Convert English into predicate first order logic.

Question

S(x), which stands for “x is a surgeon” P(x, y), which stands for “x is a patient of y” R(x), which stands for “x recovers” H(x), which stands for “x is happy”

1Express, as a formula in first-order predicate logic (not clausal form), this statement S1: “A surgeon with no patients is happy.”

2. Express, as a formula in first-order predicate logic (not clausal form), this statement S2: “A surgeon is happy if all her patients recover.”

For the first one, I have my answer as: 1. ∀(x) ∃x. (S(x)∧ ¬P(y,x) → H(x))

For the second question, I have my answer as: 2. ∀(x) (H(x) → R(x))

Are these correct? If not, what are the answers?

Answer 1

No, these are not correct. Rather than give the answers directly, I will try to write out the meaning of the two statements in a way that can facilitate your re-attempting them.

$1)$ "A surgeon with no patients is happy" means: for any person $x$, if $x$ is a surgeon and there is no person $y$ for which $y$ is a patient of $x$, then $x$ is happy.

$2)$ "A surgeon is happy if all her patients recover" means: for any person $x$, if $x$ is a surgeon and - if for every person $y$ for which $y$ is a patient of $x$, we have that $y$ recovers - then $x$ is happy.

As an example of why your answers are incorrect, you have the following for $2)$:

∀(x) (H(x) → R(x))

This says,

For any person $x$, if $x$ is happy, then $x$ recovers.

There is no mention here of surgeons, there is no mention here of patients, and you want the implication to be that patient recovery (for all relevant patients) implies happiness for the surgeon. None of this is accomplished in your proposed formulation, so take a look at the rewording at the top of this post, and give it another shot!

Answer 2

Going by your latest attempts as stated in the comment to Benjamin's answer:

For 1), you say:

$\forall x (S(x) \land \neg \exists y P(y,x) \rightarrow H(x))$

Basically ok, but I would really recommend adding some parentheses:

$\forall x (\color{red}(S(x) \land \neg \exists y P(y,x)\color{red}) \rightarrow H(x))$

For 2), your latest attempt is:

$\forall x ((S(x) \land \exists y (P(y,x) \land R(y))) \rightarrow H(x))$

This is very close, but the surgeon wants all of her patients to recover in order to be happy. So:

2. $\forall x ((S(x) \land \color{red}\forall y (P(y,x) \color{red}\rightarrow R(y))) \rightarrow H(x))$