The reductio ad absurdum rule discharges an assumption from a derivation of contradiction and concludes . It is derivable using negation introduction followed by double-negation elimination:
(1)
Conversely, if RAA is supplied, double-negation elimination follows from this natural deduction derivation:
1. ¬¬φ                  premise
2.   ¬φ                 assumption [i]
3.   ⊥                  ¬E, 1, 2
4. φ                    RAA, 2–3, discharging [i]
Thus replacing double-negation elimination by RAA leaves these classical deductions available. RAA is not being assumed as an intuitionistic rule.
Use disjunction introduction, negation introduction and conjunction introduction in the following natural deduction proof. Indentation marks the assumptions discharged at their closing lines.
1. ¬(φ ∨ ψ)             premise
2.   φ                  assumption [i]
3.   φ ∨ ψ              ∨I-left, 2
4.   ⊥                  ¬E, 1, 3
5. ¬φ                   ¬I, 2–4, discharging [i]
6.   ψ                  assumption [j]
7.   φ ∨ ψ              ∨I-right, 6
8.   ⊥                  ¬E, 1, 7
9. ¬ψ                   ¬I, 6–8, discharging [j]
10. ¬φ ∧ ¬ψ             ∧I, 5, 9
The two separate temporary assumptions establish the two required negations; neither is retained in the conclusion.
This instance of Peirce's law uses the supplied classical double-negation elimination rule. First derive while assuming , 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.
Write the new exclusive or connective as . Give it the two introduction rules
(1)
and the elimination rule
(2)
In elimination, discharge the two pairs of case assumptions; retain the assumptions needed for the major premise and the shared context . These are natural deduction rules for exactly one of , not merely at least one.
For a syntactic check, derive from by : in the first case introduce and then the left disjunct; in the second case introduce and then the right disjunct. Conversely, from that disjunction use disjunction elimination; conjunction elimination extracts in the first branch and in the second, so the respective introduction rule yields in both. This proves equivalence to the ordinary truth-functional specification and explains both the introduction and elimination rules.
Use the exclusive-or natural deduction rules defined in part (i). The only extra lemma needed is double-negation introduction, derived from the given rules by assuming under , obtaining contradiction, and discharging that assumption. The full proof incorporates that derivation:
1. ¬P ⊕ Q               premise
2.   ¬P                 first ⊕E case assumption [i]
3.   ¬Q                 first ⊕E case assumption [j]
4.   P ⊕ ¬Q             ⊕I₂, 2, 3
5.   ¬¬P                second ⊕E case assumption [k]
6.   Q                  second ⊕E case assumption [l]
7.   P                  ¬¬E, 5
8.     ¬Q               assumption [m]
9.     ⊥                ¬E, 6, 8
10.  ¬¬Q                ¬I, 8–9, discharging [m]
11.  P ⊕ ¬Q             ⊕I₁, 7, 10
12. P ⊕ ¬Q              ⊕E, 1, 2–4, 5–11,
                        discharging [i], [j], [k], [l]
In the first case the second introduction rule is instantiated with right operand . In the second case the first introduction rule requires , which is exactly the derived line 10.

Ancestors (10)

  1. Digital systems, linear algebra, introduction to formal proof
  2. 2018
  3. Past exam of the year 1 of the computer science course of the University of Oxford
  4. Year 1 of the computer science course of the University of Oxford
  5. Computer science course of the University of Oxford
  6. Undergraduate course of the University of Oxford
  7. Course of the University of Oxford
  8. University of Oxford
  9. List of universities
  10. Home

View article source

Discussion (0) New discussion

There are no discussions about this article yet.

Articles by others on the same topic (0)

There are currently no matching articles.
See all articles in the same topic Create my own version