### Pure Second-Order Logic with Second-Order Identity

Alexander Paseau
Source: Notre Dame J. Formal Logic Volume 51, Number 3 (2010), 351-360.

#### Abstract

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.

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Primary Subjects: 03B15
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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

### References

[1] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
Mathematical Reviews (MathSciNet): MR1059055
Zentralblatt MATH: 0697.03022
[2] Denyer, N., "Pure second-order logic", Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 220--24.
Mathematical Reviews (MathSciNet): MR1167978
Zentralblatt MATH: 0760.03001
Digital Object Identifier: doi:10.1305/ndjfl/1093636099
Project Euclid: euclid.ndjfl/1093636099