What About The Other Statement?

Here is a more formal version:

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Part I - Curry's Paradox

Let us explore a proposition A defined as (if A then B). The following argument holds:

1. A := if A then B [definition]
2. if A then A [Rule of Assumption]
3. if A then (if A then B) [definition of A]
4. if A then B [contraction]
5. A [definition of A]

So, A is true.

Part II - Pi Han's Paradox

Let us rephrase the statements in the problem as following:

1. p := if p then (not q) [definition]
2. q := if q then (not p) [definition]
3. p [by part I]
4. q [by part I]
5. (not p) [modus ponens using 1 and 3]
6. (not q) [modus ponens using 2 and 4]

But we just proved that p and q both are both true and false.

But that is a contradiction

We have four cases to consider:

Case 1 : Statement 1 is true, Statement 2 is true as well.
Case 2 : Statement 1 is true, Statement 2 is false.
Case 3 : Statement 1 is false, Statement 2 is true.
Case 4 : Statement 1 is false, Statement 2 is false as well.

Suppose Case 1 is true,
then the latter part of Statement 1 implies that Statement 2 is false.
However, this contradicts the fact that Case 1 is true, which is absurd.
Hence, Case 1 cannot be true.

Suppose Case 2 is true,
then the latter part of Statement 1 implies that Statement 2 is false.
Statement 2 is false. That means, in Statement 2, the antecedent "statement 2 is true" is false. Because the antecedent is false, the whole statement is true.
But this implies that Statement 2 is true as well, which is absurd.
Hence, Case 2 cannot be true.

Case 3 is similar to Case 2,
because we can just flip the Statements 1 and 2 and we're back to Case 2.
Hence, Case 3 cannot be true.

Suupose Case 4 is true,
this implies that statement 1 is false.
Statement 1 is false. That means, in Statement 1, the antecedent "statement 1 is true" is false. Because the antecedent is false, the whole statement is true.
Hence, Case 4 cannot be true too.

Since we have exhausted all possible cases, and none of these cases produces a logical conclusion, then there is no solution to this puzzle, or equivalently, "This is an impossible scenario".

Food for thought : Is Curry's paradox applicable here?

If the first statement was true

• The second will be false

If the second statement was true

• The first would be false

It can't be logical

They both can't be true together

Hence it's Impossible to tell

Assume (1) is true. Then (1)'s antecedent is true, and so is its consequent, which states: (2) is false. Therefore, (2)'s antecedent is true and its consequent is false, which is a contradiction: (1)'s consequent contradicts (2)'s antecedent. So this is impossible.

Assume (1) is false. Then (1)'s antecedent is true and its consequent is false. But (1)'s antecedent contradicts our assumption, so this is impossible as well.

The same assumptions and reasoning can be applied to (2), since (2) is the same statement, to show that "this is an impossible scenario."