# nLab second-order logic

## Idea

Second-order logic(SOL) is an extension of first-order logic with quantifiers and variables that range over subsets of the universe of discourse, and hence is a higher-order logic of stage 2.

An important fragment is Monadic Second-order logic (MSOL), where second-order quantification is restricted to second-order unary relations between subsets i.e. MSOL quantifies only over set predicates e.g. $\forall X.\varphi(X)$ but not $\forall X\forall Y .\varphi(X,Y)$.

## Remark

As SOL permits characterization of mathematical structures up to isomorphism, it is sometimes promoted as a contender to set theory for the foundations of mathematics (cf. references).

## References

• Wikipedia, Second-order logic

• Stewart Shapiro, Do Not Claim Too Much: Second-Order Logic and First-Order Logic , Phil. Math. 3 no. 7 (1999) pp.42-64 . (pdf)

• Jouko Väänänen, Second-order Logic and Foundations of Mathematics , BSL 7 no. 4 (2001) pp.504-520. (ps)

• Jouko Väänänen, Second-order logic, set theory, and the foundations of mathematics , Ms. (pdf)

Last revised on July 6, 2014 at 00:12:45. See the history of this page for a list of all contributions to it.