More law resources on TSR
1. Non-cognitivism is not the view that moral truths cannot be proved; and even if it were, it does not follow from the fact (if it is one) that moral truths cannot be proved that there are no moral truths. Lots of things are difficult to prove but that doesn't mean there's no truth to be had in the first place. (There's a truth about what you had for lunch last Wednesday but you'd be hard pressed to prove it.) Non-cognitivism is the view that first-order statements about morality (e.g. murder is wrong) do not express propositions, and therefore cannot be true or false. (Compare anti-realism, which is the view that there are no mind-independent facts about morality.)
2. Practically all aspiring and first year law students seem to be moral relativists; that is, they believe that what is morally right for one person may be morally wrong for another, that morality is all 'just a matter of opinion'. Most of those likely to read your essay will consider this a highly implausible view (in my opinion, with good reason). This doesn't mean you shouldn't argue for it - no views are 'off limits' - but it does mean that it needs arguing for. The trouble with your argument here is that you use the supposed truth of moral relativism to argue against a moral statement. Of course, all statements that offer purported truths about morality can be undermined if moral relativism is true. But that's quite a cheap way of undermining such statements. And since the premise (moral relativism) is an independently implausible one, it makes for an underwhelming argument. Let me reiterate that you are in excellent company in your relativist views - 90% of students sitting the LNAT will agree with you. But that's why an argument that assumes moral relativism and uses it as the central premise in an essay about morality will not look all that impressive.
I only have a problem with your thesis statement. Firstly, the thesis statement should be the last sentence of the first paragraph and so that is worth noting.
Secondly, your thesis statement is vague and ambiguous... The use of the word "nonsensical" as the foundation of your thesis is not only absurd but also unacceptable.
Your thesis statement ought to be more specific and telling.
Great work and success