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Why is first order logic preferred to higher order logic in practice?

  • A.First order logic is decidable whereas higher order is not
  • B.It has a complete proof procedure and higher order logic does not
  • C.Higher order logic cannot express anything first order cannot
  • D.Higher order formulas cannot be written down finitely

Answer: B

Completeness means every entailed sentence is provable, which higher order logic lacks under the standard semantics, so automated support is far weaker there. Higher order logic is strictly more expressive and perfectly writable, and neither is decidable. Choosing the formalism for which mechanical support exists, rather than the most expressive available, is the same reasoning that produced description logics at the other end of the scale.

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