• TheMadFool
    6.5k
    The problem is that theCurrySentence() function can't be evaluated until the antecedent is evaluated. But the antecedent can't be evaluated until theCurrySentence() function that it calls is evaluated. So nothing gets evaluated and no truth value isAndrew M

    Well, the paradox rests on self-reference and I don't have a clue why computers can't handle self-reference. However, humans fare better at it, hence the paradox.

    Doesn't the equality cause for an infinite regress in variables or am I not understanding?ep3265

    No. It doesn't P1 := (P1 > P2)

    There's no infinite regress which would require some kind of nesting which I don't see.
  • ep3265
    54
    Saying something's equal to me would be considered nesting right?
  • TheMadFool
    6.5k
    Saying something's equal to me would be considered nesting right?ep3265

    Not really. If I remember correctly, for something to be nested every previous step must occur, wholly, in the step that follows.

    This is nesting:
    (a) = b
    ((a) = b) = c
    (((a) = b) = c) = d


    How can we do the above to P1 := (P1 > P2)?
  • Andrew M
    1k
    Well, the paradox rests on self-reference and I don't have a clue why computers can't handle self-reference. However, humans fare better at it, hence the paradox.TheMadFool

    Computers can handle self-reference as long as the self-reference eventually terminates.

    The Curry sentence is not well-defined due to the non-terminating self-reference. Treating it as if it were well-defined (and thus evaluable as either true or false) is what leads to paradox.
  • ep3265
    54
    I'm not quite sure what the := means. Is it just equals?
  • TheMadFool
    6.5k
    I'm not quite sure what the := means. Is it just equals?ep3265

    := means definition

    x := y means x is defined to be another name for y
  • TheMadFool
    6.5k
    Computers can handle self-reference as long as the self-reference eventually terminates.

    The Curry sentence is not well-defined due to the non-terminating self-reference. Treating it as if it were well-defined (and thus evaluable as either true or false) is what leads to paradox.
    Andrew M

    Kindly explain the difference between terminating and non-terminating self-reference in re to Curry's paradox.

    The Curry sentence is this P1 := P1 > P2. How is it not well-defined. There are no syntactical or semantic errors as far as I can see.
  • Andrew M
    1k
    Kindly explain the difference between terminating and non-terminating self-reference in re to Curry's paradox.TheMadFool

    What I mean by terminating self-reference is that there exists a method for evaluating the sentence in a finite number of steps. Consider the following sentence:

    This sentence has five words

    That sentence is self-referential and the self-reference terminates because a procedure can be defined that counts the number of words in the sentence. So the sentence is well-defined and has a truth value (true in this case).

    The Curry sentence is this P1 := P1 > P2. How is it not well-defined. There are no syntactical or semantic errors as far as I can see.TheMadFool

    First, for a more familiar example, consider the Liar sentence:

    This sentence is false

    That sentence has a non-terminating self-reference. To evaluate the full sentence, it is first necessary to evaluate the referent of "This sentence", which is "This sentence is false". So the sentence expands to:

    "This sentence is false" is false

    But now to evaluate the inner sentence, it is first necessary to evaluate the referent of "This sentence". And so the expansion of the sentence continues and doesn't complete in a finite number of steps. That is, the sentence is not well-defined and fails to be truth-apt.

    Now if a formal system allows this kind of sentence and assumes it is truth-apt, then the result is trivialism which is generally not what is wanted. So the formal system must be fixed in some way to avoid that result (such as by excluding that type of sentence, or by limiting explosion as with paraconsistent logics).

    The same situation occurs with the Curry sentence:

    If this sentence is true, then Germany borders China

    Similar to the Liar sentence, this expands to:

    If "If this sentence is true, then Germany borders China" is true, then Germany borders China

    Continuing in this vein, the expansion also fails to complete in a finite number of steps. That is, the sentence is similarly not well-defined and fails to be truth-apt. As with the Liar sentence, the formal system must be fixed in some way to avoid trivialism. That's easier said than done. Nonetheless, the non-terminating self-reference is the crux of the problem.
  • TheMadFool
    6.5k
    Continuing in this vein, the expansion also fails to complete in a finite number of steps.Andrew M

    But the Curry statement does terminate. It is self-referential but doesn't result in an infinite loop.
  • Andrew M
    1k
    But the Curry statement does terminate. It is self-referential but doesn't result in an infinite loop.TheMadFool

    It doesn't terminate. Here's the expansion:

    1. If this sentence is true, then Germany borders China

    expands to

    2. If 'If this sentence is true, then Germany borders China' is true, then Germany borders China

    expands to

    3. If "If 'If this sentence is true, then Germany borders China' is true, then Germany borders China" is true, then Germany borders China

    expands to

    4. ...

    and so on indefinitely.
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