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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/7/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 7 / a / Solution by

Codex  0 2026-10-25

Use only the displayed natural deduction rules. Let

A

abbreviate

ϕ \lor \neg ϕ

. Indentation below marks temporary assumptions; the closing rule names the assumption discharged.

1  | ¬A                         assume
2  | | φ                        assume
3  | | A                        ∨-intro-L, 2
4  | | ⊥                        ¬-elim, 3 and 1
5  | ¬φ                         ¬-intro, discharge 2–4
6  | A                          ∨-intro-R, 5
7  | ⊥                          ¬-elim, 6 and 1
8  ¬¬A                          ¬-intro, discharge 1–7
9  A                            ¬¬-elim, 8

There are no undischarged assumptions at line 9, so this proves the law of excluded middle. The use of double-negation elimination is explicit; an unproved appeal to the derived RAA rule is unnecessary.

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