Biconditional Indicator: "either or, but not both"

miriaml7

I have struggled with translating biconditionals, so I have been drilling this skill using Manhattan Prep's "5lbs. Book of LSAT Practice Drills." Basically, you're given a conditional and asked to translate it. So far I have been getting them correct or have been able to identify where I went wrong. However, there's one particular part of the drill that I am stuck on, and was hoping someone could help me figure it out.

The statement reads, "If the trip includes Burkin Faso, it will include either Mali or Ghana, but not both."
My train of thought:

-"if" is Gl, sufficient
-"either or, but not both" is a biconditional indicator

BF- <----> M or G

/M and /G <-----> /BF

However, the answer key in the book says that the correct translation is:

BF------>M or G
M and G------>/B
/M and /G ------>/B

Why would this statement not be considered a biconditional? And why are there three possible answers?

Thanks in advance!

Logician

Hi @miriaml7, The answers provided in the back of the book are most certainly correct. Here's why:

1) M and G -----> /B : this is a correct translation because if both M and G are present, we can say with certainty that B cannot appear (as B necessitates one, not both of either M or G).

2) /M and /G -------> /B : this is also a correct translation because if we don't have M nor G, then we can say with complete certainty that we cannot have B, as once again, B necessitates either M or G.

3) B -----> M or G : This one is a little trickier to see as there are two types of "or" relationships and they are pretty much context dependent. One is the exclusionary or, which is the one noted above, we know this is an exclusionary "or" because of the "but not both" that has been added on, meaning we must see one at the exclusion of the other. The other "or" is inclusive and means at least one, at most two. for example /A ------> B.

The reason your translation (BF <----> M or G) is incorrect is because it does not accurately reflect the stated relationship. If all I told you was that M or G appeared, you could not conclude that B must appear. That's because B is sufficient to trigger either M or G but M or G alone is not sufficient to trigger B, that is why this is not a bi-conditional relationship.

Alternatively you could diagram this relationship as B -----> (M <------> /G).

I hope this helps!

miriaml7

@Logician I greatly appreciate your detailed response! I guess my lesson is that I shouldn't take the biconditional indicators verbatim. Any suggestions on how to distinguish between a biconditional indicator and a regular logical indicator?

Logician

@miriaml7 No problem! Well I think perhaps the most important thing is to be cognizant of the context. Yes, you can more or less rely on the indicator language to discern the type of relationship, but as you've witnessed there are situations where that alone will not suffice. So when I say pay attention to the context, let me give you an example of where this might apply.

ex. Tom must go to either Italy or France at midday tomorrow. this is a good example of the importance of context, if we solely paid attention to our indicators, we'd translate this is as
"/I ----> F". however, this would be an incorrect translation because it leaves open the possibility of there being both F & I, in other words, Tom going to both France and Italy at midday tomorrow, which we know is impossible because he cannot be in two places at the same time (unless of course we've already cracked the secret to human cloning). So, although the situation I presented does not indicate common bi-conditional language, we can say that based on the context it is implied. This is because we know Tom MUST go to either Italy or France, yet he cannot go to both at the same time, hence " I <-----> /F".

Conversely, If i told you John must buy milk or cookies at the store, this implies he must buy at least one of the two, but there's nothing excluding him from buying both. Thus, this would be a regular inclusive "or" relationship. /M -------> C.

Now, Going back to the question you presented yesterday, the statement reads "If the trip includes Burkin Faso, it will include either Mali or Ghana, but not both." Here you need to pay attention to the language more carefully, "IF the trip includes Burkin Faso" this is how we know it is not a bi-conditional, because it's not the case that is MUST include Burkin Faso. If the statement "Burkin Faso attend's the trip" were added, we could then go ahead and deduce a bi-conditional between M and G, as we would know that one of them MUST attend.

So the take away is pay attention to the both the language and the context of information being presented to you as they both impact the way it's translated into logic.

miriaml7

@Logician You are incredible!!! Thank you so much!!