Embedded Conditionals Issue

The 180 Bro_OVO

This is a two part question in reference to the embedded conditionals video.

https://7sage.com/lesson/mastery-embedded-conditional/

Part 1:

Put the following sentence into Logic and contrapose it:

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

SPW --> (~FA-->~FB )
which more clearly is:

SPW and ~FA --> ~FB

contrapositive:

FB --> ~SPW or FA

1) Is that correct?
2) If it is correct, that makes for an odd result. If the flowers do blossom, then the seeds were NOT planted or the fertilizer is applied. That doesn't seem to make any sense at all. What am I missing here???

Part 2:
Down in the comments someone asked:
What if the parenthesis are around the 1st and 2nd elements? ie. [A -> B] -> C.

JY responded with:

Original: (A–>B ) –> C
Contraposed: /C –> /(A–>B )
Group 3: /C –> /(/A or B )
De Morgan’s: /C –> (A and /B )
Simplified: /C –> A, /C –> /B

Or alternatively, we could have applied
Group 3 translation rule first: (/A or B ) –> C
Simplified: /A –> C, B –> C

Where did the Group 3 come from????

I got lost in his explanation there.
Can someone answer that question or maybe explain it in a different way. I feel like it's very obvious, I'm just not seeing it right now.

Thank you!

Creasey LSAT

For part 1, FB --> ~SPW or FA is correct. It also makes sense. We know from the original conditional statement that if seeds are planted in the winter, the only way flowers can blossom is with fertilizer being applied. If there's no fertilizer and the seeds were planted in the winter, we can kiss the prospect of seeing flowers blossoming goodbye. The contrapositive tells us that flowers are blossoming, so what does this imply? Well, either the seeds were not planted in the winter (maybe they were planted in the spring, summer, or some other time when fertilizer is not needed and a different set of conditions are at play...) OR fertilizer was applied (in which case it would be possible for the seeds to have been planted in the winter), OR, perhaps both.

DumbHollywoodActor

@"The 180 Bro_OVO" Where did the Group 3 come from????

Essentially, any conditional statement, A-->B, can also be stated as an "or statement" by negating the sufficient (Hence, Group 3) and turning the arrow into an "or", /A or B. Hopefully, you can see why:

All Jedis use the Force (J-->F) is logically equivalent to We have a non Jedi or a Force-user (/J or F). Keep in mind from the "or statements" in the in/out curriculum that this is an "inclusive or", that we could have non-Jedi and Force users as well. See https://7sage.com/lesson/not-both-v-or-truth-tables/ if what I said made no sense.

Q.E.D

Be careful about the scope. You showed a proof to this effect:

(A->B)->C
....
~C->A; ~C->~B

I gather that, in the discussion you alluded to, JY was simply showing hypothetical results for a situation where you start with that particular formula. For safety, I just want to point out that 'A->B->C' is ambiguous between '(A->B)->C' and 'A->(B->C)'. These are absolutely not the same thing.

Because ambiguity can lead to misinterpretation, it's important to avoid sentences like the following.

A & B -> C
C -> A & B
A & B or C
A or B -> C

These are just examples of sentences that mean different things depending on how you bracket them. The brackets set the scope of each truth function (and, or, not, ->), so these are called scope ambiguities.