Understanding Curry's Paradox

Question

I was trying to understand Curry's Paradox on wikipedia.
https://en.wikipedia.org/wiki/Curry%27s_paradox

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?

Answer 1

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.