# Difference between “ proof by reductio ad absurdum” and “proof by contradiction”?

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).

• – R. Suwalski Aug 28 '17 at 15:49
• 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

## 1 Answer

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 $\lnot A$".