Contrapositive of embedded conditionals?

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nessa.k13.0

It depends on the conditional statement. Do you have an example @Thoughtful ?

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TimLSAT180

I'll take a stab at this because I came across the exact same problem when I doing an LG section for PT32 (Game #1 Rule#3). So, basically you want to think of everything that comes before the "unless" as A and everything that comes after "unless" as B: A unless B. So we have A(If the teacher warned the students, then the students did not listen) unless B(the fire alarm was quiet). So we know that for "unless" statements, we translate it into /B -> A. So, then we have /B(If the fire alarm was not quiet), then (if the teachers warned the students, then the students did not listen). I hope this helps! @"nessa.k13.0" can also add on to this if I missed anything.

Cant Get Right

I think this is a structure that is actually really interesting. So:

(A --> --> C

This means that if A and B exist in this conditional relationship with each other, then C. We're actually free to play with the A --> B conditional however we like. We can confirm A or deny B thus triggering the conditional, or we can deny A or confirm B and make it irrelevant. It doesn't actually matter how we manipulate that conditional as long as the conditional relationship exists. C is necessary as long as the A and B remain in the conditional relationship.

So the contrapositive:
/C --> /(A -->

The negation is the relationship between the terms within the embedded conditional--not necessarily the terms themselves. This means that if we deny C, A and B do not exist in their conditional relationship. There is actually no reason though that we couldn't have /C, A, B. It's just that the A and B would retain no conditional relationship with each other and their presence with /C would be coincidental. Once you pile language on top of this logical structure, it's easy to see how crazy difficult something like this can be!

Cant Get Right

Lol, it turned my "B)" into a little guy.

lenelson2

lol, I was wondering where that smiley face came from. Thanks for the explanation!

DumbHollywoodActor

@"Cant Get Right" said:
I think this is a structure that is actually really interesting. So:

(A --> --> C

This means that if A and B exist in this conditional relationship with each other, then C. We're actually free to play with the A --> B conditional however we like. We can confirm A or deny B thus triggering the conditional, or we can deny A or confirm B and make it irrelevant. It doesn't actually matter how we manipulate that conditional as long as the conditional relationship exists. C is necessary as long as the A and B remain in the conditional relationship.

So the contrapositive:
/C --> /(A -->

The negation is the relationship between the terms within the embedded conditional--not necessarily the terms themselves. This means that if we deny C, A and B do not exist in their conditional relationship. There is actually no reason though that we couldn't have /C, A, B. It's just that the A and B would retain no conditional relationship with each other and their presence with /C would be coincidental. Once you pile language on top of this logical structure, it's easy to see how crazy difficult something like this can be!

You've invented your own form of lawgic, emoti-lawgic. LOL!

nessa.k13.0

@"Cant Get Right" said:
Lol, it turned my "B)" into a little guy.

Hahaha I was wondering if I was missing something at first

zkchrumz

edited:

@Thoughtful said:
If the teacher warned the students, then the students did not listen unless the fire alarm was quiet.

WS -> (SL -> FQ)

I read this a bit differently than the others. (Chances are good I'm missing something.) I see:
IF the students were warned AND IF the fire alarm was NOT QUIET/was loud --> the students did not listen.

WS + ~FQ --> ~SL

In the same way that @"Cant Get Right" mentioned, the C here (the fire alarm being quiet) changes the whole equation. If the fire alarm IS QUIET, then the sufficient does NOT activate, we have no idea if the kids were listening; the conditionals here do not apply. If the fire alarm IS NOT QUIET, then the conditionals apply.

I see it this way by looking at the 'unless' as a sufficient condition indicator (unless = if not). So even before this conditional comes into play, we need to look to the quietness of the fire alarm. If it is NOT quiet, then the conditional applies. If it IS QUIET, then we are missing a piece of sufficiency and cannot trigger the conditional, as we are failing the sufficient condition. The items may or may not be present, we just cant actually PROVE that they are or are not.

Basically the logic is similar, I just find it easier to look at a compound sufficient. There may, however, be circumstances that my way fails, logically. Any thoughts?

tanes256

This has nothing to do with the OP but how do we use emojis?? I've been dying to know!

CinnamonTea

Great question. I also would like a bit more clarification on this. If the conditional we are using above is:

"If the teacher warned the students, then the students did not listen unless the fire alarm was quiet".
WS -> (SL -> FQ)

Then, according to the core curriculum, it would turn be equivalent to:
WS and SL ---> FQ

So what would the contrapositive be? Would it be ~FQ ---> ~WS or ~SL? (Using DeMorgan's Law?) Why?

Q.E.D

The scope ambiguity is killing me.

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CinnamonTea

Example statement from PT 62.S4.Q18: "If there are sentient beings on planets outside our solar system, we will not be able to determine this anytime in the near future unless some of these beings are at least as intelligent as humans."

Diagram: SB ---> (D ---->IAH)
SB = Sentient Beings
D = Determine
IAH = Intelligent As Humans

CinnamonTea

According to JY's lesson on Embedded Conditionals in "Advanced Logic" (CC),
The original statement would be:
1) SB ---> (D ---> IAH)

Which is logically equivalent to:
2) SB and D ---> IAH

What is the contrapositive of the "logically equivalent" statement above (#2)? Would it then be:
3) ~IAH ---> ~SB or ~D?

Would that be a valid contrapositive, considering #1 above is equal to #2, and the contrapositive of #2 is #3?

danielznelson

Hey! I think I can provide a satisfactory answer here.

The embedded conditional is effectively SB --> X, with X representing (D ---> IAH).

When we take the contrapositive, what are we taking the contrapositive of? X (D --> IAH) or SB --> X?

If we're trying to apply the contrapositive to the entire relationship, the end result, instead of SB and D --> IAH, be:

SB and /IAH --> /D

Check out the lesson below at around 4:30 to get JY's explanation as to why this is.

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

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