implication versus conjunction correctness in FOL?

Question

I've just started learning FOL and I'm really confused about whether to use conjunction or implications. For example, if I want to represent

some students who answer the easiest question do not answer the most difficult

I came up with several solutions that seem equivalent to me.

1) ∃x. (student(x) Λ solve(x, easy) Λ ¬solve(x, hard))

2) ∃x. (student(x) -> (solve(x, easy) Λ ¬solve(x, hard)))

3) ∃x. ((student(x) Λ solve(x, easy)) -> ¬solve(x, hard)))

Can anyone explain which is correct and why the others are wrong?

Answer 1

It depends on the structure on which you evaluate your formulae. For simplicity I would introduce $3$ predicates $\mathsf{student}, \mathsf{solve\_easy}, \mathsf{solve\_hard}$. (The parameterized solve works but I think it is a little confusing)

If the universe of your structures contains both students and non-students (which I assume because you introduced the predicate $\mathsf{student}$) then:

• Formulae $1$ is correct. It states the existence of a student who answers easy questions but not difficult ones.
• Formulae $2$ is not correct. Assume that students solve no questions at all and there is at least one non-student. By the definition of implication the non-student satisfies $\mathsf{student}(x) \rightarrow \varphi(x)$ for any $\varphi$. Hence, the formulae is satisfied but it shouldn't be.
• Formulae $3$ is not correct. Assume again that students do not solve any questions and that there is a non-student.

If all elements in the universe are students all $3$ formulae are correct.

Answer 2

One way to look at it is to translate the logical statements into more everyday language, and see what they actually say.

1) There exists someone who is a student, answered the easy question, and didn't answer the hard question.

2) There exist someone who, if they are a student, answered the easy question and not the hard one (but if they're not a student we know nothing about what questions they answered).

3) There exists someone who, if they are both a student and answered the easy question, did not answer the hard question (if they are not a student, we do not know which questions they answered; if they are a student but did not answer the easy question we do not know if they answered the hard one).