No man (or woman) is an island

I was reading a satirical article in the New Yorker, "College Essay" by Christopher Buckley, and came across the following statement:

It was a seventeenth-century English-person John Donne who wrote, “No man is an island.” An excellent statement, but it is also true that “No woman is also an island.”

Now, leaving aside the satire and references to Poets, I want to concentrate on these two lines:

"No man is an island"
"No woman is also an island"

The writer is trying to say that neither men nor women are islands, but due to the improper placement of "also", the meaning changes into something funny, or atleast there is some oddness.

(1) What exactly is the unintended funny meaning or oddness ?
(2) How could the second line be stated to mean that neither men nor women are islands, while retaining the structure ?

Here, it is ok:

"Man is an animal"
"Woman is also an animal"

But if negation is involved, how to say it grammatically, while retaining the structure ?

I think this works:

"No man is an island"
"No woman is an island either"

Solution 1:

"Also, no woman is an island." does it. Or, "No woman is an island, also."

The problem is that "no" has the force of "not"+"any": "Not any woman is an island", where "not" applies to the entire remainder of the sentence with the subject "any woman". In a predicate logic rendering, "not" would correspond to sentence negation and "any" would correspond to the existential quantifier.

In the context given, you want "not" to be within the scope of "also", yet if "also" is part of the sentence which is negated, that will not be the case. So "also" has to be removed from that sentence by being made a modifier of the sentence, either before it or after it.

Solution 2:

The normal understanding of "No man is an island" is that there is not any man who is an island (lives in complete isolation).

However, the specific wording of "No woman is also an island" may (humorously) cause the reader or listener to reconsider the meaning of the first sentence to mean there is an entity "No man" who actually is an island, thus allowing for the second sentence to be understood to mean that there is another entity "No woman" who is also an island.

Solution 3:

I think it would work much better with "either", so

"No woman is an island either."