• Michael
    13.9k
    Is this statement false? If I've done the truth table right, then it means that the first line of the proof is wrong.Brendan Golledge

    The first line is a definition, not a premise, and so not truth apt. It is simply saying this:

    Let "A" mean "if A is true then B is true".
  • Brendan Golledge
    63
    If definitions aren't subject to truth apt, then can I say, "Let 'X' mean a married bachelor," and that this sentence is not truth apt?
  • Michael
    13.9k
    If definitions aren't subject to truth apt, then can I say, "Let 'X' mean a married bachelor," and that this sentence is not truth apt?Brendan Golledge

    The sentence isn't truth apt, but "married bachelor" is a contradiction.

    But neither "if this sentence is true then I am 30 years old" nor "if this sentence is true then I am not 30 years old" is a contradiction, or at least not obviously so.
  • Lionino
    849
    the paradox is right there in the initial version of Principia Mathematica; that is, an "invalid" statement was implied by the formalisation of mathematics in a first order logic. It looked as if the whole edifice would collapseBanno

    So what should be done instead?
  • TonesInDeepFreeze
    2.3k
    When people argue that the paradox is explained away by saying that the sentence is not meaningful, they are overlooking at least this: In formalized mathematics, it needs to be mechanical to check whether a given string is or is not a well formed sentence. If "meaningful" is not definitely specified so that it is mechanically checkable for "meaningfulness" then that criterion is not usable for formal languages.

    Also, Tarski's undefinability theorem shows that there is no definition of a truth predicate (per the standard model for the language of arithmetic) in the language of arithmetic. The proof makes use of "This sentence is false" by showing that a truth predicate would allow, via Godelization, the formation of the sentence, which would be both true and false in the model, which is impossible since, for any given model, there is no sentence that is both true and false in that model.
  • Lionino
    849
    Between the black and white of hand-holding or handing a 900 page pdf without context, there are many shades of grey.
  • TonesInDeepFreeze
    2.3k
    What invalid statement was implied by mathematics formalized in first order logic? By definition, all non-logical axioms are invalid (i.e. not validities, not true in every interpretation). And unrestricted comprehension implies Russell's paradox, which is not 'this sentence is false'.
  • TonesInDeepFreeze
    2.3k
    Someone said 'This sentence is false' doesn't indicate what sentence is being referred to.

    In 'This sentence is false', 'this sentence' is referring to 'This sentence is false'.

    Not recognizing that is just waving away the problem.
  • Count Timothy von Icarus
    1.8k


    Ignore that anyhow, it's just work around and epicycles. This is the true solution.
  • Lionino
    849
    Thank you, I will add that one to my non-reading list as well.
  • Apustimelogist
    309
    The way I see it, these paradoxes show in a nice way how all truth is an idealization. Why? Because we never have direct objective access to some kind of objective truth. The best we have is a kind of proxy that involves sequences of events with some kind of formal structure, enacted by some kind of observer. In other words, we can't talk about truth, we can just act: say things or assert things or behave in ways that seem coherent in the context of our environment. Our talk about truth cannot be separated from this: from how truth-talk, inference and observation occurs. There needs to be certain assumptions filled to make this stuff valid and "coherent"... but they can be broken.

    The subject matter of truth is an idealized model of the "truth" process, of the process of these acts, inferences, observations. But because it is idealized, the required assumptions are only implicit. I like to think the best model is a communication process: an environment provides a signal and the observer responds with an appropriate signal (this in itself is an idealization ignoring how or the means by which the observer validates its signal to itself, in terms of use). We also have state transitions: producing a signal implies a change of state in the observer / environment.

    Maybe not a universal explanation, but I think a big feature for many notable paradoxes... the major assumption broken is the underlying assumption that for communication to occur there must be a clear object/subject divide. To talk about the environment, the environments behavior must be completely independent of the signal the observer uses - something which seems like our assumptions about objectivity in the world. The world or things exists or have truth objectively and independently regardless of the signal the observer produces to respond to it in the communication game - context independence. If we see the observer's state transitions as being induces by the environment then the world can provide a signal which induces an observer state transition to a matching signal (which contains information of what is the case) and that is that. The communication problem is finished and the observer state will be stable so long as the environment doesn't change.

    But if the world is dependent on the signal then stability is broken because, like the observer changes their signal due to the environment, now the environment changes due to the observer's signal. The observer's "truth" description of the environment then induces the environment to change its signal. If the signals that the environment and observer can make are all easily distinguishable and unique to every possible new situation, then stability is lost and the environment and observer will keep changing, inducing changes in each other. Like a (idealized) mirror scenario: you hold up a signal of what you see in the mirror, but the second you hold up your signal, the mirror image has changed to you holding a signal. So to communicate what you see, you must now hold up a signal which is about you holding a signal... but that changes the mirror image again. The stability of the observer's signal is impossible because any signal they make changes what they are observing which then changes their signal and it goes on and on. It seems to me, all these paradoxes are only ever made salient when you talk about their consequences in a sequential fashion like this which never stabilizes. Obviously the mirror alludes to how self-reference is a special case of this.

    The "truth" then cannot be beyond the assumptions and processes that embody how observers enact their epistemic behaviors. And once the formal or mechanical or physical scaffolding that supports that fails then so does notions of truth. Without the strong object-subject divide enforcing context-independence then it is impossible for the signals of the environment and observer to match up at any point in time (denoting stable, coherent communication). This doesn't necessarily need to be between one observer and environment (or an observer and themself) either but maybe networks of observers communicating to each other in a way that context-independence fails for all of them because the signals directly influence other signals in the network of communication.

    The fact that these paradoxes can occur just reflects the mechanical or formal capabilities of the system being described and the same which underwrites any communication process - sometimes the system cannot settle in its dynamics. Our notions of truth or even the notion of communication are secondary; communication as we commonly see it assumes that there is an objective fact of the matter about a meaning of what these signals represent which is somehow beyond the formal or physical mechanics of the situation. In reality, the mechanics are all there is, just like the epistemic activities of humans is nothing more than brain behaviour, or maybe sequences of experiences - depending on how you want to view the mind. Our everyday coherent notions of truth or communication require constraints beyond what is constrained in these mechanics and so it is no surprise they sometimes fail.

    I think probably another reason why things like these fail sometimes, which I won't go much further into, is something which I crudely refer to under the umbrella of factorization assumptions. which sometimes we cannot but help make, but also may make things like computation or representation easier. For instance, If we want to make statements about the world, and that information is in both the word meanings AND syntax, well they cannot be independent when it comes to any kind of truth descriptions of the world (but maybe speech isn't just about truth so I am deliberately ignoring part of the picture). However, for whatever reasons, whether trivial or sophisticated, syntax and semantics are obviously independent, which allows you to make nonsensical statements - e.g. colorless green ideas sleep furiously - which again reflect the fact that the mechanics and formal constraints on truth are much tighter than the constraints on the systems which enact or embody these things for us. Its interesting that when you look at machine learning and occasionally computational neuroscience, things like factorization assumptions that enforce statistical independence are used because it makes inference easier... however, it also amounts so something like making a deliberately false assumption about the data. You can even see this kind of thing in a slightly different way in moral statements: many statements are very coarse like "stealing is bad" but in reality we all know that whether stealing is bad depends on the situation - its different in every scenario (e.g. what if stealing was involved in some operation which was about national security) - but because there are regularities we can assume we can make the simplifying statement "stealing is bad" is independent of the situation.
  • Pantagruel
    3.2k
    Practically speaking, no one would ever utter the phrase "this sentence is false". What they might do is assert "I am telling you the truth". Which, if you do not believe them, translates into "this sentence is false." This would maintain the integrity of the logical form (the meaning of the sentence would contradict its sense), and remove the apparent paradox (by way of the assumed premise that people actually say things for a reason).
  • Brendan Golledge
    63
    I think these paradoxes can be solved by using a truth table on the definition.

    If you say, "X is false", clearly that could be represented as X -> F. Then if you say that X is "this sentence," then you could write something like "X <-> (X -> F)". This is very similar to the sentence used in Curry's paradox: "X <-> (X -> Y)", where Y is any arbitrary statement.

    If you do the truth table for X, Y, (X -> F), & (X -> Y), then you see that the definitions are simply false. "X <-> (X -> F)" is exactly backwards, so that "NOT X <-> (X -> F)" is a tautology. "X <-> (X -> Y)" is only true if X = T and Y = T.

    Michael said earlier that a definition is not truth apt. I can see how that would be the case if you defined an entirely new variable, such as Z <-> (X -> Y). However, since you are setting X equal to itself, you can do a truth table on it.

    I remember hearing that if a system contains a contradiction, then anything can be proven. So it makes sense to me that the premise in Curry's paradox contains a contradiction, hence its ability to prove any arbitrary statement.
  • Michael
    13.9k
    Michael said earlier that a definition is not truth apt. I can see how that would be the case if you defined an entirely new variable, such as Z <-> (X -> Y). However, since you are setting X equal to itself, you can do a truth table on it.Brendan Golledge

    These are two different sentences that you seem to be confusing:

    1. X ≔ (X → Y)
    2. X ↔ (X → Y)

    In ordinary language, these mean:

    1. "X" means "if X is true then Y is true"
    2. X is true if and only if (if X is true then Y is true)
  • Brendan Golledge
    63
    So what's the difference? One is a definition and the other is a logical equivalence? I don't think you can get away with any arbitrary definition. If I define X to be equal to 2, then it must be that X is also equal to 2. I feel like you are just playing semantics. If I define X to be NOT X, then that is a contradiction. C := "If C, F" is also a contradiction, if F is false.
  • Michael
    13.9k
    I don't think you can get away with any arbitrary definition.Brendan Golledge

    You seem to misunderstand what is happening here.

    Take the English language sentence "this sentence is English". To better examine this we decide to translate it into symbolic logic. To do that we have to do something like the below:

    S ≔ E(S)

    Now take the English language sentence "this sentence is French". In symbolic logic this is:

    S ≔ F(S)

    Now take the English language sentence "this sentence is true". In symbolic logic this is:

    S ≔ T(S)

    Now take the English language sentence "this sentence is true and English". In symbolic logic this is:

    S ≔ T(S) ∧ E(S)

    Now take the English language sentence "this sentence is true and French". In symbolic logic this is:

    S ≔ T(S) ∧ F(S)

    Regardless of whether or not the right hand side is true, these are the accepted ways to translate an ordinary language (self-referential) sentence into symbolic logic.

    See also here.
  • TonesInDeepFreeze
    2.3k
    To add to the above:

    If X := X->Y then X <-> (X->Y).

    But we don' t have the converse that if X <-> (X->Y) then X := X->Y.

    So X := X->Y is not equivalent with X <-> (X->Y).

    So we can't dispense the paradox by incorrectly saying that it reduces to X <-> (X -> Y).
  • Banno
    22.9k
    The way I see it, these paradoxes show in a nice way how all truth is an idealization.Apustimelogist

    Trouble is, that's just an idealisation.

    man-cutting-the-branch-sitting-on-illustration-by-frits-ahlefeldt1500-square.jpg

    Happy sawing.
  • Joshs
    5.1k

    The way I see it, these paradoxes show in a nice way how all truth is an idealization.
    — Apustimelogist

    Trouble is, that's just an idealisation.
    Banno

    One can deconstruct idealizations without one’s deconstruction itself being an idealization.
  • Banno
    22.9k
    Of course - at the expense of not telling the truth.
  • Joshs
    5.1k
    ↪Joshs Of course - at the expense of not telling the truthBanno

    Can one put into question the notion of god’s eye truth without that questioning itself being assumed to be oriented by a god’s eye perspective?
  • Banno
    22.9k
    ...god’s eye truth...Joshs

    This sentence is six words long.

    I don't much care what god thinks. There are true sentences, contra , and, it seems, your good self.
  • Moliere
    3.9k


    Yes. And No.

    Honestly my Big Brain project is seeing how it might be possible to unite both of those big-azz books.

    I still have work to do in both, though. And they ain't epicycles, either of them.



    They (the sentences which are true) are pesky, though.
  • Brendan Golledge
    63
    To add to the above:

    If X := X->Y then X <-> (X->Y).

    But we don' t have the converse that if X <-> (X->Y) then X := X->Y.

    So X := X->Y is not equivalent with X <-> (X->Y).

    So we can't dispense the paradox by incorrectly saying that it reduces to X <-> (X -> Y).
    TonesInDeepFreeze

    I can do the same thing without equating := to <->


    You can use a truth table to prove NOT X <-> (X -> F).
    (X -> Y) <-> (X -> F) in the case where Y is false, so this applies to Curry's paradox as well as "this sentence is false".

    Then you take your definition X := (X ->F) and substitute NOT X for the second part.

    Then you get X := NOT X

    Clearly, there has to be something wrong with that definition.
  • Michael
    13.9k
    Then you get X := NOT XBrendan Golledge

    Yes, that's how the sentence "this sentence is not true" is translated into symbolic logic.
  • TonesInDeepFreeze
    2.3k
    You can use a truth table to prove NOT X <-> (X -> F).
    (X -> Y) <-> (X -> F) in the case where Y is false, so this applies to Curry's paradox as well as "this sentence is false".
    Brendan Golledge

    Yes:

    |- ~X <-> (X -> F)

    If Y is false then (X -> Y) <-> (X -> F) is true.

    That's not Curry's paradox.

    Then you take your definition X := (X ->F) and substitute NOT X for the second part.Brendan Golledge

    Who does that? You? Did someone previously define?:

    X := (X -> F)
  • Apustimelogist
    309


    Trouble is, that's just an idealisation.Banno

    Honestly, I feel like my views on the world would actually be less consistent if I didn't think that my views or the things I said did not suffer those kinds of qualities of idealization or related issues. Part of the central basis of my views is that what we do or say is at the mercy of the constraints of how our minds, brains work as computational systems. It would actually be not as coherent if I didn't think these kinds of things to my own mind and thoughts, beliefs, theories all the time. Why would I be exempt from things I apply to the rest of the academic and cognitive world in its entirety?

    The idealization thing is only an issue for people with a certain kind of goal here... which I do not think I share.

    There are true sentencesBanno

    Yes, when you agree to play the game in the right way. Even better when you ignore the parts where it breaks down.
  • Banno
    22.9k
    Well, why not play the game?

    For my part the notion of "objective" truth causes more problems for those of a philosophical bent than it heals. In particular, there are folk who supose that because they cannot access "objective" truth (whatever that is), that there are no truths at all.

    But it ain't so.

    ...when you agree to play the game in the right way.Apustimelogist
    Are you saying it is better to play the gamein the wrong way?

    Why not play it in the right way, or at least, work out what the rules are...?
  • Brendan Golledge
    63
    You can use a truth table to prove NOT X <-> (X -> F).
    (X -> Y) <-> (X -> F) in the case where Y is false, so this applies to Curry's paradox as well as "this sentence is false".
    — Brendan Golledge

    Yes:

    |- ~X <-> (X -> F)

    If Y is false then (X -> Y) <-> (X -> F) is true.

    That's not Curry's paradox.

    Then you take your definition X := (X ->F) and substitute NOT X for the second part.
    — Brendan Golledge

    Who does that? You? Did someone previously define?:

    X := (X -> F)
    TonesInDeepFreeze

    To add to the above:

    If X := X->Y then X <-> (X->Y).

    But we don' t have the converse that if X <-> (X->Y) then X := X->Y.

    So X := X->Y is not equivalent with X <-> (X->Y).

    So we can't dispense the paradox by incorrectly saying that it reduces to X <-> (X -> Y).
    TonesInDeepFreeze


    It seems to me that everybody is being super-pedantic about this. I am studying formal logic informally (without being in a class), so I'm not surprised if I'm not using some symbols correctly. However, the logic should still work

    Who does that? You? Did someone previously define?:

    X := (X -> F)
    TonesInDeepFreeze

    "X -> F" is supposed to mean, "This sentence is false." "X := (X -> F)" is supposed to mean "This sentence says, 'This sentence is false'."

    That's not Curry's paradox.TonesInDeepFreeze

    I've seen in multiple sources that Curry's paradox is defined as X := (X -> Y), and some of them then change it to X <-> (X -> Y).

    If X := X->Y then X <-> (X->Y).TonesInDeepFreeze

    You yourself said that this is allowed, so I don't know why you are arguing with me about this.


    I am new for formal logic, but I understand algebra just fine. If I define Y := X + 1, then it is impossible to say that Y is false, because Y has no outside definition. However, if I define X := X + 1, then this obviously involves a contradiction. It seems the same ought to apply to formal logic. I do not see how you guys can argue about this so much.

    Maybe the difference is that I come from a physics background rather than a math background. If I can make the math give me the answer I want, then I think it must be right. What I'm doing here gives me the answer I want, because the truth table for "This sentence is false" shows that X <-> NOT X, which is the same answer you get by working through the paradox with human language. In Curry's paradox, the truth table gives that the sentence is self-contradictory if the assertion is false, which resolves the paradox. It seems to me that mathematicians get stuck on arbitrary definitions & distinctions, like := vs <->, even if doing so makes everything harder and nothing easier. If the proof of Curry's paradox is correct, then we get that logic is broken, because there is a paradox. However, using a truth table to check the definition shows that the definition is contradictory, and thus there is no paradox. It seems bizarre to me that people are arguing with me that I can't check the definition for consistency when doing so makes everything so simple.
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