Possible argument refuting idealism?

Idealism: All that exists is mental, i.e. exists in (a/my/some/the) mind.


Possible argument refuting idealism:

(i) Language serves the purpose of communication.
(ii) Communication presupposes the existence of a mind other than my own.
(iii) Therefore, minds other than my own exist.
(iv) If minds other than my own exist, there is a world independent of my mind .
(v) Therefore, a world independent of my mind exists.

What do you think of this argument? The argument relies on the closure principle for knowledge:

If a subject S knows that p, and p entails q, then S knows that q.

The argument can be restated:

If S knows language, and language entails other minds, then S knows other minds.

If S knows other minds, and other minds entail an external world, then S knows an external world.

It follows from this argument that ''all that exists'' is not mental, since the argument affirms the existence of things (other minds, an external world) that are not present to my mind.

I think the idealist could question (i) and/or (ii).