I had a few questions about translating sentences into symbolic logic. I feel I am adding on conditionals where I do not need to be. My book only provides answers for a few selected exercises. I'm wondering if these sentences are correct I'm wondering if there is a better way to rewrite them. I feel ok on recognizing key words for quantifiers, but I think I may be falling into a cause and effect trap. Where I keep looking for a causal relationship. My book warned about that early on so if I'm doing so I don't want to develop it into a habit.
a) Anyone who has bought a Rolls Royce with cash must have a rich uncle.
∀x ((R(x)^C(x)) -> ∃y U(y,x) ^ W(y))
R: Stands for bought rolls royce
C: Bought Rolls Royce in case
U: Is uncle to
W: Wealthy
b) If anyone in the dorm has the measles, then everyone who has a friend in the dorm will have to be quarantined.
∃x ((D(x) ^ M(x)) -> ∀y(D(y) ^ ∃z (F(y,z) -> Q(y)))
D: Lives in dorm
M: Has measles
F: Is friend of
Q: Quarantined
c) If nobody failed the test, then everybody who got an A will tutor someone who got a D.
∀x (~F(x) -> ∀y A(y) -> ∃z(D(z) ^ T(z,y)))
F: Failed the test
A: Made an A
D: Made a D
T: Be tutored by
d) If anyone can do it, Jones can.
∀x (D(x) -> x = jones, D(x))
D: Can do it
e) If Jones can do it, anyone can.
∀x ( x = jones, D(x) -> D(x))
D: Can do it
a. Is good, although the choice of predicates is a bit strange, given that both R and C express that a Rolls Royce was bought; were these predicates given to you?
b. Antecedent is good, but two things go wrong in the consequent: first, you have an existential there with a conditional inside ... That is almost always bad news, and it is indeed bad news here. Second, your sentence is teying to quarantine anyone who lives in the dorm and has a friend (that is otherwise unrelated to the whole scenario). instead, you need to quanrantine anyone with a friend that lives in the dorm. So it is the friend that lives in the dorm, but the person to be quarantined need not live in the dorm. here is what you get:
$\exists x (D(x) \land M(x)) \rightarrow \forall y (\exists z (D(z) \land F(y,z)) \rightarrow Q(y))$
c. Antecedent should just be $\forall x \neg F(x)$... I.e the rst of the sentence is not within the scope of the $\forall x$. but the art about tutoring someone with a D does need to be in the scope of the $\forall y$. So:
$\forall x \neg F(x) \rightarrow \forall y (A(y) \rightarrow \exists z (D(z) \land T(y,z)))$
d. You can't use a comma here (same problem in e.). Also, your sentence is trying to say that Jones is the only one that can do it, i.e. that if there is anyone that can do it, then that person must be Jones, but the English sentence is saying instead that if there is one person that can do it, then it is Jones (persumably because because Jones is 'most capable' of anyone) ... But this does not rule out that there can be multiple eopl that can do it. So, instead you want:
$\exists x D(x) \rightarrow D(jones)$
e. $\forall x (D(jones) \rightarrow D(x))$ works, but I think a bit more readable is:
$D(jones) \rightarrow \forall x D(x)$