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What is a logical equivalence?

  • A.Two sentences that can be derived from one another
  • B.Two sentences with the same number of symbols
  • C.Two sentences that are both valid or both unsatisfiable
  • D.Two sentences true in exactly the same models

Answer: D

Equivalent sentences have identical truth in every model, so either may replace the other anywhere without changing what is entailed. Mutual derivability coincides with it in a sound and complete system. Symbol counts are irrelevant and the last option is far too weak. Equivalences such as De Morgan's laws and the distributive laws are the tools by which a formula is rewritten into normal form, so knowing them as equivalences rather than as inference rules is what licenses the rewriting.

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