# A and ~A in logical proof

In my logic class last semester, we went over proofs with the rules of induction and replacement. In a couple of the exercises, I noticed something. In each of the exercises in question, all of the letters in the conclusion weren't in any of the premises. So how I went about getting to conclusion is that I eventually got to some letter by itself (like A) and the opposite (~A). I then added the entire conclusion to the A through addition and used the ~A to prove it using a disjunctive syllogism.

My question is that if I am able to have both A and ~A in the same proof, wouldn't that mean that the argument is invalid because there is a contradiction?

Here is an example of one of the problems that I came across and my proof for it (which it said was correct): • As per @Nick R's answer, the premises of the argument are contradictory, i.e. they are not (jointly) satisfiable. Consider 3rd premise: ~O; it is true when O is false. Then consider 2nd premise: W v O; if O is false, in order for it to be true we must have W true. Thus: O is false and W true. Finally, consider 1st premise, i.e. the conjunct W ⊃ O: with W true and O false it is false. Thus, it is impossible that all three premises are simultaneously true. May 28 '16 at 9:37
• In conclusion: no wonder that in the derivation you can find a contradiction : O and ~O. It was already there from the beginning... May 28 '16 at 10:57

I believe you may be confusing the notions of validity and soundness.

A valid argument can certainly derive a contradiction. For example, a (valid) reductio ad absurdum argument, sometimes called proof by contradiction, is a valid argument that derives a contradiction.

The example you give is a valid argument. However, the fact that it allows you to derive a contradiction means that it is not sound. The conclusion one can draw is that at least one of the premises must be false.