The question asks to construct a derivation of the argument if it's valid, and otherwise to provide an arithmetical interpretation which shows it's invalid. Am I doing this correctly?
Argument: "All good critics like every poet mentioned in the lecture. No good critic likes Edgar Guest, although Edgar Guest is a poet. Therefore, Edgar Guest was not mentioned in the lecture."
Let $C(x)\leftrightarrow x\ is\ a\ good\ critic $
$L(x,y) \leftrightarrow x\ likes\ y$
$P(x) \leftrightarrow x\ is\ a\ poet$
$M(x) \leftrightarrow x\ was\ mentioned\ in\ the\ lecture$
$g = Edgar\ Guest $
I symbolise the premises of the argument as:
Premise 1: $\forall x(C(x)\rightarrow \forall y(M(y)\wedge P(y) \rightarrow L(x,y)))$
Premise 2: $P(g)\wedge\forall x(C(x) \rightarrow \neg L(x,g)) $
The conclusion is $\neg M(g)$
In my first attempt, I tried to prove that the argument is true by assuming the negation of the desired conclusion, $M(g)$, and finding a contradiction. However, it only led me to conclude from the premises that $\forall x(M(g) \rightarrow \neg C(x))$
So I tried to find an interpretation which makes the argument invalid and found the following:
$C(x) \leftrightarrow x < 1$
$P(x) \leftrightarrow x\ is\ prime$
$M(x) \leftrightarrow x=2$
$L(x,y) \leftrightarrow x <y $
Essentially I chose an interpretation of $C(x)$ so that implications in which $C(x)$ is an antecedent are always true because $C(x)$ is always false. This allows the first premise to be true. Note that this also makes the second term of the second premise true. I then choose an interpretation of $P(x)$ and $M(x)$ so that $P(g)$ ("2 is prime") is true while $\neg M(g)$ ("2$\neq$2") is false. Is this solution fine?