Convert sentences to predicate logic

Question

When converting sentences to predicate logic what do imply the following sentences?

One of neighbors smokes. Does this mean exactly one or at least one?

OR

There is a neighbor who smokes. Does this mean exactly one or at least one?

Answer 1

As to the FIRST question: "One of neighbours smokes" is not English. Do you mean "One of the neighbours smokes"? Or "One neighbour smokes"? In fact it makes little odds as far as the first question is concerned. Consider these conversations.

(A)

"No one round here smokes"

"No, you're wrong. One of the neighbours smokes, to my certain knowledge, and I suspect another does.

Plainly, here 'One of the neighbours smokes' does not mean exactly one neighbour smokes ... I am explicitly allowing that it might be true that more than one smokes.

(B)

"Exactly how many of your neighbours still smoke?"

"I happen to know the answer to that! One of the neighbours smokes

Plainly here I do mean that exactly one of the neighbours smokes. Different conversational contexts, same words, different messages. [We'd get the same with "one neighbour smokes".]

And of course it is exactly because of this sort of phenomenon in every day English that for some purposes -- where clarity and lack of ambiguity are important -- we replace English with unambiguous sentences framed in the artificial language of predicate logic. So we shouldn't expect that there is always a nice single content-independent way of rendering English into predicate logic.

As to the SECOND question, it is perhaps more difficult to think of contexts where 'There is a neighbor who smokes' is naturally construed as meaning there is exactly one. But it isn't impossible, especially if you allow the article "a" to be heavily emphasised. But the same general point applies.