Nancy is a faculty member.
If you are on the payroll, then you are being paid by the school. If someone has a job at a school, then they are on the payroll. All faculty members have a job at a school. One can either be a faculty member or a teacher. Every teacher has students. If Nancy is a teacher, then they are on the payroll. If Nancy is not a teacher, then they are not paid by the school.
∀x (OnPayroll(x) → PaidBySchool(x)) ∀x (JobAtSchool(x) → OnPayroll(x)) ∀x (FacultyMember(x) → JobAtSchool(x)) ∀x (FacultyMember(x) ∨ Teacher(x)) ∀x (Teacher(x) → HasStudents(x)) Teacher(nancy) → OnPayroll(nancy) ¬Teacher(nancy) → ¬OnPayroll(nancy)