Pi Han's Paradox

piHan

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