If $(p\implies (q \implies r)$ and $q \implies r$ then does $r \implies q$?

Question

The question is like this

The original Question:

By defining appropriate propositions, show that the following inference is valid. You will lose points if you do not give reasons for every statement you infer.

• If the prime interest rate goes up, then it is sufficient that unemployment goes down for prices to rise.

• However, unemployment goes down only if the prime interest rate goes up.

• Therefore, if prices do not go up, then unemployment does not go down.

-if P then Q for R (for example)

If the prime interest rate goes up, then it is sufficient that unemployment goes down for prices to rise

-However, Q if P

-Therefore, if $\neg R$ then $\neg Q$

So far, I have deduced

first statement $$p\implies(q\implies r)$$

second statement

$$p\implies q$$

and so it modus ponens can be used to deduce

$$q\implies r$$ $$q$$ $$\therefore r$$

which proves that

$$r\implies q$$

inturn proving the hypothesis

$$\neg r\implies \neg q$$

is this correct or have I gone horribly wrong at some point?

Answer 1

You don't have the symbolization correct. Yes, the statement

If the prime interest rate goes up, then it is sufficient that unemployment goes down for prices to rise

Does indeed translate to $P \rightarrow (Q \rightarrow R)$, assuming you are using:

$P$: Prime interest rate goes up

$Q$: Unemployment goes down

$R$: Prices rise

However, with this, the statement

However, unemployment goes down only if the prime interest rate goes up.

translates to $Q \rightarrow P$, rather than $P \rightarrow Q$

You seem to be unfamiliar with the difference between '$P$ if $Q$' and '$P$ only if $Q$', so here is example.

We know that someone can be a bachelor only if they are unmarried. However, not everyone who is unmarried is a bachelor. Being unmarried is therefore not a sufficient condition for being a bachelor, but it is one of several necessary conditions for being a bachelor, for one also has to be male, and of adult age. So, 'bachelor only if unmarried' does not translate into $Unmarried \rightarrow Bachelor$. However, it does translate into $Bachelor \rightarrow Unmarried$, exactly because being unmarried is a necessary condition for being a bachelor!

In general, '$P$ only if $Q$' translates to $P \rightarrow Q$, which is the converse of '$P$ if $Q$'

With the corrected symbolization, you should be able to prove the desired result.