# Parentheses in Conditionals

Mr. Nance,

Could you please explain how the placement of the parentheses is determined in Test 1, Form A, #12 of Intermediate Logic? My student wrote “(M ⊃ P) ⊃ ~C,” but the answer key says “M ⊃ (P ⊃ ~C).”

The original proposition is,

“If we see a movie then if we eat popcorn then we do not eat candy.”

This proposition has the overall form pq, where p is the antecedent, “We see a movie” (abbreviated M) and q is the consequent. This consequent is another complete conditional: “If we eat popcorn then we do not eat candy.” This is the (P ⊃ ~C). Because it is a complete proposition in itself, this consequent gets placed in parentheses.

It will be important later to note that propositions of the form p ⊃ (qr) are equivalent to propositions of the form (p • q) ⊃ r. The given proposition could be understood in this way. “If we see a movie and eat popcorn, then we do not eat candy.” Notice in this form, “We see a movie and eat candy” is the antecedent, and it is a complete proposition in itself, and thus gets placed in parentheses.

Blessings!