Talk:Curry's paradox/Archive 1

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)[reply]

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)[reply]

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)[reply]

> 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."

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)

The logic, and/or the description, and/or the example is completely broken here.

The sentence doesn't say it's true.
There isn't a paradox. If this sentence is true, then Santa Claus exists. But since Santa doesn't exist, then the sentence is false. 82.21.228.223 02:26, 31 January 2007 (UTC)[reply]

You are misunderstanding something. As the article says: assume that the sentence is true. Then the hypothesis of the "if" statement is true, so the conclusion, that Santa Claus exists, is also true. This shows that, if we assume the sentence is true, then Santa Claus exists. But that's exactly what the sentence itself says - so the sentence is true. This is not directly because the sentence says its true - it's because we can prove that the sentence is true the same way we prove any other if/then statement to be true. CMummert · talk 02:34, 31 January 2007 (UTC)[reply]

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)[reply]

How about the FSM? —Preceding unsigned comment added by 67.158.31.66 (talk • contribs) 22:53, 27 October 2007 (UTC)[reply]

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)[reply]

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)[reply]

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)

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)[reply]

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:

If «if « ... », then Santa Claus exists», then if « ... », then Santa Claus exists.
If «if « ... », then Santa Claus exists», then if «if « ... », then Santa Claus exists», then Santa Claus exists.
If «if « ... », then Santa Claus exists», then Santa Claus exists.
If «...», then Santa Claus exists.
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)[reply]

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)[reply]

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)

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.

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)[reply]

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).

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Curry's paradox sentence is true if Santa Claus exists. X = (X → true) so X = true. If Santa Claus does not exist then Curry's paradox sentence is contradicting itself. X = (X → false). The (X → false) part there is common way to define negation using implication (X → false) = ¬X. So on case Santa Claus does not exist the sentence is equivalent to ... X = ¬X and that is called liar's paradox? 217.159.142.14 01:34, 29 January 2007 (UTC)[reply]

Begging the question

I apologize for changing the natural language description under the impression the author was begging the question. I missed the transition away from supposing the statement's correctness. Alas, I cannot think of a better way to phrase it to avoid such misunderstanding in the future. 198.205.33.93 19:52, 18 December 2006 (UTC)[reply]

Don't worry about it. It's very hard to grasp what the paradox is, and unsurprisingly, people have very often tried to "fix" the explanation. Well-meant contributions are always welcome here. :-) —Ruakh_TALK 22:38, 18 December 2006 (UTC)[reply]

Explanation?

Is what's really happening is that we are evaluating the validity of the implication (->)? As follows: An implication P->Q is true except in the case when P is true but Q is false. However, in such a case "this sentence would NOT be true", and so P is false. Thus, (P->Q) is true. If so, then we derive from the statement that Q is true. CDaMama 15:52, 10 February 2007 (UTC)[reply]

Yups. :-) —Ruakh_TALK 18:27, 10 February 2007 (UTC)[reply]

Recursion

If this statement is true, then Santa exists.

Lets write this as A→B, where

A is 'this statement is true', and B is 'Santa exists'

In order to investigate the accuracy of A→B, we first need to establish whether A is true or false.

A, remember is 'this statement is true'. What statement? The statement 'A→B'.

Oh, so in order to investigate the accuracy of A→B, we first need to establish the accuracy of A→B

Thats like saying 'you can't eat your pudding until you have eaten your pudding'

Deglog 17:53, 20 February 2007 (UTC)[reply]

I agree with the above point, and further think it provides a solution to the paradox. If we evaluate X = (X → Y) as a computer programmer would, not as a mathmetician would (that is to say, procedurally rather than holistically) we see that you cannot assign a result to X until you first evaluate (X → Y). To do that, you need to know the value of X. You cannot know the value of X until you evaluate (X → Y). So you are stuck in an infinite loop. 72.208.56.148 14:32, 28 July 2007 (UTC)[reply]

Extremely brief proof by reductio

Maybe worth adding?

B <-> (B -> A)

Assume ~B on the right side of the equivalence. Then the implication is true by its definition. Then B is true on the left side. Contradiction. Same goes if starting with ~B on the left side. If ~B on the left side then the implication is false, and then its antecedent is true by definition. Contradiction. Therefore, B must be true, and we have B, B -> A therefore A. Kronocide 18:59, 3 March 2007 (UTC)[reply]

Hmmm.

Now I am no genius by any means nor a mathmetician or logician. However I have a problem with this paradox, perhaps those much smarter than I can explain it to me.

It seems in an attempt to bring this paradox outside of semantics, the statement is simplified in order to avoid the troublesome task of defining the terms "Santa Claus" and/or "exists". The phrase is meant to be representative of any false statement, as any false statement could take its place very easily and the supposed paradox would remain.

The thing that is bugging me is if we simplify the entire statement into easily understood terms it would really be just like this:

True=False if True=True

Given that, it seems that this is no paradox, it is simply a blatent contradiction and falls into the realm of nonsense. Now as I said, I am no genius and I could be mistaken but if I am mistaken, I'm certainly not sure how...

12.18.155.206 21:04, 3 March 2007 (UTC)James[reply]

I don't see how the statement simplifies into those "terms". CMummert · talk 00:51, 4 March 2007 (UTC)[reply]

I'm sorry, I probably should have walked through my thought process a bit better.

The phrase "Santa Claus exists" is representative of any false statement which I changed into simply "False". This is necessary to do otherwise it is easy enough to say that Santa Claus does in fact exist and the paradox disappears completely into semantics.

An If/Then statement is always true because it does not give you a chance to make it false. That isn't one of your options. What could possibly be untrue about such a statement? Consider if Santa Claus did not exist (he does you know...) The statement still isn't false because it didn't say that he does exist, it made it conditional. Above, I just called it "True".

This might make more sense...

If True Phrase is True Phrase then True Phrase is False Phrase.

which is just as good as writing...

Blue is red.

It's not a paradox it's just nonsense. And as I said before, I'm no genius so it's very likely that I'm not thinking about this from the correct angle and I am completely open to being corrected.

131.191.87.67 01:43, 9 March 2007 (UTC)James[reply]

Eh? "If A then B" can certainly be false. Consider "If the sky is blue, grass is red"; this statement is false, because the sky is blue, but grass isn't red." —Ruakh_TALK 05:08, 9 March 2007 (UTC)[reply]

I stand corrected 131.191.87.67 07:53, 9 March 2007 (UTC)James[reply]

From false follows anything (even the false statement itself)

It is possible to deduce anything from a false statement. So the discussion (of ther verbal) paradox should discuss what happens if we assume that the statement to start with is wrong. if then still the existence of Santa follows then there is a huge paradox (and maybe Santa exists). Otherwise it is as much a paradox as 0=1 , assume this to be true --> 1=2 --> 0=1 SEE, it is true whow! Andreask 04:56, 5 June 2007 (UTC)[reply]

Re: "So the discussion (of ther verbal) paradox should discuss what happens if we assume that the statement to start with is wrong.": There's no need. A statement of the form A → B is necessarily true if A is false, so to test it, we only need to examine the case where A is true. (See Wason selection task.) —Ruakh_TALK 15:12, 5 June 2007 (UTC)[reply]

And if I may add ...

if the Moon were made of cheese there'd be little Green Men on it by now.

A + B → C is how I would want the above statement to be inferred and conveyed. What context would the above statement be appropriate? Answer: a ridiculous context.

Not to be facetious, rather to introduce another dimension, forum or platform to place these "paradoxical" statements and others, such as barbers who shave others or not.

Zero (0) and Unity (1) denote nonexistence and existence of a statement, in the sense of applicability within the world and the times one lives within. All paradoxical statements are rendered Zero → null, in effect an absurdity. No sense speculating on how many angels can fit on the head of a pin, fella's. Not to imply one should dismiss paradoxes, though to convey a sense of "irrelevance."

From Russell's paradox - Wikipedia

6 Applications and related topics

The Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick.

A statement rendered Unity → 1, 2, 3 ... ∞, is immediately applicable in context to the world one finds themselves living in. Valid for no other reason than for perceived, practical utility. Perhaps in application for designs of IC logic gates.

These thoughts here need further clarification. Paradox is an inappropriate word to label these type of statements with. Paradoxical statements are not recognized or identified to the level of absurdity that they ought to be, I suppose is what prompts what I write now.

Mergatroidal 01:25, 26 September 2007 (UTC)[reply]

I think you may have it wrong. "If the Moon were made of cheese there'd be little Green Men on it by now." is not a Curry paradox because it has no self-reference. You need to say, "If this statement is true, then the Moon is made of cheese." ←Ben^B4 08:01, 26 September 2007 (UTC)[reply]

This'll make your head explode

if this sentrence is true, then it is false. —Preceding unsigned comment added by 67.158.31.66 (talk) 22:51, 27 October 2007 (UTC)[reply]

Yes - I noted the statement Furthermore we could substitute any claim at all for "Santa Claus exists" in the article, and immediately came up with If this sentence is true, then this sentence is false. Indeed, it is unpleasantly head-bending! --Kay Dekker (talk) 16:03, 23 April 2010 (UTC)[reply]

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 03:54, 10 November 2007 (UTC)[reply]

The Mathematical section is incorrect (non-paradoxical)

Step 2 says:

2. X → (X → Y)

   substitute right side of 1, since X = X → Y

for (X,Y) truth pairs,

X is true if and only if we are in the set ((1,0),(1,1)). X → Y is true if and only if we are in the set ((0,0),(0,1),(1,1)).

These are far from equivalent. —Preceding unsigned comment added by 207.171.191.60 (talk) 06:32, 15 December 2007 (UTC)[reply]

It is right. X is defined as X → Y (see the lead paragraph of that section), and so in your notation we see easily the only solution is (1,1). But the argument does not rely on these truth tables or on any sort of principle that every statement is either true (1) or not true (0). --192.75.48.150 (talk) 19:30, 17 December 2007 (UTC)[reply]

There is no paradox.

The sentence "If this sentence is true, then Santa Claus exists." is false. No paradox:

The argument for this paradox is fallacious. If a sentence gives results according to reality doesn't mean it's not false. "If cats are mammals, then China exists." gives correct results, even though it's obviously false. Let's say we have a glass full of water and this sentence: "If this glass is full of water, then Santa Claus exists." The sentence is false. We'll empty the glass, so we have an empty glass and still the same sentence. "If this glass is full of water, then Santa Claus exists." This sentence is still false, because when we'll add water again, we'll not create Santa Claus. --88.101.76.122 (talk) 12:02, 6 April 2008 (UTC)[reply]

You are mistaken, at least when using the formal mathematical definition of "if". "If A, then B" is always true if A is false. "If this glass is full of water, then Santa Claus exists" is a true statement if the glass is empty. If you fill it up, then it becoms a false statement. --Ashenai (talk) 13:51, 7 April 2008 (UTC)[reply]

I think 88.101 brings up a good point you're missing. Written another completely equivalent way, his statement is "cats are mammals therefore China exists" or even "china existing is a consequence of cats being mammals." Is the statement true? If you followed the system of logic down to the letter, then yes. If asked in conversation, my gut reaction would probably be "no, preposterous." Here, there seems to be a divide between intuition and logic. --Traversc (talk) 01:31, 29 June 2008 (UTC)[reply]

I agree, that would be a fallacy of relevance. But it seems to me that there is no step of the form "If cats are mammals, then China exists." Certainly there is none in the formal derivation. --EmbraceParadox (talk) 02:50, 29 June 2008 (UTC)[reply]

The "step" lies in the assumption that vacuously true statements ARE true. 88.101's argument is this: if P and Q have no relevance to one another, the veracity of P shouldn't imply anything about Q. Let's say P = cats are mammals and Q = China exists. P->Q reads "China existing is a consequence of cats being mammals" which isn't clear (to me) that it is true. So if we say that P->Q can be false, if A and B are true, that doesn't imply A->B is true, which means A=A->B can be false. --67.49.155.91 (talk) 06:18, 29 June 2008 (UTC)[reply]

Yes, I know. I rewrote the article somewhat since the time that 88.101 said so. The fact is, Curry's paradox can be derived without truth tables. Statements such as "China's existence follows from cats being mammals" are sometimes considered a type of fallacy of relevance. Relevance logic is designed specifically to exclude fallacious conclusions like those. But, Curry's paradox can still occur, if self-referential sentences are allowed. The formal mathematical proof definitely does not contain any such fallacies. But perhaps the intuitive proof could be clearer. For instance A->A is NEVER a fallacy of relevance: "If the Curry sentence is true, then the Curry sentence is true" - seems we can't argue with that! But the Curry sentence says that if the Curry sentence is true, then Santa Claus exists. So we go from "If the Curry sentence is true, then the Curry sentence is true" to "If the Curry sentence is true, then if the Curry sentence is true, then Santa Claus exists." And this is not a fallacy of relevance either. And so on. --EmbraceParadox (talk) 01:52, 30 June 2008 (UTC)[reply]

That's not correct, since A = A->B where B is NOT A. There's no getting around the fact that A contains a proposition. --Traversc (talk) 09:02, 1 July 2008 (UTC)[reply]

I'm not claiming A=A->A, I'm just claiming A->A is true! I mean, isn't it? If anything is a relevant implication, then A->A is - no matter what the truth of A. Conceivably there is a way to deny A->A (I don't know what that might be), but it's not going to be on grounds that A is irrelevant to A.

And then, of course, since A->A is true, and A=A->B, then A->(A->B) should also be true. And so on, just like in the article. Cheers, --EmbraceParadox (talk) 14:44, 1 July 2008 (UTC)[reply]

But the statement '(A=A)->B' is still a *statement*. It can be verified to be *not false*, but it can't be considered *true*, surely. The flaw in the paradox is to assume that 'not proven to be false' is the same as 'proven to be true'. 'Logical implication' has always bothered me this way. --Natecull (talk) 01:00, 18 November 2008 (UTC)[reply]

Recent Revisions on Natural Language Explanation

The recent revisions to the natural language explanation, which have totally eliminated the older version of the section, seem nonsensical. Regardless of whether it is a more accurate version, though, the author needs to explain why he/she made the change in the first place. 75.182.82.35 (talk) 01:28, 5 May 2008 (UTC)[reply]

I also prefer the version that uses modus ponens, rather than the current version using negation. It isn't necessary to have any sort of negation operator in a language for the paradox to hold, so it's really not the best way to explain the paradox. — Carl (CBM · talk) 01:35, 5 May 2008 (UTC)[reply]

The comment in the edit performed 20:19, 3 June 2008 claims that "If X then Y" does not mean "Y, or not X" in natural language. I must respectfully disagree. Run the truth tables and compare. —Preceding unsigned comment added by 163.231.6.66 (talk)

That's formal logic, not natural language. --99.245.206.188 (talk) 03:01, 15 February 2009 (UTC)[reply]

Definitions of 'if/then'

This paradox makes no sense to me. It suggests that there is something wrong with the way predicate logic is usually formulated, and in particular, how it treats the concept of 'if/then' compared to the natural language (or even programming language) idea of the term.

Claims of the form "if A, then B" are called conditional claims. It is not necessary to believe the conclusion (B) to accept the conditional claim (if A, then B) as true.

That right there is a problem, to me. A claim of the form 'if A, then B' in natural language is still *a claim about B*. It need not be accepted as true unless it's true; it's an assertion, on exactly the same order as just saying 'B'.

Why does logic treat a claim like this as something different, a kind of meta-statement which can be proved within the system? --Natecull (talk) 01:22, 18 November 2008 (UTC)[reply]

Consider the sentence

If I go to the mall, then I will buy shoes (*)

This sentence can be true even if I don't know whether I am going to the mall. But the sentence (*) does not imply that the statement "I will buy shoes" is true, only that if I go to the mall then I will buy shoes. It may be that, since I don't have much money, I don't really expect to go to the mall, but as long as I am committed to buying shoes when I do go then the sentence (*) is still true. This analysis is entirely in natural language. — Carl (CBM · talk) 04:10, 18 November 2008 (UTC)[reply]

The current santa example also makes no sense. A flying man delivering presents does not lead logically to any statement about Santa - the statements are independant. Are there any natural language examples that actually have this paradox in (in which the first stement is actually logical?)YobMod 11:55, 18 March 2009 (UTC)[reply]

The sentence with the word "flying" in it is not the example of the paradox. the paradoxical sentence is: "If this sentence is true, then Santa Claus exists." Every natural language sentence of the form "If this sentence is true, then (...)" is provable, regardless whether the phrase (...) is true or false. That's the paradox. — Carl (CBM · talk) 12:08, 18 March 2009 (UTC)[reply]

In naive set theory

The version I've always seen defines X as

X\ {\stackrel {\mathrm {def} }{=}}\ \left\{x\mid (x\in x)\to Y\right\}.

(notice that it's a lowercase x on the right-hand side.) Is there some reason that capital X appears on both sides? This makes it a self-recursive definition, which IMO defeats the point of the section. -- Walt Pohl (talk) 16:03, 18 November 2009 (UTC)[reply]