I have this natural language sentence: "On the table there is exactly one book." Suppose I have this unary predicate - $s(x)$ - which means: "$x$ is a book on the table."
In my textbook this sentence is written in a formal language like this: $\exists x(s(x)∧\forall y(s(y)\to(x=y)))$ .
My question is: Can I write $\forall x$ istead of $\exists x$ ? Why not?
On
$\exists x~(s(x)\wedge\forall y~(s(y)\to x=y))$ says: "some book is on the table and there is no other book on the table."
$\forall x~(s(x)\wedge\forall y~(s(y)\to x=y))$ says: "all books are on the table but they are all the same book," that is: "there is only one book and it is on the table."
$\forall x~(s(x)\to\forall y~(s(y)\to x=y))$ says: "if there is a book on the table, then it is the only one on the table."
That would be a little different. Ultimately, I think it comes down to "There is exactly one book, and it is on the table." The answer in the book includes the possibility that there are other books in the universe that aren't on the table.