Logic / Unit 1 Logic 022 worked example, 60 to 90 seconds
Testing Validity by Counterexample
The idea
Testing an argument for validity by counterexample means stripping it to its bare form, then building a second argument with that same form whose premises are all true and whose conclusion is false.
In practice
Take this argument: some athletes are vegetarians; all cyclists are athletes; so all cyclists are vegetarians. First put it in standard form. Then replace the content with letters, keeping the structure intact: Some A are B. All C are A. So all C are B. Now hunt for new terms that make every premise come out true and the conclusion come out false. Let A be mammals, B be whales, and C be dogs. Some mammals are whales — true. All dogs are mammals — true. All dogs are whales — false. The form permits true premises alongside a false conclusion, so the original argument is invalid.
Where people slip
A counterexample has to share the argument's form, not its subject matter. Swapping in new content while quietly altering the structure tests a different argument and settles nothing about the original. And failing to find a counterexample never establishes validity — it records only that you have not found one yet.
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