Conditional Operator Question

Question

T(x,y) = Student x has taken class y

I need to translate the following statement into a quantified expression:

Everyone besides Sam has taken at least two different classes.

My initial thought was:

$$\forall x \exists y \exists z [(x\neq Sam) \leftrightarrow (T(x,y)\land T(x,z)\land (y \neq z)) $$

But I am being told that

$$\forall x \exists y \exists z [(x\neq Sam) \rightarrow (T(x,y)\land T(x,z)\land (y \neq z)) $$

Is the correct answer.

I am a little confused why the biconditional doesn't work here. The way I see it is, if there are two sets of students, those who have taken at least two classes and those who have not, then if you know a student has not then it must be Sam and only Sam and, if you know a student has taken at least two classes you know that it is not Sam.

Answer 1

The problem is that there are two ways to read this sentence in English.

1. Take the set of students excluding Sam and call it $S$. Everyone in $S$ has taken at least two classes. We aren't commenting on Sam though.

2. Take the set of all students and call it $S$. Everyone in $S$ has taken at least two classes, except for Sam, for whom it is not the case that they have taken at least two classes.

Your translation captures the meaning of (2). The translation using implication captures the meaning of (1).

In ordinary conversation, rules like Grice's maxims apply, which can rule out weak interpretations like (1) that are less informative than competing interpretations like (2).

When introducing folks to logic though, people will frequently present you with exercises where assumptions that are unremarkable ordinarily are not supposed to be translated into the resulting symbolic claim.

Answer 2

"besides" $\neq$ "except". I would agree with the suggested solution that "besides" perhaps pragmatically implicates, but does not logically entail, "only those who are not": A sentence like "Everyone besides Sam has taken at least two different classes; Sam himself even has taken five" is not contradictory, so the reverse direction of the implication $(x \neq Sam) \leftarrow \ldots$ indeed should not hold.