Reductio ad Absurdum: Proofs, Logic, and Historical Uses

All flashcards

Reductio ad absurdum

A strategy for supporting a claim by showing that its contrary leads to an unacceptable consequence or contradiction.

Principle of non-contradiction

A proposition and its negation cannot both be true.

Empirical absurdity versus formal contradiction

An empirical absurdity conflicts with observation; a formal contradiction derives incompatible claims from assumptions.

Double-negation stability

A proposition is stable when its double negation implies the proposition: $\neg\neg P \Rightarrow P$.

Existence proof by contradiction

Assume no object has the required property, then derive a contradiction to establish that such an object exists.

Quiz questions

1. To establish $\neg P$ by refutation, which proof pattern is appropriate?Answer: Assume $P$, derive falsehood, and conclude $\neg P$.
2. Which rule expresses classical double-negation elimination?Answer: $\neg\neg P \Rightarrow P$
3. Which statement describes intuitionistic logic's treatment of reductio?Answer: It permits proving $\neg P$ by deriving falsehood from $P$, but not unrestricted double-negation elimination.
4. In the standard proof that $\sqrt{2}$ is irrational, what contradicts the assumption that $a/b$ is in lowest terms?Answer: Both $a$ and $b$ must be even.
5. How does resolution-based automated theorem proving test whether a conclusion follows from hypotheses?Answer: It adds the negated conclusion to the hypotheses and searches for a contradiction.