Talk:Curry's paradox

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Natural language Section contradictory

• As before, imagine that the antecedent is true — in this case, "this sentence is true".

ok so now we are counting the sentence as true

• Does Santa Claus exist, in that case?

the consequent must be true

• Well, if the sentence is true, then what it says is true: namely that if the sentence is true, then Santa Claus exists.

correct

• Therefore, without necessarily believing that Santa Claus exists, or that the sentence is true, it seems we should agree that if the sentence is true, then Santa Claus exists.

yes we can assert that if the antecedent is true, then the consequent must be true

• But then this means the sentence is true.

what does? the whole argument just broke down. To say that we assert the validity of a conditional is not the same as saying that a conditional is true. We have agreed that if A then B but simply saying that we agree to the validity of conditionals in general does not mean that we agree that this conditional is true. In other words, a conditional is not true because it is a conditional.

• So Santa Claus does exist.

well, the argument has fallen apart so this conclusion is in question —Preceding unsigned comment added by Sdoherty777 (talk • contribs) 02:27, 4 January 2010 (UTC)

Because we can obtain the proof of the consequent by assuming the truth of the antecedant, this means that we can prove the overall conditional statement. In turn, that means that the overall condition statement is true.

It may be easier to look at it in symbolic form. Suppose that the sentence "A" says "If A is true then B is true". To prove "A" you would assume the antecendent ("A is true") and prove the consequent "B is true". But assuming "A" is true means you are also assuming "If A is true then B is true", so you can use just modus ponens to obtain the consequent. This is because assuming the antecedent of the particular self-referential statement "A" causes you to also assume the truth of "A" itself, because of the self-reference. — Carl (CBM · talk) 03:09, 4 January 2010 (UTC)

Could someone please explain to me how this is significant? Reading from above, to prove "A" you assume the antecedent true, which because it's self-referntial leads to "B" being true. Contrastingly, to disprove "A" you would assume the antecedent false, which due to self-reference leads to "B" being either true or untrue. I don't understand how it can be considered paradoxical because the proof can be applied to any situation, where the disproof could as well. -Greg —Preceding unsigned comment added by 124.198.166.97 (talk) 11:12, 8 November 2010 (UTC)

Once we have established the paradox, it doesn't matter if we can do other things as well. For example, we can say that if x = 3 then 2x = 6. The fact that we could also study the case when x is not 3 doesn't affect the validity of the original conclusion that if x = 3 then 2x = 6. (Also, you cannot disprove a conditional statement by assuming the hypothesis is false, but that is another matter.) — Carl (CBM · talk) 12:51, 8 November 2010 (UTC)

Carl: I think you need to explain first why you cannot prove a conditional by assuming the hypothesis is false, otherwise the natural language example is not doing its job. --Gak (talk) 15:36, 5 August 2011 (UTC)

Logic Table Analysis

Take 2:

If "this statement" called (A) is true, then "another statement" called (B) is also true.

We write this as:

A ↔ (A → B)

Typically you do not put implies or if and only if within a logic table. The reason being these operations need to be true for all legal values, not simply for the current values being considered. However, if we have two arbitrary statements, A and B, that (A → B) is only true, if and only if ((A and B) or (not B)) is true for all legal sets of A and B values. Listing ((A and B) or (not B)) will allow examination of (A → B).

Likewise, for two arbitrary statements A and B, ( A ↔ B ) is only true if A=B for all legal sets of A and B values. One can also list A = ((A and B) or (not B)) in a table to examine (A ↔ B).

A comprehensive logic table with no constraints on A and B would look like this:

A	B	(A and B)	(A and B) or (not A)	A = (A and B) or (not A)
T	T	T	T	T
T	F	F	F	F
F	T	F	T	F
F	F	F	T	F

Now what follows next depends on how we wish to classify our original statement. If we classify it as the definition of A, then all values in our table must conform to that definition. Any rows where A = (A and B) or (not A) is false needs to be excluded to conform to our definition of A. After removing the excluded rows, our logic table then simplifies to:

A	B	(A and B)	(A and B) or (not A)	A = (A and B) or (not A)
T	T	T	T	T

Thus, we have proven that given our definition of A, then B MUST be true. So hurray, Santa does exist!

Stop your celebrating. This result is simply a result of accepting a bad definition. Instead, consider A as statement that needs to be tested. Since we have no a priori constraints, we cannot drop out values from the table. Instead, we note that A = (A and B) or (not A) is not true for all A and B values. Consequently, we must conclude (A ↔ ( A → B)) is a false statement.

I see no paradox here. Only a warning to be careful about what definitions you accept.Bill C. Riemers, PhD. (talk) 22:08, 4 March 2010 (UTC)

The paradoxical situation is that "If this sentence is true, then B" is a perfectly good English sentence, and if it is examined with the usual meanings of English it turns out that not only can we prove that the sentence is true, we can use the truth of the sentence to prove B is true.

That is not paradox, that is just pointing out English is a imprecise language to use for mathematical proofs.Bill C. Riemers, PhD. (talk) 23:07, 4 March 2010 (UTC)

So you are right that the propositional formula A ↔ (A → B) is not in general true for all truth values of A and B. But if C is the particular English sentence "If this sentence is true then B" then C ↔ (C → B) is true, by the definition of C and the usual semantics for "if/then" in mathematical English. — Carl (CBM · talk) 22:25, 4 March 2010 (UTC)

Exactly my point. It is the acceptance of a sloppy definition that makes it true. If we only accept it as a statement to be tested, we do not have a problem. A more careful wording of our result would be, "Given our definition of C, B must be true." What is needed to avoid such problems is to recognize any definition that places constraints on other items in the system is really an axiom or a postulate. Bill C. Riemers, PhD. (talk) 23:16, 4 March 2010 (UTC)

You're missing the point; C is a perfectly reasonable English sentence, not a definition. It exists because of the way English grammar is already defined. Moreover, the logical analysis of the sentence is carried out using exactly the sort of logical deduction that is commonly employed. Unlike "This sentence is false", which has no determinate truth value, the sentence C does have a truth value: it is true, and that's the problem. For philosophers and for people who study the semantics of natural languages, it's a serious issue if those semantics appear to be inconsistent. You asked why people care about the paradox, and that's the reason. — Carl (CBM · talk) 02:22, 5 March 2010 (UTC)

If C is not a definition, then how can a step of your proof be "by the definition of C"? The proof only works if you assert you can apply the statement because it is required as part of the definition of C. Even the set theory version of the problem is the same way. If you take away the "by definition step" you have no proof. BTW. In the correct way to address the set theory version of the paradox is the same way. Instead of taking X = ... as a definition of X, just take it as a statement and try and determine if set X exists. Only the set theory version is a bit more complicated. However, I the existence of X can only be taken as a postulate except in specialized cases of Y. In essence every definition is also a postulate or an axiom.Bill C. Riemers, PhD. (talk) 04:06, 5 March 2010 (UTC)

The difference between a definition and a statement, is a definition a defining statement assumed to be true, without proof. A definition may be applied within the steps of a proof, without first proving the definition. A statement is something that needs to proven or disproven, before it can be used in the steps of a proof.

You don't need to be self referencing to prove non-sense with a paradox. Consider the fun you could have with the following definitions:

Santa is a guy with flying reindeer who really exists.

X is a number between 1 and 2 that is greater than 3.

If a definition is flawed, the conclusions are flawed. Bill C. Riemers, PhD. (talk) 04:23, 5 March 2010 (UTC)

And here comes the resident Ph.D.-in-unrelated-field, who in his sheer genius, has managed to fart around on wikipedia fore 30 minutes, and disprove a paradox that was developed by one of the greatest minds in logic and has confounded the entire field for going on a century. What's more, he's so brilliant he could do it from a mere summary, rather than from reading any of the actual papers. Give this man a Nobel! You have not managed to square the circle, my friend. You are just reinventing old errors.24.179.56.142 (talk) 04:33, 15 May 2011 (UTC)

Burrrn! —Keenan Pepper 17:28, 26 May 2011 (UTC)

Edit 2010-11-4

Cleared up the ambiguity on the natural language section. The previous description offered no explanation as to why it results in 'Santa Claus' existing or elaborating on how it is a paradox. — Preceding unsigned comment added by 124.198.175.237 (talk • contribs) 08:19, 4 November 2010

I undid the edit because several of its claims are incorrect. The sentence is not just "surprising"; it's a genuine paradox. And it proves that Santa Claus actually exists, not just that he exists hypothetically. — Carl (CBM · talk) 10:58, 4 November 2010 (UTC)

I rewrote that section, and changed the example to something more temperate. The old section was overgrown with digressions and, you're right, did not explain how the paradox is derived. — Carl (CBM · talk) 11:36, 4 November 2010 (UTC)

This appears to be true only if you aim to prove it to be true. If you aim to prove it true you assume the hypothesis to be true, which results in the conclusion being true. If you aim to prove it false, you assume the hypothesis is false, which results in the conclusion being false. How is this significant? —Preceding unsigned comment added by 124.198.180.140 (talk) 03:20, 6 November 2010 (UTC)

You do not prove a conditional statement false by assuming that the hypothesis is false. However, it doesn't really matter how you would prove it false; the paradox alrady occurs just from thinking about how we would prove the conditional statement is true. — Carl (CBM · talk) 12:10, 6 November 2010 (UTC)

please delete or

please delete or provide a reason why this is notable enough for inclusion and rewrite it in a less in-universe way. because this is actually not a paradox.

"if this sentence is true, Germany borders China" is simply true. if the sentence is true, than Germany borders china, but if it is not, Germany doesn't border China. of course, it is not true, but the sentence doesn't claim to be true. so this is not a paradox. the truth of the first part is not a condition of the sentence itself.

if i say "if 4 is a symbol meaning "five", than we have 4 fingers on our hands" is also simply true. the sentence is true, but we still do not have 4 fingers on our hands. the sentence doesnt claim we have. just like the previous sentence does not claim Germany to border China. no paradox · Lygophile has spoken 01:56, 8 February 2011 (UTC). ps. for fun, lets consider the reverse: "if Germany borders China, this sentence is true" :/

The paradox is that (1) we can prove, using only accepted techniques, that "if this sentence is true, Germany borders China" is a true sentence. The sentence doesn't directly claim to be true, but as the article explain we can prove that it is true. And (2) from the truth of that sentence, we can prove "Germany borders China". This is a paradox because we would not expect widely accepted techniques to lead to a contradiction. — Carl (CBM · talk) 02:11, 8 February 2011 (UTC)

then how come "if Poland was called China, than Germany borders China" isn't a paradox as well? or how about "if Germany borders China, than China borders Germany"? why are they not paradoxes just as much?· Lygophile has spoken 00:19, 10 February 2011 (UTC)

It's not clear from what you're saying that you've read the article carefully. The article explains in detail how the self-referential nature of the sentence leads to the paradox. The point is that even if we assume that "if Poland was called China, than Germany borders China" is true, it does not tell us Germany borders China. But we can prove that "If this sentence is true, then Germany borders China" is true, and the truth of that sentence then allows us to prove that Germany borders China. — Carl (CBM · talk) 01:06, 10 February 2011 (UTC)

i have read it, and it does not make clear why the same thing would not aply to my sentences. Germany borders Poland, so if we assume that Poland is called China, then Germany borders China. except it doesn't. i don't see the difference.· Lygophile has spoken 01:57, 10 February 2011 (UTC)

If Poland is called China, then Germany borders China. Poland is not called China, then Germany doesn't necessarily borders China. "If this sentence is true, then Germany borders China" is true (which is proved in the article), and because it refers itself, it also means that Germany actually does border China. There's the difference.79.182.215.95 (talk) 02:07, 19 July 2012 (UTC)

Where I find the problem is at the fact that Germany bordering China has nothing to do with that sentence, despite what the sentence claims. "If this sentence is true, then Germany borders China". What? No! Germany borders China if and only if they are geographically located next to each other; the conclusion does not follow from the premise and the IF .. THEN implication suggested is clearly false. A more accurate sentence would have been "regardless of whether this sentence is true or false, Germany does not border China". I don't see a paradox because the whole sentence seems to me false, therefore the conclusion does not even apply. However, seeing how so many people appear confused by such a silly past-time, I guess an article clarifying how this is not a paradox would not be overdue. 62.37.5.52 (talk) 09:35, 1 September 2012 (UTC)

It must be hard for you. You know, being braindead. — Preceding unsigned comment added by 85.242.102.172 (talk) 08:50, 14 February 2013 (UTC)

Now that's just silly. A braindead person cannot type, and therefore your assertion is clearly false. Since you don't provide any other arguments, your contribution becomes meaningless. 80.103.84.224 (talk) 12:04, 23 March 2013 (UTC)

"traduction"

2nd sentence:

It is a traduction in minimal logic of Russel's paradox (naive set theory), or Gödel sentence (proof theory).

Is 'traduction' a word? I can't find any reference to it in English, but it's the French word for 'translation' apparently -- is this an error? Or is it meant to read:

It is a translation in minimal logic of Russel's paradox ...

BrideOfKripkenstein (talk) 06:27, 16 March 2011 (UTC)

I also found that sentence very unclear, even if traduction is replaced by translation. I just removed the sentence. — Carl (CBM · talk) 10:51, 16 March 2011 (UTC)

In Romanic languages it is used. Probably someone made a wrong "traduction". Anyway, I must agree with @CBM it is a confusing sentence that should be deleted. In fact, the whole article is confusing and for a logic-related article, there's nothing logical about it (see "This is not a paradox" below). — Preceding unsigned comment added by JMCF125 (talk • contribs) 14:18, 12 March 2013 (UTC)

That awkward moment when...

Far too many Wikipedia users genuinely think this is a paradox. It is not a paraodx, delete the article. 203.206.11.64 (talk) 13:20, 28 July 2011 (UTC) Harlequin

The paradox only exists within a naive calculus of logic. Putting it into natural language immediately demonstrates why the calculus is flawed, so the example is good. However, hiding the explanation in formal rhetoric does not help the reader to understand the flaw in the calculus. I have therefore rewritten the explanation in natural language, which is what the section heading claims to do. Hopefully this answers Harlequin's point, which is validly made. --Gak (talk) 15:34, 5 August 2011 (UTC)

Um, what's the antecedent of "this"?

This article is unnecessarily hard to read because of ambiguous antecedent choices for "this" in the example sentence. It seems like the first of these two readings is the one that's meant:

1. If this conditional statement is true, then Germany borders China.
2. If the antecedent of this conditional statement is true, then Germany borders China. — Preceding unsigned comment added by 208.26.194.131 (talk) 00:54, 28 October 2011 (UTC)

If B is true, then B must be true

Can't this be more simply represented by

If B is true, then B must be true

Since the "this" in "If this sentence is true, then B is true" means B is true? Or putting another way "Assume what i say next is true, B is true". --TiagoTiago (talk) 11:09, 1 November 2011 (UTC)

But B has a different value in both instances; "this" cannot simultaneously be what you are saying now and what you will say next. Let's see both sentences:

If this sentence is true, then [it must be true that] Germany borders China.

If B₁ is true, then it must be true that B₂ (rearranged from yours).

B₁ = "this sentence"

B₂ = "Germany borders China"

B₁ ≠ B₂

B₂ does not follow from B₁ because B₁ has nothing to do with B₂, they are unrelated concepts. Let us not forget that there is a verb "to border", included in B₂. What does it mean? It means that the subject is adjacent to the object; if X borders Y, it implies that X is located next to Y. Now trace the shortest line you can find between Germany and China. You will find that it goes through Poland, bits of Belarus, Ukraine, Russia, Kazakhstan, a bit of Uzbekistan and Kyrgyzstan before reaching China. Clearly, then, they are not adjacent.

Note how the meaning implied by the verb "to border" does not make any reference to anyone's sentences being true or false; it is an unrelated concept. Therefore, the link attempted to be established between the sentence being true and Germany bordering China is not a true link. It is not true that Germany borders China if this or that sentence is true; Germany borders China if they are both adjacent. Since the link is not true, the whole sentence is not true; B₁ is not true. And since B₁ being true is the premise on which the consequence relies, the consequence does not even apply. Therefore, we are not dealing with a paradox here; simply with a false statement.

But people can call it whatever they like, as in so many other cases. Don't people call chemical units "atoms", despite the tremendously obvious fact that they are not indivisible, and therefore not atoms? 80.103.84.224 (talk) 12:59, 23 March 2013 (UTC)

Please provide the standard solution of the paradox

The standard solution is: do not mix language with metalanguage 123unoduetre (talk) 21:06, 18 July 2012 (UTC)

Actually, this is not a solution... this paradox can be demonstrated by formal logical language too... 79.182.215.95 (talk) 03:03, 19 July 2012 (UTC)

If it can be demonstrated by formal logic, can you show it and contribute to the article? The current "logic" is:

1. X → X

rule of assumption, also called restatement of premise or of hypothesis

2. X → (X → Y)

substitute right side of 1, since X is equivalent to X → Y by assumption

3. X → Y

from 2 by contraction

4. X

substitute 3, since X = X → Y

5. Y

from 4 and 3 by modus ponens

What happens is that step 2 can't be right, because the assumption isn't. A wrong assumption may (and in this case, does) generate false results. X is false, therefore ¬X → ¬X and the rest is flawed. JMCF125 (talk) 21:43, 23 April 2013 (UTC)

This is not a paradox

This reminds me of the ontological argument of St. Anselm. Of course that if A implies B, and the whole thing is true, then A is true, and therefore B is true. But if A is not true, nothing is said. It can also be proven by contraposition that this is false: "If this sentence is true, then Germany borders China" is equivalent to "If Germany doesn't border China, then this sentence isn't true.". As such, I do not ask to delete this article but to include it in a list of false paradoxes.

All the explanations made in here suppose the whole thing or "this sentence is true" is true, and then become amazed by the fact it is true. There is the similarity with the ontological argument, it supposes A without actually proving it. JMCF125 (talk) 14:03, 12 March 2013 (UTC)

The argument does not have any extra assumption in it, which is why it is a paradox. The argument proves that "if this sentence is true, then Santa Claus exists" is actually true, by proving that if the hypothesis is true, then Santa Claus exists. The ability to prove this is the source of the paradox. — Carl (CBM · talk) 00:37, 16 March 2013 (UTC)

But to prove that the sentence is true you have to prove the sentence is true! If the if clause is true, then the sentence is true, but what if it isn't true? If it isn't true it isn't true. There is an assumption made that the whole thing is true; "if" means "assuming that". There is no paradox nor proving here, just strange logic. JMCF125 (talk) 19:12, 23 April 2013 (UTC)

JMCF, I understand what you're saying. There is a paradox here, but it's a little hard to see it from the way the article is written. Maybe this will help. Let X be the sentence "If X is true, then Santa Claus exists". At the beginning, we don't know if X is true or false. But it's fun to think about what would happen if X were true. So hypothetically, what would happen if X were true? Well, in that case, we would believe X, and also believe its first half, so we would have to conclude Santa Claus exists.

So now, we still don't know if X is true, and we still don't know if Santa Clause exists, but we do know one thing: if X is true, then Santa Claus exists. We definitely know that. Because we imagined what would happen in that scenario, and concluded Santa Claus would exist in that scenario. So we can definitely state that if X is true then Santa Claus exists. In other words, the sentence "If X is true then Santa Claus exists" is definitely true. But wait! That definitely-true sentence has a name. It's called X. So we know that X is definitely true.

That's very strange. We started out not knowing anything for sure, and ended up knowing that X was definitely true. And then from that, we can derive that Santa Claus exists. So now we definitely know Santa Claus exists. Houston, we have a problem.

The interesting thing is that we didn't start off believing X. We simply thought about what would happen if it were true, and wrote that conclusion as a conditional. But it turned out to be identical to X. So then we had to believe X itself. And that leads to problems. That's the paradox. — Preceding unsigned comment added by 70.113.33.136 (talk) 05:24, 30 April 2013 (UTC)

Thanks, but I had discussed with CBM (you can see the discussion in his talk page), and he had already convinced me of the paradox. Here's the proof I propose (and may include in the article): First, let's see we're dealing with material conditional so we can build the following truth table, where P(x) is the logical value/predicate function of the proposition x:

P(p)	P(q)	P(p → q)
0	0	1
0	1	1
1	0	0
1	1	1

The last 2 cases are pretty obvious, however the others are not quite. Let's take a non paradoxical statement, for expl., "All oranges are fruits". In this case, p is "x is an orange" and q is "x is a fruit (for all 'x')", and p → q is "If x is an orange then x is a fruit (for all 'x')".

Let's apply it to an orange (case P(p) = P(q) = 1): an orange is an orange? Yes. An orange is a fruit? Yes. If an orange is an orange then an orange is a fruit? Of course, and therefore the last case is true.

Now with an apple (case P(p) = 0 and P(q) = 1): an apple is an orange? No. An apple is a fruit? Yes. If an apple is an orange then an apple is a fruit? As an apple is not an orange, but even if it was, it would still be a fruit, so although p is false, the overall statement (p → q) is true.

And let's take a rock (case P(p) = 0 and P(q) = 0): a rock is an orange? No. Is it a fruit? No. If a rock is an orange then a rock is a fruit? Yes, because a rock is not an orange, and as such the consequent, q, does not necessarily (and in orange/fruit case, not ever) follow.

Therefore, the case P(p) = 1 and P(q) = 0 would only happen (proving that not all oranges are fruits) if there was an x that was an orange but that was not a fruit. As there isn't any non-fruity orange, "All oranges are fruits" is true.

Knowing all this, I've come up with the following proof of Curry's Paradox:

Definitions:

p:=(P(p→q)==1) (in English: p is defined as the truthness of the statement itself)

q can be anything, as long as it is false (otherwise all the propositions could be true at the same time, and there would be no paradox)

Proof itself:

1. ⊢(p → q) (assuming the original statement)
2. P(p → q) = 1 (result of the above assumption, the statement is true)
3. P(q) = 0 (q is false)
4. p (p is defined exactly as p → q)
5. q (by modus ponens of 1. and 3.)
6. P(q) = 0 (that can't be, q is false)
7. P(p) = 1 ∧ P(q) = 0 (2 statements before repeated)
8. P(p → q) = 0 (if p is true and q is false, the overall statement is false) (you can begin right before this, with ⊢¬(p → q))
9. P(p) = 0 (p is defined as the statement being true, if it isn't, then p is false)
10. P(p → q) = 1 (if they're both false, then the overall statement is true, and were back in the beginning, actually 2.)

All p, q and p → q must be true and false (at different times). This works because we can be sure q is false, as it is unrelated to the sentence, while p depends on the truthness of the statement, and the statement on the truthness of p and q. Should I include the above proof? How can I improve it? (it does look kind of confusing, so I'm positive improving will be required or this sentence is false) Please share your opinion. JMCF125 (talk) 17:39, 1 May 2013 (UTC)