Talk:Curry's paradox

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)

To say that we assert the validity of a conditional is not the same as saying that a conditional is true. -- Yes it is. 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. -- Yes, of course it does. Asserting P is the same as asserting that P is true. Asserting that A implies B is the same as asserting that "A implies B" is true. -- 71.102.133.72 (talk) 04:42, 15 September 2014 (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)

i don't see the difference. -- So what? People who can grasp basic logic do. -- 71.102.133.72 (talk) 05:04, 15 September 2014 (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)

Where do stupid and ignorant people get the idea that, if they don't get something, the problem is with that thing, and not with them? -- 71.102.133.72 (talk) 05:04, 15 September 2014 (UTC)

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.

Sorry, but your personal opinion on this matter (which happens to be wrong and based on extremely muddled thinking and a lack of basic knowledge of logic) does not determine the content of Wikipedia. Numerous reliable sources say it's a notable paradox, therefore Wikipedia should describe it as such. (And this talk page is not the place to analyze the paradox or to try to explain it to arrogant dunderheads who think they know better than the world's top logicians.) -- 71.102.133.72 (talk) 04:58, 15 September 2014 (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)

This article looks like someone intelligent and knowledgeable wrote it and then some incompetent ignoramus came along and messed it up. -- 71.102.133.72 (talk) 05:11, 15 September 2014 (UTC)

The world's logicians -- you know, reliable sources, people who know stuff, people who aren't stupid and ignorant -- think it's a paradox. Therefore Wikipedia properly calls it a paradox and provides the analysis from those reliable sources. People who aren't happy with that should delete their accounts. -- 71.102.133.72 (talk) 05:10, 15 September 2014 (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)

Um, "If this sentence is true" is not a sentence and so obviously (duh) is not the antecedent of "this". The antecedent of "this sentence" is obviously (duh) the entire sentence, from the initial capital letter to the final period. (duh) -- 71.102.133.72 (talk) 05:15, 15 September 2014 (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)

Look, Haskell Curry and the rest of the world's logicians know what they're doing. This talk page is for improvement of the article, not for ignorant dunderhead crackpots to put forth their own original "research" on how the sentence should be phrased, whether its a paradox, or whatever else comes to their incompetent little minds. -- 71.102.133.72 (talk) 05:19, 15 September 2014 (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)

I was wrong about this. Anyway I'll continue with my previous logic, to prove my previous conclusion wrong:

1. ¬X → ¬X

2. ¬X → ¬(X → Y)

(X is (X → Y))

3. ¬X → ¬(¬X ∨ Y)

(material conditional equivalence)

4. ¬X → X ∧ ¬Y

(simplification of the above expression, can be proved with some truth tables)

5. ¬X → X

(by conjunction elimination)

And a contradiction is found, proving X has to be true and false, and that there is a paradox. Perhaps this should also be included in the article, to conclude the paradox circle. JMCF125 (talk) 18:31, 1 May 2013 (UTC)

Please provide reliable sources for you pet theory, rather than calling upon unnamed persons to insert your pet theory into the article. -- 71.102.133.72 (talk) 05:25, 15 September 2014 (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)

Reliable sources, including all the world's top logicians, say it's a paradox. It doesn't matter how many ignorant dunderheads who can't reason their way out of a paper bag disagree -- editors' opinions aren't relevant at Wikipedia. Should I include the above proof? -- No, no original research should be included in any Wikipedia article. -- 71.102.133.72 (talk) 05:30, 15 September 2014 (UTC)

The reason why people believe this is not actually a paradox

I now understand the paradox, but from the comments here, and from the article itself, it is not very clear (as it was not to me, although the article is better at this moment than back then). What must be included is a formal logic proof (as the two I presented) to show that it cannot be false. If one just proves it isn't true you say, "Well, then it's false". It's important to notice it cannot be false either (otherwise it wouldn't be a paradox), basing on the clear definition of the material conditional. JMCF125 (discussion • contribs) 11:47, 11 May 2013 (UTC)

It is just endless recursion. It should be possible to substitute "this sentence" with the sentence. So if it is true that if it is true that if it is true that... You'll never get to the second concluding part of the sentence. Logicians seem to want to jump this infinite gap and then act all confused. To just assume this eternal attempt at a complete sentence can accept a finite value like true or false seems to be the nub of the problem. 89.247.73.51 (talk) 12:03, 17 October 2015 (UTC)

Seems to be a falsehood, not a paradox

In logic, asserting a statement implicitly asserts its truth. Lets call the statement X. So we are asserting that X is true. If the statement is $X\implies Y$ then clearly asserting X is asserting Y. So if Y is false and you assert it obviously you get a contradiction.

I dont see the issue here. It's asserting a statement that is false. It's lieing.

Saying "if this statement then Y" is equivalent to stating Y, as implicitly "this statement" is deemed true.

The confusion is that in natural language, using "motivational reasoning" we dont read it like that. In human logic the "if" prefix immediately introduces a question as to the truth of X, because we understand that the person making the statement is not certain of the truth of the statement. If they were certain why didnt they just say Y. So obviously the person is questioning the truth of X. The hearer no longer understands X as being asserted as true. Humans read this as,

$true\implies Y\lor false$

Which is a perfectly harmless statement. The problem arrises from mixing up two styles of reasoning.

• Logical reasoning says "I dont care about your motivation, what are you saying.".
• Motivational reasoning says "forget what you are saying (your probably lying anyway), what are you trying to achieve".

Such is the joys of being a strange ape like creature with a predispossession for lieing. Feeling versus thinking in Myers-Briggs.

Thepigdog (talk) 13:43, 23 May 2013 (UTC)

You're right, asserting a false statement yields false. The problem is that if you assert the supposedly true statement that is asserting the opposite of a false assertion, it doesn't yield true either. See my comments above (I have thought what you are thinking) and the reasons I eventually, against my previous arguments, gave that this is a paradox (it is not false or true as "this sentence is false", although that is a different kind of paradox). JMCF125 (discussion • contribs) 14:47, 23 May 2013 (UTC)

I guess the statement is not true or false then. Its truth value admits no solution. But as soon as you assert the opposite then X is a different statement.

Either

$X=(X\implies Y)$

or

$X=\not (X\implies Y)$

X can't represent both statements. Thepigdog (talk) 23:51, 23 May 2013 (UTC)

X does not represent both, X represents the first. Let me quote a comment I already wrote above (the first comment is from back when I would have agreed with you):

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)

I was wrong about this. Anyway I'll continue with my previous logic, to prove my previous conclusion wrong:

1. ¬X → ¬X

2. ¬X → ¬(X → Y)

(X is (X → Y))

3. ¬X → ¬(¬X ∨ Y)

(material conditional equivalence)

4. ¬X → X ∧ ¬Y

(simplification of the above expression, can be proved with some truth tables)

5. ¬X → X

(by conjunction elimination)

And a contradiction is found, proving X has to be true and false, and that there is a paradox. Perhaps this should also be included in the article, to conclude the paradox circle. JMCF125 (talk) 18:31, 1 May 2013 (UTC)

JMCF125 (discussion • contribs) 18:23, 24 May 2013 (UTC)

Sorry I misunderstood the original statement of the paradox. The article says "In this context, it shows that if we assume there is a formal sentence (X → Y), where X itself is equivalent to (X → Y) ..." I believe the meaning is, "Consider the expression (X → Y). Lets name this expression as X."

This only asserts the statement,

• X = (X → Y) (because you are defining X to be equivalent to the statement (X → Y) )

Then using some logic identities you get,

• (X ∧ (¬X ∨ Y)) ∨(¬X ∧ X ∧ ¬Y)

so

• X ^ Y

The implicit axiom is that you may name any expression. Maybe the naming axiom is that you may only name an expression, if you have a new unique name, not already used. Only then may you assert that the new name = the expression.

Anyway indescriminate naming could create all sorts of simpler falsehoods,

• Name e to be 3 * e.
• Name X to be ¬X.

If the language really had a "This Statement" function built into the language then the naming would be legal. But the "This Statement" function is not OK, because it does not return the same value each time it is called.

(talk • contribs) 17:20, 5 June 2013 (UTC)

Anyway that is my non technical view of the problem.

Thepigdog (talk) 16:15, 4 June 2013 (UTC)

I think there should be no axiom to avoid a recursive definition. "X = ¬X" is like the paradox of "This statement is false." BTW, you can name some "e" to be "3*e", notice: "e = 3 * e <=>e - e = 3e - e <=> 0 = 2e <=> e = 0" (the thing that makes this look paradoxal is that you can't divide by "e").

Also, there's no need to be sorry, and I do not divide between technical and non-technical views. I, as you can see, have also failed in understanding this at first. That's why I'm here discussing; to help others know this and improve the page. JMCF125 (discussion • contribs) 16:11, 5 June 2013 (UTC)

"This statement" would need to be a parameter variable passed to the statement. Let T be "This Statement". Then consider a statement, $\lambda T.(T\implies Y)Z$. Within that statement I have a free value X that I wish to be equal $(T\implies Y)$.

I start by asserting

• $\forall K\forall Z\exists X\lambda T.(X=K)Z$

Can prove this is true by letting X = K. Next let $K=(T\implies Y)$ <---- This is the false step

• $\forall Z\exists X\lambda T.(X=(T\implies Y))Z$

Rearanging,

• $\forall Z\exists X\lambda T.XZ=\lambda T.(T\implies Y)Z$

Let Z = X,

• $\exists XX=(X\implies Y)$
• $\exists XX\land Y$
• $Y$

Or even simpler

• $\forall Z\exists X(X=(Z\implies Y))$

Can prove this is true by letting X = (Z \implies Y). If the statement is true for all Z then choose Z = X,

• $\exists XX=(X\implies Y)$
• $\exists XX\land Y$
• $Y$

Now I am really confused. Thepigdog (talk) 17:52, 5 June 2013 (UTC)

I think you are overcomplicating. I don't think we need to bring here another types of logic, that just adds up to the confusion. The "this statement" function would be an identity function, and thus return always the same result. JMCF125 (discussion • contribs) 18:42, 5 June 2013 (UTC)

This Statement returns the statement in which it is called. But it is not in any way parameterised by this statement. Therefore it is either a global function or a variable set up in the context. It cant be a global function as it returns a different value if called for different statements. So it must be set up in the context.

The above examples dont rely on any special definition of "this statement". I think they create the same paradox by explicitly setting up "this statement" variable in the context.

Thepigdog (talk) 09:22, 6 June 2013 (UTC)

I think you're right, they do produce the same paradoxes. Nonetheless they're not required for them to make sense. <This statement>'s definition is <this statement>, and as such no function is required (although, as you insist, it can be created); an implicit set up of the definition is the very definition, that's enough. JMCF125 (discussion • contribs) 15:30, 9 June 2013 (UTC)

Yes I have read plato.stanford.esu web site about a version of the proof which names "this statement" indirectly. It talks about a T-schema that allows you to indirectly name a statement. I think I was wrong in my argument about global functions.

My naive solution

• Axiom of naming. Any expression may be given a name (say x) is replaced with;
  • For all y in {true, false} there exists x such that x = y.
  • I.e. if the value y does not exist you can't name it.
• Now take z to be some self contradictory statement
  • This statement is false
    • The value of this statement is the solution of the equation $x=\not x$ which is neither true or false (does not exist).
  • This statement implies false
    • The value of this stament is the solution to the equation $x=(x\implies false)$ which is neither true or false (does not exist).
• These expressions dont evaluate to true or false so the axiom of naming does not apply. This breaks the naming step.

A T-schema [x] is a name for a value in {true, false}. So the naming may only occur if the value is in {true, false}.

We could take x = " $([x]\implies false)$ " where [x] means evaluate x (turn the text into logic and evaluate it). But again we would conclude that [x] does not exist, therefore we cannot assert the existance of the name [x].

In short, naming something cannot implicitly imply it's existance.

Thepigdog (talk) 06:06, 10 June 2013 (UTC)

Where do people get the idea that this talk page is a place to analyze the statement or to dispute whether it's a paradox, as if their opinions matter, rather than Wikipedia being drawn from reliable sources? I guess from the same stupidity and arrogance that makes it so difficult for them to understand why an obvious paradox is a paradox: If the sentence is not true, then it is equivalent to false -> P, which is always true ... so the sentence is provably true. From which P, whatever it is, follows ... a paradoxical result.

In logic, asserting a statement implicitly asserts its truth. Lets call the statement X. So we are asserting that X is true.

This is pathetic. Simply writing down a proposition does not assert its truth.

-- 71.102.133.72 (talk) 05:39, 15 September 2014 (UTC)

Some suggested changes for clarrification only

Proof that the sentence is true

The following analysis is used to show that the sentence "If this sentence is true, then Germany borders China" is itself true. The quoted sentence is of the form "If A then B" where A refers to the sentence itself and B refers to "Germany borders China". The usual method for proving a conditional sentence is to show that by assuming that hypothesis (A) is true, then the conclusion (B ) can be proven from that assumption. Therefore, for the purpose of the proof, assume A. Because A refers to the overall sentence, this means that assuming A is the same as assuming "If A then B". Therefore, in assuming A, we have assumed both A and "If A then B". From these, we can obtain B by modus ponens. Therefore, A implies B and we have proved "If this sentence is true then Germany borders China" is true. Therefore "Germany borders China", but we know that is false, which is a paradox.

The paradox

In a naïve logic, the sentence itself, denoted A, is true. The sentence is of the form "If A then B". Then "A = (if A then B)" is true. But this reduces to "If A then B" is true. So A is true. We then apply modus ponens to show that B is true; but this is impossible, because B is "Germany borders China", which is false.

Thepigdog (talk) 03:39, 15 June 2013 (UTC)

New section

I propose a new section for clarification purposes. I would to group,

• Natural language
• Formal logic
• Naive set theory
• Combinatory logic

under a section Statements of Curry's Paradox

With some opening words like "The Paradox may be expressed in natural language and in various mathematical forms.".

Also I would like to add a section Language Capabailities for Expressing the Paradox under deiscussion and add in the following section Existential Problem.

If no objecttions I will proceed in about a weeks time.

Existence Problem

This paradox is similar to,

• Liar Paradox
• Russel's Paradox

in that each paradox attempts to give a name for something that does not exist. These paradoxes all attempt to give a name or representation to a solution to the equation,

X = ¬X

The paradox arises by naming or representing an expression of the form ¬X to be X. In the case of Curry's Paradox, the negation is constructed from implication,

X = X → false = ¬X ∨ false = ¬X

The domain of a boolean variable X is the set {true, false}. However neither true or false is a solution to the above equation. So it must be wrong to assert the existance of X. Modern approaches to these paradoxes denie the right to name such an expression, or they extend the domain.

Thepigdog (talk) 11:19, 19 June 2013 (UTC)

More Minor Clarifications

Clarify,

2 The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences;

This is not explicitly used in the proof. I propose removing it.

1 The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");

as

# The language must contain an axiom or apparatus which simultaneously names and asserts the existence of an expression in the language. This allows the expression to refer to, and talk about, itself (such as quotation marks, names, or expressions like "this sentence");

Down further I would like to add.

An expression in a language differs from a statement in that a statement is implicitly true. That is, asserting a language expression makes it a statement. Asserting a statement says that the value of the expression is true.

Resolution in Logic

I am also uncomfortable with the "Resolution in Logic" section. I think it could be misread as saying that there is some particular difficulty with Currys Paradox which no one can resolve. That is probably not what the original auther intended.

I propose some rearrangement, and something like,

The resolution of Curry's paradox (and other similar paradoxes) in logic is difficult, because logic needs axioms from which set theory and the rest of mathematics may be built. From one perspective, logic and then set theory may be considered as the languages from which mathematics boot straps itself. So even though the problem can be described quite simply, a detailed discussion detailed is limited to a technical discussion of the axioms of logic.

Thepigdog (talk) 12:20, 29 June 2013 (UTC)

Another addition

Banning "this statement" from the language may prevent Curry's paradox from being formed. However the same problem may be formed in many other ways. For example if we an add an Eval function to logic that takes a string and converts it into a logical expression, then consider the string,

s = "Eval(s) → false"

then the expression,

Eval(s) = Eval(s) → false = ¬Eval(s)

again gives Currys's paradox. To handle such a situation the boolean type needs a value representing an error situation.

Thepigdog (talk) 18:09, 29 June 2013 (UTC)

The Situation with Currys Paradox is much worse than I thought

From what I can understand researchers are suggesting fairly desperate solutions to Currys Paradox. To me the paradox arises immediately you allow the construction of an expressions whose value does not exist. This is really easy to do in natural language and any even small addition to logic allows the construction of such value. I think pure predicate logic may not allow the construction of the paradox, but I am not sure.

In computer programming if given a value that does not fit the domain we would throw an exception. The equivalent in a functional language would be to return an error value. This would require extending all domains with,

• Unknown - I dont know
• Error - The value does not exist.

These values dont so much extend the domain as provide an "out" when things get difficulty. Unknown indicates that the value is not available yet. So the only valid response is to wait.

Error indicates that an infinitely recursive function has been called, or an expression does not have a value in the domain, or the input data violates the assertions made about the data. The only valid response is to terminate.

Questions:

• Would somebody knowledgable indicate if such an approach is possible.
• Can Currys Paradox appear in pure logic.

Thepigdog (talk) 19:56, 29 June 2013 (UTC)

Derivation of Combinatory Logic paradox expression

Translation of

$r=(\lambda x.xx\to y)$

then define,

m A B = A → B

the translation is,

$T[\lambda x.(m(xx))y]$

$ST[\lambda x.(m(xx))]T[\lambda x.y]$

$S(ST[\lambda x.m]T[\lambda x.(xx)])T[\lambda x.y]$

$S(S(Km)T[\lambda x.xx])T[\lambda x.y]$

$S(S(Km)(ST[\lambda x.x]T[\lambda x.x]))T[\lambda x.y]$

$S(S(Km)(SII))(Ky)$

= S (S (K m) (S I I)) (K y)

Thepigdog (talk) 12:17, 1 July 2013 (UTC)

Untyped lambda calculus

The comment added under Statement of Curry's paradox/Lambda calculus needs clarification.

It also does not fit well with the structure of the article, and its dramatic purpose. Currently the paradox is shown first, which involves the reader in wanting to understand the resolution.

I am hoping when the author has time that an explanation may be added, and that perhaps some restructuring may be made. An explanation of how Currys paradox is handled by "proofs as programs" would be valuable.

Thepigdog (talk) 10:53, 21 May 2014 (UTC)

Lambda calculus: What does the arrow mean?

In the expression

r = ( λx. ((x x) → y) )

I don't understand what the arrow (→) means. It doesn't seem do be defined in the linked article Lambda calculus either.

Is it just a variable name? Then, I think, it would be more clear if the expression is written like

r = ( λx. ((→) (x x) y) )

Is it a part of an extension of the lambda calculus, a link or explanation is needed.

Also, I don't really understand how Lambda calculus expressions can be interpreted as propositions.

Baum42 (talk) 16:56, 8 June 2015 (UTC)

Agreed, I'm trying to figure that out as well. The arrow is clearly supposed to be the "implies" symbol (which is certainly not a variable!). Although implication can be interpreted as a function, if there's a technique for this in lambda calculus itself, that doesn't seem to be part of most introductions to the lambda calculus, so it should be clarified here (assuming it's even correct).

The other issue with the lambda calculus section is that it doesn't mention Curry's fixed-point combinator, which, according to the Fixed-point combinator page, is typically used when implementing Curry's paradox in the untyped lambda calculus. This seems like a fairly major oversight. 2601:283:8102:2C0:64A6:58AB:600E:59B2 (talk) 07:09, 4 January 2016 (UTC)

Additional thoughts: → may be intended to be something like the "pairing combinator," a function that resolves to one of two values depending on whether its input is 0 or not. I.e., ((x x) → y) reduces to y IFF (x x) is true. But in this case there should be some alternative expression to which it resolves when (x x) is false. Additionally, the principle of explosion is invoked for the case where (r r) is false, but it's not clear why "(r r) is false" would be considered a contradiction, so it's not clear that the principle of explosion can be invoked. 134.217.237.30 (talk) 21:35, 6 January 2016 (UTC)

Purported resolution in natural language

Please provide a source for the purported resolution in natural language that has been re-introduced to the article. The claim that the sentence "creates a falsehood if B is false" makes little sense - what does it even mean for a sentence to "create a falsehood"? Similarly, the claim that no value for A satisfies "if A then false" is false, because in natural language "if this sentence is true, then false" gives such an A - that is why we have this paradox! In any case, without a source, it is impossible to improve the paragraph to state whatever is actually intended, and WP:V requires a source when a claim is challenged. I will pause for several days to allow a source to be provided. — Carl (CBM · talk) 10:11, 25 August 2015 (UTC)

Natural language section is only understandable to those who understand it

Relax. I'm not going to say that there is no paradox. However, the discussion in the natural language section is confusing and misleading for people who have no understanding of formal logic or lambda calculus. It doesn't bring them to a point where they can learn why the paradox is important. It is easy for people to see how the sentence is wrong. Germany does not border China, therefore the sentence "If this sentence is true, Germany borders China". They fixate on that bit and can't wrap their minds around the important part.

First, I would suggest linking to the Conditional Proof page to provide some evidence for the technique of assuming the antecedent. Then instead of simply saying "and by modus ponens", actually spell it out for them. "Modus ponens" might as well be "abra cadabra" as far as the average reader is concerned. The key thing that I think should be emphasized is that we assume the antecedent. However, the antecedent is the whole sentence. Since the antecedent is the whole sentence the consequent must follow. By assumption ;-). This is clearly "a bad thing", but that is the whole reason why this paradox is important. We have a logical system/lambda calculus that allows this set up.

The last bit is the place where I think this article is falling down the most. The previous people who commented on this article are pretty much all wrapped up in this misunderstanding. They feel if you just restate the sentence, you can avoid the paradox. Which is true, but misses the point entirely. I suggest they are missing the point because it was never stated in the article: I can prove things that aren't true because my logic calculus has a problem.

Finally, providing some historical context on how the paradox arose and what changes were made to various logical systems to deal with the problem would really go a long way towards making the issue understandable. It's actually why I came here and was quite disappointed. 59.85.105.186 (talk) —Preceding undated comment added 14:22, 18 October 2015 (UTC)

Thank you - these are very helpful comments. I will try to implement them over time. The historical part is, unsurprisingly, the most difficult. The other changes you suggest should be easier to put in place. — Carl (CBM · talk) 22:44, 18 October 2015 (UTC)

Natural Language

OK I don't mind at all if you don't want to use this. But just deleting the section without improving it is a bit annoying. This is not an argument about logical deduction systems. It is an argument about how you would understand this in English, or some other natural language system.

The paradox starts by asserting a statement of the form

If A then B

Where A refers to the whole statement.

A = (If A then B)

But if B is false there is no solution for A. All this is explained earlier.

In natural language we are allowed to use natural arguments, that are understandable without a formal system.

If (neither true nor false) then B

is not a valid statement in natural language and can be ignored.

I am more than happy for someone to describe this better.

What follows is a slightly expanded version of what was there before.

3.1 Resolution in natural language

Consideration of the sentence "If A then B" where A refers to the sentence creates a falsehood if B is false, because there is no value for A that satisfies the expression A = "if A then false". If you put A = true you get,

true = if true then false ... and this is false.

If you put A = false you get,

false = if false then false ... and this is false.

So when we use the statement,

If A then B

Natural language logic assumes that all statements are either true or false. But A is not, so the rest of the argument is invalid because it is arguing from an expression that has no possible value (does not exist) when B is false.

Thepigdog (talk) 02:57, 2 November 2015 (UTC)

Math StackExchange post to help clarify lambda/combinator sections

I've opened a math.stackexchange question to attempt to understand Curry's paradox better; specifically, the article's explanation of how the paradox can be formulated using combinators or lambda calculus don't make sense to me (see the section on lambda calculus above).

http://math.stackexchange.com/q/1602611/52057

134.217.237.30 (talk) 17:41, 8 January 2016 (UTC)

Weird example

Glad i'm not the only one that is confused by the example given to explain this. "If this sentence is true, then Germany borders China.". It seems like this sentence should read more like, "This sentence is true, so Germany borders China." -or- "Germany does not border China, so this sentence is false.". This might be going over my head, but from my understanding from reading previous topics, is that this is better understood with Math than through sentence examples. That sentence example might make sense to some people on here, but it seems, IMO, that the section explaining the example could be worded better for a reader. DrkBlueXG (talk) 21:07, 7 June 2016 (UTC)

No implication operator in lambda calculus

There is NO implication function according to the Curry-Howard correspondence.

• https://en.wikipedia.org/wiki/Church_encoding
• Church Booleans
• If

Can't this be used to construct the implication operator?

Thepigdog (talk) 04:24, 10 September 2016 (UTC)

Example given not actually a paradox

Techincally, Germany does border China. Since Germany has an embassy in the capital of China, and all embassies are considered to be the territory of the embassy's country, we can conclude that Germany borders China, and therefore the example given in the article, If this sentence is true, then Germany borders China, is not actually an example of Curry's paradox. 130.126.255.117 (talk) 05:29, 2 November 2016 (UTC)

A Better Example of the actual paradox

If this sentence was typed, there are 5 quarters in a whole. 205.142.232.18 (talk) 22:50, 13 December 2016 (UTC)

What is contraction?

The formal logic proof has a step that uses "contraction" as its justification. I was not familiar with contraction, and the linked page was not at all helpful. I suggest that additional steps be added as follows:

x -> (x->y) .... (already in the proof)

(-x) -> (x->y) .... (if the antecedent of a conditional is false, then the conditional is true)

x OR -x .... (law of the excluded middle)

(x->y) OR (x->y) .... (I think you can do this substitution, where each side of the previous step is replaced using of the two steps before it)

(x->y) .... (Is there a name for the rule that "a OR a implies a"?)

This returns us to the proof.

Another way to accomplish this would be to appeal to the Distributivity of Implication (https://en.wikipedia.org/wiki/Distributive_property#Truth_functional_connectives)

x -> (x->y) .... (already in the proof)

(x->x) -> (x->y) .... (distributivity of implication)

TRUE -> (x->y) .... (law of identity)

(x->y) .... (modus ponens)

This returns us to the proof. — Preceding unsigned comment added by 63.138.92.98 (talk) 13:27, 24 April 2017 (UTC)

Vandalism

5.249.127.17 (talk) removed a ton of material anonymously, without discussion.

I suggest that he log in as a user. Roll back the changes. Then justify the changes here in talk. Then he can re-apply them.

Thanks for your co-operation.

Thepigdog (talk) 01:17, 19 May 2017 (UTC)

I'm afraid I have to disagree. For one, the user also added material. More to the point, the edits have edit comments explaining them, and the removed material was not cited to reliable sources. I suggest reading these edits as WP:CHALLENGEs. They should not be undone without addressing the issues raised by the IP contributor. Paradoctor (talk) 06:43, 19 May 2017 (UTC)

Dispute

The link at the top of the resolution section links here, but "here" didn't exist until I made this section. Anyway, the section needs reliable sources so I'm opening this discussion in hope of finding some. Hopefully I'll be back with some. HueSurname (talk) 14:34, 2 February 2021 (UTC)

"Curry`s paradox" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Curry`s paradox and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 October 15#Curry`s paradox until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 1234qwer1234qwer4 15:36, 15 October 2022 (UTC)

white text on black background: some displayed material is not visible

I read this page using white text on a black background, which is generally fine for Wikipedia. Some text isn't visible to me, for instance the displayed material after "by examining the set":

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

I don't know what's going on here. I presume that this material is being displayed in black. 2A02:A475:62C9:1:BDBA:6E29:FF67:98FA (talk) 07:02, 17 December 2022 (UTC)

The formal sentential logic proof isn't sentential logic or formal

I hate to add to this discussion page, but if this is to be an educational page I think it's important that the formal sentential logic proof that's provided isn't formal or sentential logic. '=:' isn't a symbol in sentential logic. It's a symbol from the metalanguage, meaning "means the same thing as" or something like that. (It's usually used to state a definition.) This also means that the proof isn't formal, since '=:' denotes a semantic concept. In fact, this paradox (my favorite one!) depends on self-reference, and you can't express self-reference using the vocabulary of sentential logic. Now try substituting material equivalence for '=:' and see what you can prove... Kronocide (talk) 22:23, 28 April 2023 (UTC)

Curry's Paradox: Either I've resolved it or I don't understand it

I've been a programmer for more than 40 years. In writing programs, I have to create comparisons and tests. Based on my background, I don't believe there is a paradox because I can resolve it. Or I'm completely missing the problem. Here is how I resolve the conflict.

I can simply say, we analyze a sentence as a whole. For example, "If this sentence is true," is a conditional clause. We do not presume it as either true or false until after the condition is evaluated. It depends on a condition antecedent to the clause. Since the secondary clause "Germany borders China" is not true, therefore the sentence is false and no paradox occurs. Now, I think the opposite might be better as an alleged paradox, e.g. "If this sentence is false, then Germany borders China." Yes, the sentence is false, but, truth is that which is concordant with reality. Since the sentence is false, none of its conclusions have any value or relevance, because they have no relation to reality, so they have no validity. Thus, the proposition ""Germany borders China" is irrelevant.

Another way to put it is if a sentence is false, that is, not concordant with reality, any claim or proposition within may be ignored. This also resolves the paradox.

An example in math is 0*. 0 times anything is 0 regardless of what number follows, therefore any digit or string of digits may be ignored. Same if the entry to the right is a parenthised expression, it may be ignored as irrelevant.

So maybe someone can show me where I have misunderstood the paradox? I resolved it in about 10 minutes, it seemed way too easy. The inverse was harder.

"Understanding of things by me is only made possible by viewers (of my comments) like you."

Thank you.
Paul Robinson Rfc1394 (talk) 12:36, 12 January 2024 (UTC)

Indeed, the explanation in section "In natural language" is pretty superfluous in that it omits the details of the "common natural-language proof techniques" mentioned in item 1. Since you are a programmer, you should be able to understand the formal proof in section "Sentential logic", which actually gives all details. I'll try to come up with a natural language version of it within the next few days. - Jochen Burghardt (talk) 19:27, 14 January 2024 (UTC)

 Done: I made a first suggestion; please check if it is comprehensible and convincing. (BTW: In your argument, the step "Since the secondary clause "Germany borders China" is not true, therefore the sentence is false" is flawed, cf. Material conditional). - Jochen Burghardt (talk) 20:02, 14 January 2024 (UTC)

Truth table treatment?

Two posters above have alluded to this idea, so maybe there is already consensus that it's not a good approach. Anyway, I suspect that truth tables are more accessible to readers than lambda calculus, combinatory calculus, etc. And it seems to me that truth tables explain what the paradox is concisely. So here is my treatment.

In general, here is the truth table for an implication $P\to Q$:

$P$	$Q$	$P\to Q$
F	F	T
F	T	T
T	F	F
T	T	T

In this case specifically, $P$ is "the sentence is true" (or "my claim is true"), and $Q$ is "Germany borders China". But $P$ holds if and only if $P\to Q$ holds. So the only row of the truth table that does not contradict itself is the fourth row, where Germany borders China. So Germany borders China.

Questions:

1. Does this treatment capture the paradox correctly?
2. Is it not rigorous enough? If its gaps were filled in, then would it be just as verbose as the other treatments given?
3. Is it original research? (I think not, because of WP:CALC.)
4. Would it improve the article?

Regards, Mgnbar (talk) 14:16, 6 May 2024 (UTC)