Indirect Proof

Theorem

Let $P$ be a proposition whose truth value is to be proved (either true or false).

There are two aspects to this:

Reductio ad Absurdum

If, by making an assumption $\neg \phi$, we can infer a contradiction as a consequence, then we may infer $\phi$.

The conclusion $\phi$ does not depend upon the assumption $\neg \phi$.

Thus a reductio ad absurdum argument for the truth of $P$ is a valid argument which takes as a premise the negation of $P$, and from it deduces a contradiction:

Proof by Contradiction

If, by making an assumption $\phi$, we can infer a contradiction as a consequence, then we may infer $\neg \phi$.

The conclusion $\neg \phi$ does not depend upon the assumption $\phi$.

Thus a proof by contradiction argument for the falsehood of $P$ is a valid argument which takes $P$ as a premise, and from it directly deduces a contradiction:

Proof

For proofs, see:

• Reductio ad Absurdum/Sequent Form
• Proof by Contradiction/Sequent Form

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.6$: The Rule of Indirect Proof
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 5$: Proof by contradiction
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic