What the Debate Club Note Licenses
A debate club posts this note before a tournament:
- Every club member who competed at the regional tournament attended at least one practice.
- Some students who attended a practice did not compete at the regional.
- No student who missed more than two practices competed at the regional.
Three students ask what the note lets them conclude. Priya is a club member who missed three practices. Dev is a club member who attended four practices. Rae attended one practice and is not in the club. Nothing else is known about who practised or who competed.
1. Priya missed three practices. Does it follow that she did not compete, and does anything follow about her practice attendance?
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She did not compete - that does follow. Rule 3 is a universal: anyone who missed more than two practices is outside the competing group. Three exceeds two, so Priya is excluded.
Nothing follows about her attendance beyond the three absences. Two different errors are available here and both should be refused:
- Rule 1's valid contrapositive is: a member who attended no practice did not compete. Running it from 'did not compete' back to 'attended no practice' reverses the arrow - that is the inverse error. Not competing is compatible with attending every practice.
- Rule 3 constrains competitors only. It says nothing about people who did not compete.
So the note fixes her status at the tournament and leaves her practice record open, apart from the three absences already given.
2. A member claims the note implies 'Some club members attended a practice and did not compete.' Assess the inference.
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It does not follow, and the gap is a single missing word: 'club.'
Rule 2 says some students attended a practice and did not compete. Students, not members. Rae is the standing counterexample: a non-member who attended a practice. The existential in rule 2 could be satisfied entirely by people like Rae, with every club member who practised going on to compete.
A model that satisfies all three rules while making the conclusion false:
- Every club member attended at least one practice and competed.
- Rae and one other non-member practised and did not compete.
Rules 1, 2 and 3 all hold; the conclusion fails. So the inference is invalid, however natural it sounds - we slide from 'students' to 'club members' because the note is about the club.
3. Which single added sentence would make the conclusion follow, and why does 'Every club member attended at least one practice' fail to do the job?
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Working addition: Every student who attended a practice is a club member.
With that, rule 2's 'some students' are members, so at least one member practised and did not compete, and the conclusion follows directly.
Why the other sentence fails: 'Every club member attended at least one practice' pushes information in the wrong direction. It tells you that members practise; it does not tell you that the people in rule 2 are members. The counter-model survives intact - all members practise and compete, non-members supply rule 2's existential - so the conclusion is still not forced.
The general lesson: to transfer an existential claim from one class to another, you need the bridge that moves the existential's subjects into the target class, not one that describes the target class from the outside.