Writing english statement into predicate logic using quantifiers.

Question

Below was the exercise problem I was solving from Discrete Mathematics by Kenneth Rosen (for preparation of GATE Exam) and I have doubt in it.

Let S(x) be the predicate that "x is a student", F(x) be the predicate "x is a faculty member", and A(x,y) the predicate "x has asked y a question", where the domain consists of all people associated with your school. Use quantifiers to express each of these statements.

(f)Some student has asked every faculty member a question.

Now my doubt is

it can be framed like there is at least one student such that for all faculty members, he must have asked them a question.

so I wrote my expression as

∃x ( (S(x) ^ ∀y ( F(y) $\rightarrow$ A(x,y) ) )

But in Rosen answer is given as below and I have 2 doubts in it.

Rosen's Ans : ∀y ( (F(y) $\rightarrow$ ∃x ( S(x) v A(x,y) ) )

Doubt 1: I think in above expression we must have "and operator" instead of "or operator" in the second part of expression which is quantified by existential quantifier and so it should be

    ∀y ( (F(y) $\rightarrow$ ∃x ( S(x) ^ A(x,y) ) )

Doubt 2: What is the difference between my answer and Rosen's answer.Which one is correct.

Please help.

Answer 1

You're right about the operator: that should definitely be a $\land$, rather than a $\lor$

Otherwise, the difference is this: you took the statement "Some student has asked every faculty member a question." to mean that it was the same student who asked every faculty member a quesrion. And, with that interpretation, your answer is correct, and Rosen's (even with the $\land$ instead of the $\lor$) would be incorrect.

However, English (like all natural languages) is ambiguous, and you can also interpret the statement "Some student has asked every faculty member a question." as saying that for each faculty member there is some student (but this time not necessarily the same one for each faculty member) that asked that faculty member a question. And if that is the intended meaning of the sentence, then Rosen's answer (again, with the $\land$ instead of the $\lor$) is correct.

Now, the latter is certainly not a very intuitive reading of the statement, and I much prefer the former interpretation, and hence your answer! However, some people do mean the latter interpretation when expressing the English statement.

Answer 2

Rosen's intended answer, $\forall y~ (F(y)\to ∃x~( S(x) \wedge A(x,y) ) )$, claims that for any faculty member there is some student that has asked them a question.   They need not be the same students.

• That is: "Every faculty member has been asked a question by some student."

Your answer, $\exists x~\forall y~(S(x)\wedge (F(y)\to A(x,y)))$ claims that some student will have asked every faculty member a question.   Which is what was required.

• That is: "Some student has asked every faculty member a question."

It is logical distinction which is oftimes misrepresented in natural language.