second-order logic

Definition

Extension of first-order logic with quantifiers and variables ranging over subsets of the universe of discourse

Notes

• in first-order logic, quantification confined to elements of domain, however in second-order logic, quantification is also permitted on relations
  • e.g. $\forall x. \exists X .\forall y. X(x,y) \leftrightarrow x=y$
• see monadic second-order logic, which requires monadic predicates
• 'In Philosophy of Logic [Quine 1970], W. V. Quine summed up a popular opinion among mathematical logicians by referring to second-order logic as “set theory in sheep’s clothing”.' (Resnik, 1998)

References

1. https://ncatlab.org/nlab/show/second-order+logic
2. https://plato.stanford.edu/entries/logic-higher-order/#SyntSecoOrdeLogi
3. Resnik, Michael D., 1988, “Second-Order Logic Still Wild”, The Journal of Philosophy, 85(2): 75–87. doi:10.2307/2026993
4. https://www-sop.inria.fr/members/Martin.Avanzini/teaching/2021/AL/slides/w2.pdf