Self referencce paradoxes
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?
Why does a self reference lead to paradoxes so many times?
Comments (63)
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.
"Self, why does refering to you always lead to contradictions?"
Eeeeehhhh.... Because I'm a vat full of it? :smile:
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.
"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".
Alas, poor VincePee! I knew him, forum members, a fellow of infinite jest, of most excellent fancy.
I know him too!.Crazy motherfucker! Let me tell you.
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.
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.
Is it self-referential? Could you explain it to me, please?
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:
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.
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.
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.
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 ..."
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.
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.
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.
I wasn't critiquing your presentation.
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.
I stand corrected! Thanks again.
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.
Hmm. A conditional is just shorthand for a conjunction: P?Q ? ~(P & ~Q)
And what of Yalbo's paradox? No self-reference there.
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?
Not all paradoxes are self-referential is all I can say.
:up:
Self-referential paradox: A person saying, "I don't exist."
1. Reflexive (implicit) affirmation: To say "I don't exist", I must exist but this is unstated.
2. Reflexive (explicit) negation: I don't exist.
Yalbo's paradox is not reflexive.
And isn’t the op about self reference paradoxes in particular?
Quoting Banno
Well remove the self-reference (reflexivity) and check if the paradox still exists
1. That sentence is false (liar paradox).
2. If that sentence is true then, Germany borders China. (Curry's paradox).
Also, not all paradoxes are self-referential.
Yes, it is.
How do you infer that?
Ok, it should read "If sentence 2 is false, some sentence after 2 is true".
Suppose some Sn is true.
So Sn+1 is false.
So there is some k > n+1 such that Sk is true.
But Sn is true and k > n, so Sk is false.
So Sn is false.
So for all n, Sn is false.
But then for all n, Sn is true.
So for all n, Sn is false and Sn is true.
How is this a paradox? IF this it true then you must exist, Definitely that is true.
But if this is false then you do exist. This doesn't seem impossible to me, there is a definite conclusion that can be reached, "This statement is false."
It's not a paradox. It's a lie. Someone saying he doesn't exist denies his own existence. So, instead of a lie, it could be denial also.
It's self-refuting, it amounts to saying: I exist AND I don't exist, a classic contradiction!
No contradiction here. Just two different meanings of "to exist".
Equivocation! Interesting, explain please.
Equivocation. That's the word I looked for! Thanks. There are two different meanings for "exist". Which ones? I can exist while not existing at the same time.
Only agents can refer to anything. (reference is a form of thought)
Agents cannot refer to themselves. (A finger can only point away from itself)
Can you please explain your reasoning.
The only definite statement I can say is that he cannot be telling the truth but from that I can infer that his statement is false or it is misunderstood. If it is a misunderstanding then there is no point into discussing it any further, at least from this perspective. So the only worthwhile conclusion is that the person's statement is false, which would mean that he does exist. and I don't see any problem with that conclusion. If I am wrong then how? Please explain.
then:
Quoting Yohan
Yes, the person who utters the words, "I don't exist" is lying but the reason why you came to that conclusion is because it's self-refuting or self-contradictory. See reductio ad absurdum.
Not much of an explanation.
Agents cannot refer to themselves. — Yohan
— Banno
I can use my finger to point at my body, but I can't point the tip of my finger at itself.
You can point it at the other tips. Assuming all tips have a common core you can learn about the pointing tip.
I can use a mirror to indirectly observe what my face looks like. Indirect self knowledge is possible. My point is that some part of the reference-er cannot refer directly at itself.
What about that mirror thing? It's pretty freaky. Just remember to account for the inversion or you'll get all fucked up. The mirror image is said to be identical, because it has the same chirality, but's really not identical because it is a reflection.
I am not the man in the mirror? I am not looking at myself looking at myself?
When you use the mirror, doesn't the tip of your finger point at itself through the means of reflection? Can't a pointing be reflected?
A mirror image of my finger is not my finger, so if we are very technical, we can still say no.
But I guess we can say this is a way the finger can indirectly point at itself, or refer to itself. But this would be me assigning meaning to my finger as pointing, and then further assigning meaning to the image in the mirror as pointing.
You would have to ask me if I am intending to point at the mirror, the imagine in the mirror, or am using the mirror to point my finger at itself.
The part that refers cannot be referred to while referring. Indeed. But what's so important about tha? Making a prediction cannot be part of that what is to be predicted.
That's right. And this is an indication of the inherent ambiguity within "pointing". To clarify, and resolve the ambiguity, we need to ask, 'what are you pointing at, and the pointer must provide further context to ensure that the person being shown the thing interprets the pointing in the same way as the person pointing.
Quoting Gobuddygo
Why is this a problem for you? I see no problem, as long as we maintain as reality, that there is always a medium between the person doing the referring, and the thing referred to. Point your left finger at your right finger, and say "I am pointing at my finger". You can be pointing at either your right or your left, because it is you who is pointing, and your finger, as the means is just a medium. When I say "I" the part that refers, myself, is doing the referring. The medium, the word "I" is just the means which I choose, like when I choose my right finger to point at my left finger. The ambiguity is a feature of the choice of medium..
I think some of the apparent paradoxes are due to talking as if self-reference were happening.
Quoting Gobuddygo
Sounds right, but kind of unrelated?
Quoting Metaphysician Undercover
Not sure what you mean by further context exactly, but yeah I guess if I am trying to communicate something to someone I have to make sure we agree on meanings, or else the meaning will only be for myself.