# What is the logical law proving “if not p then q” is equivalent to “p or q”?

I know that (¬p → q) ≡ (p v q) from comparing the truth tables. But is there a law that states this? Something like the laws of propositional logic: idempotent, associative, commutative, distributive, etc.?

The rule is called material implication in classical logic. Here's Wikipedia's description:

In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and that either form can replace the other in logical proofs.

In more detail one has (¬p → q) ≡ (¬¬p v q) by material implication. By double negation elimination one can change ¬¬p to p.

This is also presented in Wikipedia's list of logical equivalences involving conditional statements.

Wikipedia contributors. (2019, July 6). Material implication (rule of inference). In Wikipedia, The Free Encyclopedia. Retrieved 00:04, August 31, 2019, from https://en.wikipedia.org/w/index.php?title=Material_implication_(rule_of_inference)&oldid=905048316

• Why do you call this classic logic? If this type of logic was not around until the 19th century. That is Aristotelian logic doesn't use this nor did medevial philosophers. – Logikal Aug 31 '19 at 1:06
• @Logikal Because I needed double negation elimination, this would not be intuitionistic logic. So I used the word "classical" logic to note that this should not be applied as a law indiscriminately across all logics. Aristotelian logic would have been ancient or term logic using categorical propositions: en.wikipedia.org/wiki/Categorical_proposition Also a natural deduction proof of material implication itself may need the law of the excluded middle although I am not sure if there isn't a way around that. – Frank Hubeny Aug 31 '19 at 2:37
• @Logikal It's just the name we have for the bivalent logic developed by Frege and Russell among others, that includes proposotional, first order and i suppose the higher order logics you'll normally see. It's "classical" to us becuase it's dominant in maths and philosophy, and the other logical systems are alternatives put forward due to problems people have with classical logic. The language is a bit strange here though; we'll say that modal logic is an extension of classical logic, but also that there's classical and non classical modal logics, so i don't think the name is treated reverently – Daniel Prendergast Sep 1 '19 at 22:07