# How do I prove :((A ⊃ B) ⊃ C) ⊃ (B ⊃ C)?

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).

• Can you edit your question to include all premises clearly and the conclusion you are trying to reach. It seems you have one proposition listed. What is there to prove? You should number the premises and identify the conclusion clearly so no one mistakes premise for the actual conclusion. Jan 21 '20 at 18:47
• Assume (A ⊃ B) ⊃ C); assume B and from it derive (A ⊃ B). Then conclude. This is quite straightforward with Natural Deduction. Jan 21 '20 at 19:02
• If the answer below is enough for you, please accept it and we can "close" the post. Feb 21 '20 at 10:28

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.