Word usage: "either" or "both"?

Question

Consider the following usages:

1. Either the book and the pen are neither in the pocket nor in the backpack of either Sally or Peter.

2. Both the book and the pen are neither in the pocket nor in the backpack of either Sally or Peter.

3. Both the book and the pen are not in the pocket or the backpack of Sally and Peter.

Which of the three sentences is correct?

What I am trying to mean is that "the book is not in the pocket of Sally; the book is not in the backpack of Sally; the book is not in the pocket of Peter; the book is not in the backpack of peter". And same thing for the pen.

Can they be improved?

The question focuses on the word usage and English grammar of "either", "neither", and "both". This is strictly not a logic puzzle because meaning is very clear: nothing is the pocket of anyone.

Answer 1

1.  Either the book and the pen are neither in the pocket nor in the backpack of either Sally or Peter.

That is incorrect. Either needs or as its correlative conjunction partner, and are should be is. Corrected:

    Either the book or the pen is neither in the pocket nor in the backpack of either Sally or Peter.

Now that means either the book or the pen is missing. If the book is missing, the pen is not. If the pen is missing, the book is not. But that is not what you are trying to say.

2.  Both the book and the pen are neither in the pocket nor in the backpack of either Sally or Peter.

That is grammatically okay. It means both the book and the pen are missing; they are not in either of two pockets or either of two backpacks. That is what you are trying to say.

3.  Both the book and the pen are not in the pocket or the backpack of Sally and Peter.

That is grammatically okay, but it means that Sally and Peter share one backpack and one pocket. That is not what you are trying to say.

If this goes beyond an exercise, you should say it in plain English:

    The book and the pen are not in Sally’s or Peter’s pocket or backpack.
    Neither the book nor the pen is in Sally’s or Peter’s pocket or backpack.

Answer 2

I beg to differ; the meaning was not clear to me at all, until I read the whole of the text of your question. (Does each of the two people have both a book and a pen, or are there only one book and one pen? And if each person has their own pocket and their own backpack, this isn't clear either.) I take my lead from your paragraph starting "What I am trying to mean".

OK, there are two items and four places. Each of your suggested sentences is a grammatically correct sentence but none of them expresses your intended meaning. Here is a way to express it:

Neither the book nor the pen is in either Sally's pocket, her backpack, Peter's pocket, or his backpack.

How did I arrive at this sentence? A statement that might be true or false about an object is the object being in one of those four places. The appropriate connective word for the places is thus "either ... or". (The word "either" is optional here, but I put it in to make my suggested sentence more like yours.) The comma before the "or" is optional.

Now that statement is false for the book and also false for the pen. That is, It's true for neither of the objects. So I start my sente ce with "Neither the book nor the pen".

I've also rephrased the list of the four places to make it clear that there indeed four, two pockets and two backpacks.

Answer 3

1. Either the book and the pen are neither in the pocket nor in the backpack of either Sally or Peter.

Either {the book and the pen are {neither in the pocket nor in the backpack} of either Sally or Peter}. The first either is an orphan. it lacks an or.

2. Both the book and the pen are neither in the pocket nor in the backpack of either Sally or Peter.

{Both the book and the pen} are {neither in the pocket nor in the backpack} of {either Sally or Peter}. This seems to work, but isn't how you say it.

3. Both the book and the pen are not in the pocket or the backpack of Sally and Peter.

This is ambiguous and logically suspect. It could be "Neither the book nor the pen are in the pocket[s] or backpack[s] of [either] Sally or Peter."