Second order logic

  • Ordinary first order predicate logic is not able take quantify over subsets of the universe even though it seems often a natural thing to do.
  • Second order logic permits existential and universal quantification over subsets (and over relations and functions) of the universe.
  • Model theory of second order logic seems very difficult. The usual results such as compactness theorem, Löwenheim-Skolem theorem, completeness theorem, etc, all fail.
  • Second order logic can be given so called "weak" or "Henkin" semantics, which makes it virtually the same as first order logic, but then the quantifiers do not range any more over all subsets of the universe.
  • Third order logic allows quantification over sets of subsets, fourth order logic allows quantification over sets of sets of subsets, etc.
    • Logic program - Helsinki Logic Group - Department of Mathematics