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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/7/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 7 / c / Solution by

Codex  0 2026-10-31

This instance of Peirce's law uses the supplied classical double-negation elimination rule. First derive

P \to Q

while assuming

\neg P

, using explosion inside a further temporary assumption.

1. (P → Q) → P          premise
2.   ¬P                 assumption [i]
3.     P                assumption [j]
4.     ⊥                ¬E, 2, 3
5.     Q                ⊥E, 4
6.   P → Q              →I, 3–5, discharging [j]
7.   P                  →E, 1, 6
8.   ⊥                  ¬E, 2, 7
9. ¬¬P                  ¬I, 2–8, discharging [i]
10. P                   ¬¬E, 9

This is a formal natural deduction proof from exactly the given premise; no unproved excluded-middle lemma is needed.

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