Is "each" an adverb, pronoun, determiner, or what else?

In a related question, John Lawler has suggested Quantifier as the designation for each:

In linguistics and grammar, a quantifier is a type of determiner, such
as all, some, many, few, a lot, and no, (but not
numerals)... that indicates quantity.

The Wikipedia article recognizes the difficulty of analyzing quantifiers in natural language:

The study of quantification in natural languages is much more
difficult than the corresponding problem for formal languages. This
comes in part from the fact that the grammatical structure of natural
language sentences may conceal the logical structure. Moreover,
mathematical conventions strictly specify the range of validity for
formal language quantifiers; for natural language, specifying the
range of validity requires dealing with non-trivial semantic problems.
For example the sentence "Someone gets mugged in New York every 10
minutes" does not identify whether it is the same person getting
mugged every 10 minutes....

Each of the six sentences in the OP share a common logical meaning:

Y received a book

Where Y is the quantified group: each of my friends.

In the first four sentences, the agent is identified with the active voice of give, while the identity of the agent is suppressed in the passive euivalent of receive for sentences 5-6. Additionally, each sentence adds or reduces identifying information without changing the common logical meaning of the sentence, but in doing this changes the interaction of the words and phrases.

1. I gave my friends a book each.

This is the logical equivalent to: I gave each of my friends a book, where:

• gave is the transitive verb


• I is the agent of the action


• a book is the theme of the action


• each of my friends is the recipient

As John Lawler explained the Q-float: "each of my friends" is transformed, eliminating the of and moving each from its normal prenominal position into an adverbial position:

I gave my friends a book each, where:

• gave is the transitive verb


• I is the agent of the action


• a book is the theme of the action


• my friends is the recipient


• each identifies ambiguously with gave or my friends

It is ambiguous because each has floated to the end of the sentence, but one can reasonably parse each as an adverb. "The grammatical structure conceals the logical structure" slightly, but the natural reader, who rarely bothers to parse consciously, can intuitively interpret the common logical meaning: Each of my friends received a book, with an added emphasis on each.

2. I gave each a book.

This is a much simpler analysis. Each of my friends is reduced to each, which is used as a pronoun recipient. Again the natural reader infers the common logical meaning: Each of my friends received a book, but they are probably expecting clearer identification in the larger context.

3. I gave each of my three friends a book.

Adding the extra modifier three does not change the quantifier function in this construction. It behaves like all, because all is also a quantifier. Because the phrase is reduced so naturally, each can be parsed as a pronoun recipient. Still the natural reader interprets the common meaning: Each of my friends received a book, with the additional information.

4. I gave a book to each one of my friends

Since quantifiers are a type of determiner, this is simply adding one determiner to another along the lines of "any other". Again, the phrase is reduced so naturally, each is parsed routinely as a pronoun. The natural reader interprets the common meaning: Each of my friends received a book, with the additional emphasis of each one.

5. Each one of my friends received a book.

The shift to the passive equivalent "received" subdues the identity of the agent, emphasizing the theme and the recipient. Once again, the natural lister automatically gets it.

6. Each friend of mine received a book.

This construction is the clearest use of each as a quantifier, making it very easy to label it with the more general tag: determiner. Each friend of mine is the logical (and linguistic) equivalent to each of my friends, so the natural reader interprets the same common meaning: Each of my friends received a book.

Conclusion:

The grammatical usage of the quantifier each may change with different constructions, but each of the uses is logically connected to its function as a quantifier.