• TheMadFool
    2.3k
    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?
  • Banno
    3.1k
    I was chatting to @Sam26 earlier about similar issues.

    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.
  • TheMadFool
    2.3k
    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.Banno

    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?
  • Banno
    3.1k
    in that there's no error.TheMadFool
    sentences about truth and falsity are not well formed in first order logic. So there is no translation of L nor L'.
  • Michael
    7.1k


    We've been over this before.

    https://thephilosophyforum.com/discussion/1047/liars-paradox-an-attempt-to-solve-it-/p1

    When you say that the statement "this statement is false" is neither true nor false you're not saying that the statement "this statement is false" means "this statement is neither true nor false". You're conflating statement A with a (different) statement about A. A and B are not logically equivalent.Michael

    You all are right.

    A: this statement is false

    A has no truth value

    So, we should be saying: "A is neither true nor false" instead of ''this statement is neither true nor false''
    TheMadFool
  • Michael
    7.1k
    Kripke's solution.
  • Banno
    3.1k
    one possibility.

    So is the metalanguage treatment.
  • TheMadFool
    2.3k
    Sorry. My memory isn't as good as it used to be.

    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?
  • Cuthbert
    216
    L is equivalent to L' because both are ill-formed. All ill-formed (non-propositional) statements are materially equivalent, because they are all devoid of truth value. So L and L' are both equivalent to 'the mome-raths are gyring in the wabe'. But we cannot use any of them in propositional logic.
  • andrewk
    1.4k
    Here is what the sentence actually means:

    ((((((((((...) 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.
  • Michael
    7.1k


    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'.
  • Michael
    7.1k
    Let's say that to be true is to refer to a fact and to be false is to refer to a fiction. The liar sentence is then "this sentence refers to a fiction".

    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.
  • TheMadFool
    2.3k
    Could you do it for me please, as I got a bit tired doing the above.

    Let me know when you're done.
    andrewk

    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.

    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'.
    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).
  • Michael
    7.1k


    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.
  • TheMadFool
    2.3k


    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?
  • TheMadFool
    2.3k
    I just looked at the dictionary and found out that "this" isn't a self-referential word.

    Google definition: used to identify a specific person or thing close at hand or being indicated or experienced.

    Close at hand isn't self-referential. It's not like saying ''I like x''. So, the liar sentence is grammatically incorrect.

    What do you think?
  • andrewk
    1.4k
    It needn't be self-referential but it may be. Close at hand includes self. I can non-self-referentially point at a pot and say 'this pot needs a scrub', or I can point at myself and say 'this Australian understands how lucky he is to live in a country in which most people don't carry guns'.

    Also, 'This sentence consists of exactly seven words' is self-referential but not viciously circular like the liar sentence. Can you see why?
  • MindForged
    315
    @OP I just go with Dialetheism, personally. Also, there is no official position on the Liars. The only agreement among logicians seems to be that no one has a proper solution yet, so if there is a solution it must be a strange one because all the obvious responses have been tried and they failed (e.g. Kripke's solution doesn't work, Tarski's infinite hierarchy of metalanguages doesn't work, etc.)

    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.
  • Michael
    7.1k
    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.MindForged

    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.
  • MindForged
    315
    How is it not in the realm of a deductive truth? It's taking bivalence and using it to derive a contradiction when applied to a certain kind of self-referential sentence. There doesn't seem to be anything about truth predication that makes it different than properties like those in the other self-referential sentences. This would seem to require abandoning Tarski's Undecidability Theorem if you really can't make this move.

    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.
  • Michael
    7.1k
    How is it not in the realm of a deductive truth?MindForged

    From what axioms and definitions can one derive "this sentence is false"? Can you set out the proof that concludes with 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.
    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.
  • MindForged
    315
    From what axioms and definitions can one derive "this sentence is false"? Can you set out the proof that concludes with the liar sentence?

    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.

    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.

    "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.
  • Michael
    7.1k


    So T(x) ≔ x and x ≔ ¬T(x). Therefore, T(x) ≔ ¬T(x). Your definitions are contradictory.
  • MindForged
    315
    I didn't define x as contradictory, I took x and applied the T-schema to produce the Liar.
  • Michael
    7.1k
    The T-schema defines T(x) as x. The liar paradox defines x as ¬T(x). These are contradictory definitions.
  • Michael
    7.1k
    The point is if your solution works for "This sentence is false" then it should dispel "This sentence is meaningless"MindForged

    Why? They're different sentences.
  • MindForged
    315
    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.

    The T-schema defines T(x) as x. The liar paradox defines x as ¬T(x). These are contradictory definitions.

    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.
  • Michael
    7.1k
    Because it's the same type of paradox caused by the same feature.MindForged

    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.").

    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: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).
  • Banno
    3.1k
    There's nothing about the liar which is any different than any other self-referential sentence.MindForged

    As Michael says, yes there is. It talks about its own interpretation.
  • ssu
    476
    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.Michael
    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.
  • Michael
    7.1k
    I think it's simply about negative self reference. The negative makes it problematical. Otherwise it would be just circular and then perhaps meaningless.ssu

    This sentence isn't 36 characters in length.
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