Say you got a formal proof:

Line # Statement                                                     Reason

1          A                                                                    Premise

2         ...

...

n         B                                                                     m, Some reason

n+1     A -> B                                                              n, Deduction Theorem.

Now we have proven: A -> A -> B, so we proved: true -> B = true (by truth table of "therefore"). So we proved A -> true (by deduction theorem). Now A could be false, then we shouldn't be able to derive "true".

23 minutes ago, Willem F Esterhuyse said:

Now we have proven: A -> A -> B, so we proved: true -> B = true

You're mangling the associativity of the operator (it's right-associative).

We've proved A -> (A -> B), not (A -> A) -> B.

In English, it's "if A, then if A then B", not "if (if A then A) then B".

Edited February 11, 20232 yr by Lorentz Jr

4 hours ago, Willem F Esterhuyse said:

A could be false, then we shouldn't be able to derive "true".

This is incorrect. Anything, true or false, can be derived from a false statement. See the truth table and the second dot here:

5 hours ago, Willem F Esterhuyse said:

Now A could be false, then we shouldn't be able to derive "true".

Your proof was contingent on A being true. Its conclusion is only valid in that case. It says nothing about the state of affairs if A is false.

I made an error: (true -> B) = B, (true -> B) not= true.

Edited February 11, 20232 yr by Willem F Esterhuyse