Embedded Conditionals -- "Carve-out" vs. Translation

BlindReviewer

Hi Everyone,

I've been going back to the core curriculum to brush up on a few things, and I realized that I rely more on what JY calls the "carve-out" method of dealing with embedded conditionals rather than the translation. (It just feels more intuitive for me.) But as I was trying to match up my "carve-out" understanding with the translations JY does, I ended up with a few questions.

Here's the link to the CC lesson with the example I'm considering: https://7sage.com/lesson/mastery-embedded-conditional/

"If the seeds are planted in the winter, then flowers will not blossom unless fertilizer is applied."

JY draws these two statements using the "and" translation:

SPW and FB -> FA

SPW and /FA -> /FB

So I like to think about this by relying heavily on the "unless" factor at the end:

1) If fertilizer is applied, then I have to negate (SPW -> /FB) into (SPW -> FB). [FA -> (SPW -> FB)]
2) If fertilizer is NOT applied, then the "carve-out" exception to the rule doesn't hold, so the relationship is still (SPW -> /FB).

My question is, aren't JY's statements (and my own) supposed to be biconditionals? For example:

1) FA -> (SPW -> FB)
2) /FA -> (SPW -> /FB) the contrapositive is (SPW -> FB) -> FA

So together, FA <-> (SPW ->FB)

Because right now, in JY's statements, if I know fertilizer is applied (FA), nothing happens. If I plug in FA, then it fulfills the necessary for the first translation and fails the sufficient for the second. But isn't it true that if FA, then it must be true that if seeds are planted in the winter, that the flowers will blossom?

BlindReviewer

Bump -- still looking for help on this!

Logic Gainz

You're essentially saying that if a necessary condition is affirmed, its sufficient condition must also be affirmed which is something we know we can't do because there could be other necessary conditions involved that aren't mentioned in the stimulus.

By applying fertilizer (FA), we can't infer that (SPW -> FB) like you mentioned, because there could be other necessary conditions involved with planting seeds in the winter in addition to fertilizer being applied that must occur before the flowers can blossom.

Maybe you have to water the plants too. Maybe you have to plow the land before you plant the seeds. There could be other necessary conditions in addition to fertilizer being applied which is why your translation doesn't work. We just know that if you plant seeds in the winter AND they blossom, along with all the other potential necessary conditions being satisfied above, we know that fertilizer being applied was among them.