talk:Material implication (rule of inference)#Gregbard's hatnote at Material conditional

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Switcheroo

As I recall, Douglas Hofstadter referred to this as the Switcheroo rule in his book Gödel, Escher, Bach. --81.138.95.57 (talk) 08:51, 12 September 2012 (UTC)

Can we not somehow justify the definition?

Rather than simply defining A \implies B \iff \neg A \lor B (or equivalently \neg (A \land \neg B)) , can this "definition" not somehow be justified by means of a formal derivation using a combination of more self-evident properties of implication, e.g. conditional proof, and the rule detachment?

Danchristensen (talk) 22:01, 16 April 2018 (UTC)

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For what it's worth, I have found it is possible to derive (in 25 lines) this "definition" using only the following rules of inference for natural deduction:

  1. Assumption
  2. De Morgan
  3. Eliminate \neg \neg
  4. Introduce \land
  5. Eliminate \land
  6. Eliminate \to (Detachment, Modus Ponens)
  7. Introduce \neg (Proof by contradiction)
  8. Introduce \to (Conditional proof)
  9. Introduce \leftrightarrow

--Danchristensen (talk) 16:00, 4 August 2021 (UTC)