First order logic - proving an argument is invalid

Question

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:

Domain: $\mathbb{Z}^+$

$C(x) \leftrightarrow x < 1$

$P(x) \leftrightarrow x\ is\ prime$

$M(x) \leftrightarrow x=2$

$L(x,y) \leftrightarrow x <y $

$g=2$

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?

Answer 1

Indeed, in your first attempt you only managed to show that $\forall x (M(g)\to\neg C(x))$, which is equivalent to $\forall x(C(x)\to\neg M(g))$ (by contraposition). This indicates already that to conclude that $g$ was not mentioned in the lecture, the universe must contain at least one $x$ that is a good critic. In other words, "there is still the possibility that $g$ was mentioned and that is in case no one is a good critic".

So, you came up with a universe/model which has no "good critics" at all, to prove that the initial argument is not valid.

Well done!