Practice question · Multiple choice
Why does an inconsistent set of premises entail every sentence?
Hints
- Apply the definition of entailment literally to a set that has no models.
- Entailment fails only when some assignment makes the premises true and the conclusion false, and here there is no such assignment.
Show the answer
D. No assignment makes the premises true, so none makes them true with a false conclusion
Why
Entailment is defined by the absence of a counterexample. If nothing satisfies the premises, no counterexample can exist, so every conclusion follows vacuously. This is the principle of explosion, and it is why contradiction is fatal to a theory.
Practise Tautology, Entailment and Consistency
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