Meaning of biconditional used in expressing this statement.

Question

This question is from Rosen:

If L(x,y): x loves y.

Use quantifiers to express:

"There is someone who loves no one besides himself or herself"

The answer given by textbook is ∃x∀y(L(x,y)↔x=y)

What I don't understand is what does the statement mean when both p and q in p <=> q are false. The statement is true, according to biconditional truth table, but what does the statement actually mean?

My answer for the question was ∃x∀y(L(x,x) ∧ ¬L(x,y) ∧ ¬(x=y))

Answer 1

Consider to set $x$ as $John$ (we "name" him).

We have:

$∀y(L(John,y) ↔ John=y)$,

that means "a person loves John iff that person is John himself".

What happens when $y$ is not $John$ ?

Well: $(John=y)$ is false and he does not love $John$, i.e. $L(John,y)$ is false also, and we know that $p ↔ q$ is true when both $p$ and $q$ are false (they are "equivalent").

What about the proposed:

$∃x∀y(L(x,x)∧¬L(x,y)∧¬(x=y))$ ?

We have that (using again $John$ as $x$): $∀y(L(John,John)∧¬L(John,y)∧¬(John=y))$.

The $∀y$ quantifier means "for all"; thus, instantiating it with $John$, we get:

$L(John,John)∧¬L(John,John)∧¬(John=John)$

that is contradictory and also false (see the part $¬(John=John)$), contrary to our intentions.