibberish.” You don’t want to make tautologies gibberish, or you throw out any hope of doing logic. Anything axiomatic is tautologous, relative to a logical system. We need tautologies.
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It seems to me that purely self-referential statements are an instance of purely self-referential entity A that is defined as ~A.
I guess truth statements are funny things because they can uniquely ‘take the reigns’ of their own truth value in a self-referential way. Since statements adhere to the law of identity, isn’t the sentence “this statement is not true” semantically equivalent to “this A is not ~A”?
I’d love to hear your thoughts.
Thanks, as always, for the post! I hope you do more with Ozy and Matt in the near future.
-Nathan
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Oops, I meant to say in my last post:
isn’t the sentence “this statement is not true” semantically equivalent to “this A is ~A”?
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I’m still bothered by ‘This sentence is true’. You say ‘there is nothing other than itself that could possibly make it true’, and that this is weird. But it’s worse than that, isn’t it? It’s not a tautology, and it’s hard to see how it could ‘make itself true’. Indeed, I am unable to think of anything that could make it either true or false. So should we say that it lacks a truth-value? No, because then it would be false that it’s true, in which case it would be false (and so would not lack a truth-value). My point, such as it is, is that this sentence isn’t all that much less problematic than the original Liar sentence. If we could figure out what’s wrong with it, maybe we could make some progress on the Liar sentence too.
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It seems to me that both sentences lack determinate truth conditions. Truth conditions being the way things would have to be for a statement to be true. A tautology is true in all conditions and a contradiction false in all. Since statements like these are necessarily true or false and statements 3 and the liar lack empirical content (easy Quine) yet neither are necessarily true or false, 3 could conceivably be either (though I have no idea what conditions would make it so (which may at a stretch qualify it for meaningless for some hardened Davidsonians)) and the liar cannot conceivably be either. it’s tough to say. As I’m writing this I feel I’ve made some incredibly obvious oversight but I don’t know what that is.
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ah yes, there was an objection to truth conditional semantics involving a machine that always produces true statements (or something of the sort), if such a thing existed, and produced 3 that would give it a truth condition. If such a machine produced the Liar we would still have our difficulty it seems.
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what about this proposed solution?
http://steve-patterson.com/resolving-the-liars-paradox/
he says somwhat like you cannot determin wether the sentence means
“this sentence” = false
or
“this sentence is false” = false
which would lead to an obvious contradiction because both would be possible.
it is also the question if the sentence “this sentence” is false actually has any meaning.
trying to wrap my head around this haha
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Hi. I don’t have any time for Steve Patterson. But in particular, his ‘solution’ is ad hoc. If ‘this sentence is false’ is meaningless for the reasons he says, then so is any case of self reference. So ‘this sentence contains five words’ is meaningless on his view, even though it is plainly meaningful and true.
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Not sure how relevant:
But what is it when I say:
a) In London, u’v got the houses of parliament, tower bridge, big ben etc.
b) In London, u’v got 2 Os and 2 Ns etc
Is this a particular branch of logic/maths?
The authur of the quran, for instance, says ,
i) ‘had the quran been from any other than God, they wud’v found much discrepancy therein’. in the b) type sense – it just so happens that the form of the wurd ‘discrepancy’ in arabic occurs only wunce in the quran.
further:
ii) ‘the likeness of jesus in God’s sight is as the likeness of Adam’ again besides meaning, the b) type sense it just so happens that the wurd ‘jesus’ occurs 25 times as does the wurd ‘adam’.
iii) ‘the number of munths are 12’ the wurd munth is mentiond 12 times.
etc
Any merit in this?
Thanks
Alif
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Alex, I saw the video that partly inspired this, with you, Ozy, and Matt, and had one question, if you please: is Matt correct that P or -P boil down to True or Not True?
It seems to me that we do, indeed, try to examine either a statement (that purports to describe the world ) as true, or not true, or sometimes false or not false, but these four categories are due to how we investigate and are not reflections of the proposition itself. If P is true, then -P must be false. We may sometimes evaluate only whether P is true (not considering whether it is false at all), as Matt argues, but this does not mean that P suddenly can have four possible values (true, not true, false, or not false), correct? Statements that describe the world (rather than simply gibberish or commands) are true or false. I don’t see how that can not be so. It seems to me that Matt may have fallen into a little trap of his own making.
Could you comment on this to help me understand?
Steven
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I find the strengthened liar’s paradox very interesting. I was reading the above comments suggesting a state called gibberish and I think you are correct to dismiss that term. However I don’t think that defeats the idea, just gets lost in additional connotations associated with the word gibberish.
What if instead we say, the statement ‘This statement is not true’ is incoherent. Meaning that the concept truth value does not apply to it. I’m thinking of a similar statement, ‘This circle has four corners.’ That may refer to an object that is depicted with four corners, however the term circle can not be a reference to an object with corners unless we modify the meaning of the term circle. So the truth value of the statement is incoherent, it is not a circle but it has four corners.
In the same way, the enhanced liar’s paradox is referencing truth as a concept in a manner where it can not be assessed. The statement reads clearly and each piece seems like a sensible placement of words, but below that the concepts can not actually match without requiring a word to be redefined.
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I made the following observation:
If we agree that “False” ≠ “Not True” (i.e. that they are different concepts), then
Q1: which of the following propositions is correct:
a) (p is True) and (p is False) is a contradiction
b) (p is True) and (p is Not True) is a contradiction
Notice that on (a) we are trying to use “False” as an inverse of “True” to consider it a contradiction
Notice that on (b) we are trying to use “Not True” as an inverse of “True” to consider it a contradiction
Notice that we agree “False” ≠ “Not True”
Q2: Can we use DIFFERENT things as EXACT inverses of the same thing to reach a contradiction…OR…is there a unique inverse that can contradict another?
Let me use uniqueness of inverses in arithmetic as an analogy:
———————————————————Analogy———————————————————-
We know that 3+(-3) = 0
However, given the equation “3 + (____) = 0” is there another (____) ≠ (-3) that you add to 3 and also gives zero?
In this particular arithmetical context the answer is no, because (-3) is unique. Meaning that any instance that we have 3 + something = 0 we can conclude that something = (-3) because of uniqueness of additive inverses. (Digression: Uniqueness of additive inverses in arithmetic follows from the properties of “+” operation: it is well defined, associative, and it is commutative with respect to inverses. Which are also properties of “And” in logic)
So I’m approaching the question in this same way
– “True” would be analog to “3”
– The conjunction “and” would be analog to “+” operation.
– “= 0” would be analog to “is a contradiction”
– we are trying to answer:
is “False” the analog of “(-3)”
or is “Not True” the analog of “(-3)”
or Both are analogs (in which case there is no “uniqueness of inverses” because “False” ≠ “Not True” and they are both inverses of “True”)
————————————————————————————————————————
How can they BOTH be EXACT opposites of the same thing, while also being different?
The point I am trying to make is that MAYBE one of those two (a,b) should not be consider a contradiction, and that this might get rid of the problem.
I believe the answer is “Only b is an actual contradiction” in the same way that we would say that:
only “True or Not True” is the actual dichotomy, while “True or False” is NOT an actual dichotomy
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Just to make my comment simpler, here is the summary:
Which of the following propositions is correct:
a) (p is True) and (p is False) is a contradiction
b) (p is True) and (p is Not True) is a contradiction
I believe the answer is “Only b is an actual contradiction” in the same way that we would say that:
Only “True or Not True” is the actual dichotomy, while “True or False” is NOT an actual dichotomy.
Remember that (by D’Morgan) all the principles of logic are actually logically equivalent to each other:
Non-contradiction ≡ Excluded-Middle ≡ Identity
~(p ∧ ~p) ≡ (p v ~p) ≡ (p⇒p)
Therefore, considering that “True or Not True” is the actual dichotomy (Excluded-Middle) it follows (by D’morgan) that b is the actual contradiction.
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The other point to all this mess is the way we understand:
The established relationship between (truth-values) & (declarative-sentences). We consider two truth-values “True” and “False”. We give them the following relationship with respect to declarative-sentences”
1) “if I say a declarative sentence, and if it is *true*, then what-the-sentence-says (i.e. THE PREDICATE) *correctly describes* the-thing-that-the-sentence-is-about (i.e. THE SUBJECT)”
2) “if I say a declarative sentence, and if it is *false*, then what-the-sentence-says (i.e. THE PREDICATE) *incorrectly describes* the-thing-that-the-sentence-is-about (i.e. THE SUBJECT)”
Immediate question: What about a declarative sentence that is NOT TRUE (instead of False). Does it mean incorrect description?…like a False declarative sentence? (I ask because False ≠ Not True)
This is important…because given my previous comments…we might not reach an actual contradiction.
Lets try two things: (A) assume the sentence is false or (B) assume the sentence is not true….and see what we can reach.
A) Assume the stronger-liar sentence is false. Then by (2) its predicate incorrectly describes the sentence, that is, the predicate “is Not true” incorrectly describes the subject “This sentence”. Does that mean that “True”…is the correct way to describe it? Im not sure, but lets say that this is the correct implocation. Then we reach the conclusion that the correct predicate to describe the subject “This sentence” should be “is True”. So we have: (p is false) and (p is true). Notice how p = “the stronger-liar sentence” in the first parenthesis and and p = “This sentence” on the second parenthesis. Assuming this is not a problem… I would still ask…do we really have a contradiction? (p is false) and (p is true) is a contradiction ? (read my previous comment to understand this question.)
Lets see if we reach the contradiction by the second approach.
B) Assume now that the stronger-liar sentence is NOT TRUE. Then we don’t have anything to keep going because we have not defined what “Not True” means for declarative sentences. False means incorrect description…but what does Not True mean?
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Hi Juan. Thanks for the thoughtful comments. There is a lot there, so please allow me to offer a brief reply to start with.
The first thing to consider is that on the classical picture, False and not-True are equivalent. This means that deriving that something is false and that it is true is the same as showing that it is true and not true, which is a contradiction. Once we move into a non-classical setting, where there are formulas which are neither truth value, then as you say they are no longer equivalent. So to begin with, deriving both true and false counts as a contradiction, and later we need to be more careful. Hopefully I was, but maybe I messed up again somehow.
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Thank you for the reply. I just want to say, you are very enjoyable to read.
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This is great text and very interesting problem.
If something comes to my mind about this subject – it will be linguistic problem.
It seems like sentence is uninformative – I tried to find some other sentences similar in form :
– This volume is big
– This sound is loud
– This weight is light
– This thought is interesting
It looks like there is lack of contetnt/context in these sentences – and truth value judgement can be added only if such content/context exist. Information carry only subject and judgement but lacks description.
Of course in case of liar paradox situation is different because sentence refers to something specific (to itself) – but it look like self-refering sentence refers to its form not to its content – can form of sentence be true or false?
It is like road sign “this sign informes” – yes it clearly inform you but does not give you any information.
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Did Matt ever reply to the revenge of the liars paradox? I’d be curious to hear what he said as I did not agree with the statements he made in the video.
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