Self-study
At a gathering at which bankers, athletes, and lawyers are present, all of the bankers are athletes and none of the lawyers are bankers.
Summary
The stimulus can be diagrammed as follows:
Notable Valid Inferences
Some of the athletes are not lawyers.
Some of the people who are not lawyers are athletes.
19.
If the statements above are █████ █████ ███ ██ ███ █████████ ██████████ ████ ████ ██ █████
a
All of the ████████ ███ ████████
Could be false. (A) confuses the necessary and sufficient conditions: we know that all the bankers are athletes, but the opposite is not necessarily true. Maybe there are 100 athletes and 50 bankers, in which case the bankers are still athletes, but many athletes aren’t bankers.
b
Some of the ███████ ███ ███ █████████
Could be false. (B) has a negation problem: our diagram shows that some athletes are not lawyers, but that’s not the same thing as “some lawyers are not athletes.” What if, for example, there are 100 athletes and 50 lawyers, and every lawyer is an athlete?
c
Some of the ████████ ███ ███ ████████
Must be true. As shown below, chaining the conditional claims yields a valid argument with the conclusion “some athletes are not lawyers.” In other words, we know that there are bankers at the gathering, and we know that those bankers are people who are athletes but not lawyers.
d
All of the ███████ ███ ████████
Must be false. The stimulus says that none of the lawyers are bankers, which, by the contrapositive, means that none of the bankers are lawyers!
e
None of the ███████ ███ █████████
Could be false. While we know that not all of the athletes are lawyers, we don’t know anything about the percentage of lawyers who are or aren’t athletes. Maybe, for example, there are 100 athletes and 50 lawyers, and every lawyer is an athlete! Or 20 of the lawyers are athletes!