The paradox that broke my brain.

I first learned about the basic version of the Liar's Paradox - "This statement is false" - when I was about 13 and thought it was the silliest thing. I thought I quickly solved it and there was nothing remotely complicated or interesting about it and continued thinking that for many years. It was only a few years ago that I learned about the true extent of the actual conundrum, and it broke my brain. For days and weeks I kept returning to it and just could not figure it out. Until I did (I think).

But let's start from the beginning, let's look first at the basic version, "this statement is false". If it's true, then what it says is accurate, but it says it's false. In other words if it's true then it's false. That's not good. If it being true is a problem then maybe it's false, but that means when it says it's false it represents the situation accurately, meaning it's a correct statement. In other words, if it's false then it's true. That's not possible either.

So we have a paradox. A statement is either true or false. But either possibility leads to a contradiction. So what's happening, is the universe about to explode? 

Not so fast - who says that's even a statement? In a proper statement all the terms are defined. "2 + 2 = 38" is a statement, a false one. But "2 aswjhyu 2 = 38" is not a proper statement if aswjhyu is not defined. It's neither true nor false. 

Let me say something even more obvious: you can't define aswjhyu in terms of something that is itself not previously defined. And it certainly would be silly to define something in terms of itself, like saying "I'll tell you exactly what shmoles are - they come out of other shmoles' eggs. Great, now you know shmoles".

But that's exactly what statement X = "this statement is false" = "X is false" is guilty of. It's defined in terms of itself. So it's not a proper statement, so it's neither true nor false. No more paradox and the universe is saved.

This seemed to perfectly resolve the whole thing when I was a teenager, and I remained blissfully unaware of any further issues for years. That is until I discovered the monstrosity that I will call the Mutant Ninja Liar's Paradox. 

This will be an interactive portion of the presentation. Take a pen and write on your hand: "There is no true statement written on my hand". Now try to answer this 

Question: is there a true statement written on my hand? 

Perhaps you see why this mutant version is much deadlier than the original version. Remember, before we were able to find a third alternative: X doesn't have to be either true or false, it could be meaningless. But now there doesn't seem to be a third option, the answer is either yes or no. For example, even if you tried to say that what's written on your hand is meaningless gibberish, then your answer to the Question would be: no, there's no true statement on my hand, only gibberish.

But now we seem to have a true inescapable dilemma: both "yes" and "no" lead to a problem. To see why, let's call what you wrote Y:

Y = "There is no true statement written on my hand."

Now let's look at both answers to the Question.

Option Yes: there's a true statement written on my hand.

Since the only thing written on your hand is, presumably, Y, that means Y is a true statement. So what Y says is accurate, which happens to be: there's no true statement written on my hand. But that directly contradicts option Yes. So if option Yes is true then option Yes is false.

Option No: there's no true statement written on my hand.

But take a look at Y. Option No is a statement that directly asserts Y, word for word. So if option No is true then Y is a true statement. But then there in fact is a true statement on my hand, namely Y! So if option No is true then option No is false.

We seem to have quite a situation on our hands: either possibility is impossible. The universe is getting ready to explode again. How do we save it? It took me a while to figure out. I don't want to spoil you all the fun by immediately blurting out my answer, so I will leave it for a future article. Meanwhile, how would you save the universe? Let me know in the comments. 

24 Comments - Go to bottom

  1. Hey Dmitriy,

    I think this is a great choice for an article. And it “interests” me that you say you have a solution. That said, I’m not so sure I understand the motive behind categorizing the liar sentence as not a statement. Firstly, it can’t literally be comparable to meaningless phrases like, “dfhsha dd”. In other words, the liar sentence is not incomprehensible, whereas I have no idea what “dfhsha dd” it’s pretty clear to me and I think to all of us, what the liar paradox sentence means. We all understand what the sentence is asking us to do, in fact you spent an entire opening paragraph explaining the requirements imposed by the sentence.

    The only problem with the sentence is that it doesn’t have a fixed truth value. Hence it seems like labeling the liar sentence a non-statement can only work if we decided to define statements as yielding a single, fixed truth value. We don’t want to go the alternative route of labeling all self-referential statements as ‘bad’, since there are plenty which are perfectly coherent and don’t create paradoxes.

    In any case, it’s quite easy to create a ‘strengthened liar paradox’ in response to such an objection. Observe:
    “This sentence is either false or a non-statement”

    If it’s true then it’s false or a non-statement (which can’t be true or false), if it’s false then it’s both true and a statement. This strengthened liar paradox is known as the ‘revenge liar’ :)

    Note that it basically works for almost any objection. Whatever reason you can think of that would make the liar sentence a ‘bad’ sentence (it’s meaningless, it’s not a statement, it’s not coherent etc...) we can just include such objections in the “bad” box.

    Then we can revise our sentence to be:
    “This sentence is either ‘bad’ or false”

    In other words, any objection which attempts to take the liar sentence out of the category of things which have truth values (e.g. statements) runs into trouble in the above way. And this will always hold true so long as our ‘bad things’ can’t be true things.

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    1. Also, I don’t see how the alternative sentence about what’s written on your hand, is superior in any way to the classic liar sentence. The difference is that the alternative sentence appears to be making direct reference to an object out in the world, “the stuff on your hand” whereas the classic liar does not.

      But I don’t see how this issue of reference bears relevance on whether a sentence can be correctly categorized as a statement. There are plenty of statements which are purely non-referential (2 + 2 = 4) but make sense. Assuming there is no platonic heaven of course.

      Finally, about your initial reasoning regarding a statement being bad if we can’t define it/understand it except in terms of itself. I would say that you are misapplying this to the liar sentence. Every word in the sentence has a definition which is not self-referential.

      Thus, this is not a problem of understanding. It is a problem of solving the task which we understand the sentence is asking us to do (find its truth value), and not of understanding what we are supposed to solve. There are plenty of normal self-referential statements, like “this sentence is shorter than 274 characters” which are true and make sense.

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  2. Hey Alex,

    >I’m not so sure I understand the motive behind categorizing the liar sentence as not a statement.

    By "statement" I just mean basically something that must be either true or false. When in logic we formulate the law of excluded middle: either A or not A, that's what I mean by a statement, those "A"s that the law talks about.

    I used "meaningless" as a loose/sloppy synonym for "not a statement", not in a precise sense. For example, "how are you?" is not a statement because it doesn't express any state of affairs that would be either true or false. But it's not meaningless.

    I discovered the Liar's Revenge, Tarski's truth predicate, metalanguages, the undefinability of truth, etc. all around the same time. That's exactly the time I spoke about when I eventually found the formulation of the paradox that broke my brain. The Liar's Revenge is not so brain damaging though, let's take a look at your explanation:

    “This sentence is either false or a non-statement”

    If it’s true then it’s false or a non-statement (which can’t be true or false), if it’s false then it’s both true and a statement. This strengthened liar paradox is known as the ‘revenge liar’ :)


    We can resolve this the same way, by saying that it's neither true nor false, i.e. it's a non-statement. And no contradiction follows from this option.

    You were asking how my mutant version is superior to the liar or Liar's Revenge. That's because in the mutant version no third option is available.

    Finally, you are saying if I understood correctly that being self-referential is not a problem for a statement or, even if it is, "this statement is false" is not self-referential. I disagree on both counts. In formal languages you can't define a statement "this statement is...", that's why Goedel had to employ a clever diagonalization trick to define a statement that is true if and only if it's unprovable. The same with Tarski's undefinability of truth, he had to use the same kind of trick to show that if a formal language contains its own truth predicate then you can find a valid statement that indirectly expresses "this statement is false".

    That's, by the way, a resolution of the Liar's Paradox for formal languages. You can, by trickery, construct these pseudo-self-referential statements, but a language can't contain its own truth predicate, so it can't express the Liar's sentence.

    The version that broke my brain is the version for non-formal languages. Here, we can debate whether self-reference is allowed for a statement to be well defined. I still say no, but this is a moot point because we can use a trick to get rid of self-reference. We don't have to do weird things like Goedel or Tarski had to do for formal languages, we can just have stuff written on my hand talking about stuff written on Dmitriy's (from ReasonMeThis ) hand.

    The important point is that we can foreclose objections based on self-reference and create a situation with only two options, not three. The regular liar and the revenge version don't accomplish that.

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    1. "I discovered the Liar's Revenge, Tarski's truth predicate, metalanguages, the undefinability of truth, etc. all around the same time. That's exactly the time I spoke about when I eventually found the formulation of the paradox that broke my brain. I discovered the Liar's Revenge, Tarski's truth predicate, metalanguages, the undefinability of truth, etc. all around the same time. That's exactly the time I spoke about when I eventually found the formulation of the paradox that broke my brain. "

      Cool! :)

      "You were asking how my mutant version is superior to the liar or Liar's Revenge. That's because in the mutant version no third option is available."

      I meant just the classic liar sentence, why is the mutant version better?

      "Finally, you are saying if I understood correctly that being self-referential is not a problem for a statement or, even if it is, "this statement is false" is not self-referential."

      I meant it's not a problem (for non-formal versions) in the sense you explained. It seemed like you were saying that the reason the classic liar is bad, but the mutant version is not, is that the former is meaningless whereas the latter is not. And I simply wondered what form this "meaninglessness" was supposed to take.

      As regard formal language statements, yes I definitely agree. And more specifically, it was type theory (by Russell) and modern set theories like Zermelo-Fraenkel set theory, which did away with all self-referential paradoxes in their constructions. But I don't find this a satisfying solution in any way as regards natural language. The above just shows that it is possible to construct a logically coherent binary truth-value formal language system. I don't think the liar paradox was ever meant to put that in doubt.

      "We can resolve this the same way, by saying that it's neither true nor false, i.e. it's a non-statement. And no contradiction follows from this option."

      I'm not sure I understand, if the statement (the whole statement) is a non-statement, then wouldn't it therefore be true (and thus contradictory)?
      For instance:
      "This sentence is either false or (neither true nor false)"

      Unless you mean that we are not even supposed to engage in the process of evaluating the sentences' truth value (because it's not a statement). But how are we supposed to know it isn't a statement if we don't first engage in such evaluation? Normally, one labels a sentence a non-statement after doing evaluation and discovering that it fails to meet certain criteria. In this case the criteria being that it is neither true nor false.

      How do you propose we determine that the above sentence is neither true nor false? Presumably by undertaking the normal truth conditions evaluation process (i.e. plugging in if x true, then y etc...). In this case we get, if x is neither true nor false, then x is true.

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  3. >How do you propose we determine that the above sentence is neither true nor false?

    One way is by knowing there are three options and proving that the first two lead to a contradiction. The third one, x is neither true nor false, doesn't seem to lead to a contradiction, because I don't see how you conclude that x is then true. If it's not a statement then you can't keep parsing it to evaluate its truth-value, it just doesn't have it.

    Another way is by an analysis similar to how in formal languages we have an algorithm to determine whether a formula is well formed, and then whether it constitutes a statement in the formal system. All of which can be done without evaluating whether it's true or false, which is often impossible to do.

    In particular, if a formula contains undefined symbols the algorithm will deem it not a statement. Similarly, if we adopt the view that just like in a formal language everything needs to be defined the same goes for natural language statements, then we can disqualify that sentence on those grounds because it refers to something not yet defined - itself.

    If we don't adopt that view then we'll just have to go with the first way, but I do like that view.

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    1. Oh, and this part confused me:

      >"You were asking how my mutant version is superior to the liar or Liar's Revenge. That's because in the mutant version no third option is available."
      I meant just the classic liar sentence, why is the mutant version better?

      I just wanted to double check that you understand my position: I think the mutant version is better because no third option is available that we can run to. Even if you disagree that the third option removes the paradox (I think it does in the cases we discussed), you would probably at least agree that there's a long conversation to be had about whether it does or doesn't. Even that alone makes the mutant version cleaner.

      It is also easier to conceptualize, it's closer to normal ordinary language: do I have something true written on my hand or not? Seems like anybody can understand that.

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  4. Hey Dmitriy,

    Sorry I'm just catching up to this.

    "I just wanted to double check that you understand my position: I think the mutant version is better because no third option is available that we can run to. Even if you disagree that the third option removes the paradox (I think it does in the cases we discussed), you would probably at least agree that there's a long conversation to be had about whether it does or doesn't."

    I'm not sure I understand. I presume the third option can't work in the liar case because you decided to exclude self-referential statements. That is, the liar can be meaningless (apparently) because it is self-referential. If this is so, then I don't see how the mutant version is not self-referential? The statement written on your hand seems to be referring to itself, namely the statement on your hand that is about the statement written on your hand.

    Statements like, "this sentence written here is less than 280 characters" are self-referential. If we think written characters on a board are capable of representation (as we all do) then the thing written on your hand can be about things (namely itself).

    The only difference between the above sentence (and the mutant version), as opposed to the liar sentence, is that the latter is devoid of content. But a proposition can have useful content, meaning it can be about something in the world, while at the same time being self-referential. By lack of content, I mean that a proposition doesn't refer to any object in the world "like the proposition 2 +2 =4". While such a proposition expresses a relation, it is not clear what such a relation is between (i.e. what is a mathematical object?).

    Similarly, the liar paradox expresses a relation between a constituent of the proposition and the whole of the proposition, but it is not known what these things are. However, as demonstrated by the above, this doesn't prima facie mean that propositions cannot lack content. Thus, neither self-reference nor lack of content appears to be fatal.

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    1. About your proposed methods to eliminate the liar revenge paradox:

      Method 1) "One way is by knowing there are three options and proving that the first two lead to a contradiction."

      I'm not sure I understand this approach. Not to be pedantic, but an option being contradictory means that it is both true and false, so how is this supposed to demonstrate that the sentence is in fact neither true nor false? Note that regular contradictory propositions are logically held to be false propositions. The statement "The snowy mountain is not a mountain" is contradictory and false.

      Method 2) "The third one, x is neither true nor false, doesn't seem to lead to a contradiction, because I don't see how you conclude that x is then true. If it's not a statement then you can't keep parsing it to evaluate its truth-value, it just doesn't have it."

      But I asked you to justify the belief that such a statement is a non-statement. Why is the burden of proof on you to do so? Because prima facie we recognize statements by whether they express truth values, and we determine this by seeing whether the supposed statement has declarative content which could possibly be true or false. Obviously the liar paradox can possibly be true or false (since it's both).

      In needing to justify that the liar revenge is a non-statement, one begs the question by starting from the position that it is not. We want to justify whether the statement is neither true nor false, and in order for us to know this it must be true that the above statement is neither true nor false. But if it is true that the above statement is neither true nor false, we can see a contradiction.

      Earlier, I proposed a criterion for evaluating whether a sentence has truth values (if it does not, we can declare it a non-statement). The normal criterion for doing so is to plug in truth values for the contents of the statement (If x is true then y is etc...) so as to determine the truth conditions. But here we see that this leads to contradiction.

      What is your proposed alternative criterion for demonstrating that the liar's revenge statement is a non-statement? You provided none, except that we are not supposed to undertake my standard method, but of course one needs some method, otherwise one begs the question.

      Method 3) "Another way is by an analysis similar to how in formal languages we have an algorithm to determine whether a formula is well formed, and then whether it constitutes a statement in the formal system."

      Okay, but this just amounts to declaring all self-referential statements invalid. However, as I already mentioned, propositions can bear content in natural languages while still being self-referential AND being only true or only false. For example, the sentence about the number of characters in the sentence expresses a proposition about itself. So it is both self-referential and carries content (it tells us useful information about itself). Hence, it seems unreasonable to go so far as to declare such statements to be non-statements.

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  5. Hey Alex,

    "I don't see how the mutant version is not self-referential? The statement written on your hand seems to be referring to itself, namely the statement on your hand that is about the statement written on your hand."

    It's not self-referential in the sense that it doesn't need to refer to an as yet undefined object. It talks about things in the actual world: my hand and scribblings on it.

    But my main point is that even if there are three options available for the statement on my hand, only two options seem to be available as answers to the Question.

    "Statements like, "this sentence written here is less than 280 characters" are self-referential."

    This is an important point to bring up regarding the permissibility of self-reference. Because it does seem like some self-referential statements make perfect sense. In my solution article I talk about the difference between sentences and statements. A sentence, like a hand, can be viewed as a thing in the world. We are not defining it into existence. For example, I am looking at the characters of that self-referential sentence you gave, and they definitely exist. So I can perfectly coherently interpret "this sentence" to be referring to the already existing, and therefore already defined, sequence of characters.

    What I claim we can't do is referring to an abstract object that we haven't yet fully defined.

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    1. Hey Dmitriy,

      Okay I understand now. This is I interesting, but a few points to say if I may.

      1) We can coherently express propositions about things which are not in the world. For instance, a proposition about a fictional entity like Harry Potter can be true or false.

      This is important because it means that propositions can be about things without having to “define them into existence”. So who says that the self-referential liar paradox is about any real existing thing?

      2) About our not fully defining an abstract entity: Yes I noticed that and I replied to your other post. To quickly sum up my objection, I noted that we can coherently refer to abstract things which we don’t have a clear grasp on (like free will). Also, there exist propositions which are totally void of content and reference but are still meaningful (like analytic propositions). So it might be that the liar paradox isn’t actually making reference to anything while still being okay.

      But this is irrelevant because it’s pretty clear to me that the liar paradox isn’t meant to be interpreted as making reference to an abstract entity. Yes, there are differences between propositions and sentences, the easiest way to think about this is that the latter are the things which refer to things, while the former are the facts being expressed by the latter.

      E.g. the proposition is the mathematical relation being expressed by the sentence written on the computer.

      We should interpret the sentence, ‘this sentence is false’ as referring to itself, namely the thing being denoted (sentence) and the thing which is making the reference (the sentence). Since a sentence is not abstract, we have no problem.

      Note that we still have a problem of self-reference however. Real existing things in the world (like sentences) can be used to express/represent other things in the world. Hence, our real sentence has the property that it can refer; it refers to itself, namely the thing with the property to refer. So we still have a problem here I would say.

      Best,

      Alex

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  6. Hey Alex,

    About 1: Some of my descriptions are imprecise for brevity's sake, or because of linguistic sloppiness on my part. By "defining something into existence" I'm not excluding referring to Harry Potter, or even to a "machelor". A machelor is defined, by me just now, as a married bachelor.

    "Machelor" is now a perfectly well defined concept, and we can make statements about machelors. In particular we can say that machelors don't exist. So we are not defining into existence an object, but a concept. There may not be a single object in existence that the concept/definition applies to.

    About 2: We have to make a distinction between the actual language we speak, which is sloppy and ambiguous, and an idealized ordinary language as I describe in the Solution article. Only in that kind of language sentences are mapped to definite precise meanings, states of affairs that are definitely either true or not true. The paradox relies on this clear dichotomy of true/false, so I am imagining that we are working with an idealized ordinary language, otherwise there's no paradox.

    Also, I don't accept that "2+2=4" is devoid of content and reference. I think it refers to some abstract entities like equivalence classes etc. But that's probably a whole other big topic.

    The rest of what you said I agree with, subject to provisos described in the Solution article: a sentence by itself doesn't mean anything, we need a pair (sentence, language) to get a meaning. This is a key element that I think helps get out of the paradox, as you already saw in the Solution article.

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    1. To clarify 1 further, I am saying we can refer to, and create a term for either:
      - an actual existing object (my hand, scribblings on it),
      - something constructed of already defined concepts (machelor)

      We can subsume everything into only the second case if we figuratively subsume actual existing objects into the category of "already defined", which kind of what I did earlier at some point when I said:

      "So I can perfectly coherently interpret "this sentence" to be referring to the already existing, and therefore already defined, sequence of characters."

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    2. Hey Dmitriy,

      I summed up my objections in a reply in the other thread. Briefly however, do you agree with me that the liar sentence refers to itself, specifically to the words being written down which would constitute:
      “an actual existing object (my hand, scribblings on it)”?

      So I’m confused why you think the liar sentence lacks a meaningful reference, or why it is distinct from the mutant version in that vital sense...

      Maybe we can talk about this in the other thread.

      Best,

      Alex

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  7. Hey Alex,

    I agree sort of, that's what I addressed with "subject to provisos described in the Solution article: a sentence by itself doesn't mean anything, we need a pair (sentence, language) to get a meaning. This is a key element that I think helps get out of the paradox, as you already saw in the Solution article."

    Do you agree with this proviso? And yes, we can continue in the other thread.

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    1. Of course I agree we need a language (and many other things) to extract meaning out of a sentence. I’m still a bit confused though, because it sounded like you were saying that an inability to refer to an actual existing thing was (one) reason the liar sentence fails.

      You wrote (regarding the mutant version):
      “ It's not self-referential in the sense that it doesn't need to refer to an as yet undefined object. It talks about things in the actual world: my hand and scribblings on it.”

      Since I assume you think the liar sentence is self-referential, it would seem that, according to the above, the liar sentence cannot refer to written words (i.e. itself). That’s because the above quotes implies that reference to an actual existing thing is a sufficient condition for legitimate reference (not self-referential).

      Let me know where I go wrong.

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  8. >Of course I agree we need a language (and many other things) to extract meaning out of a sentence.

    For the purposes of these articles I have defined a language to be something that maps sentences to statements, so only a language and nothing else is needed in addition to a sentence.

    About whether the liar referring to itself, my view is:
    - the liar statement can't refer to itself for reasons we discussed (similar to how you can't mathematically define X in terms of X)
    - the liar sentence can't refer to itself because a sentence is just characters, a second ingredient is needed for semantics
    - a pair (S, L), where S is a sentence and L is a language can refer to S.

    So the mutant version can be reformulated without using my hand:
    S = "this sentence is not translated into a true statement by L"
    L = some idealized version of English

    And the solution would be the same as I described in the solution article. Does that clarify my view?

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    1. To clarify, the version with referencing "my hand" is much easier to conceptualize and think about I believe than the version I just gave, even if both represent the paradox well.

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  9. Hey Dmitriy,

    It's not really clear to me what your theory of language is, and therefore I find it quite difficult to grasp your deductions throughout the other post. In any case, I still think the problems I summarized in the other post remain, specifically the problem of justification. If we adopt the view that all self-referential statements are bad, this seems to conflict with the notion that certain statements of self-reference like "this sentence has more that 2 words" are perfectly valid. So, it's not enough to say that in my language this sentence is actually bad, because we want to fix this in English. And in English the above sentence seems perfectly coherent.

    Moving on, like I said its not so clear to me what you think a statement is, is it something like what philosophers term a proposition (a non-linguistic entity), is it linguistic and if so what is its ontological status? It also seems ontologically extravagant to introduce a third category between the sentence and the facts of the world; why do we have to do this exactly?.

    Further, what is the relation between sentences and statements? If a language is that which maps the former to the latter, is it safe to say that sentences are not actually about the facts of the world, that they are just about statements? If that's false then why do we need statements at all (we can just bypass statements and use sentences to talk about the world)?

    Finally if it's true, then it's not clear to me that we have this problem of "the liar statement can't refer to itself for reasons we discussed (similar to how you can't mathematically define X in terms of X)". Because if a statement is in an ontological class of its own (an abstract entity), then we can just point to whatever you think that thing is, and say this is what the liar statement is referring to. If this is problematic, on account of the ontological status of a statement being unclear; then this seems equally troublesome for all sentences (since all sentences refer to statements).

    Answering all these questions will help me understand why exactly you think the paradox is solved.

    One more tidbit I'd like to concentrate on. Regarding, "the liar sentence can't refer to itself because a sentence is just characters, a second ingredient is needed for semantics
    - a pair (S, L), where S is a sentence and L is a language can refer to S."

    I think it would be more appropriate to say that L is a necessary condition for S' meaning. It can't literally be true that (S,L) are the things which refer to S' contents. It's not the case that the sentence "snow is white" along with the entire English language are referring to the referents of 'snow' and 'white'. Rather, it's just the sentence which is making the above reference, and the English (idealized English) language is a necessary condition for such reference/meaning to take place.

    However, just because L is a necessary condition for S, doesn't mean we have to include it in S (the revised liar sentence). There are many necessary conditions required to give the liar sentence meaning, like the proposition that if something is true then it is not false. But we somehow don't find it necessary to write out the liar sentence as, "this sentence is false only if (all the necessary conditions are met).

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    1. By the way, I can see you've been reading a lot of Tarski :)

      However, I don't think it's satisfying to just give a similar account to Tarski' formal language involving languages and meta-languages, which would fix all the paradoxes of self-reference. It's pretty clear that you can construct such a language, but it's also pretty clear that natural languages are not such entities.

      Even Tarski thought that English and all natural languages are semantically closed (and thus inherently prone to such paradoxes), meaning that we need to construct a more formal language of logic if we want to get at 'truth'.

      In a more general sense, I would say that I think we encounter serious problems when we attempt to import such solutions into our semantically closed natural languages. The reason for this is that English and similar languages are very semantically rich. I don't want to get too much into the philosophical weeds here, but this program (attempting to logicize natural languages) was undertaken by certain analytic philosophers long ago. It is now widely conceded to have failed.

      In part this is because we have to make too many radical changes, which don't comport with what we seem to mean. But it is also because making language more precise has the unfortunate side effect of making it less meaningful (able to capture less of the world). The more precise a language becomes, the more it becomes reduced to talk about the atomic constituents (or aggregates of such) of the world.

      Such a thing quickly becomes unworkable and excessively complicated, note the example of my listing all the necessary conditions involved in the liar sentence. Also note the tension between a semantically closed natural language being able to capture more (it can talk about itself), and a more formal version being able to talk about less.

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  10. Hey Alex,

    I can see that I haven't successfully communicated the concepts I am using.

    A statement is a synonym for a proposition. It seems I need to improve the section of the solution article where I talk about the differences between sentences and statements and give examples of how the same sentence can be translated into completely different statements (meanings).

    >Further, what is the relation between sentences and statements? If a language is that which maps the former to the latter, is it safe to say that sentences are not actually about the facts of the world, that they are just about statements?

    I don't understand the second question very well. Yes, a language provides a map from the realm of "scribblings" to the realm of meanings, so yes, from the former to the latter. The sample sentence that means completely different things when interpreted in Finnish vs Fakish was supposed to illustrate the need for such a structure. So I wouldn't say sentences by themselves are about anything. The above-mentioned sample sentence seems to demonstrate that.

    Your next paragraph perhaps presupposes a different answer to the above.

    >I think it would be more appropriate to say that L is a necessary condition for S' meaning. It can't literally be true that (S,L) are the things which refer to S' contents. It's not the case that the sentence "snow is white" along with the entire English language are referring to the referents of 'snow' and 'white'.

    I am not sure I understand what you mean by "L is a necessary condition". A necessary or sufficient condition is a state of affairs, but L is a map, as we discussed. You couldn't say that sin(x) is a necessary condition for something, right?

    There's no canonical map from the set of sentences to the set of statements - each language maps it them differently. But there IS a canonical map from the set of pairs (S, L) to the set of statements - since such a pair determines a statement, L(S).

    In other words, S alone doesn't have a meaning, you need both S and L, which is another way of saying you need a pair (S, L). That's why I said that S alone doesn't refer to anything but (S, L) can.

    Even more briefly, you need both for semantics.

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    1. Hey Dmitriy,

      “ The sample sentence that means completely different things when interpreted in Finnish vs Fakish was supposed to illustrate the need for such a structure”

      That we need such a structure is not the same thing as stating that it is the structure (S, L) which refers to x or means x. It’s one thing for x to be necessary for y to have a meaning, and another thing for x to be that which is doing the meaning (along with y).

      The above only establishes that we need something outside the characters of the sentence if we wish the sentence to have meaning. But this only goes so far as to give us necessary conditions for a sentence having meaning. I agree that these necessary conditions include the existence of the English language (among other things).

      However, I don’t see how you’ve established that the sentence on its own can’t be the thing which has meaning or does the referencing. In fact, as I mentioned, that can’t literally be true for L. If L means the (idealized) English language, we surely don’t mean to say that a sentence plus the entire language is the thing which means x.

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    2. Addendum:

      For clarification, I am assuming that a proposition is just a fact ‘in the world’. Although it’s controversial whether propositional entities are linguistic entities; I’m assuming that a proposition is just the fact in the world (this eliminates any need to add a third category). In that sense, there is a simple [sentence > fact in the world (e.g. 2 + 2 = 4)] relationship.

      It sounds to me from your declarations about statements that you want to introduce a third layer between the facts of the world and the sentence. You say you want a statement to just be a proposition, but it sounds from your descriptions like statements are actually supposed to be linguistic in nature. That follows from your point that the liar sentence can’t be referring to sentences (since it has to refer to statements).

      But if sentences have propositional content then of course they can refer to objects in the real world, since they are about the real world (the facts of the world, i.e. propositions). Naturally, a sentence is an object in the real world.

      Thus, my many questions about why we appear to need this ‘third layer’.

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    3. Confusingly I wrote:
      “ Although it’s controversial whether propositional entities are linguistic entities”

      When of course I believe the exact opposite! Propositions are definitely non-linguistic, and that is a pretty uncontroversial statement. In fact, I meant to that there is great controversy over what propositions really are.

      If they are facts, thoughts in our head or some kind of abstract entity etc...

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  11. Hey Alex,

    since we are talking about concepts from the solution article, I will reply there.

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