1. Joined
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    14 Feb '18 11:316 edits
    I am trying to understand Fitch's paradox of knowability but don't understand it because I don't see any paradox there!
    The 'paradox' is supposed to be a challenge to the 'knowability thesis', which states that every truth is, in principle, knowable.
    But is that "knowable" only in the present or does that include the future? I initially naturally assumed obviously it must mean the latter but then later got confused into thinking from what wiki says that it may mean only in the present!

    Wiki says;
    https://en.wikipedia.org/wiki/Fitch%27s_paradox_of_knowability
    "...
    Proof (of the paradox);
    Suppose p is a sentence that is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence "the sentence p is an unknown truth" is true; ..."

    OK, so far so good.
    But then it says;

    "...and, if all truths are knowable, it should be possible to know that "p is an unknown truth". ..."

    OK, But is that supposed to be "possible to know that "p is an unknown truth" in the present or in the future?

    Then it says;

    "...But this isn't possible, because as soon as we know "p is an unknown truth", we know that p is true, ..."

    At this point it looses me completely;
    How does us knowing "p is an unknown truth" logically imply that "we know that p is true"?
    Where is the contradiction in us knowing "p is an unknown truth" and "we DON'T know that p is true"?
    Please somebody enlighten me...
  2. Joined
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    14 Feb '18 15:26
    Originally posted by @humy
    I am trying to understand Fitch's paradox of knowability but don't understand it because I don't see any paradox there!
    The 'paradox' is supposed to be a challenge to the 'knowability thesis', which states that every truth is, in principle, knowable.
    But is that "knowable" only in the present or does that include the future? I initially naturally assumed obv ...[text shortened]... ing "p is an unknown truth" and "we DON'T know that p is true"?
    Please somebody enlighten me...
    'p is an unknown truth' automatically and tautologically implies 'p is true'.

    That meand that if we know that 'p is an unknown truth' is a true statement, we van immediately, without any more data, conclude that 'p is true' also holds.

    Basically, it's the same reason why, if we know that 'x is a purple cow', we also, immediately, know that 'x is a cow'. It only appears less obvious because it mentions truth and is used in a meta-argument about knowledge and truth; but the latter statement does follow as immediately from the former in p's case as much as in x's.
  3. Joined
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    14 Feb '18 18:562 edits
    Originally posted by @shallow-blue
    'p is an unknown truth' automatically and tautologically implies 'p is true'.
    Not sure but somehow I naturally assume, given the context, that what they mean by
    "... we know "p is an unknown truth" ..."
    is
    "...we know we don't know whether p is true..".
    which wouldn't imply "p is true" but perhaps you might be right.

    ANYONE;
    Is shallow-blue interpretation correct here and mine incorrect or vice versa?
  4. Subscribermoonbus
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    14 Feb '18 19:051 edit
    'p is true' reduces to 'p'.

    'p is an unknown truth' reduces to 'p'.

    When is irrelevant in the timeless present.

    Shallow Blue is correct.
  5. Joined
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    14 Feb '18 19:276 edits
    Originally posted by @moonbus
    'p is true' reduces to 'p'.

    'p is an unknown truth' reduces to 'p'.

    When is irrelevant in the timeless present.

    Shallow Blue is correct.
    Then, if that is correct, when they said (in the link);

    "... we know "p is an unknown truth" ..."

    that above statement is nonsense because it is a self-contradiction!
    Because, if that interpretation is correct, then what they mean by
    "... we know "p is an unknown truth" ..."
    is that BOTH we know p is true AND we know p is UNknown to us to be true!
    BUT, then, "we know p is true" contradicts "p is UNknown to us to be true".
    Have I got that right?
    I ask because, if so, I think it is a strange thing for them to say as it just seems far to 'self-evident' to me that it cannot be true.
    And I think that is the reason why I naturally assumed (apparently erroneously) that what they mean by
    "... we know "p is an unknown truth" ..."
    was
    "...we know we don't know whether p is true..".
    as that wouldn't then be so obviously a self-contradiction!

    But I think what you both say must be right (I see no reasonable alternative explanation) so I must thank you both for enlightening me 🙂
  6. Subscribermoonbus
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    14 Feb '18 20:46
    If we know that p (is true), then it is not an unknown truth. That is obvious.
  7. Standard memberDeepThought
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    15 Feb '18 05:15
    Originally posted by @humy
    Then, if that is correct, when they said (in the link);

    "... we know "p is an unknown truth" ..."

    that above statement is nonsense because it is a self-contradiction!
    Because, if that interpretation is correct, then what they mean by
    "... we know "p is an unknown truth" ..."
    is that BOTH we know p is true AND we know p is UNknown to us to be true! ...[text shortened]... ht (I see no reasonable alternative explanation) so I must thank you both for enlightening me 🙂
    The traditional definition of knowledge is that it is a justified belief that is true. So it's clear that a sentence can be true without the contents being known as the justification can be missing. I think that there's a theory that there's a diamond in the centre of Jupiter. There isn't any good evidence for this, it's just a theory. So it may well be true that there's a diamond there but, even if it is true, the sentence: "There is a giant diamond at the centre of Jupiter." can't count as knowledge because it's not properly justified. So a sentence can be true, but not known to be true, there isn't a contradiction there.

    I have two problems with Fitch's paradox, the first is that although the assertion that "P is unknown but true", implies that P itself is true, I don't think that that makes P known, it just implies its truth - it doesn't justify it properly, except as a sort of logical juggling act.

    The other problem I have is that it seems to me to depend on an unjustifiable assertion of the truth of the wrapping sentence. If P is unknown then I don't see that the sentence "P is an unknown truth" can be assigned a truth value reliably. The truth of "P is an unknown truth", depends on whether P is true or not, but that isn't known for sure, so the truth of the wrapping sentence isn't either. Otherwise one could prove that there is a giant diamond at the centre of Jupiter simply by asserting that it's an unknown truth that there is a giant diamond at the centre of Jupiter.
  8. Subscribermoonbus
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    15 Feb '18 05:54
    Originally posted by @deepthought
    The traditional definition of knowledge is that it is a justified belief that is true. So it's clear that a sentence can be true without the contents being known as the justification can be missing. I think that there's a theory that there's a diamond in the centre of Jupiter. There isn't any good evidence for this, it's just a theory. So it may well ...[text shortened]... by asserting that it's an unknown truth that there is a giant diamond at the centre of Jupiter.
    Fitch's so-called paradox is pretty feeble. It is a self-reflexive muddle; weak logicians often get into a twist over self-reflexiveness. Another example of a self-reflexive muddle is "this statement is false." The intuitively obvious and correct reply is that there is no statement there at all, there isn't any there there, so there nothing to be either true or false.
  9. Standard memberDeepThought
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    15 Feb '18 06:491 edit
    Originally posted by @moonbus
    Fitch's so-called paradox is pretty feeble. It is a self-reflexive muddle; weak logicians often get into a twist over self-reflexiveness. Another example of a self-reflexive muddle is "this statement is false." The intuitively obvious and correct reply is that there is no statement there at all, there isn't any there there, so there nothing to be either true or false.
    I kind of like the liar paradox, but yes it's internally inconsistent. Interestingly the intuitionistic equivalent of these, "This sentence is not provable.", works fine and Godel's famous theorem depends on showing that such a statement exists in any formal system of arithmetic complicated enough to allow multiplication. So whether the paradox is trivial or not seems to depend on which modal logic is in play.
  10. Joined
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    15 Feb '18 11:05
    Originally posted by @moonbus
    Fitch's so-called paradox is pretty feeble. It is a self-reflexive muddle; weak logicians often get into a twist over self-reflexiveness. Another example of a self-reflexive muddle is "this statement is false." The intuitively obvious and correct reply is that there is no statement there at all, there isn't any there there, so there nothing to be either true or false.
    That's not really the fault of Fitch's paradox. The feebleness is in the assumptions; what Fitch's does is demonstrate the feebleness of knowability theory, not its own.
  11. Subscribermoonbus
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    15 Feb '18 12:31
    Originally posted by @deepthought
    I kind of like the liar paradox, but yes it's internally inconsistent. Interestingly the intuitionistic equivalent of these, "This sentence is not provable.", works fine and Godel's famous theorem depends on showing that such a statement exists in any formal system of arithmetic complicated enough to allow multiplication. So whether the paradox is trivial or not seems to depend on which modal logic is in play.
    Agreed, natural language is too complicated to prove an equivalent of Goedel’s theorem, at least formally, but it might still be case that although all knowledge is true(by definition), it were nonetheless not the case that all truths were knowable.
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