In statement logic, a counterexample to an argument is a row with true premises and a false conclusion (Validity by Truth Table). In predicate logic it is a world with true premises and a false conclusion. Building one is how you show a predicate argument is invalid, using what you learned in Truth in a Small World.
Key idea
Why it works
Worked examples
Worked example: The converse of an A statement
- We need every $F$ to be $G$, but some $G$ not to be $F$.
- World: one object $a$ that is $G$ and not $F$.
- Premise: true (there are no $F$ to fail). Conclusion: false ($a$ is $G$ but not $F$).
Answer: Invalid: the one-object world is a counterexample.
Worked example: Two separate witnesses
- World: $a$ is $F$ only; $b$ is $G$ only.
- Both premises are true; nothing is both, so the conclusion is false.
Answer: Invalid.
Worked example: In the picture
- The conclusion $\exists x Cx$ must be false: no circles.
- $\forall x(Cx \to Rx)$ is then true automatically.
- $\exists x Rx$ needs something red.
Answer: Any world with no circles and at least one red object, such as one red square.
Common mistake
- Make the conclusion false first: that usually fixes a lot.
- Then add or adjust objects until each premise is true.
- Check everything again at the end.
Sometimes no world works, however hard you try. That is a hint, though not a proof, that the argument is valid. The next unit gives the tools to prove it: rules that let you reason from quantified premises directly.
Counterexample worlds are the predicate-logic twin of the short truth table method: start from what the counterexample needs and fill in the rest.