Unless as a Contrapositive Must Distribute!

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theZeigs

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Unless as a Contrapositive Must Distribute!

Postby theZeigs » Thu Apr 05, 2018 9:51 pm

Greetings all,

I think I've stumbled onto something that I haven't been able to find anywhere else on these forums, or in any other prep material (at least explicitly spelled out). But, it seems important enough to bring up. The rule is this:


When you have an unless, and an "AND" or "OR", you need to distribute the "unless" or "IF NOT" across these two (or more) clauses, and invert the "AND" to "OR" or "OR" to "AND."


I discovered this when trying to look at contrapositives.

DIAGRAM:

Unless A and B, no C:

~(A + B) --> ~C
= ~A OR ~B --> ~C
C --> A
AND
C--> B

When you distribute the "IF NOT" you flip "AND" to "OR" and vice versa.

Example 1:
It’s a beautiful day and I want to do something outside, rather than go to the store. I could do something if it rained yet was warm, or if it was cold but dry. So, unless it is raining and cold, I will not go to the store.

You would think to phrase this as:

"If not rain and if not cold, then I won't go to the store"
~rain + ~cold --> ~store

Taking the contrapositive:
Store --> rain
OR
Store --> cold

...but this is not true!!! Just look at the statement, it even contradicts precisely how I set it up!

Example 2:
I will not laugh unless I find something funny and am smiling

Again, you'd think:
"If not find funny and if not smiling, am not laughing" :
~smiling + ~find something funny --> ~laugh

Laugh --> smiling
OR
Laugh --> find something funny

...but again, here, common sense would tell you that this isn't right.

Example 3:
Here's one that really cements it, IMO:
One will not become president unless they are over 35 years old and born in the USA.

"If not over 35 and if not born in USA, then not president"
~over 35 + ~born in USA --> -president

Contrapositive:

president --> over 35
OR
president --> born in USA

But we know that, even though OR means "both COULD be true," here, BOTH MUST BE TRUE!

But, this even works on "OR" statements:

EXAMPLE 4:

Unless it is cheap or I really like the band, I won’t go see the concert


"If not cheap or if not like band, then won't go see concert."
~cheap --> ~concert
OR
~like band --> ~concert

Contrapositive:

concert --> cheap tickets
AND
concert --> liked the band

Again, this is obviously not what I meant.

THE SOLUTION:
"Unless" means "if not," and the "if not" should apply and distribute to the "and" or "or" statement:

"if not (raining + cold)"
"if not (funny + am smiling)"
"if not (over 35 + born in USA)"
"if not (cheap tickets OR like band)"

When you do this, you would then turn the "and" to make it "or," and turn "or" into "and." Then, your contrapositive would become "and" or "or" again:

store --> rain + cold
laughing --> funny + am smiling
president --> over 35 + born in USA
go to concert --> cheap tickets OR like band



PT 39, S2, Q11:
A gift is not generous unless it is intended to benefit the recipient and is worth more than what is expected or customary in the situation...
...[not relevant here, but so that someone doesn't think I'm cherry picking language, the rest of the stimulus is...]...a gift is selfish if it is given to benefit the giver or is less valuable than is customary.

Stem: Which one of the following judgments most closely conforms to the principle above?

I would have diagramed, initially:
~intended to benefit recipient + ~worth more than expected/customary --> ~generous

Which controposes to:
Generous --> intended to benefit
OR
Generous --> worth more than expected/customary

But again, this is not what the English means… it has to be both intended to benefit the recipient AND worth more than expected or customary!

So, I take my alternate, where you distribute:

~intended to benefit recipient ---> ~generous
OR
~worth more than expected/customary --> ~generous

then you have:

generous --> intended to benefit + worth more than expected
which can be written:
generous --> intended to benefit
AND
generous --> worth more than expected

The correct answer choice is… “Olga gives her daughter a computer as a graduation gift. Since is the gift that all children in Olga's family receive for graduation, it is not generous.” i.e. since ~worth more than expected/customary --> ~generous

Note that Manhattan explains this by splitting:

generous --> intended to benefit
generous --> worth more

As I had it above. But at no point do they teach you to "distribute" across when using "UNLESS"

Hope this helps someone. This was my hour and a half of studying for the night. FYI, if you liked this post, see some of my older posts also dissecting words for logic. Need to update those, but the ideas are still there.

yezwaj

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Posts: 1
Joined: Mon Nov 06, 2017 9:38 pm

Re: Unless as a Contrapositive Must Distribute!

Postby yezwaj » Sun Apr 15, 2018 5:00 am

Thank you -- this was extremely helpful.



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