Practice question · Multiple choice
"Every student who studied passed" is true in a class where nobody studied. Why does the universal claim survive a domain with no qualifying members?
Hints
- Rewrite it as: for every student x, if x studied then x passed.
- Ask what a counterexample would have to look like, and whether one can exist here.
Show the answer
A. Because an implication with a false premise is true, so it holds vacuously.
Why
Written out, the claim is an implication per student, and a counterexample would be someone who studied and failed, with nobody studying, none exists. It is the same vacuous truth that makes ∅ a subset of every set. The practical warning is that it hides failure: a test asserting 'every returned record matches' goes green when the query returns nothing at all.
Practise Predicate Logic and Quantifiers
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