How do I prove, :((A ⊃ B) ⊃ C) ⊃ (B ⊃ C), using symbolic logic derivations where ⊃ represents a conditional i.e. A ⊃ B = A implies B?
The first line of my derivations is the assumption, (A ⊃ B) ⊃ C. The second line is a second sub derivation, B. However I don't know how to get C out of my second sub derivation so I can return to the original assumption with (B ⊃ C).
Keep your eye on the goal and work backwards.
You seek to derive
C under the assumptions of
(A ⊃ B) ⊃ C and
B. So if you may somehow derive
A ⊃ B then you may use modus ponens (aka conditional elimination) to reach the goal.
| |_ (A ⊃ B) ⊃ C
| | |_ B
: : : :
: : : :
| | | A ⊃ B
| | | C
| | B ⊃ C
| ((A ⊃ B) ⊃ C) ⊃ (B ⊃ C)
Now, you already know that a conditional may be introduced when you can derive the consequent under assumption of the antecedent.