Double not arrows for use with certain conditionals

[lsatmaniac]

Dear Powerscore,

I'm currently going through the LGB and I'm in chapter 5, grouping, and I find myself writing out additional double not arrows in the margins as I read along. For example: on Page 271, I read, "[there are only 2 groups] If P is in group 2, then Q must be in group 2". Then I write out the formal logic: P2 --> Q2, Q1 --> P1, then, thinking about the implications, I write out,"Q1<--/--P2" next to the formal logic, as its the only preclusion given by this rule. Do you think something like this would be useful during logic games or just be a waste of time/energy?

LSATmaniac

[Nikki Siclunov]

Hi lsatmaniac,

You are, of course, correct in your deduction. Given the original rule (P2 Q2), and its contrapositive (Q1 P1), we can safely infer that Q1 P2. However, this is not terribly useful in this case. Why? Because it merely suggests a prohibition: You're basically saying that if Q is in 1, then it cannot be true that P is in 2 (and vice versa). This doesn't need repeating: you already know from the original rule what must be true if either Q is in 1, or else if P is in 2. Positive rules (mandates) are usually more helpful than negative rules (prohibitions). This is particularly true in dual-value grouping game systems, where the contrapositive of a conditional rule is a positive statement, as you noted yourself. So, while your double-arrow notation is not wrong, it is probably not terribly useful.

If I were you, I'd focus my attention on whether P and Q must always be in the same group. They do not: placing P in 1 and Q in 2 does not violate the original rule, so it is possible that they end up in two different groups.

Compare this to a similar rule, where your inference would actually be quite useful:

P1 Q2
(Contrapositive: Q1 P2)

Here, it is clearly impossible for P and Q to be placed in the same group as each other. So, we can infer that P Q. This inference is useful, because it doesn't tie P or Q to any particular group: it applies to the relationship between them regardless of which group either of them occupies.

Does this make sense? Let me know.

Thanks!