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 secondorder identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with secondorder 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. © 2010 by University of Notre Dame. All rights reserved.
CITATION STYLE
Paseau, A. (2010). Pure Second-Order Logic with Second-Order Identity. Notre Dame Journal of Formal Logic, 51(3), 351–360. https://doi.org/10.1215/00294527-2010-021
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