Self-study
Support If you climb mountains, you will not live to a ripe old age. ███ ███ ████ ██ █████ ██████ ███ █████ ██████████ ██████████ ██ ███ ████ ██ █ ████ ███ ████ ███ ████ ████ ████ ██████
Chaining Conditionals
As a climber who will apparently die young and not-bored, I quite enjoy this question. This stimulus chains two conditional statements together, yielding a valid inference. There are a few different ways to think about the argument’s structure, but no matter what you’ll have to do some contrapositive shenanigans. I’ll present one translation here in both English and formal logic, explicating which moves I’m making:
English
P1: If you climb, you won’t live.
P2: If you aren’t bored, you must have climbed.*
________
Con (P2 + P1): If you aren’t bored, you won’t live.**
Con (Contra): If you lived, you were bored.***Logic
P1: Climb → /Live
P2: /Bored → Climb*
________
Con (P2 + P1): /Bored → /Live**
Con (Contra): Live → Bored***
*Different methods of translating “unless” click for different people in different contexts. Here, I’d argue the more intuitive translation is actually “if you don’t climb, you’re gonna be bored.” Then again, the version presented above makes for a cleaner conditional chain – we can just link P2 right up with P1.
**We reach this conclusion by forming a conditional chain from premise 2 to premise 1.
***To match the stimulus’ wording, we need to take the contrapositive of the inference we drew from our conditional chain.
13.
Which one of the following ████ ███████ █████████ ███ █████████ ██ ███ ████████ ██████
a
If you do ███ ███ ██ █████ ███ ████ ███ █████ ███ ██ █████ ███ ███ ████ ███ ██ ████ ██ █████ ██ ███ ██ ███ █████ ███ ██ █████ ██████████ ███ ████ ███ ██ █████
(A) is wrong because the word “must” in this context introduces a value judgment – it makes a normative statement about what you should do. The stimulus’ claims are all descriptive – they simply comment on what is.
You should aspire to notice this difference rapidly and without the need to diagram anything.
b
If you do ███ ████ █████ ███ ████ ███ █████ ███ ████████ ███ ███ ████ ██ █████ ████ ████ ██████ ███ █████ ██████ ███ ████████ ███████████ ██ █████ ███ ████████ ███ ████ ████ ██ █████ ██ ███████ █████
(B) is wrong because of the conceptual shift between “enjoyment” and “relaxation.” These concepts aren’t the same thing, so they can’t form a conditional chain.
You should aspire to notice this difference rapidly and without the need to diagram anything.
c
If you work ███ ████ ██████████ ███ ████ ███ ███████ ████ ██████ ████████ ███ ███ ████ ███████ ████ █████ ████ ██████ ███ ████ ███ ████ ██████████ ██████████ ██ ███ ███████ ████ ██████ ████████ ███ ████ ████ █████████ ████ █████ █████
(C) mirrors the stimulus’ conditional chain. Like the stimulus, it also uses the word “unless” in its second premise, which makes it a lot easier to match up with whatever diagram or mental model you built from the stimulus. Here’s a side-by-side comparison of the English translations:
Stimulus
P1: If you climb, you won’t live.
P2: If you aren’t bored, you must have climbed.
________
Con (P2 + P1): If you aren’t bored, you won’t live.
Con (Contra): If you lived, you were bored.(C)
P1: If you work, you won’t get better at guitar.
P2: If you fulfilled your civic duty, you must have worked.
________
Con (P2 + P1): If you fulfilled your civic duty, you didn’t get better at guitar.
Con (Contra): If you got better at guitar, you must have neglected your civic duty.
And here’s the logic:
Stimulus
P1: Climb → /Live
P2: /Bored → Climb
________
Con (P2 + P1): /Bored → /Live
Con (Contra): Live → Bored(C)
P1: Work → /Guitar
P2: Civic → Work
________
Con: Civic → /Guitar
Con (CP): Guitar → /Civic
d
If you do ███ ██████ ███ ████ ███ ██ █ ████ ████████ ███ ███ ████ ██████ █████████ ██████ ██████ ███ ██████ ██████████ ██ ███ ██████ ███ ████ ███ ████ ██████ █████████ ███████
(D) fails when its second premise repeats “not train” as a sufficient condition. This makes it impossible to form a conditional chain using (D)’s premises.
English
P1: If you don’t train, you won’t be good.
P2: If you don’t train, you’ll get exhausted.Logic
P1: /Train → /Good
P2: /Train → Exhausted
From there, (D)’s conclusion is just the inverse of premise 2 – it negates both the sufficient and the necessary condition.
e
If you spend ███ ██ ████ ██████ ███ ████ ███ ██████ ████████ ███ ███ ████ ██████ ██████ ██████ ███ █████ ███ ██ ████ ██████ ██████████ ██ ███ ██████ ████████ ███ ████ ███ ██████ ███████
(E) is wrong because its conclusion has one mistakenly negated term. Premises 1 and 2 are fine – they create a conditional chain from not-hungry to spend to not-wealthy. Rather than validly concluding “if you’re wealthy, you must be hungry,” however, (E) concludes “if you’re wealthy, you must not be hungry.” Here’s the logic:
P1: Spend → /Wealthy
P2: /Hungry → Spend
________
(Valid Con: /Hungry → /Wealthy
(Valid Contra: Wealthy → Hungry)
(E)’s Con: Wealthy → /Hungry