When should we use proximity rule in "either/or", and "neither/nor"?

Solution 1:

The “proximity rule” you are referring to is that when you have a compound but disjunctive subject, the verb agrees in number with the closer — or in the case of three or more, the closest — of the subjects.
All these are correct, and :

• Either my brother or my sister is going to get it.
• Either my brothers or my sisters are going to get it.
• Either my brother or my sisters are going to get it.
• Either my brothers or my sister is going to get it.

• Either I or my brother is going to get it.
• Either my brother or I am going to get it.

• Either my brother or you are going to get it.
• Either you or my brother is going to get it.

• Either my brother or thou art doomed.

But it is often better to rephrase:

• Either my brother is going to get it, or my sisters are.
• Either my brother is going to get it, or I am.
• Either you are going to get it, or my brother is.
• Either my brother is going to get it, or you are.
• Either my brother is doomed, or thou art.

And then there was nor

This same rule applies (well, or can apply) to neither/nor sets as much as it does for either/or instances. So in all the examples just provided, you can change all instances of either into neither and of or into nor, and the verb remains unchanged. So:

• Neither my brother nor my sister is going to get it.
• Neither my brothers nor my sisters are going to get it.
• Neither my brother nor my sisters are going to get it.
• Neither my brothers nor my sister is going to get it.

All that being said, one can also find examples in the works of writers of renown where neither is used with a plural verb despite both elements being singular themselves.