6.2.11 Recursion in the Equivalence Rules

For the convenience of the user, the equivalence rules work recursively. For example

• ~(~P v Q) v (~R v S)
• ——–
• (P → Q) → (R → S)

This is an example of using implication recursively. Recursion only works if the rule is being used the same way. For example, removing multiple parentheses with association is fine, however adding and removing parentheses with association is not.

Commutatvitity and idempotence work differently than the others when it comes to recursion. If commutativity is applied to a connective, then no parts of that connective, or parts of those parts, and so on, can be used in commutativity. However, other parts from the sentence can be rearranged. The same goes for idempotence.