# What does this statement imply? (formal logic)

"Pia hits only if Mia misses, unless, if Mia misses, then Pia misses too."

Does this statement imply that Pia misses? and how can this alternatively be answered by using the rule of interchange?

Also, how would you formalise the statement using truth-functional connectives and sentence variables?

• Assuming "misses(x)" means "¬hits(x)", then "(¬¬hits(mia) ⇒ (hits(pia) ⇒ ¬hits(mia))) ∧ (¬hits(mia) ⇒ ¬hits(pia))" seems like the most sensible transcription. Commented Mar 9, 2018 at 12:32
• Just using "X, unless, if Y, then Z" as "(¬Y ⇒ X) ∧ (Y ⇒ Z)" and "M only if N" as "M ⇒ N". Commented Mar 9, 2018 at 12:39
• Though maybe "X, unless, (if Y, then Z)" would be a better bet. Commented Mar 9, 2018 at 16:17

I would symbolize as follows: (P & ~M) --》 (~M & ~P).

The consequent call also be written as ~(M v P) instead.

SYMBOL KEY: P = Pia hits.

M = Mia hits

~= Not

V = Either . . . Or.

---》= IF . . . THEN.

The main connective above is the conditional. I tried using the conjunction as the main connective and derived only false values. The truth table for the above symbolization is identical to standard conditionals.

I would not say Pia misses is implied. All that is said is there is a relationship in a specific order that should be recognized.

One could also use this context: (P & ~M) V (M V P).

This would yield a standard disjunction truth table.

Defining “miss” as “non-hit”, the premises say:

1. Only if M non-H, then P H.

2. Unless if M non-H, then P non-H.

The combined premises read like this: if Mia misses, then Pia both hits and does not hit. As written, these statements look mutually contradictory.