Talk:Curry's paradox

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You obviously seem to disagree that Curry's paradox is indeed paradoxical. In modern mathematics, you are right: the strategy cannot be used to create a paradox since it cannot be formulated without using unrestricted comprehension. The article already states this clearly, so I removed your duplicate effort. However, there might still be something to say about the application of the paradox to the natural language (namely, we should mention the existence of an implicit subject upon which we are or are not mistaken), but I think your attempt was too confusing to be left on the real article. Please discuss your ideas here so we can get them polished before posting them.    — Gelisam

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[edit] Mistaken

The subject of the mistake changes as the passage progresses, from being mistaken about Santa Claus to (not) being mistaken about the validity of the statement about his own knowledge. Mathematically, I'm sure this would not arise (I hope this would not arise), but is there any way someone could clear up this fudge? It makes out the paradox to be semantic nonesense.

equivocation


Isn't the paradox resolved by stressing the difference between language and meta-language?--SurrealWarrior 00:58, 27 July 2005 (UTC)

You are correct. This is indeed the case. Its the same problem as the sentence "This statement is wrong". Mixing two different levels, one is the actual statement and one the language formualting it.
However, Wittgenstein pointed out that in Russell's type theory, the statement "all sets can only be members of sets of a higher type" is meaningless, since you can't talk of all sets in any language. Since that was exactly what Russell attempted to say, the problem persists... :-) Kronocide 20:28, 28 January 2006 (UTC)

Something should be done about this natural language explanation. It's totally nonsensical. This seems especially important since it's at the top of the page, and therefore likely to be read before the mathematical explanation. It could lead people to think that Curry's Paradox is just plain silly, and dismiss it before they get to the important bit.

> Abelard: "If I'm not mistaken, then Santa Claus exists."
> Eloise: "I agree: if you are not mistaken then Santa Claus exists."

Fair enough so far.

why is this fair enough? it's a meaningless statement; i don't know why eloise is even agreeing with it. --dan 18:46, 4 July 2006 (UTC)

> Abelard: "You agree: what I said was correct."
> Eloise: "Yes."

Again quite true.

> Abelard: "Then I am not mistaken."
> Eloise: "True."

Not mistaken in saying what, exactly? Abelard is not mistaken in saying "if I'm not mistaken, then Santa Claus exists". Fair enough. This does not in any way imply that the original statement is true. It does not prove whether he's mistaken or not in the original statement. It's just a play on words, a riddle of sorts. Merely a way of using an ambiguity in natural language to confuse people sufficiently that they'll believe something that's blatently illogical.

> Abelard: "If I am not mistaken, then Santa Claus exists. I am not mistaken. Therefore, Santa Claus exists."

Eloise: "Erm, no. Not at all. Just shut up would you Abelard? You're giving me a headache."

[edit] natural language explanation

I agree with the author above regarding the natural language explanation in that the argument as written does not read as a paradox at all, just a silly play on words. If there is a better way to write this up, I would enjoy reading it; as of right now I do not see a paradox.

I have to agree. I think the trouble is that "If I'm not mistaken" kills the self-reference, because
  • It is a common phrase meaning roughly "I think that"
  • Even if taken literally, "I" is a person and not a sentence.
-Dan 15:05, 20 December 2005 (UTC)
Hmm, it turns out that "If I'm not mistaken" was due to me, nearly three years ago. I repent. Let's change it. -Dan 16:58, 20 December 2005 (UTC)
Fixed. I also folded in Loeb's paradox, refactored the page somewhat, solved the halting problem, and fetched your slippers. -Dan 17:39, 23 December 2005 (UTC)

[edit] The wrong example to use?

I myself don't believe in Santa but does anyone feel a better example could be used than Santa? - Andrew Northall 08:47, 28 December 2005 (UTC)

[edit] In Islamic logic

Let us denote by ALLAH the proposition to prove, in this case "ALLAH exists". Then, let QURAN denote the statement in question, which asserts that ALLAH follows from the truth of itself. Mathematically, this can be written as

QURAN = (QURAN → ALLAH), and we see that QURAN is defined in terms of itself. The proof proceeds:

1. QURAN → QURAN

identity

2. QURAN → (QURAN → ALLAH)

substitute right side of 1, since QURAN = QURAN → ALLAH

3. QURAN → ALLAH

from 2 by contraction

4. QURAN

substitute 3, since QURAN = QURAN → ALLAH

5. ALLAH

from 4 and 3 by modus ponens

A particular case of this paradox is when ALLAH is in fact a contradiction. Then QURAN becomes QURAN = (QURAN → false), or equivalently (QURAN = ¬QURAN), which is exactly the liar paradox. Ohanian 10:00, 6 January 2006 (UTC)

Har har. Personally I don't see what's wrong with "Santa Claus". The point is of course that it doesn't matter what you put in there. -Dan 14:43, 6 January 2006 (UTC)


The point I'm trying to show is that if you are a non-muslim, you would have no problem up to step 3.

But you would not be able to accept step 4. Hence whatever that links step 3 to step 4 must be wrong. That linkage is:

QURAN = (QURAN → ALLAH)

Thus the statement above must be false.

ie. "QURAN is true is equvalent to if QURAN is true then ALLAH is true" must be false.Ohanian 06:57, 9 January 2006 (UTC)

If you reject that a sentence can say "If this sentence is true, then....", then the proof doesn't go through. Is this what you meant? In fact yes, most systems of mathematical logic do not admit (explicitly) self-referential sentences at all. Natural language, of course, has self-reference. I expanded the Discussion section to cover this, and ALLAH knows better. -Dan 14:37, 9 January 2006 (UTC)


[edit] Solution to Curry's Paradox

What do we mean when we say

A → B

We mean the following:

Suppose there is a statement S1 which is being declared as true

S1 ≡ A → B

thus

content_of(S1) returns "if A is true then B is true"

value_of(S1) returns true by virtue of declaring the statement S1 as true

value_of(A) returns unbinded

value_of(B) returns unbinded


Now using this we can look at the curry's paradox

A = A → B

which is equivalent to statement S2 which is being declared as true

S2 ≡ S2 → B

thus

content_of(S2) returns "if S2 is true then B is true"

value_of(S2) returns true by virtue of declaring statement S2 as true

value_of(B) returns true by modus ponens

Hence proving that the statement "A = A → B" is the same as the statement "B"

So if you are asserting "A = A → B" is true then you are asserting "B" is true.

Ohanian 02:14, 12 January 2006 (UTC)


We take A to be a self-referential sentence. If we reject self-reference, that's the end. But if we accept self-reference, then "A = A → B" is not something which needs to be asserted, it is in the structure of the sentence. For example we can write "If this sentence is true, then Santa Claus exists" as an infinitary sentence «If «if «if « ... », then Santa Claus exists», then Santa Claus exists», then Santa Claus exists».
(Using « » now to make grouping clear.) We don't need propositional variables at all:
  1. If «if « ... », then Santa Claus exists», then if « ... », then Santa Claus exists.
  2. If «if « ... », then Santa Claus exists», then if «if « ... », then Santa Claus exists», then Santa Claus exists.
  3. If «if « ... », then Santa Claus exists», then Santa Claus exists.
  4. If «...», then Santa Claus exists.
  5. Santa Claus exists.
Justification:
  • The rule of identity allows us to deduce, without any assumptions, any sentence of the form «if «blah», then blah» (and line 1 is of this form).
  • Line 2 is the exact same as line 1, only the second ... is written out one more level to make the next step clear.
  • The rule of contraction, which, assuming a sentence of the form «if «blah», then if «blah», then neener» (and line 2 is of this form), allows us to deduce the sentence «if «blah», then neener» (and line 3 is the correct result).
  • Line 4 is the exact same as line 3, only the ... is written out one less level to make the final step clear.
  • The rule of modus ponens, which, assuming two sentences, one of the form «if «blah», then neener» (and line 3 is of this form) and the other of the form «blah» (and line 4 is of this form) allows us to deduce the sentence «neener» (and line 5 is the correct result).
QED. -Dan 16:01, 12 January 2006 (UTC)

A natural language version of Curry's paradox might be:

If everything in this box is true, then Santa Claus exists.

which is equivalent to statement S1 and S2 which is being declared as true

S1 ≡ A → B

S2 ≡ A = A → B

thus

content_of(S1) returns "if A is true then B is true"

value_of(S1) returns true by virtue of declaring statement S1 as true

value_of(A) returns unbinded

value_of(B) returns unbinded

content_of(S2) returns "A is equivalent to if A is true then B is true"

value_of(S2) returns true by virtue of declaring statement S2 as true

Next we can manipulate the statements

S2 ≡ A = A → B

S2 ≡ A = (A → B ) → B by expanding the definition of A

S2 ≡ A = S1 → B

S2 ≡ A = true → B by virtue of S1 being true

S2 ≡ A = B which is binded to true

S2 ≡ A = true

value_of(A) returns true

value_of(B) returns true

Ohanian 22:02, 16 January 2006 (UTC)

I suspect we are talking across each other. Do you understand the point I am making, which is that A = A → B is not something which has to be assumed? I must admit I do not understand the point you are making. What are content_of() and value_of()? What exactly are you trying to show? -Dan 02:12, 17 January 2006 (UTC)
I am not assuming A = A → B merely that it is a statement S2 in a computative model being declared as a true or valid statement by which further manipulation is permitted. In this computative model, a variable X is either unbinded or binded to true or binded to false. The statement (container) S2 is a container which holds a statement. The value_of() function returns the current value of a variable. Ohanian 08:24, 17 January 2006 (UTC)
I'm not sure I understand this model. How did we get value_of(S1) = value_of(S2) = true? How did we go from "S2 ≡ A=B" to "S2 ≡ A=true (fourth last line)? And what is the point you are making? -Dan 14:44, 17 January 2006 (UTC)

[edit] Box on "In Natural Language"

I just reverted a change from February 5th, which claimed it improved the blockquoting. While it made things look better visually, it removed the 'box' mentioned in the argument, turning it into nonsense.

[edit] Sentence formatting

Stop changing the example sentence to monospace type. Monospace type is good for a few rather limited uses, such as including programming source code, or perhaps for referring to letter shapes -- but it's NOT good for example sentences. Futhermore, "enclosing in a box" is one of the stupidest reasons for using monospace type, since material within a <pre>...</pre> element is NOT enclosed in box on many Wikipedia "skins". I see no box there, because of the skin I have chosen in my user preferences. If you want to enclose it in a box, use a table with borders, but NOT monospace type!! AnonMoos 20:27, 8 February 2006 (UTC)

Hello, I was the one responsible for this in the first place. Thanks for pointing this out, and forgive my ignorance. I have no knowledge of Wikipedia skins. I have taken your suggestion. I hope what I have done is now "skin-invariant". -Dan 20:57, 8 February 2006 (UTC)
I did it the second time, and I also didn't know about monospace type not having a box on some skins. Sorry about that. On the bright side, Dan's new formatting looks great. -MauricioC 14:53, 10 February 2006 (UTC).

[edit] Hold on, I think I get it now

The slip says that if everything in the box is true, then Santa exists. If we grant that the slip is true, then it would follow that everything in the box is true, and that means that Santa Claus exists. But, if we suppose that it is false, then nothing is proven.70.25.138.179 04:06, 3 July 2006 (UTC)

We seem to agree that if we grant that everything in the box is true, then Santa Claus exists. Yes? Maybe it would be clearer if the slip said "If we grant that everything in the box is true, then Santa Claus exists." Thoughts? 192.75.48.150 17:09, 4 July 2006 (UTC)

[edit] Explanation in terms of redudancy

It could also be said that Curry's Paradox is redundant, as, in the box above, "Santa Claus is real" is a statement, so, in effect, Santa Claus is only real if Santa Claus is real.

Sounds like we are trying to substitute for "this sentence" to go from "if this sentence is true, then Santa Claus is real" to "if Santa Claus is real, then Santa Claus is real" yes? But "this sentence" was not "Santa Claus is real", it was "if this sentence is true, then Santa Claus is real". 72.137.20.109 17:33, 15 July 2006 (UTC)

I think that's a valid simplification of the paradox, within my current understanding. The most likely explanation is that my current understanding is flawed, but let's see if we can get away from the box.
Analyzing some arbitrary assertion, A:
The content of assertion A is simply that, assuming A, A is true.
An axiom of logic states that A implies A, for any arbitrary A.
Therefore, A follows directly from the axiom, since its only assertion is true axiomatically.
Am I missing something here, or does the presence of some statement B just confuse the issue? Isn't the real problem that we assume an absence of extra-logical content while doing formal logic, then impose it again once the proofs are done?--Joel 05:17, 22 October 2006 (UTC)

[edit] simplifying and removing self-reference

i think i have a simpler way of saying all this, instead of having one box/statement, or having nested statements, or all that malarky. with the one statement in the box thing, you basically have (i don't know the symbolic logic, so i have to write it out):

A = if A is true, then B is true
B = santa exists, or whatever else you want

but instead, you can expand it out, remove the self-reference, and make it less confusing for people like me:

A = C is true
B = santa exists, or whatever else you want
C = if A is true, then B is true

bam. --dan 04:03, 19 July 2006 (UTC)

Bam! This is like what is sometimes done for the Liar's paradox. Instead of "This sentence is false." have "The following sentence is false. The preceding sentence is true." --Different Dan 192.75.48.150 14:17, 24 July 2006 (UTC)

[edit] Not equality

A=A→B is not what the sentence says... that should be A→A→B. "this sentence is true if and only if santa clause exists" (which wold be A=A→B) is obviously false at first glance. —The preceding unsigned comment was added by Brilliand (talk • contribs) .

I think you're misunderstanding. The sentence says A→B, where A is the sentence. Put another way, A=A→B. Do you follow?
If B is the statement "Santa Claus exists", then A = A→B is a symbols-only way of saying A is the quasi-statement "if this sentence is true, then Santa Claus exists". The point is, A can't be false, because then it would be vacuously true; so A is true, so B must be true — i.e., Santa Claus exists.
Ruakh 16:31, 20 September 2006 (UTC)

[edit] thoughts

IIRC, starting with the assumption the premise is true and looking for inconsistancies is only one way to attempt a proof; you can start with the assumption the premise is false and see if that leads to error.

Starting with the assumption the slip lies (and simplifying a little), we get:

If this slip is true, santa exists

this slip is not true

therefore santa may or may not exist (we don't have proof of the converse)


which proves nothing. The only way it proves anything is if you start by assuming the slip is true, which means santa exists; that's part of your assumption. how is anything being deduced? It's like a statement saying "God exists." You only belive it if you want to.... Kuronue 18:02, 26 September 2006 (UTC)

Not quite. By definition, statement of the form "if A, then B" is true if A is false (in which case it's vacuously true) or if B is true (in which case it's trivially true). In the case of "If this slip is true, Santa exists," if we accept it as a valid statement, then either it's false or it's true. If it's false, then it's vacuously true, which is a contradiction; so, it must be true. If it's true, then it's not vacuously true, so it must be trivially true: Santa exists. (The solution to the paradox is to recognize that it's not a valid statement from which you can draw logical inferences; self-referential statements defy the rules of logic, and cannot be permitted.) Ruakh 18:27, 26 September 2006 (UTC)


If it's false than it's true.... oi. Logic hurts my head.I understand that if it's true it's not vacuously true because if B is true, the slip is correct, therefore, A is true. But if the entire statement is false, that means the contents of the box have nothing to do with santa, right? because you can't assume the converse, right? Kuronue 15:05, 27 September 2006 (UTC)
Assuming the box is true is not a matter of proof by contradiction here, or a matter of taste, it is a matter of "what if" or conditional proof (very lame stub page, sorry). Assume I am 60 years old, and that you are 80 years old. 60 is less than 80, therefore, I am younger than you. Now, you might object that I might not be 60, or that you might not be 80, so you don't accept the conclusion. But you do accept I have proven the statement "if I am 60, and you are 80, then I am younger than you", right? 192.75.48.150 18:04, 27 September 2006 (UTC)
No, you misunderstand. There is indeed a proof by contradiction here.
There's a limit to how many different ways I can explain it, but let me try a different way.
Consider the statement, "if {Jessica is a man ← protasis}, then {Santa Claus exists ← apodosis}." Jessica isn't a man, so the protasis is false, so the statement is true. We still don't know anything about Santa Claus, though.
Now consider the statement, "if {this statement is true ← protasis}, then {Santa Claus exists ← apodosis}." If this statement is false, then the protasis is false, so the statement is true; this is a contradiction. So, the statement is not false; it must therefore be true. If the statement is true, then its protasis is true, so by the (true) statement, Santa Claus exists, Q.E.D.
The solution to the paradox is that the statement is neither true nor false; it's simply not a valid statement. In general, self-referential statements (such as "this statement is false") are invalid, and modern logical systems don't include them.
Do you understand now?
Ruakh 18:21, 27 September 2006 (UTC)
I understand just fine. I was actually responding to her, not you. Incidentally, Curry's paradox is not dependent on boolean logic, or truth tables, or indeed any form negation at all! So it's not fundamentally a proof by contradiction. 192.75.48.150 18:37, 27 September 2006 (UTC)
I'm sorry, you're completely right; I misunderstood what you were trying to say. Your explanation below makes more sense. I framed it as a proof-by-contradiction because I think that's easier to follow, but you're completely right that the paradox doesn't rely on it. Ruakh 03:58, 3 October 2006 (UTC)

[edit] Removed explanation

There are three possible solutions to this paradox. The first would be that the statement is correct, and by it being the only “thing” in the box, the clause "Santa Claus exists" would be also true, leading to the conclusion that Santa Claus does in fact exist. The second solution is the statement is false and that Santa Claus does not exist. The third would be that statement is just as the second, false, and that Santa Claus does exist. If the statement is wrong, just as it is in the two possibilities above, the existence of Santa Claus is conjectural and cannot be proven.

I also agree that this is mistaken. It also looks like the misunderstanding comes from the truth table, which I'm going to remove. You might not (yet) agree that the statement is true, or that Santa Claus exists, but you seem to admit that IF the statement is true, THEN Santa Claus exists. Right?

Of course, that's what the statement said. So it's true after all. So Santa Claus exists.

I thought this was clear, but perhaps not. Maybe I'll do a bit of digging to find something clearer. 192.75.48.150 18:49, 2 October 2006 (UTC)

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