About Dropping Parentheses

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Although I am not planning to test you on the correct application of the parenthesis-dropping convention (text, pp. 9-10), the exercises in the text often use it. Here's a quick guide to putting back in parentheses that have been dropped.

• Rule 0: The outside parentheses on wffs always get dropped, so you can always add them (exception: remember that there are no outside parentheses that accompany ~).
• Rule 1: Never add a parenthesis that comes between a ~ and anything else.
• Rule 2: You may add parentheses around & or v and the wffs they connect.
• Rule 3: After you have finished using Rule 2, if the wff contains both → and ↔ place parentheses around → and the wffs it connects.

These three rules will get you through most of the problematic cases.

Example 1

~P v Q → R

Add parentheses around the v, but don't come between ~ and anything:

(~P v Q) → R

Add a pair on the outside:

((~P v Q) → R)

Example 2

Q & ~R ↔ P v S

Add parentheses around & and v, but don't come between ~ and anything:

(Q & ~R) ↔ (P v S)

And add them on the outside:

((Q & ~R) ↔ (P v S))

Example 3

P → Q & R ↔ P & Q → R

First use Rule 2:

P → (Q & R) ↔ (P & Q) → R

Now, since thre is nothing left to apply Rule 2 to and there is a ↔ in the wff, apply Rule 3:

(P → (Q & R)) ↔ ((P & Q) → R)

Finally, apply Rule 0:

((P → (Q & R)) ↔ ((P & Q) → R))