Mr. Nance,

I have a question on Intermediate Logic, Exercise 17a, problem 5. To justify the conclusion (L • M) ⊃ N, the answer key says to use the rule of commutation from (M • L) ⊃ N. But the rule of commutation says (p • q) ≡ (q • p). How can I use that rule without switching the propositions, but switching the letters inside of a proposition? For example, in step 3, they are switching the propositions and not the letters inside the parentheses.

The rule of commutation allows you to switch any two propositions around a disjunction or a conjunction wherever they occur. Thus it can apply to the entire proposition, or just inside parentheses in a compound proposition. This is because commutation is a rule of replacement.

Thus, if in a proof you had the proposition ~A ∨ (B • C), the rule of commutation could apply to the entire proposition and change it into (B • C) ∨ ~A, or it could be applied to the proposition inside parentheses, and change it into ~A ∨ (C • B).

Blessings.