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I always thought that both “proof by reductio ad absurdum” and “proof by contradiction” mean the same, but now my professor asked this question on my homework and I don't know.

I believe that in both cases you assume the negation of the conclusion and develop a contradiction through the premises. This will imply the conclusion. Today I have a meeting with the assistant professor so I can clarify this, but I really would like to know what you guys think, or if possible it would be great if you point me into some good references.

UPDATE:

I just came from my extra help and the assistant professor explains the difference this way:

Reductio ad absurdum: $$ \vDash [\neg p\to(q\wedge\neg q)]\to p$$

Proof by contradiction:

$$ \vDash [\neg (p\to q) \to (r\wedge \neg r)]\to (p\to q)$$

And the examples of application were these:

Using proof by contradiction: $\sqrt2$ is irrational.( First suppose it is rational and derive a contradiction).

Using proof by reductio ad absurdum: If $f$ is differentiable on $(a,b)$ then $f$ is continuous on $(a,b)$. ( First we suppose that $f$ is differentiable on $(a,b)$ but not continuous on $(a,b)$ and derive a contradiction).

Novato
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  • See https://philosophy.stackexchange.com/questions/561/what-is-the-difference-between-reductio-ad-absurdum-and-proof-by-contradictio –  Aug 28 '17 at 15:49
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    I always thought of it as: reductio ad absurdum is proving $\lnot \phi$ by assuming $\phi$ and proving $\bot$ or $\psi \wedge \lnot \psi$. Proof by contradiction is proving $\phi$ by assuming $\lnot \phi$ and proving $\bot$ or $\psi \wedge \lnot \psi$. So, for example, in intuitionistic systems, reductio ad absurdum is still valid, but proof by contradiction wouldn't be as all it proves is $\lnot \lnot \phi$. (Not sure if this is officially valid, though.) – Daniel Schepler Aug 28 '17 at 15:55
  • Usually (but the distinction is not so "stable") the proof by contradiction is of the form: "if from $A$ a contradiction follows, then $\lnot A$ can be inferred". In the indirect proof the assumption is $\lnot A$ and the conclusion inferred (through the contradiction) is $A$. – Mauro ALLEGRANZA Aug 28 '17 at 15:58
  • Possible duplicate of Difference between proof of negation and proof by contradiction – Clement C. Aug 28 '17 at 16:08
  • Ok. So "reductio ad absurdum" is synonymous with "indirect proof", and "proof of negation"? – Novato Aug 28 '17 at 16:17
  • I regard all these names as referring to the same basic proof technique: if some assumption leads to a contradiction, then you can reject (negate) the assumption. If people make any distinction at all here, it will be highly pedantic and not something that is commonly agreed upon. – Bram28 Aug 28 '17 at 19:44
  • By what you've written above, it looks like proof by contradiction is the special case of reductio ad absurdum where the proposition is an implication. In practice they are the same because there are always some conditions we assume to be true before we derive the absurd conclusion based on the fact that some proposition is false.. – John Douma Aug 28 '17 at 22:51
  • The first example you give is not an example of either rule you give. Your second rule is just an instance of the first rule, and the first rule is just a different way of writing double negation elimination. Your first example is simply a proof of a negation, which doesn't require double negation elimination per Daniel Schepler's comment. See http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/ The distinction between proof of a negation and proof by contradiction (i.e. double negation elimination) is hardly a minor or "pedantic" one. – Derek Elkins left SE Aug 28 '17 at 23:53

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Regarding the rule of indirect proof:

"if from assumption $\lnot A$ a contradiction follows, we can infer $A$",

we can see:

Sometimes the nomenclature RAA is used; it stands for reductio ad absurdum, the mediæval Latin name of the principle. [...] A genuine indirect proof in propositional logic ends with a positive conclusion.

The principle is equivalent to Double Negation elimination.

If we agree with this approach, proof by contradiciton is more general, because it applies also to inferences with negative conclusion, licensed by the principle of Negation Introduction:

"if from assumption $A$ a contradiction follows, we can infer $\lnot A$".

Mauro ALLEGRANZA
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