I don't like potatoes or ice-cream [closed]

You could try

"I like neither potatoes nor ice cream"

though it sounds somewhat old-fashioned.

The presence of a negation makes all the difference!

The sentence is interpreted as:

I don't (like (potatoes or ice-cream)). -> I don't (like potatoes or like ice-cream).

This logic can be represented with and instead of or, if we use the negation twice:

I don't like potatoes and I don't like ice-cream.

Without a negation, this would go like:

I (like (potatoes or ice-cream)). -> I (like potatoes or like ice-cream).

Which may be valid, but puzzling. Your version with hate doesn't change that.

This looks like (a variation on) negative raising.

In short, your original sentence seems to convey exactly the meaning you intended.

How about simply using "both":

I hate both potatoes and ice-cream.

While this serves to prove that you hate potatoes and ice-cream too, it doesn't have the subtle implication that can sometimes arise out of using "and" like you pointed out:

To use "I don't like potatoes and ice-cream" would be correct but ambiguous as it appears to imply they have to be served together to illicit dislike.

You have analysed your query sentence incorrectly. If you are determined to expand it, it becomes something like

I don't like either potatoes or ice cream.

There is actually nothing wrong with the original format of the sentence, which is both well-formed and idiomatic. Trying to expand it is over-thinking what is basically a very simple and straightforward structure.

You are right that

"I don't like potatoes, and ice cream"

is wrong (not to say grossly unidiomatic). However,

"I don't like potatoes, and I don't like ice cream"

is acceptable as a way of emphasizing your dislike of both these commodities.

Yet another simple way of fixing the sentence:

I don't like potatoes or ice cream either.

Google Books has a number of results for "or with him either", sometimes with a coma before "or".