Talk:Material implication (rule of inference)

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)

Article should be restored to a link
Prior to 2012 this article was merely a link to Material conditional. (Actually it should be the other way round: logicians usually refer to this logical operation as material implication.) Then some creative editor dreamed up the idea that material implication must be a rule of inference. This is simply false. Material implication, also known as classical implication, is a logical connective that is distinguished from other implications (such as intuitionistic implication and relevant implication) by virtue of satisfying Peirce's Law. When interpreted in two-valued logic material implication P &rarr; Q is false when P is true and Q is false and otherwise is true; as such it is equivalent to not-P or Q when negation and disjunction are present. Wikipedia should not be promoting the misleading idea that material implication is a rule of inference, which has no basis in logic. Vaughan Pratt (talk) 20:35, 21 October 2013 (UTC)
 * I'm afraid this is a rule of inference, although (I believe) rarely under the name "material implication". — Arthur Rubin  (talk) 10:46, 23 October 2013 (UTC)i
 * On the contrary what you're referring to is a logical equivalence. Every logical equivalence vacuously gives rise to a derived rule of inference in which either side of the equivalence can be substituted for the other.  Because it is vacuous one never dignifies any particular logical equivalence as a rule of inference in its own right.  Conversely no rule of inference that has its own name gives rise to a logical equivalence.  You will not find any reputable source that refers to this particular logical equivalence as a rule of inference.  Furthermore material implication is neither a logical equivalence nor a rule of inference, it is an operation and it is simply wrong to call it anything else. Vaughan Pratt (talk) 20:56, 28 October 2013 (UTC)
 * Vaughan Pratt is correct. This article is completely broken.  For correct usage see https://www.britannica.com/topic/implication#ref289368 Cerberus (talk) 00:38, 29 February 2020 (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)