Indirect Proof Walkthrough
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Indirect proof assumes the denial of what it wants, derives an explicit contradiction from that assumption, then discharges it — the contradiction showing the assumption could not have held.
Premises: A wedge B, and tilde B. Prove A. Line three: assume tilde A, indenting from here. Line four: B, from lines one and three by disjunctive syllogism, since the assumption denies the left disjunct. Line five: B dot tilde B, from lines four and two by conjunction. That line is an explicit contradiction, written out rather than merely noticed. Discharge. Line six: A, citing lines three through five as an indirect proof. Now the honest observation: disjunctive syllogism on lines one and two proves A in a single step, with no assumption at all. Indirect proof was not needed here, which is exactly why it is the right example to learn on — the machinery can be watched without a hard problem competing for attention.
The contradiction has to be written as a line, not merely spotted. A proof that stops at the moment two lines look incompatible has skipped the step that licenses the discharge, and skipping it becomes a habit that fails on longer proofs where the incompatibility is several moves apart.
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