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.