Deciding between implication and conjunction

Question

This is one of the solved problems in Velleman's How to prove book:

Analyze the logical forms of the following statements: 1) John likes exactly one person.

Let L(x, y) mean “x likes y,” and let j stand for John. We translate this statement into symbols gradually:

(i) ∃x(John likes x and John doesn’t like anyone other than x).

(ii) ∃x(L( j, x) ∧ ¬∃y(John likes y and y $\ne$ x)).

(iii) ∃x(L( j, x) ∧ ¬∃y(L( j, y) ∧ y $\ne$ x)).

But when I try to think this in implication terms, that doesn't sound wrong:

(i) ∃x(If John likes x then John doesn’t like anyone other than x).

(ii) ∃x(L( j, x) -> ¬∃y(John likes y and y $\ne$ x)).

(iii) ∃x(L( j, x) -> ¬∃y(L( j, y) ∧ y $\ne$ x)).

So, how do logicians decide when to use implications versus when to use conjunction ? Is there any general guidelines for avoiding this pitfall ?

Answer 1

But the statement clearly states that there does exist someone that John does, in fact, like. Using implication doesn't guarantee the existence of anyone that John likes.

All your interpretation says is:

IF John likes some person x, then he likes no one else.

That tells us absolutely nothing about the possibility that John doesn't like person x. E.g. Consider two possible scenarios that would each make your statement true:

• John doesn't like anyone, including person $x$, or

• John likes everyone except person $x$

because in each case, the antecedent to your implication is false.

What conjunction allows us to assert that, in fact, there does exist someone x that John likes, AND it so happens that there is no one else (other than that x) that John likes.