Past exam of the computer science course of the University of Oxford / 2015 / Knowledge representation and reasoning / 1 / a

The condition about taking the course restricts which students the first-order sentence talks about. Within that condition we need both an inclusive logical disjunction and a prohibition on satisfying both alternatives:The logical conjunction on the right expresses exactly one of the two categories. It does not prohibit unrelated individuals from belonging to both categories: the restriction applies to the students in the antecedent.

The supervisor can depend on the student, so the existential quantification belongs inside the universal quantification:The first-order sentence permits different students to share a supervisor. It requires neither that the supervisor be different from the student nor that the supervisor be a faculty member, since neither additional condition was given.

Using the supplied counting function, an appropriate first-order sentence isHere the counting comparison expresses the threshold and the existential quantification expresses teaching at least one course. The word “if” supplies this direction of logical implication; it does not assert that these are the only reasons a faculty member can be busy. With the intended meanings of the nonlogical symbols, no further counting axioms are needed for this translation. If those meanings were unspecified, the displayed first-order sentence alone would not define arithmetic or connect the counting function to the supervision relation.