# Indirect Proof with contradiction in premises

The premises are as follow:

``````1. A
2. A > ~B
3. B
4. C
``````

The conclusion is `(A & B) & \~C`

My teacher proceed to use solve it using Indirect Proof.

``````5. ~[(A & B) & ~C ]      AP
6. ~B                    1,2 MP
7. B & ~B                3,6 Conj
8. (A & B) & ~C          5-7 IP
``````

If such proof is valid, that means if there are premises with contradiction, I can create any sort of conclusion with IP, even false argument like `A & ~A`. If so, would that make IP not very strict/complete system?

Is there any rule that required AP must be used within the skeleton of IP that result in the reaching the IP conclusion? I think this can prevent creating false argument.

If such proof is valid, that means if there are premises with contradiction, I can create any sort of conclusion with IP, even false argument like A & ~A. If so, would that make IP not very strict/complete system?

If the premises entail a contradiction, then you can validly derive anything from them. After all, anything is at least as true as a contradiction.

An Indirect Proof operates as follows: When a contradiction can be derived from an assumption of a negated statement, then we may infer the statement. This is also known as Reduction to Absurdity (or Reductio Ad Absurdum to be fancy). (Assuming `~P` reduces to an absurdity, therefore `P`.)

``````|  |_ ~P
|  |  :
|  |  B & ~B
|  P
``````

Now, the issue you have is that in your teacher's proof, the contradiction is not derived from `~P` itself, but from the premises. This is okay. In a Natural Deduction indirect proof we do not care about the origin of the contradiction, just that it can be validly derived within the context of the assumption. So this is quite valid:

``````|  B           Premise
|_ ~B          Premise
|  |_ ~P       Assumption
|  |  B & ~B   & Introduction
|  P           IP
``````

Now, there is a similar rule, Proof Of Negation (or Negation Introduction): When a contradiction can be derived when assuming a position, then we may infer the negation of that position. This leads to the equally valid inference:

``````|  B           Premise
|_ ~B          Premise
|  |_ P        Assumption
|  |  B & ~B   & Introduction
|  ~P          ~I
``````

So `B & ~B` entails not only `P`, but `~P` too!

Well, that just means that a contradiction will entail a contradiction. Meh. There's no problem with that.

However, this is why we prefer to avoid having contradictions in premises. They do not make anything derived from them invalid, but the proofs are quite vacuous.

Remark: This is for Classical Logic. There is a proof system called Relevance Logic which does require that the conclusion of a sub-proof be relevant to the assumption.

Exactly. The premises 1,2 and 3 entail a contradiction: B & ~B.

Ex falso (aka: Principle of explosion) is a valid rule of inference: we can derive a formula whatever from a contradiction.