• VincePee
    84
    Many paradoxes arise when a self reference is involved. In the realm of sets in particular. Sets containing itself are (in)famous.

    Why does a self reference lead to paradoxes so many times?
  • SolarWind
    127
    Ask yourself! :)
  • baker
    2.5k
    Why does a self reference lead to paradoxes so many times?VincePee

    Because there are actually two statements in place, but just one of them is put into words, while the other one is implied or otherwise needs to be discerned from the context of the first.
  • VincePee
    84
    Ask yourself! :)SolarWind

    "Self, why does refering to you always lead to contradictions?"

    Eeeeehhhh.... Because I'm a vat full of it? :smile:
  • Jack Cummins
    3.5k

    Perhaps the idea of self is problematic I some ways. It is is a concept which is at the juncture of phenomenology and the whole nature of the interpretation of experiences.I wonder if self and others is at the core of many of paradoxes, because paradoxes are about apparent contradictions. However, there may be underlying aspect of 'truth ' which can be seen as inherent in paradoxes. The idea of self and other minds may have some relevance with regard to ideas of self reference, but this would probably need to be backed up within the context of a specific understanding of philosophy.
  • TonesInDeepFreeze
    873
    Sets containing itself are (in)famous.VincePee

    "There is a set that is a member of itself" is not in and of itself contradictory.

    The famous contradictory statement is "There is a set of all sets that are not members of themselves".

    But that does not even require the notion of 'set' or 'member': For any 2-place relation R, we have the theorem of logic "It is not the case there is an x such that for all y, y bears R to x if and only if y does not bear R to y".
  • T Clark
    6.3k


    Alas, poor VincePee! I knew him, forum members, a fellow of infinite jest, of most excellent fancy.
  • MikeBlender
    31
    Alas, poor VincePee! I knew him, forum members, a fellow of infinite jest, of most excellent fancy.T Clark

    I know him too!.Crazy motherfucker! Let me tell you.
  • Hermeticus
    58
    On some very basic level, we define things through what they aren't. We need to compare X with not X to know what X is.

    If you were born with no senses, with no way to compare yourself to the outside world, you would have no concept of self.

    1 (or any number) in math is useless by itself. It needs the entire set to put it into context.
    1=1 is true but it says nothing about what 1 stands for at all. With 1+1=2 though, by comparing it with something else, we suddenly know what both of them stand for.

    Even the whole set of numbers only becomes useful when it's put into the framework of language.
    1+1=2 says nothing about our world. The statement by itself represents nothing other than the statement itself. But if I put one and one unit of apple together and I then have two apples, it gives a physical purpose to the formula as well as making apples countable.

    Our reality, or at the very least our perception, is a system of relations. Without this interdependence nothing really works or makes any sense.
  • TheMadFool
    11.9k
    Basically, self-referential paradoxes require two ingredients.

    1. Affirmation (usually implicit) of something that's essential, existentially. e.g. existence is implicit in I.

    2. Negation (invariably explicit) of that which has been affirmed in 1 e.g. I don't exist.

    The classic example is the liar paradox: This sentence is false.

    3. Implicit affirmation: The sentence is true.

    4. Explicit negation: The sentence is false.
  • Banno
    14.2k
    Curry's paradox does not require negation.
  • MikeBlender
    31
    "Math has a fatal flaw" Youtube video on intuitionists, formalists, Hilbert, Gödel, etc.
  • TheMadFool
    11.9k
    Curry's paradox does not require negation.Banno

    Is it self-referential? Could you explain it to me, please?
  • MikeBlender
    31
    Curry's paradox: If chicken curry doesn't contain curry it is still a curry chicken.
  • Banno
    14.2k
    A challenge. Probably not.

    Let P be the statement "If P is true then fish need bicycles".

    P is either true or false.

    If it is true, the antecedent is true - the antecedent just is P.

    But if it is false, then the antecedent is also false - the antecedent just is P.

    Take a look at the truth table for implication:

    KiJ8A.png

    The bottom two lines show the cases in which the antecedent is false. In both cases, the implication is true.

    But the implication just is the consequent.

    So if the antecedent is false, then it is true.

    There's a start.

    So... the last two lines lead to contradiction.

    And line 2 is also a contradiction; P is both true and false. SO it's out.

    All that is left is line 1.

    And if line 1 is true, then Q is true, and fish need bicycles.

    Or whatever other thing you might put in it's place.
  • ssu
    4.5k
    Why does a self reference lead to paradoxes so many times?VincePee

    Because you can make negative feedback loops, which math & logic have a problem with.

    Basically one gets the problem easily when thinking of algorithms and computers.

    Now you cannot program a computer, that basically follows algorithms with the command:

    "Do something else, not written on this program".

    The self-referential part is that the computer is following this program. Then it would have to do something not in the program. It cannot, because it's following algorithms. There are no instructions how to "do something else", because if there was, it thus then would be written in the program.
  • Olivier5
    2.9k
    Bateson wrote about how paradoxical injonctions can make people skizophrenic. The same apply IMO to self-contradicting philosophies: they can literally make you mad if you believe in them.
  • TheMadFool
    11.9k
    The truth table doesn't match what's going on in Curry's paradox.

    X = If this sentence is true, Japan is on the moon.

    Assume X is true; the antecedent is also true.

    Ergo, the antecedent is (assumed) true AND the conditional itself (X) is (assumed) true; Japan is on the moon follows.

    Self-reference but no negation and yet, a paradox.
  • Srap Tasmaner
    3k
    Let P be the statementBanno

    I've spent way too much time in old logic textbooks. Some of them introduce this sort of thing rather casually -- "You know about assigning variables from your math classes; here we do it with English sentences" -- while others devote entire chapters to attempting to formalize -- okay, how do I complete this sentence?

    Can I say, "formalize statements like the above"? Is it a statement? In what language? Is it the same language the rest of the proof is in? Or a meta-language? Which one is P part of? Is "Let P ..." part of the proof?

    Or do I have to say "formalize a sequence of symbols like the above"? Only if you head this way -- dreaming of a reduction of semantics to syntax -- you end up facing a rather unpleasant choice between saying that the logic we're building is just a bit of math, a rule-governed domain of symbol manipulation that has nothing to do with reasoning in a natural language, or saying with Montague that natural languages are in fact formal languages, that linguistics is in fact a branch of mathematics.

    My meta-logic -- unlike Curry's! -- is weak, so this is probably all bollocks. I'm just pointing out that the interesting part of what's going on here is certainly not in the perfectly routine application of standard inference rules, but in the line at the top we pass right over without thinking, that innocent little "Let P ..."
  • Banno
    14.2k
    You want a formal proof, there are plenty of examples online. I was explaining to @TheMadFool, who's not keen on formality.

    The odd consequences for logic are the very point of working on paradoxes.

    The point of bringing in CUrry's paradox was to counter Mad's suggestion that it was self-reference and negation that resulted in paradox. It isn't.

    Next comes Yablo's Paradox, which doesn't even use self-reference.

    Putting your finger on what it is that brings about the paradox is the fun bit.
  • Banno
    14.2k
    SO let's have a go at Yalbo's paradox.

    Consider the sequence:

    1. Every sentence after this one is false
    2. Every sentence after this one is false
    3. Every sentence after this one is false
    4. Every sentence after this one is false
    .
    .
    .

    If sentence 1 is true, then sentence 2 is false. If sentence 2 is false, sentence 3 is true.

    But if sentence 1 is true, sentence 3 is false.

    Alternately, if sentence 1 is false, then some sentence after 1 is true. Take that true sentence, wherever it is, and apply the same process as above. If any sentence in the sequence is true, then the next sentence to it is false. and the sentence after it, true.

    And we have a paradox without self-reference.

  • Banno
    14.2k
    ↪Banno The truth table doesn't match what's going on in Curry's paradox.TheMadFool

    One way to think about it is, if P just is P⊃Q, then the only line in the truth table that is not a contradiction is line 1; if line 1 is true, anything follows from P.
  • Srap Tasmaner
    3k
    You want a formal proof, there are plenty of examples online.Banno

    I wasn't critiquing your presentation.
  • TheMadFool
    11.9k
    ↪Banno The truth table doesn't match what's going on in Curry's paradox.
    — TheMadFool

    One way to think about it is, if P just is P⊃Q, then the only line in the truth table that is not a contradiction is line 1; if line 1 is true, anything follows from P.
    Banno

    Yes! Thanks for making me look up Curry's paradox. I had heard of it but never really took time out to study it until you brought it up. It seems Curry's paradox is connected to a lot many other ideas, especially in math.
  • TheMadFool
    11.9k
    The point of bringing in CUrry's paradox was to counter Mad's suggestion that it was self-reference and negation that resulted in paradox. It isn't.Banno

    I stand corrected! Thanks again.
  • Banno
    14.2k
    SO the next step is to work out what else they have in common...?
  • TheMadFool
    11.9k
    SO the next step is to work out what else they have in common...?Banno

    Oh! Well, if you want my opinion, I'd say that self-referential statements are prone to paradoxes because, it can be a part of itself (Curry's paradox is a case in point); reminds me of Russell's set that contains itself. When that happens, what's affirmed/denied of the part becomes affirmed/denied of the whole. That should lead to paradoxes, especially with conditional statements since by assuming the part we also assume the whole, the net effect of which is to generate a modus ponens syllogism.

    Too, there seems to be something odd about conditional statements because in one, if you notice, both the antecedent and the consequent are themselves statements and yet, we don't consider a conditional as a compound statement like we do a conjunction or a disjunction. I'm not sure how relevant this is though.
  • Banno
    14.2k
    e don't consider a conditional as a compound statement like we do a conjunction or a disjunction.TheMadFool

    Hmm. A conditional is just shorthand for a conjunction: P⊃Q ≡ ~(P & ~Q)

    And what of Yalbo's paradox? No self-reference there.
  • TheMadFool
    11.9k
    Hmm. A conditional is just shorthand for a conjunction: P⊃Q ≡ ~(P & ~Q)Banno

    IF this sentence is true THEN, Germany borders China.[from Wikipedia]. This conditional is logically equivalent to (p -> q = ~p v q):

    This sentence is false OR Germany borders China.

    Now, "this sentence" doesn't seem to refer to "IF this sentence is true THEN, Germany borders China."

    Moreover, "this sentence is false" is the liar paradox sentence. Hmmmmm.... :chin: See :point: Self-reference Paradoxes

    What's happening? Any ideas?
  • TheMadFool
    11.9k
    And what of Yalbo's paradox? No self-reference there.Banno

    Not all paradoxes are self-referential is all I can say.
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