'Both', 'each', or 'either', when there can only be two of something?

Question

a) He had a gun in both hands.

b) He had a gun in each hand.

c) He had a gun in either hand.

In a) there are two hands but only one gun. Clear.

In b) there are two hands and two guns, one per hand. Clear.

You would find c) in sentences questioning whether or denying that he had a gun in one hand or the other, or even two guns, one per hand. Clear too.

But how about these?

c) There were trees on both sides of the street.

d) There were trees on each side of the street.

e) There were trees on either side of the street.

Two sides, trees on one side and trees on the other side. Why the different determiners then? What shades of meaning?

Answer 1

There is no difference between your examples, and all are perfectly grammatical.

The key difference between these three determiners is that both takes a plural verb, but both each and either (usually) take a singular even when used distributively in the sense of all or every, not just when used disjunctively:

1. Both takes either two nouns each of either number, or else a single plural noun in a sense limited to two. Its verb is always plural.

   • Both cars were totalled.
   • Both your mother and I are ready.
   • Both parents were present.

2. Each takes a singular noun or else a partitive constructive selecting from a plural set. Its verb is always singular but its sense is always distributive. Although it is not limited to senses where only two possibilities can apply, when it is used with one member of a natural dual such as hands or eyes, then its sense is distributive to both of these.

   • Each member is expected to vote.
   • Each hand holds a knife at ready.
   • Each contestant has taken his seat.
   • Each of the many options was investigated.

3. Either works mostly like each but it can be either distributive and so plural in sense yet singular in construction, or else disjunctive and therefore usually singular in sense. But in present-day use it is more strongly limited to two of something. In the distributive sense it means each/both of the two. It occasionally selects from the disjunction of more than two items, or from two items one or both of which is itself plural. It can also occasionally select one element from a list of more than two.

   • Either parent is enough for this.
   • Either your mother or your father has to sign this.
   • From either earlobe dangles a flashing diamond earring.
   • Either Welshmen or Scotsmen are expected to win.
   • Either Larry, Moe, or Curly is in trouble.

References

Using either to distribute to two things where it means each of the two, both of the two falls under this sense from the OED:

B. adj. (attributive) (in later use, determiner).
I. With distributive or universal meaning.

1. With singular noun.
   a. Each of the two. Frequently in either side.
   See also either way at way n.¹ and int.¹ Phrases 7a(c).
   • 1914 K. L. Bates Chaucer's Canterbury Pilgrims iii. 84
     An altar stood below: on either hand A priest with roses crowned, who held a myrtle wand.
   • 1943 Triumphs of Engin. 53
     Generating plants exist on either bank.
   • 2015 K. Kwan China Rich Girlfriend ɪɪ. vii. 177
     As the wing doors rose, they saw that the driver's seat was in the center of the car, with a passenger seat flanking either side.

It used to also be used with a plural noun here, but that is now considered nonstandard.

Answer 2

I will eventually answer your question.

However, I want you to know that the example which uses the word "trees" is more likey to make people feel happy than the example using the word "guns."

A large portion of the human population enjoys looking at living trees and touching living tree bark.

Reading a large number of examples of English phrases containing words like "gun", "weapon", "die", "blood", etc cetra is more likely to increase a readers risk of becoming clinically depressed in the future than if they read example using words without negative connotations.

Now, I will answer your question.

Ambiguities in the English language can somtimes be resolved by employing mathematical methods.

Let σ be the sentence, "There were trees on both sides of the street".

σ has almost the same meaning as the following mathematical statement:

Let s be a street.

By the definition of street, s has exactly two sides.

Denote the sides of street s as side x and side y such that x is distinct from y.

There are trees on x and there are trees on y.

The string "There were trees on each side of the street" has a meaning similair to the mathematical example provided above.

However, the string, "there were trees on either side of the street" is problematic.

It is more correct to write, "either there were trees on one side of the street or there were trees on the other side of the street"

If the error (noise factor) is small, then fluent users of the English language can discern what the meaning of a misspelled sentence is.

Suppose that the street under consideration has a West side and the street under consideration has an East side.

The following examples all have the same meaning:

1. You can put the trees on either side of the street.
2. You can put the trees on west side of the street or you can put the trees on the East side of the street.
3. Either (you can put the trees on the west side of the street) or (you can put the trees on the east side of the street)

Most people limit the amount of time they spend thinking about it.

People guess what the meaning of sentences are within a reasonable degree of aproximation.