I was trying to understand Curry's Paradox on wikipedia.

The example given is:

To produce Curry's paradox, as described in the two steps above, apply this method to the sentence "if this sentence is true, then Germany borders China". Here A, "this sentence is true", refers to the overall sentence, while B is "Germany borders China". So, 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". Therefore B is true, by modus ponens, and we have proven "If this sentence is true, then 'Germany borders China' is true." in the usual way, by assuming the hypothesis and deriving the conclusion.

Now, because we have proved "If this sentence is true, then 'Germany borders China' is true", then we can again apply modus ponens, because we know that the claim "this sentence is true" is correct. In this way, we can deduce that Germany borders China.

I went through the formal proof as well, but what I fail to understand is that we still haven't proved that the sentence A is True, just assumed it to be True.
Since the resulting statements give us a paradox, it follows that the statement A can't be True in the first place. Where am I getting this wrong?

You can see for more detailed analyses the complete SEP's entry on Curry's Paradox and its formalization in Deriving the Paradox :

"Curry’s paradox”, as the term is used by philosophers today, refers to a wide variety of paradoxes of self-reference or circularity that trace their modern ancestry to Curry. Curry’s paradox arises in a number of different domains. It can take the form of a semantic paradox, closely akin to the Liar paradox. Common truth-theoretic versions involve a sentence that says of itself that if it is true then an arbitrarily chosen claim is true. The paradox is that the existence of such a sentence appears to imply the truth of the arbitrarily chosen claim.

The key point is the possibility of producing a

Curry sentence : Let π be a sentence of the language of T. A Curry sentence for π and T is any sentence κ such that κ and κ→π are intersubstitutable according to T.

This can be done (see Wiki's example) with a self-referential language, able to express the statement "This sentence is true".

This can be done with “naive” truth theories, as well as with “naive” set theories (i.e. those featuring unrestricted set abstraction).

For set theories, Curry's (first) method can be read as a version of Russell’s paradox, while for truth theories Curry’s (second) method is a variant of a well-known semantic paradox, Grelling’s paradox.

It is well-known that, for "standard" logics, like e.g. first-order logic, self-reference can cause troubles.

• So basically, it boils down to whether or not I can construct a sentence of the form "This sentence is True", in the theory? Aug 30, 2018 at 13:10
• @novice - Yes... Aug 30, 2018 at 13:15
• Could you please elaborate on this: "Let π be a sentence of the language of T. A Curry sentence for π and T is any sentence κ such that κ and κ→π are intersubstitutable according to T." - how does this lead to - "This can be done (see Wiki's example) with a self-referential language, able to express the statement "This sentence is true"." Is this self referential statement the only statement which can lead to a Curry Statement? Aug 30, 2018 at 13:24
• @novice - see the Wiki's entry you have quoted : we have a sentence : "If this sentence is true, then Germany borders China" where A, "this sentence is true", refers to the overall sentence, while B is "Germany borders China". Thus we have that A := (A → B) and we can substitute A in place of (A → B). The key-point is that A is at the same time referring to (A → B) and part of (A → B). Aug 30, 2018 at 13:34
• I think I have the gist of it now, thanks a lot! Aug 30, 2018 at 13:35

Here is a more intuitive version of the paradox:

(1) "If I am not mistaken, then Santa Claus exists.”

Is this a valid statement? Yes, because it says that, if it is true, then what it says is true. Fair enough. So I can say:

(2) I am not mistaken to say, “If I am not mistaken, then Santa Claus exists.”

To answer the question, we conclude from (2) that (1) has no mistake, and therefore Santa exists.

But this is not right. The mistake in (1) concerns the existence of Santa Claus. The (first) mistake in (2) concerns the conditional in (1). These two mistakes cannot cancel each other.

The sentence (2) can be rephrased as:

(3) It is true to say, “If it is true that Santa exists, then Santa exists.”

This by no means proves that Santa exists.

It is just like me saying: “If what this printer print is valid, then you owe me \$1000.” And then I print: “You owe me \$1000”. Obviously, you don't owe me anything! The value of this document is contingent on the validity of my sentence, which is contingent on being true. And I provide no proof whatsoever that what I am saying is true.

Or to say: “If Joe is right, you owe me \$1000,” then Joe says: “You owe me \$1000.” Does that make you indebted to me with \$1000? Of course not. Because what Joe says is contingent on my statement, which is contingent on being true!

It is not difficult, in my opinion, to resolve this paradox. What (1) really says is:

(4) If this sentence is true, then it is true.

This is a tautology in disguise.

• Welcome to SE. I think I follow your answer. But I'm not sure it answers the question. You don't explain what the questioner's mistake is. Feb 25, 2023 at 13:12