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.
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1282137987
Digital Object Identifier: doi:10.1215/00294527-2010-021
Zentralblatt MATH identifier: 05773616
Mathematical Reviews number (MathSciNet): MR2675687