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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):

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    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. – Mauro ALLEGRANZA 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... – Mauro ALLEGRANZA May 28 '16 at 10:57
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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.

See the IEP article on validity and soundness for more details. See also this article on the reductio ad absurdum method.

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Indeed, when premises contradict, any conclusion (even a false one) is true. This is due to the principle of explosion (ex falso quodlibet):

The principle of explosion ... is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction.

We can rewrite several premises P1, P2, ..., Pn as one: P = P1 ∧ .. ∧ Pn. If the premises contradict each other, this conjunction will be false.

Therefore, for any Q the implication P → Q will be true. Hence, any conclusion Q follows from P. This is due to the truth table of implication, where a false antecedent always give a true outcome.

In proof methods, we abstract from the actual content of premises and conclusions. The principle of explosion is applicable whenever a premise is false. So, also from the premise "France has a king" we can derive anything, even though it does not have the form A ∧ ¬A. Therefore, the fact that the premise contains both A and ¬A is insignificant with respect to the proof method and therefore with respect to the validity of the argument.

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