Advanced Conditional Logic Questions

[k.shaulis]

Hi everyone-- So I've been working on logic game strategy recently, and I came across a couple questions I had in the conditional logic section of the Manhattan LSAT Logic Games book.

First, #5 states "F cannot be selected unless H is also selected."

The Diagram for this one turns out to be F-->H, with the contrapositive being -H-->-F. I originally had down H-->F as my diagram. Is that a sufficient/necessary issue? I think that's where I went wrong on that one, but wanted to double check.

My other question is with #6, which states "L is selected if, and only if, M is also selected." From that, it follows that M--> L (contrapositive being -L-->-M), but apparently L-->M and its contrapositive are valid as well. I know it has something to do with the "if, and only if" part of the statement, but could someone explain to me exactly why that works? I'm a bit confused.

Thank you so much! Prepping for the October LSAT and stressed/excited/hopeful/anxious/all that jazz. But mainly stressed haha.

[timmydoeslsat]

First, #5 states "F cannot be selected unless H is also selected."

The Diagram for this one turns out to be F-->H, with the contrapositive being -H-->-F. I originally had down H-->F as my diagram. Is that a sufficient/necessary issue? I think that's where I went wrong on that one, but wanted to double check.

Unless introduces a necessary condition. We leave this necessary condition as it is presented, be it positive or negated.

The portion before the unless is a sufficient condition. We negate this portion. Sometimes you have a situation where nothing comes before the word unless, at which point you continue to interpret the part immediately after the word unless as the necessary condition. After this part, usually given by a comma, the sufficient condition will be there, and we will negate it.

Example:

Unless B happens, A happens.

~A ---> B

A cannot happen unless B cannot happen.

A ---> ~B

So in the example above:

F cannot be selected unless H is also selected

F ---> H

My other question is with #6, which states "L is selected if, and only if, M is also selected." From that, it follows that M--> L (contrapositive being -L-->-M), but apparently L-->M and its contrapositive are valid as well. I know it has something to do with the "if, and only if" part of the statement, but could someone explain to me exactly why that works? I'm a bit confused.

The statement presents "if" and "only if."

We know that "if" presents a sufficient condition.
We know that "only if" presents a necessary condition.

This means that using the two in unison will mean that we have both occurring. The easiest way to notate this is to use a biconditional arrow. (<--->)

Example: A if and only if B

A <---> B

Not B if but only if C

~B <---> C

If but only if.....If and only if....mean the exact same thing. The words and & but mean the same thing logically.

In case you got lost with how we arrived at the biconditional arrows, look below:

A if and only if B:

What do we know? We know if B then A.

B ---> A

We also know that A only if B.

A ---> B

As you can see, we can combine those. A <---> B

We know that A and B are both sufficient and necessary for the other variable. We also know that the absence of one variable will lead to the absence of the other variable.

[k.shaulis]

Thank you so much! That helps a lot!