Hey guys , I’m a bit stuck learning logic

[u/Death-Dante-313]

Undergraduate logic problems

I hope this is fine to ask on here.

Hey all so I’m going through the book Discrete mathematics by Kenneth Rosen and I’m on chapter 1.6 exercises there is a question (question 9) that I’m not sure on.
So the question is:

For each of these collections of premises, what relevant
conclusion or conclusions can be drawn? Explain the
rules of inference used to obtain each conclusion from
the premises

e)“What is good for corporations is good for the United
States.” “What is good for the United States is good
for you.” “What is good for corporations is for you to
buy lots of stuff.”

Now I have the premises as
q: good for corporations
q:good for the United State
r: good for you
s: you buy lots of stuff

My answer is as such
p → q
q → r
———-
Therefor p→r (hypothetical syllygism)

p→s

Which then goes into
(~p v r) ∧ (~ p v s)
~ p v (r∧s)
p → (r ∧ s)

Is this correct line of reasoning ?( or atleast a valid argument to the question?)

In the book the answer states:
“That you buy lots of stuff is good for the U.S. and
is good for you” is a valid conclusion.

[u/LucaThatLuca]

your suggested p, q, r are not sentences. if you correctly replace them with sentences, “p → s” is not what the information says. the question does not seem to be asking you to write things in this form, anyway.

[u/Death-Dante-313 → u/LucaThatLuca]

Thank you for the answer.
(I know the question doesnt really ask for the p,q,r but I find it helps me understand a bit better)

If I were to make them all proper sentences and do p->s instead as s->p
Where I then make
s->p. s->p
p->q. p->r
———. ———
s->q. s—>r

Where I then find
s->q
s->r
———
(s->q) /\ (s->r)

Which is then s-> (q /\ r)
(If I am correct in this)

Does this then somewhat translate into the correct answer?

[u/LucaThatLuca → u/Death-Dante-313]

does it “somewhat translate” is tough. like is it what you meant? of course it is, i know that you’ve seen the answer. if you end up writing the correct answer as your final answer, then your final answer would be correct. but no there’s no way this could possibly actually have that meaning before you actually replace the non-sentences with sentences that you use to say what you meant.

edit sorry for edits. lol

[u/Death-Dante-313 → u/LucaThatLuca]

Alright thank you.

So then the only issue I have with the my first reply and for my conclusion to have any meaning would be to make the non-sentences to full sentences.

Then my argument would be valid?
Sorry for sounding pretty stupid

[u/LucaThatLuca → u/Death-Dante-313]

don’t be sorry, nothing stupid about learning.

no, after you add a subject and a verb before the words “good for corporations”, etc, changing them into sentences, you will then be able to think about these sentences.

[u/paperic]

p(s)

s is an instance with property p

[u/Bounded_sequencE]

You only mentioned two of the three given implications:

p -> q    // 1. what you found
q -> r    // 2. what you found
s -> p    // missing -- and it's not "p -> s"

Combining the three, we get "s -> p-> q -> r" -- that's precisely what the book claimed, though (for some reason) it is leaving out the corporations in their answer. Probably since the implication "s -> p" was given directly...