Symbolizing the statement "Everyone likes at least two people".

Question

Working on the book: Daniel J. Velleman. "HOW TO PROVE IT: A Structured Approach, Second Edition" (p. 80)

Example 2.2.2

2. Everyone likes at least two people.

The symbolisation given in the book is: $$ \forall x \exists y\exists z(L(x,y) \land L(x,z) \land y \neq z) $$

My symbolisation is: $$ \forall x \exists y\exists z(L(x,y) \land L(x,z) \land x \neq y \land y \neq z \land x \neq z) $$

In that same page, the author specifies that the statement means that everyone likes at least two different people. The symbolisation given by the author does not out rule the possibility that a person loves himself/herself, I think, so I added $x \neq y$ and $y \neq z$.

Is my interpretation correct?

Answer 1

The symbolization given by the author does not out rule the possibility that a person loves himself/herself,

Certainly, but that is not a restriction that is indicated by the statement "Everyone likes at least two people". So why would you think that you need to add it?

Just translate what is literally said.

Answer 2

If the statement said, “Everyone likes at least two other people,” yours would be the correct interpretation. Since it doesn’t, $\forall x \exists y\exists z(L(x,y) \land L(x,z) \land y \neq z)$ is the correct one. By tautology, everyone is a person and is thus included in the set of all people from which the aforementioned “two people” are being drawn. By contradiction, no one is another person, because that would mean $\exists x(x\neq x)$!

Now, in ordinary English communication, it could be reasonable to read, “Everyone likes at least two people,” as implicitly excluding the case that you like yourself. This is because it’s pretty normal to interpret “like” as specifically referring to interpersonal affection rather than to self-esteem or narcissistic vanity. But in logic problems like this, we don’t do that, because that would be more mathematically complicated. It would require that $L(x,x) \rightarrow \bot$ be part of the definition of the $L$ predicate, and nothing to that effect is specified in the natural-language statement.