Which one of the following, if substituted for the constraint that if R stays open, then M must also stay open, would...

Hayley on August 1 at 08:59PM

These kinds of questions

I have a really hard time with these types of questions. Any suggestions with how to attack them?

2 Replies

Ashlyn on August 21 at 03:32PM

Also wondering this ^^

Irina on August 21 at 06:36PM

@hayley & @ash,

Great question. The substitution questions are certainly some of the more challenging ones, so if you are short on time, it might be better to skip them from a time management perspective. One way to approach the substitution question is to see if the new constraint results in the same inference as the original one.

This game is governed by the following rules:

(1) N & R cannot both stay close
N close - > R open
R close - > N open

(2) N open -> L open
~ L open - > ~N open, i.e. L close -> N close

(3) R open - > M open
M close -> R close

(4) L& R cannot both stay open
L open ->R close
R open-> L close

We can combine these rules into the following chains:
R open ->L close->N close ->M open
L open -> R close -> N open

Now, the question asks us what rule would have the same effect as rule #3 - if R is open then M stays open. Notice that that is the only rule that governs M, so without it, we would not know if M has to stay open or close, so any substitute rule must also involve M.

Let's look at the answer choices:

(A) If L closes -> M must stay open
L close -> M open

We can see per rule (2) that if L close -> N close,
then per rule (1), N close -> R open,
and per rule (3), R open - > M open.

A new rule is equivalent to this chain: L close -> M open, and is thus the correct answer choice.

Let's briefly look at other answer choices:

(B) If L closes R must stay open

That's just an inference from rules (1) & (2) above:
L close -> N close per rule (2) - > R open per rule (1), but it does not tell us anything about M.

(C) If R closes, then L must stay open

This is an inference from rules (1) & (2) again:

R close -> N open per rule (1) -> L open per rule (2), but it fails to involve M.

Notice that (B) & (C) are contrapositives and are thus logically equivalent statements, which means we could also eliminate (B) & (C) based on this fact alone since we cannot have two correct answer choices.

(D) If L stays open, then M must close.

L open -> M close

This is an invalid inference, we know that L open -> R close per rule (4) -> N open per rule (1) but rule (3) only tells us that R open -> M open, and the only valid contrapositive if M close -> R close. We cannot infer that R close -> M close and hence L open - > R close - > M close as this rule suggests.

(E) If M stays open, then N must close.

This is also an invalid inference. Rule (3) is:
R open -> M open equivalent to:
M close -> R close

We cannot infer anything from M staying open.

Does this help?

Let me know if you have any further questions.