PHIL 250: Introduction to Symbolic Logic

tennant.9@osu.edu

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RULE of CLASSICAL REDUCTIO AD ABSURDUM (CR)

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______(i)

	¬A
	:
	:
	⊥

_______(i)

A

Explanation in words: In order to prove a conclusion A, one may assume its negation ¬A 'for the sake of argument', and derive absurdity (⊥), as indicated by the descending dots. In taking the indicated step of classical reductio, one thereby rests the conclusion A on that combination of assumptions, other than ¬A, from which absurdity has been deduced. The assumption ¬A is thereby discharged.

Since this is neither an introduction nor an elimination rule, there is no rule 'corresponding' to it! The classical rule of reductio ad absurdum (CR)'symmetrizes' the operation of negation: it enables one to deduce A from ¬¬A. (The converse implication can be proved using only the introduction and elimination rules for negation.)

Compare and contrast CR with the introduction rule for negation, and see how the negation sign has changed position. In negation introduction, the negation sign is dominant in the conclusion; whereas in CR, the negation sign occurs in the (eventually discharged) assumption for reductio.

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[¬I] [∧I] [∨I] [→I] [∀I] [∃I]
[¬E] [∧E] [∨E] [→E] [∀E] [∃E]
[EFQ]
[LEM] [Dil] [CR] [DNE]

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