• TonesInDeepFreeze
    2.3k
    super-pedanticBrendan Golledge

    Curry's paradox has a very technical context. To understand it properly requires being very careful in the formulations.

    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'."
    Brendan Golledge

    'X - > F' means "X is false".

    'X := X -> F' means that X is the sentence 'X -> F'.

    But no one asserted that X is the sentence 'X -> F'. Indeed, X is not the sentence 'X -> F'.

    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).Brendan Golledge

    What sources?

    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.
    Brendan Golledge

    I am not arguing about that.

    Again, we have:

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

    but we do not have:

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

    So we don't have:

    X := X -> Y iff X <-> (X -> Y)

    If I define Y := X + 1Brendan Golledge

    It depends on the context. For example, it could be in a computer programming language or something. But in this context, we would not write that. (I could explain why, but it's another subject).

    then it is impossible to say that Y is falseBrendan Golledge

    It's impossible to say Y is false (for given values of X and Y) because, as it seems, you're using 'Y' as variable ranging over numbers not sentences.

    the truth table for "This sentence is false"Brendan Golledge

    It's not apparent what such a truth table would be for such self-referring sentence.

    If the proof of Curry's paradox is correct, then we get that logic is brokenBrendan Golledge

    The logic isn't broken. In English, we can make such utterances. But in the logic, we are not allowed to define a sentence symbol that way. (The following part I'm not well versed enough, so take it with a grain of salt.) But with such things as arithmetization, we can form certain sentences that are "self-referring". In those cases, where Curry's paradox can be performed, we find not that the logic is broken but that the particular theories in which Curry's paradox occurs are inconsistent. (Again, I'm not real clear on that, so take it with a grain of salt.)

    I think that in this discussion, we're assuming that there is some context in which we can justify the definition of 'X' and we're reasoning from that assumption. Upon specifying a justifying context, we would then look for the import of the contradiction in that context.
  • unenlightened
    8.7k
    the truth table for "This sentence is false"
    — Brendan Golledge

    It's not apparent what such a truth table would be for such self-referring sentence.
    TonesInDeepFreeze

    Ah, if you guys had only participated in my thread on The Laws of Form, you would have discovered that such self contradictory sentences are formed by "re-entry" or recursive definition, and result in truth values that oscillate in time.

    Logic is static, and does not deal well with time, but presumes an unchanging block of eternal truth. But sometimes the cat is on the mat, and sometimes the cat is not on the mat. Cats are fickle.
  • Apustimelogist
    309


    Well, why not play the game?Banno

    Yes, I do play the game, there's no other choice, but there's always a caveat. I think its impossible to view the world outside of some particular perspective and so in that sense I would say that our notion of objective truth is an idealization. We might say there is an objective way the world is but I don't think there is a single perspective-independent way to characterize it. If I were to say there are objective truths, I don't think I would be able to give a satisfying characterization without caveats. I don't want to conflate my belief in an objective state of the world and my ability to articulate things about them because the latter is something I cannot do.

    Are you saying it is better to play the game in the wrong way?Banno

    No, I mean right way in the sense of avoiding and ignoring caveats which makes the veracity of "truths" seem obvious.
  • Banno
    22.9k
    I think its impossible to view the world outside of some particular perspective and so in that sense I would say that our notion of objective truth is an idealization.Apustimelogist

    I'll not disagree with you about "objective" truth. I don't think the notion of much use. By talking to each other we can remove biases of perspective. There are true sentences about how things are. And overwhelmingly, we agree as to what is true and what false. The places we disagree tend to be either misunderstandings or differences in what one should to do about how things are.

    Consider how much agreement was involved in your simply reading that paragraph.
  • Lionino
    849
    The word "paradox" comes from ancient Greek "para" (= besides, contrary to) + "doxa" (= opinion). Indeed, it indicates something that exists or happens which is contrary to what one expects or believes to be true or happen. For example, a paradox would be raining without any cloud in the sky. Yet, it is possible, if there are very strong winds that bring rain from some other place than where we are.Alkis Piskas
    :up:
    People underestimate the usefulness of etymology and dismiss it as "etymological fallacy" after a 5 minute reading session. But given some background facts about some of those who underestimate it, it does not surprise me at all.
  • Alkis Piskas
    2.1k
    People underestimate the usefulness of etymology and dismiss it as "etymological fallacy" after a 5 minute reading session. But given some background facts about some of those who underestimate it, it does not surprise me at all.Lionino
    This is very true. However, etymology in English --and I believe other languages too-- is often complex and even useless. This is not the case with ancient Greek and Latin, however. Esp. in Greek, one can undestand the meaning of a word just by its etymology.

    But what are we talking about? People ignore or even hate dictionaries in general. People don't like definitions. This is what I learned from this and othe similar places. All the more about etymology.
  • Lionino
    849
    However, etymology in English --and I believe other languages too-- is often complex and even uselessAlkis Piskas

    It is true, and that is what I meant with "background facts".

    People ignore or even hate dictionaries in general.Alkis Piskas

    And that is exactly when philosophy becomes affectation. Boom, title drop.
  • Alkis Piskas
    2.1k
    People ignore or even hate dictionaries in general.
    — Alkis Piskas
    And that is exactly when philosophy becomes affectation
    Lionino
    Well said, Lionino. :up:
  • Apustimelogist
    309

    Yes, I think our main difference is that you want to hang on to the idea of true sentences and I do not really care for that.
  • Banno
    22.9k
    if you like. Folk to attach too much to truth. There are true sentences.
  • ENOAH
    55
    Am I viewing the problem too simplistically? Isn't the point of the liars paradox, and all others, not in trying to make sense out of them, but in the very thing they expose? That is, that we can never conclude absolutes in our ideas, thoughts, Language, when our strongest tool for such conclusions (Reason/Logic) are threatened by fallibility. The mathematical paradoxes (forgive me, I am far from expertise) like Russell's and Gödel's expose that same threat, I.e., we cannot, using our most powerful language, conclude absolutely.
  • TonesInDeepFreeze
    2.3k


    There is no 'Godel's paradox'.

    Anyway, as best I understand your question, the answer is 'no'.
  • ENOAH
    55


    Thank you for the clarification on Godel. I see that.


    My question was twofold.
    1. Isn't it futile to "make sense" of paradoxes?
    2. Dont paradoxes expose the limitation(s) of Logic (here, in pursuit of absolutes)?

    I'll presume your answer is no; and therefore, you think that we can make sense of paradoxes, or, at least that it's a worthwhile pursuit; and, either they don't expose the limitations of logic or logic is not limited in its pursuits.

    If that's the case, I'm interested in understanding your reasons.
  • TonesInDeepFreeze
    2.3k


    1. I don't know what you mean by ""make sense" of".

    2. Frege's system was taken to be a derivation of mathematics from logic alone. Russell's paradox showed that Frege's system was inconsistent. But does that mean that there can't be a derivation of mathematics from logic alone? Whitehead and Russell offered a system that was an attempt to derive mathematics from logic alone, but their system was not logic alone. But does that mean that there can't be a derivation of mathematics from logic alone? The Godel-Rosser theorem may discourage us even more from thinking that we can derive mathematics from logic alone. So, can we derive mathematics from logic alone? I think the preponderance of philosophers of mathematics think we cannot, but there are dissenters.
  • ENOAH
    55

    I see the relevance of your point, though indirect. It remains, then, the answer to whether or not, as you put it, we can derive mathematics from logic alone, requires at the very final stage at least, a leap from logic. Some say no, dissenters say yes, in the end, a leap must get one there.
  • TonesInDeepFreeze
    2.3k


    The leap is in the form of axioms.

    Just to be clear, the logic system itself is not contravened. Rather, we add non-logical axioms. We add axioms that are consistent, thus true in some models, but that are not logically true, thus not true in all models.
  • ENOAH
    55

    Good point. So simple, but something I haven't thought about properly. One I'll consider more thoroughly. Especially its implications on my personal interests about human Mind and the status/role/nature of logic. Thank you.
  • Faust Fiore
    8
    There are many fancy logical treatments of this paradox.

    The sentence does not express a statement that has a truth value, which means that it does not express a statement at all. There are countless sentences like that. One could devise countless sentences like that. A sentence can be vapid and nonsensical, and "This sentence is false" is one of them.

    One need not consider vapid claims, and one need not construct logical proofs of their vapidity.
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