A question about the liar paradox
The Liar Paradox = L = This sentence is false.
According to common interpretation if L is true then L is false and if L is false then L is true. A contradiction and so the paradox.
How do we make sense of this paradox?
As we can see L can't be true OR false. So, it must be neither. So, L is equivalent to L' = This sentence is neither true nor false. L = L'. Am I correct?
L' is an odd creature.
It is exactly what the logic entails - that L can't be either true or false.
But L' = L and L' is true.
So, L must be true. In other words the Liar statement is TRUE.
Where is the flaw in my logic?
According to common interpretation if L is true then L is false and if L is false then L is true. A contradiction and so the paradox.
How do we make sense of this paradox?
As we can see L can't be true OR false. So, it must be neither. So, L is equivalent to L' = This sentence is neither true nor false. L = L'. Am I correct?
L' is an odd creature.
It is exactly what the logic entails - that L can't be either true or false.
But L' = L and L' is true.
So, L must be true. In other words the Liar statement is TRUE.
Where is the flaw in my logic?
Comments (102)
One simple rejoinder is to reject the liar statement as incapable of an interpretation. That is, to say it has the form of a sentence in English but where it appears to be about something - itself - it fails.
If you like, the liar has the correct structure for a statement but fails to be a proposition because it cannot be either true nor false.
I was thinking that too. It's the official position isn't it - that the liar sentence is NOT a proposition?
I still can't understand that the equivalent sentence L' = This sentence is neither true nor false is TRUE.
What I noticed is that when something goes wrong with the semantics we can make sense of it by examing the syntax.
Yet, L is grammatically correct in English. So, I was hoping to see an error in its logical formulation. The only type of logic that seems applicable to the liar statement is propositional logic and in that there's no error.
Perhaps, assigning truth values to statements is redundant. We don't say ''God exists is true/false''. It's assumed that everyone is talking about the truth.
Could the liar sentence simply be a semantically empty sentence. Something that can be expressed in correct grammar but is logically empty of meaning.
I wonder what follows in terms of linguistic implications and even for logic too?
sentences about truth and falsity are not well formed in first order logic. So there is no translation of L nor L'.
We've been over this before.
https://thephilosophyforum.com/discussion/1047/liars-paradox-an-attempt-to-solve-it-/p1
Quoting Michael
Quoting TheMadFool
So is the metalanguage treatment.
If you have the time, can you explain to me why L is NOT equivalent to L'?
Let me try. L' seems to be implied by L. But the converse isn't true. Am I right?
1. L -> L'...This follows logically right?
2. L' -> L...This doesn't follow. ''This sentence is neither true nor false'' doesn't imply ''This sentence is false.'' So the equivalence L = L' is false.
Let's take L' alone for the moment.
L' = This sentence is neither true nor false. It's a compound statement (unlike? L).
L1 = This sentence is not true
L2 = This sentence is not false
L' = L1 & L2
But L1 = This sentence is false
L2 = This sentence is true
So, L' has, within it, the liar statement L1.
Yet, L' makes sense, albeit in a weird way, L' is in fact neither true nor false but it has the liar sentence (L1) within it.
What do you think this implies?
((((((((((...) is false) is false) is false) is false) is false) is false) is false) is false) is false) is false
Only I haven't quite finished writing it yet.
To make it completely clear, explicit and unambiguous, we just need to replace that ellipsis '...' by what it means, which is '(...) is false'. We need to keep doing that until there are nor more ellipsis's left.
Could you do it for me please, as I got a bit tired doing the above.
Let me know when you're done.
L: "this sentence is written in English"
L': "L is true"
L is not equivalent to L'
L: "this sentence is false"
L': "L is neither true nor false"
L is not equivalent to L'.
We have two options:
1. "this sentence refers to a fiction" refers to a fact.
2. "this sentence refers to a fiction" refers to a fiction.
Are either of these problematic?
Although my personal opinion is closer to Kripke's. Without some evaluable fact about the world, the sentence being either true or false (referring to a fact or a fiction) is meaningless.
It's an infinite loop right? Flip-flops between true and false. Basically a contradiction. How does paraconsistent logic handle the liar sentence. I hear paraconsistent logic tolerates contradictions.
Quoting Michael
L' is NOT L is neither true nor false
L' = This sentence is neither true nor false
L -> L' but the converse isn't true. So, there is no equivalence (in my opinion).
These are two different sentences:
1. This sentence is false
2. This sentence is neither true nor false
That 1 is neither true nor false isn't that 1 = 2.
Compare with:
1. This sentence is written in English
2. This sentence is true
That 1 is true isn't that 1 = 2.
I see certain avenues for investigation:
1. Self-reference. There are other self-referential sentences that don't make sense. One I can think of is "Everything is relative" or "nothing changes" etc. There's something peculiar about self-reference that needs to be investigated.
2. Redundancy of truth value declaration. I mean we never say "1 = 1 is true". We simply say "1 = 1". We don't declare truth values explicitly. Making a statement assumes you're telling the truth. I don't know if this has anything to do with linguistics or not but it seems our language can't cope that well with falsehoods.
3. Syntax-Semantics. I've observed that when there is a problem with semantics, the problem, ceteris paribus, lies in the syntax. There's something wrong with the syntax of the liar statement but I'm not sure of it.
Do you see any other areas that might yield a solution to the paradox?
Google definition: used to identify a specific person or thing close at hand or being indicated or experienced.
[B]Close at hand[/b] isn't self-referential. It's not like saying ''I like x''. So, the liar sentence is grammatically incorrect.
What do you think?
Also, 'This sentence consists of exactly seven words' is self-referential but not viciously circular like the liar sentence. Can you see why?
I don't think that really works. There's nothing about the liar which is any different than any other self-referential sentence. E.g. "This is an English sentence", "This sentence has five words", etc. For your solution, it seems to generate another Liar, e.g.
"This sentence is incapable of interpretation"
Which, as with the Liar, must be the case of itself. Meaning it's true and it's incapable of interpretation. These Revenge Paradoxes incline me to think such solutions as this, or to call the Liars "meaningless" (or else "neither true or false", like Kripkke's) can't work. They either eat themselves (so to speak) or their solution seems to indicate that perfectly sensible sentences are lacking propositions. On the face of it, nothing seems wrong with saying the Liar is truth-apt. After all, even to me the Liars are contradictions, meaning they have to at least be false.
This issue isn't with self-reference but self-referential truth predication (without some further addition, like "this sentence is written in English and is true"). It's meaningless.
Statements being true mean either that some empirical fact obtains (e.g. with "it is raining") or that it deductively follows from some set of axioms and definitions (e.g. mathematics). There's nothing like this in the case of the liar sentence.
And besides which, isn't your solution subject to the same revenge, e.g.
"This sentence is meaningless"
I don't think meaninglessness is really truth predication, so it seems immune to that objection. But it obviously just generates the paradox again since that new Liar is meaningless, and because it says of itself that it's meaningless, it's also true.
From what axioms and definitions can one derive "this sentence is false"? Can you set out the proof that concludes with the liar sentence?
Quoting MindForged
I don't understand how this relates to the liar paradox. "this sentence is false" and "this sentence is meaningless" are two different sentences. I'm saying that the former cannot have a truth value because it having a truth value doesn't mean anything. I'm not saying anything about the latter. It, too, might be a problematic sentence, but there's no prima facie reason to believe that a solution to one must also be a solution to the other.
Just take the T-schema. 'x' is true just if it is the case that 'x'. So if you have some proposition as follows:
¬True(x) <=> x
You get a liar. To expand it to a more proper argument, we have to recognize the use of "Capture" and "Release" (there are other rules involved of course) as constituting the T-schema:
1) True(L) ? ¬True(L) (Excluded Middle)
2) True(L)
3) L (release)
4) ¬True(L) (definition of L)
5) ¬Tr?ue(L) ? True?(L) (adjunction)
There are other ways of course.
"This sentence is meaningless" is just another Liar paradox, I'm aware they're different sentences. The point is if your solution works for "This sentence is false" then it should dispel "This sentence is meaningless", but it doesn't. So at best, if it does work, it's incomplete and can't really be a general strategy for eliminating Liar paradoxes. Prima facie, if the problem is the same (the contradiction) and the features that give rise to it are the same (the self-reference and certain properties predicating to give rise to a contradiction) then the solution ought to be the same. Otherwise the solutions look ad hoc at best.
So T(x) ? x and x ? ¬T(x). Therefore, T(x) ? ¬T(x). Your definitions are contradictory.
Why? They're different sentences.
Because it's the same type of paradox caused by the same feature. If a purported solution dissolves one version but not another it simply isn't a solution. Problems with the same apparent flaw should be solved the same way, otherwise the answer is completely ad hoc. Or heck, we can just use:
"This sentence is either false or meaningless."
Which forces the contradiction again.
No, T-schema defines a true proposition as being such just if x is the case. It's not defining itself as a contradiction, it's deriving a contradiction by taking the "capture and release" rules and other basic principles and applying them to a proposition that asserts its own falsity. This isn't the only way to run the argument, but here it is again:
1) True(L) ? ¬True(L) (Excluded Middle)
2) True(L)
3) L (release)
4) ¬True(L) (definition of L)
5) ¬Tr?ue(L) ? True?(L) (adjunction)
L is simply defined as asserting its own falsity, which produces the liar as above.
Which is what? Certainly not self-reference because, as you mention, there are self-referential sentences which don't pose a problem. Something else about the liar paradox (in conjunction with self-reference) causes the problem, but it isn't a given that this "something else" is the same thing for both "this sentence is false" and "this sentence is meaningless" (or "this sentence is either false or meaningless.").
Quoting MindForged
I'm not saying that the T-schema defines itself as a contradiction. I'm saying that the T-schema defines T(x) as x and that this definition of T(x) is inconsistent with the liar paradox's definition of x as ¬T(x).
It can't both be the case that T(x) means x and that x means ¬T(x).
As Michael says, yes there is. It talks about its own interpretation.
I think it's simply about negative self reference. The negative makes it problematical. Otherwise it would be just circular and then perhaps meaningless.
Note that not every time you get into a what Russell called a vicious circle with negative self reference. Just look at the incompleteness results of Gödel and Turing etc.
This sentence isn't 36 characters in length.
It is the same thing, specifically asserting its own untruth (or if you want to go deeper, both versions of the paradox I gave make use of what Graham Priest calls the "Enclosure Schema"). Both "false" and "meaningless" are untrue, and it is that untruth which the sentence asserts of itself, which causes the paradox.
What you said earlier was:
This is a mistake, The T-schema (if accepted) just says the to predicate truth on 'x' is just to say that 'x' is the case; it's a biconditional, not equality, so they have the same truth-value. The liar simply takes a proposition which self-refers regarding some property of itself, and that entails the contradiction. Just take a look at the fuller argument I gave earlier. The proposition "L" is defined as asserting it's own untruth, but never is "L" defined as asserting its own truth.
T(x) has the same truth-value as 'x' under the T-schema, so either one an derive the other. The Liar in the format is that "~T(x)<=>x". I think if we went with your response we couldn't even accept Tarski's undefinability theorem.
Being true is not like being red.
"This sentence is true".
It's odd I guess, since nothing really happens if you attribute truth or falsity to it.
"This sentence is valueless"/"This sentence is neither true nor false"/"This sentence is either false or neither".
I'd forgotten about the objections to Tarski's work. Isn't there a reply to these objections in terms of ruling out such sentences as ill-formed?
My objection was that he was incorrect, because that's not how they were defined. I took the T-schema and so said that "True(x)" has the same truth-value as just asserting that "x". And so the Liar (as per Tarski) can be expressed as "~True(x) <=> x" (The truth of x's negation has the same truth value as asserting that x)
I mean, if "~True(x) <=>x" then "x <=> ~True(x)" (it's a biconditional after all)
Not as Michael rendered: "True(x) <=> ~True(x) (that's not how the T-schema is defined, x and True(x) are inter-derivable).
I agree. It is not the self-reference alone that is the problem.
My interpretation is that the problem is that the sentence refers to its own meaning, as assessing Truth requires first assessing meaning. So one has to work out what it means before one can work out what it means - there's the problem right there.
That's why sentences like 'This sentence is written in English' or 'This sentence has ten words' are not viciously circular. They are self-referential but the reference is to the sentence's syntax, not to its semantics (meaning). So one needs to only observe the sentence's syntax, not its semantics, before one works out its semantics.
I suppose it's that the sentence's truth value is referential that is the problem. Whereas in the examples of the last paragraph the truth value is not self-referential.
Self-referentiality need not be a problem. In mathematics, we hit self-referentiality when we have 'x', whose value we want to find, appearing more than once in an equation. Sometimes we can solve that by rearranging the equation so that 'x' appears only once, on one side. We call that 'making x the subject of the equation'. For example we can solve the equation 'x = 1/x' by re-arranging it to be 'x^2=1', so that x = sqrt(1), with solutions x=1 or x=-1.
Sometimes we can't do that but we can still guess solutions or find them by numerical analysis, as with equations like 'x = sin x'. What that involves is basically guessing a solution, seeing how it goes, and then refining it if necessary.
That approach works with the sentence 'This sentence is True' (quoted by ). We can't solve it deductively because of the vicious circle on truth value, but we can guess that the answer (solution) might be 'True' and when we substitute that for 'This sentence' in the sentence, we find that we get 'True is true', which works, and which is consistent with our tentative hypothesis (guess) that the sentence is true.
But it doesn't work with 'This sentence is false' because if we guess 'false' and substitute, we get the sentence 'False is false', which is true, which contradicts our tentative hypothesis that the sentence is false.
And if we guess 'true' and substitute, we get 'true is false', which is false, which contradicts our tentative hypothesis that the sentence is true.
So we can't guess a solution either.
My analogy is that deductively working out the truth value of a sentence, by interpreting it and assessing its truth value, is analogous to making 'x' the subject of the equation and working out the value of the other side. We can do that for 'That pot is black', 'This sentence has ten words' and for 'x=1/x', but not for 'This sentence is true', 'This sentence is false' or 'x=sin x'.
And guessing a truth value and testing how it works is analogous to guessing a value of 'x' and testing how it works. We can do that for 'x = sin x' and for 'This sentence is True' and find a truth value / x value that works, but we can't do it for 'This sentence is false' because none of the values we can guess work.
Your two premises (1 & 4) are contradictory. (5) is the least of your worries (runs for bomb shelter...)
But that seems to suggest that we can't immediately say there's a flaw in this sort of thing, otherwise it seems a bit unprincipled (it has all the same features as the Liar). So when you say you can't guess a solution, well, the Dialetheist is liable to just say you can guess a solution, but the solution is that it's both true and false (pick whatever semantics your prefer in order to spell out how this is this case).
Well said. It's a statement without a truth value and hence not a proposition.
It's wonderful that logic throws up these little puzzles. That it's not as crystal-clean as some pretend.
How's that?
I think the T-schema about as close to a definition of "...is true..." as we can get. Don't you go getting me all worked up now...
x <=> ¬True(x)
The liar sentence, as usually given, is
'This sentence is false'
But if we are being excruciatingly literal-minded, it is a simple, false sentence, because a sentence is a bunch of words and 'false' is a truth value, so the two are not the same thing (a bunch of words is not a truth value), and should not be connected by the word 'is', which implies identity.
Hence we interpret it as meaning:
'there exists x that is the truth value of this sentence and x = False'
If the truth value of the sentence is 'true' then, under the axiom schema of substitution (see I9 from here), we can substitute 'true' for the words 'the truth value of this sentence', without changing the truth value of the sentence. That gives us
''True is the truth value of this sentence and True = False'
whose truth value is False, because the second conjunct is.
I just noticed that the sentence
'This sentence is True'
or more literally, per the post I just made:
'There exists x such that x is the truth value of this sentence, and x=True'
has two different solutions. It is satisfied not only by the hypothesis that x exists and is True, but also by the hypothesis that x exists and is False.
Try it and see!
That's made my day.
If you are being "excruciatingly literal minded" then you wouldn't substitute the truth value in for the referent of the sentence. The truth-value is part of the sentence that's being referred to, that's the Liar. X is not defined as the value "false", X is defined as asserting that "~True(X)".
I had assumed we were discussing within that context, along with the inconsistencies and explosions that inevitably flow from that. Why do you think that involves denying the possibility of self-reference? My expansion of the sentence to the more formal version above is following how Russell expands 'The present king of France is bald' in his theory of definite descriptions, not seeking to forbid self-reference. The aim is to make explicit the implicit assertions hidden within a definite description.
The fact that my expanded sentence still contains the word 'this sentence' should be sufficient to demonstrate that the operation did not banish self-reference.
Quoting MindForged
I'm afraid I don't know what you are referring to with the words 'the Liar'. And also, I'm afraid I can't make anything of your first sentence. In my understanding, a sentence does not have a referent, it is names or symbols that have referents.
The liar does not tell the truth so we must negate whatever the liar says regardless of any statement's he says. It does not matter if the content of his statement is true or false, its must be negated. So, the truth value of ...'not "This sentence is false"' is not decidable as such.
The Belgian surrealist painter René Magritte presented a somewhat similar paradox for images in his work:
The Treachery of Images
This Is Not a Pipe
The text seems to be declaring its referential superiority, except that as part of the work as a whole without the image the text would not have the same meaning, while without the text the image still is referential.
The basic issue is that the liar sentence requires an infinite recursion to ground its referents.
So it fails to state anything, not even a contradiction.
Ah my mistake, I must have misread something during my last response. When I read this:
I understood it as saying that "false" was not part of the sentence being referred to ("should not be connected by the word 'is' "). So if we're going with Russell's theory of descriptions (not sure I accept it), I think an initial issue is that there is not quantifier in the Liar sentence. Note the argument I gave prior was just a propositional logic (where it is usually studied IIRC). What is actually used is predication, so the recourse to quantification seems suspect to me, if not simply misrepresentative of how the paradox is formed.
It also seems like you're conjunction is a bit odd.
Is what is referred to by "this sentence" is the entire conjunction or just the first conjunct? (I think it's the latter) And does this actually dissolve the paradox? I mean, if 'x' is "false" it seems like the following happens:
"False is the truth value of this sentence and false=false"
Which just seems like the Liar paradox put in a conjunction. Of course the conjunction comes out as necessarily false because the Liar paradox (a contradiction) is the first conjunct.
Basically I was saying that the Liar is making a predicate of falsity of itself. And I meant to say the "subject" of the sentence, not the referent.
It is an incomplete thought.
"This sentence is false". False about what?
"I am lying". Lying about what?
Thanks. Dialetheism it is then.
Thanks for the post. It's an interesting connection. Something denying itself. It happens in some mental illnesses too. I think it's called nihilistic delusion where a person basically thinks ''I don't exist''.
But, ''this'' isn't like ''I''. If we stay true to the definition of the word then ''this'' doesn't apply to itself and it should for the liar paradox to be one.
Of course we could invent a self-referential word e.g. ''thes'' and define it as such and the paradox would appear.
If one were to be as exact as possible the definition of ''this'' doesn't include self-reference. It is grammatically incorrect (I'm not a linguistic expert).
However, people do use ''this'' as you have (''this Australian needs a bath'' :D) but note that such forms of language are classified as referring to oneself in the third person. It isn't completely an instance of self-reference. People would find it odd to hear someone refer to himself in the third person.
So, I still think the liar sentence is grammatically incorrect.
However, as I mentioned above we could invent a self-referential word like ''thes'' and the liar paradox still is a problem.
What do you think?
Quoting Andrew M
I have a gut feeling the liar paradox is important. It must mean something. I just don't know what it is.
Are you sure it can do that validly? The linked page states the lemma with a premise that restricts it to first-order languages, which I expect would rule out its use in a T-schema environment which I believe is higher order.
I was going to check the proof to see if that premise is actually used, but I got tired and didn't, so I'm hoping maybe somebody else did. It would be unusual to state a premise that was not used though.
A statement with no context is meaningless.
Put the statement in context and then we can evaluate what it means/says.
Here's context: A person walks up to a stranger on a city street and says, "I am lying", and then walks away.
"This sentence is false," is only self-referential on the sentence level. "This" on its own refers to nothing at all; it's a determiner in the noun-phrase "This sentence", and that nounphrase is also not self-referential (It can't be because a noun-phrase can't be the referent for "this sentence").
Finally, the syntax can only tell you that "this sentence" refers to a sentence that the speaker indicates. The sentence is not inherently self-referential. You could point to any other false sentence while saying this. There's nothing in the syntax, though, that prevents you from picking the sentence the nounphrase occurs in, making that sentence (but not the nounpharse itself, much less "this" alone) self-referential. The liar sentence is perfrectly grammatical, and the syntax is pretty much irrelevant, except that it allows the sentence to have a self-referential interpretation.
Formally, "This sentence is false," is self-referential under the liar-interpretation because the sentence's subject refers to the sentence it is a subject of. To be able to do this, the subject cannot refer to itself (and thus be self-referential on its own).
The liar sentence is perfectly grammatical.
Yes, but it doesn't ever give you a grounded truth-apt subject. To determine the truth of the liar sentence first requires determining the truth of the subject ("This sentence"). That requires substitution with the original liar sentence and so on ad infinitum. There is no final truth-apt subject to ground the liar sentence.
So the liar sentence fails to assert anything about a truth-apt subject and so isn't itself truth-apt. If you disagree, then what do you think is being asserted?
Quoting TheMadFool
The liar sentence shows that not all sentences that appear to meaningfully assert something actually do so. It's the linguistic equivalent of a mirage.
Hah, it's not that simple but I confess I find the view persuasive.
I admit I'm too lazy (and busy, about to go to work) to check, so I just jumped on Wiki real quick and it seems to say the same:
That's just assuming the Kripke's solution and it fails for the same reason. This notion of groundedness can be just as easily used to restate the paradox:
This sentence is ungrounded.
Which must be true as per the original Liar, and thus is true and not grounded. Or else it's false and not grounded. Or else we must say that this notion of groudedness does not do the work you would need it to, because it still throws up the contradiction in the "revenge" paradoxes. The predicate "is false" must be part of the project of determining the truth of the Liar sentence, otherwise your only recourse seems to be assuming that self-reference isn't an allowed move.
What's being asserted is that the sentence itself (falsity predicate included) is not the case.
The T-schema is an attempt to define truth. What do we mean when we say that a sentence is true? To say that "'T' is true" is true if "T" is true, which you are claiming is what the T-schema is saying, doesn't answer this question.
What the T-schema is saying is that "'T' is true" means "T"; that "'it is raining' is true" means "it is raining". And what does "it is raining" mean? That can be explained by referring to some empirical state of affairs.
But take a sentence like "this sentence is true". We can use the T-schema to say that "'this sentence is true' is true" means "this sentence is true", but what does "this sentence is true" mean? Unlike "it is raining", we can't refer to some empirical state of affairs. The "is true" in "this sentence is true" isn't saying anything. Just as the "is false" in "this sentence is false" isn't saying anything. Truth-predication in these cases is a category error.
Um, that's not what I said, even in your quote of me. I said:
IOW, that 'x' and "True(x)" are logically equivalent, they have the same logical value/truth-value according to the T-schema. That seems indistinguishable from what you said so I don't follow what you're saying here, we said the same thing as far as I can tell.
"This sentence is true" under the T-schema would be logically equivalent to saying the sentence is the case: True(x) <=> x
The same for the Liar under the T-schema. The sentence "This sentence is false (or untrue)" is logically equivalent (same truth-value) to the previous is the case: ~True(x) <=> x
Isn't this simply the Tarski Undefinability Theorem?
And what does it mean for the sentence to be the case? Again, unlike something like "it is raining", it isn't explained by referring to some empirical state of affairs, and unlike something like "2 + 2 = 4", it isn't explained by referring to the axioms and definitions of mathematics.
It doesn't mean anything for sentences like "this sentence is true" or "this sentence is false" to be or not be the case, and so it doesn't mean anything for sentences like "this sentence is true" or "this sentence is false" to be true or false.
This sentence is the case
This sentence is not the case
They don't actually say anything.
You said that "True(x)" and "x" have the same truth value. I assume "True(x)" means "'x' is true"? So "'x' is true" and "x" have the same truth value. Which means that "'x' is true" is true iff "x" is true.
But the T-schema is saying more than this. It's trying to explain what it means to have a truth-value.
I don't know if it means that, because notice how the T-schema is setup:
True(x) <=> x
If it were simply to explain the meaning if truth within logic, wouldn't the equality sign be used (such statements are in some sense metalogical)? The T-schema uses a biconditional (an operator within a logical system), which just means the proposition and the proposition predicated as true have the same truth-value. After all, it goes both ways. "True(x) <=> x" and "x <=> True(x)". In a sense, they are the same, but really I think it's simply that they are inter-derivable.
That quote is from a different article (this one), and what it refers to as 'the one stated above' is not the Diagonal Lemma.
The article goes on to say that Tarski's Undefinability Theorem - the theorem the article is about - says there cannot be a formula in the relevant language L that defines T*, the set of true formulas in (L,N). I interpret that as implying that the 'Liar sentence' is not expressible in a formal language, even one with 'sufficient self-reference', because there is no 'True(...)' formula or predicate.
At the end of the article it says
I am interested in exploring this more. I do not have much familiarity with Tarski's work.
Introducing "...is the case" adds nothing, but gives the illusion of explanation. In particular it gives an unwanted air of correspondence to Tarski's schema.
If you know that the subject sentence on the left and on the right are the same, you can use it as a definition of "...is true".
So "it is raining" is true IFF it is raining
can give us a definition of truth; but only if we have a clear understanding of meaning. Easy when the sentences are the very same. What of:
"Hab SoSlI' Quch!" is true IFF your mother has a smooth forehead
That's the other use. If you are not so sure of the meaning of "Hab SoSlI' Quch!", you can take truth as given and use a T-sentence to set out the translation.
That's Davidson's plan.
What you can't legitimately do is use a T-sentence to define both meaning and truth - at least, not for the same sentence.
Tarski actually intended for that. From The Semantic Conception of Truth:
This is why I asked you before if your view was closer to Ramsey's or Tarski's. Tarski's is a correspondence theory, just rephrased to be more precise and clear, whereas Ramsey's is a true redundancy theory.
In other words, Tarski proved that sentences that look like Liar sentences are really just meaningless confusions arising from a failure to be sufficiently formal.
Do you agree with that interpretation?
I think Kripke grants that the liar sentence is a meaningful assertion but that it just lacks a truth value (and so therefore has some third value). Whereas I am claiming that the liar sentence isn't a meaningful assertion at all because it fails to meet the logical criteria for one. A bit like the sentence "the tree is false".
Quoting MindForged
That sentence fails for the same reason as the liar sentence. We can all agree that that sentence is ungrounded. But, being ungrounded, the sentence itself doesn't meet the logical criteria required for a meaningful assertion. So you can't then treat it as if it does.
That is, the sentence appears to be asserting something about itself. But it is not, despite surface appearances. Whereas our assertions about the sentence are truth-apt as long as we're not asserting that the sentence is true or false.
That is the sense in which the liar, truth-teller and revenge paradoxes are like a mirage. There appears to be water there, and it makes us think about water, but appearances are sometimes deceiving. There's no water there.
Not even wrong. :-)
Kripke is the one who came up with this groundedness solution as far as I know, you seem to have presented his analysis of it. It is not stated to have a 3rd value, it's interpreted as a failure of the Law of the Excluded Middle. And that comparison seems disanalogous. Trees don't even have the appearance of a truth-apt object, whereas even you seem to agree that the Liars at least appear as if they are truth-apt.
No no, the notion of groundedness refers to, essentially, hacking off the truth-predicate. The predicate "is grounded" (and its negation) aren't truth predicates so it's not subject to the same criticism, unless you are arguing that self-reference is itself not an allowed thing to do in language. If you agree the sentence is ungrounded, that entails that it is true, which contradicts being ungrounded. This is why Kripke's solution isn't very popular nowadays (even if there is much to commend about his attempt). If our assertions about the sentence are truth-apt there is no functional difference between the sentence referring to itself and saying exactly what we say about it. Kripke, who developed this resolution, also agreed that his solution is probably subject to the revenge paradox I gave.
They aren't mirages, they're contradictions (I showed a rendition of the argument earlier in the thread). Also, the truth teller isn't really a paradox.
What they both have in common is that the full sentence appears truth-apt (since it has a subject and a predicate) until, of course, the content of the sentence is analyzed and the subject is found to not support the predication. It's a category mistake (as Michael earlier noted).
Quoting MindForged
Then we are using "grounded" in a different way. I mean that the subject is resolved and supports the predication, whatever it may be. In this case, the subject doesn't support the grounded predication and so the sentence isn't truth-apt. (BTW, this was essentially Gilbert Ryle's solution to the liar-style sentences rather than Kripke's.)
Quoting MindForged
Self-reference is generally fine. For example, "this sentence has ten words". The truth or falsity of this doesn't depend on the subject being truth-apt, only that its words can be counted. That is a valid predication and so the sentence is truth-apt.
Quoting MindForged
It doesn't entail that since the sentence doesn't support truth predication (because, in turn, the subject of the sentence doesn't support grounded predication). But you're treating it as if it does.
Why is the mirage there in the first place?
I'll accept that because ''this'' may be defined to self-refer.
The form: [subject] [copula] [adjective]. X is Y. And, ordinarily, it is appropriate for the adjective ("true") to be predicated of the type of thing that the subject is (a sentence). It's just that in this particular case it isn't, given that predicating truth of a sentence is only meaningful when affirming some other predicate (like "is red" or "follows from the axioms"), whereas the recursive nature of the liar sentence means that it isn't (hence the term "ungrounded").
You can rephrase the liar sentence:
"The sentence I am uttering right now is false."
"What I'm in the process of saying right now is false."
What matters is that the subject of the sentence refers to the sentence it occurs in. No single component of the sentence need be self-referiatial by itself for that to happen.
I don't understand why you want to define "this" to self-refer.
Not necessarily:
1. 2 is true
2. 1 is false
True, you can rephrase this in many ways. What I'm addressing is the connection between syntax and self-reference that TheMadFool is trying to establisch here:
See Number 3.
The difference between your example and the single-sentence versions lies in the type of reference, I think.
Your example is endophoric (1. is cataphoric and 2. is anaophoric). The single-sentece versions are exophoric: you reference an object in the real world, which just so happens to be the sentence in question. I'm not sure any of this makes a difference, but if it does, that would be *very* interesting, though.
I agree that your example is a category mistake, but that's because it's trying to predicate truth on an object which cannot bear it, whereas sentences (such as what the Liar refers to) are arguably truth-bearers, so your analogy seems flawed.
Then this does appear to be a rejection of self-reference (I'm not familiar with Ryle's attempt at resolving it). But it seems spurious since it's treating truth and falsity predicates differently than other predicates. No one denies that "This sentence has five words" or "This sentence is in English" by saying the predicates cannot be supported. These seem to be sentences, and if sentences are the correct objects to bear truth I'm not sure how Ryle's solution addresses that. There's also Quine's formulation (which lacks the demonstrative "this") which I think was a response to Strawson:
"yields a falsehood when proceeded by its negation" yield a falsehood when proceeded by its negation.
But the point I was making was that both sentences are constructed the same, the only difference is their respective predicates. The subject of each sentence makes reference to a sentence (themselves) and applies some predicate to them. The predicate "has X words" is allowed, what distinguishes that from "is false"?
But why doesn't it support it? And also, "grounded" isn't a truth-predicate. Prior you said that ungrounded sentences are meaningless, yes? "This sentence is ungrounded" is not applying a truth-predicate to itself, it merely asserts (as the solution purports) that it's ungrounded (which means it's true). And since it doesn't make reference to a truth-predicate it seems like that objection is flawed.
Here is Ryle's argument which I think explains the issue well:
Thus the liar sentence is not truth-apt. It doesn't actually assert anything.
Quoting MindForged
The ungrounded sentence has the infinite recursion problem as well and so also doesn't actually assert anything. As with the liar sentence, it's a category mistake to say that it is true (or false).
So the issue is with impredication (what I earlier called self-predication), which is how I construed this resolution earlier. I don't think this works for a number of reasons. The sentence "The tallest person in this room" is just as impredicative as the Liar, as its subject depends on a set of which it is itself a member. Or even just take my earlier example, "This sentence is an English sentence." Ryle's solution makes that sentence a nonsense sentence, since we run into his so-called "name-rider problem":
"This sentence is an English sentence."
What sentence?
"The current sentence {The current sentence [The current sentence...
This is just as is impredicative as the Liar (the only difference is the predicate they put onto themselves), and so as with Russell's version of type theory, this renders scores of seemingly comprehensible sentences into nonsense. And I suspect this is why Ryle's solution isn't mentioned very often nowadays since it eliminates self-reference entirely.
The issue as I see it is not impredication, but whether the sentences in question have a truth-apt use.
"This sentence is an English sentence" would ordinarily be unpacked as, "The sentence 'This sentence is an English sentence' is an English sentence". The inner sentence is not being used as an expression but is only being mentioned. If it were used as an expression, then infinite recursion would result.
Now consider a similar unpacking for the liar sentence, "The sentence 'This sentence is false' is false". For the outer 'false' to be predicable of the inner sentence, the inner sentence must be an expression. But since it is only being mentioned, it doesn't support truth predication. So it's a category mistake. Whether a category mistake or an infinite recursion, no truth-apt use is available for the liar sentence.
But your quote of Ryle said that the issue was the use of impredicative definitions:
A far as I can tell, Ryle's argument is that the sin of the Liars family of paradoxes is that they make use of impredicative definitions, they are part of the thing they are defining, and that by doing this you can never get down to a truth-apt sentence because it the subject expands indefinitely.
That's not how Ryle's analyzed the Liar though. His claim is that the impredication never gets anywhere, and so when run against "This sentence is an English sentence", it would, when you ask for the "namely-rider" come out (as per your Ryle's quote): "Namely, the current sentence{namely, the current sentence etc. That supposedly unbridgeable gap to the verb is, on Ryle's account the problem and impredication is his diagnosis of the cause.
That's not Ryle's analysis then, at least not from what you quoted (Ryle's said that the sentence cannot even get to the verb, so the inner sentence would expand indefinitely when you try to identify the subject). Your analysis here doesn't make sense to me. Try this: "The sentence 'Snow is white is true' is true". The inner sentence is clearly true, we can predicate truth there. If "snow is white" is true, we can validly assert that "The sentence 'snow is white is true' is a true sentence". The outer sentence is just the metalanguage to the object language of the inner sentence. Ironically enough, doing this in natural language allows for Liar sentences to be validly formed (as per Tarski), because you used English as both the metalanguage and the object language.
Ryle is arguing against cyclic expressions (fillings of their own namely-riders), but he is not arguing against mentions of the referring expression (where quotation-marks have to be employed). As he says in the same paper:
Quoting MindForged
Yes, an infinite expansion results if the subject is always a truth-evaluable expression (as is indicated with the nested brackets). But that's not how we ordinarily use that sentence. Instead the referring expression is only mentioned (which I unpacked and indicated with quotation-marks in my previous post), not used as an expression. That's the use-mention distinction.
As explained in my previous post, that specific use would result in a category mistake for the liar sentence, since a mention of the referring expression would not be truth predicable.
Quoting MindForged
That's fine. There's nothing wrong with nested expressions. The problems only arise with cyclic expressions.
The Liar isn't cyclic though. The subject of "This sentence is false" is the same as the subject of "This sentence is an English sentence".
I don't follow you here. Both expressions have the same subject (the very sentence itself), so if the expansion occurs on one I can't see what feature doesn't cause it in the other. You say it's a use-mention error, but how so? Your rendering of the Liar came out as:
"The sentence 'This sentence is false' is false"
And then you concluded that
But this applies exactly the same to the "The sentence 'This sentence is an English sentence' is an English sentence". The inner sentence is only being mentioned, so is it a category mistake? I don't think it is. This is not a use-mention issue, the inner sentence is capable of being true even if it's only being mentioned. "2+2=4" is true, for example. Your rendering of the Liar is fine for my purposes though. If "This sentence is false" is a false sentence, then it is also a true sentence and that's the contradiction!
I don't follow you here. The sentence is structured no differently. What makes one a nested sentence while the other commits a use-mention error?
I don't really understand your "revenge paradox".
As I understand it, the paradox of the Liar Paradox is that IF "This sentence is false" is true then it's false, and if false then it's true.
But this is not the same for "This sentence is meaningless", which if true is meaningless, and if false is meaningful. I don't see a paradox here. It is false.
But to take the meaningless case, it's simple why it's a Liar sentence for the same reason as the above (although I need to change it a bit since I think you're right). If "This sentence is either false or meaningless" is true, the sentence is both meaningless and true or true and false, which are contradictions. After all, presumably the point of labeling the Liar "meaningless" is to deny the Liar a truth-value so one can escape the situation where picking one truth-value gets you the other one too. But that means that it both lacks a truth-value (because it's meaningless) and is true (has a truth-value). Otherwise it's both true and false again. And if that sentence is itself meaningless, it's true and meaningless.
This is the key issue. My claim is that a sentence is only capable of being true or false if it is used (i.e., expressed).
Consider the sentence, "'Snow' has four letters and is cold". Snow is mentioned, but that mention is not something that can be cold, only the snow itself is. So the "is cold" predication is a category mistake (specifically, a use-mention error). However we could apply an interpretive rule and say that in such circumstances, the "is cold" predication disquotes the mention and so is really saying that snow is cold. This would unpack as, "'Snow' has four letters and snow is cold". Such a rule would tolerate the above sentence and allow it to be truth-apt.
Now compare that with "'2+2=4' has three numbers in it and is true". My claim is that the mention of '2+2=4' is not something that can be true, but the expression (or use) of '2+2=4' is. If so, then the truth-predication disquotes the mentioned expression and uses it. This would unpack as, "'2+2=4' has three numbers in it and 2+2=4".
Since "This sentence is an English sentence" doesn't contain a truth-predicate, the referring expression is only mentioned, not used (i.e., only the surface aspects of the sentence are referred to). Whereas in the liar sentence, the truth-predication disquotes the mention and uses the referring expression. Thus it is cyclic.
That's not really the sort of sentence I used. I didn't say the Liar was " 'This sentence' is false". "This sentence" cannot have the property of truth. But I think it's worse for your approach if you apply that interpretive rule. Then the expression just becomes the Liar sentence because the falsity-predicate disquotes the phrase "this sentence" and transforms it into " 'This sentence' is false is false and this sentence is false." That's just a conjunction of Liar Paradoxes.
I know that's the contention but you haven't explained why truth and falsity predicates are subject to a different set of rules than other predicates which can apply to quoted sentences. We both agree the expression that is quoted can be attributed truth. The difference seems to be (correct me if I'm wrong) that you think the following quote cannot be predicated truth while I think it can:
" '2+2=4' is true"
On your view, does this work? Can you predicate truth of a mentioned expression? I think you can, provided the mentioned expression is the sort of thing that can bear truth. Some mentions are truth-y, the expression mentioned just has to have the right kind of structure. The Liar has that structure. This solution is the same sort of solution Kripke tried and I don't think it works.
OK, I think this runs into an issue I mentioned earlier to another user (assuming that user wasn't you, too lazy to check). Take the following:
This sentence is true.
Now that's an odd sentence, but it doesn't even have the appearance of a paradox, unlike the Liar. Under the rule you mentioned, it comes out as
"The sentence 'This sentence is true' is true and this sentence is true"
Well, that conjunction is obviously true under this interpretive rule, both conjuncts come out as true because the mentioned sentence is transformed into a use of the quoted expression. But that means that impredicative truth assignment cannot be sufficient to say the Liar is a category mistake. And note, Ryle specifically calls out impredication as the issue here (just look at the passage you quoted, he names "Impredicability").
There needs to be a way to convert a mention to a use and using the subject-predicate form is an efficient way to do that.
Quoting MindForged
Yes. To say "'2+2=4' is true" is to say "2+2=4". That is, truth predication of a mentioned expression uses the expression. So it's the inverse of quoting.
Quoting MindForged
The truth-teller sentence cycles for the same reason as the liar sentence. Here's the iterative unpacking (the => is just an arrow to indicate the transformation steps):
"This sentence is true" => "'This sentence is true' is true" => "This sentence is true" (truth predication rule). So we have a cycle.
Compare with unpacking the liar sentence:
"This sentence is false" => "'This sentence is false' is false" => "Not ('This sentence is false' is true)" => "Not (This sentence is false)" => "Not ('This sentence is false' is false)" => "'This sentence is false' is true)" => "This sentence is false". So we also have a cycle.
And compare with the English sentence:
"This sentence is an English sentence" => "'This sentence is an English sentence' is an English sentence" => true. So it's truth-apt.