 Solution

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.