grammar - Is this usage of "last" correct?

Let's say I have just written an argument for something in a text, and I then add: "Last argument implies that..."

Is that correct? Or should I add "That/the last argument implies that" or "The previous argument implies that..."

What is more correct, and more importantly, more formal? Thanks in advance.

Answer

Last and next are sometimes used as determiners in deictic time expressions (cf. this):

Last week I went to the store.
This week I went to the store.
Next week I'll go to the store.
I ate pizza last time.
I'll eat pizza this time.
I'll eat pizza next time.

But your noun phrase last argument is not a deictic time expression, so in that phrase last cannot be a determiner. It must instead be an attributive adjective, modifying the following head noun. That means your noun phrase last argument is incomplete—it needs a determiner, and the appropriate one to add is the:

The last argument implies that …

Previous is never a determiner. Like last, it would be attributive in your example, and again that would leave the determiner slot empty. Let's add the:

The previous argument implies that …

Either of these would work. I think last is neutral with respect to formality, while previous is markedly formal. Please see the comments for additional discussion.