Levels of Recursion

Haskell Curry’s paradox, titled “Curry’s paradox,” is often stated in formal logic as follows:

let A = (A–>B)          ! A is a boolean assigned the value “if A then B,” in words this means that “A is true if A’s being true means that B is true”

A –> A

A –> (A–>B)             ! Substitution

A && A –> B

A –> B

A                                      ! Substitution

B                                       ! modus ponens

The logical error lie in the ignorance of levels of recursion.  In reality, there is no such thing as letting A1 = (A1–>B), because that is an in equality.  A1 does not equal A1–>B, it equals A1.  The algebraic analog of this principle would be something like “x=x+1” for which there is no real solution, and seeing as there is no such thing as imaginary logic, at least not yet (though there should be), the expression is utter nonsense in logic.  The first statement can remain in the syntax that it is currently in; however, it should be realized that what that statement implies is “Let A1 = (A2–>B).”  Thus the recession in logic works just the same as it does in algebraic recursive sequences (i.e. we never define “a sub n” in terms of “a sub n” but rather in terms of “a sub n plus or minus some integer value”).

Thus the proof is disproven as follows:

let A1 = (A2–>B)

A1 –> A1

A1 –> (A2–>B)

A1 && A2 –> B              ! This simplifies no further.

This understanding is essential to the functionality of logic and is very relevant allover the place.

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