Logic / Unit 5 Logic 214 worked example, 60 to 90 seconds
Proof Using Replacement Rules
The idea
Replacement rules let a statement be rewritten into an equivalent form anywhere it appears, which often opens a route that the implication rules alone would not reach.
In practice
Premises: tilde A wedge B, and tilde B. Prove tilde A. First route. Line three: A horseshoe B, from line one by implication, since a disjunction with a negated left side is equivalent to a conditional. Line four: tilde A, from lines three and two by modus tollens — the conditional with its consequent denied yields the antecedent denied. Second route, same premises. Line three: tilde A, from lines one and two by disjunctive syllogism, since denying one disjunct leaves the other standing. One line instead of two, no rewriting needed. Both proofs are complete and both are correct. Replacement rules do not merely permit proofs; they permit choices between them, and the shorter route is not always the one that presents itself first.
Where people slip
Replacement rules work in both directions and on any part of a line, while implication rules run one way and only on whole lines. Rewriting a fragment of a line with an implication rule produces a step that no rule licenses, and the error is quiet because the resulting line often happens to be true.
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