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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 ?

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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.

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  • $\begingroup$ Thanks, but how does that opens the possibility that John doesn't like anyone ? Does that open the possibility when antecedent is F and consequent is T ? $\endgroup$
    – Sibi
    Aug 23, 2014 at 13:12
  • $\begingroup$ If John doesn't like some person x, the implication tells us nothing whatsoever. From a false antecedent, everything is implied. $\endgroup$
    – amWhy
    Aug 23, 2014 at 13:17
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    $\begingroup$ In general, for $\exists$, we have conjunction. And for $\forall$ we have implication. The order of the occurrence of the quantifier, and their respective scopes also are factors when translating to symbolic form. Note that in this case, Velleman uses conjunction as the main connective (given the leading existential quantifier for $x$), and also uses conjunction for the connective for the "inner" existentially quantified $y$. $\endgroup$
    – amWhy
    Aug 23, 2014 at 14:42

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