The rules can be understood through relations between classes.
Universal negative: no overlap; exclusion holds in both directions.
Universal affirmative: the first class is wholly included in the second; reverse inclusion need not be total.
Particular affirmative: at least one member is shared; overlap is symmetric.
Particular negative: some members of the first class are outside the second; nothing follows about members of the second being outside the first.
These four relations explain the rules rather than merely listing them.