Notre Dame Journal of Formal Logic

Pure Second-Order Logic with Second-Order Identity

Alexander Paseau

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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.

Article information

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

Dates
First available in Project Euclid: 18 August 2010

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

Subjects
Primary: 03B15: Higher-order logic and type theory

Keywords
second-order logic nth-order logic elimination of quantifiers compactness decidability of validity model completeness

Citation

Paseau, Alexander. Pure Second-Order Logic with Second-Order Identity. Notre Dame Journal of Formal Logic 51 (2010), no. 3, 351--360. doi:10.1215/00294527-2010-021. http://projecteuclid.org/euclid.ndjfl/1282137987.


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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.
  • [2] Denyer, N., "Pure second-order logic", Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 220--24.