# Meaning of biconditional used in expressing this statement.

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))

• – Shaun Jun 19 '17 at 12:28
• Fix $x$ as John; we have : $∀y(L(John,y) ↔ John=y)$. It means: " a person loves John iff that person is John himself". – Mauro ALLEGRANZA Jun 19 '17 at 12:32
• When $y$ is not John ( $\lnot (John = y)$), he does not love John ($\lnot L(John,y)$). – Mauro ALLEGRANZA Jun 19 '17 at 12:33
• wow! thank you so much! very helpful, quite cleared the haziness in my head. One question, how would you recommend I could improve my understanding of logic? Also, If you'd like I could accept your answer if wrote one. @MauroALLEGRANZA – momo Jun 19 '17 at 12:47
• Reading some logical textbook... :-) See e.g. Peter Smith, An Introduction to Formal Logic. – Mauro ALLEGRANZA Jun 19 '17 at 12:49

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").

$∃x∀y(L(x,x)∧¬L(x,y)∧¬(x=y))$ ?

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.