Question regarding contrapositives

[kyleenloe]

How detrimental is flipping the order of the contrapostives? For example, Steve will be team captain unless Rasheed is team captain. (substitute * for crossing out the letter)
R ----> *S
S ----> *R
In my mind I read that as if R no S, and If S no R. However, on the flashcards, the answer is *S ----> R *R -----> S

Another: Only if Hannah is hired then Isaac will be hired. In my mind I read that as If I then H, contrapostive If no I (*I----> *H) then no H. Flashcard answer: *H ----> *I - same answer, just flipped. How detrimental is this flipping? If it's crucial I put them in the right order, is there a tip or trick you can give me to divert my brain from wanting to continue this flipping?

Thanks!

[Steve Stein]

Hey Kyleen,

That's a good question—you definitely need to put those in the correct order! To clarify the issue, let's consider this real-world example:

• You won't win the lottery unless you play.

If we apply the Steve/Rasheeed analysis, we will come to the invalid conclusion that:
play win (this wrongly suggests that playing the lottery guarantees a win)

Instead, we can apply PowerScore's Unless Equation: the condition that follows "unless" is the necessary condition, and the other condition gets negated to become the sufficient:

You won't win the lottery unless you play: win play (if you win the lottery, you must have played)

And the contrapositive: play win (if you don't play the lottery, you cannot win the lottery)

Let's apply this analysis to your example:

• Steve will be captain unless Rasheed is captain

Again, what follows the unless (Rasheed becoming captain) is the necessary condition, and the negated version of the other condition is the sufficient.

Steve Rasheed (if Steve is not captain, Rasheed will be captain)

And the contrapositive: Rasheed Steve (if Rasheed is not captain, Steve will be captain).

Conditional reasoning can be tricky at first, but it gets easier with practice. Please let me know whether this is clear—thanks!

~Steve

[kyleenloe]

Yes! Thank you for your clear explanation