Fitch's paradox of knowability

Discussion in 'Philosophy' started by vostibackle, Feb 8, 2009.

  1. #1 vostibackle, Feb 8, 2009
    Last edited by a moderator: Feb 8, 2009
    Suppose we try to formalize a logic of knowledge. Take the following axioms to be true (phi and psi stand for sentences which can be true or false and known or unknown):

    1) If we know that "phi and psi" is true, then we know that "phi" is true and that "psi" is true.
    For example, if we know that "the sky is blue and the grass is green", then we know that "the sky is blue" and we know that "the grass is green."

    2) If we know that "phi" is true, then "phi" is true.
    This is just part of the definition of "knowledge." In order for a belief to count as knowledge, that belief must be true.

    3) If "phi" is provable from no assumptions, then "phi" is necessarily true.
    This just says that if we can prove that "phi" is true without making any extra assumptions, then it's logically necessary that "phi" is true, i.e. that it's logically impossible that "phi" be false. This seems obvious.

    4) If it's necessarily true that "phi" is false, then it's not possible that "phi" is true.
    Again, this seems obvious—if it's logically necessary that "phi" isn't true, then it's not logically possible that "phi" is true.

    OK, so. It can be proven that any formalization of knowledge which obeys those 4 axioms falls prey to Fitch's paradox of knowability, which states that
    So, you have three choices:
    1) There exist truths which can never, in principle, be known.
    2) All truths are, in fact, known.
    3) one of the 4 axioms listed is false

    What do you think?
  2. If truth is nothing more than a word, and words aren't the truth (what they symbolize/refer to), then no amount of words could reduce the complexity that already exists. But we still try, don't we?

    Where does our reductionist bent arise from?!

    Makes me think nothing is complex, just like everything.
  3. I think to deny that the paradox applies to real knowledge, to say that this is just a formal game which doesn't say anything about "real" truth, you have to claim that there is some kind of mismatch between the semantics of formal epistemic logic and real truths.

    To do this, you'd really have to deny one of the axioms of formal logic, though there are more to choose from than just the 4 I listed. It's just that the others are all so obvious that they can't be reasonably denied, I think.

  4. Wrong, to say anything about "the truth", you have to start with the a priori premise that "the truth" exists outside of the words.
  5. Right, well, if you assert that truth doesn't exist, you've really denied that the axioms of formal logic apply to the truth, just by denying the existence of truth itself. That's a method I hadn't thought of.

    But to me, sentences like "if A is true and if A implies B, then B is true" have a real semantic meaning. You may be right that this is an a priori premise... though I hope not :).
  6. lol No, but seriously, you're not getting off that easy, buster. :p If I may be right, does that preclude you from being right as well? If so, aren't you just trying to weasel in this idea of singular "truth"?
  7. I just don't know... I've always considered it obvious that sentences about truth have a real semantic value. I'd never thought about how to justify that before. I really have no idea how to start. I imagine I'll be doing a lot of thinking about it thanks to you :D.

    I would have thought you would be delighted by the conclusion that there exist unknowable truths. For people like me who tend toward verificationism, it's really bad, but I think it's aesthetically pleasing—human intellect is necessarily limited in the sense that there are true things which we can never possibly know. It gives me a sense of how vast the cosmos are, and how our knowledge can't ever really capture it all.
  8. It sounds like you were trying to reduce it all, rather than illuminate what parts you are blessed to have even a temporary (situational) shot at grasping. I used to reduce more, because I thought "truths" were unmessy, concise, and logical... But you know, sometimes easy questions don't have easy answers:


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