Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 51, Number 3 (2010), 351-360.
Pure Second-Order Logic with Second-Order Identity
Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with second-order identity is equivalent to a member of a circumscribed class of formulas. As a corollary, pure second-order logic with second-order identity is compact, its notion of logical truth is decidable, and it satisfies a pure second-order analogue of model completeness. We end by mentioning an extension to nth-order pure logics.
Notre Dame J. Formal Logic Volume 51, Number 3 (2010), 351-360.
First available in Project Euclid: 18 August 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03B15: Higher-order logic and type theory
Paseau, Alexander. Pure Second-Order Logic with Second-Order Identity. Notre Dame J. Formal Logic 51 (2010), no. 3, 351--360. doi:10.1215/00294527-2010-021. http://projecteuclid.org/euclid.ndjfl/1282137987.