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
51(3):
351-360
(2010).
DOI: 10.1215/00294527-2010-021
[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. MR1059055 0697.03022[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. MR1059055 0697.03022
[2] Denyer, N., "Pure second-order logic", Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 220--24. MR1167978 0760.03001 10.1305/ndjfl/1093636099 euclid.ndjfl/1093636099
[2] Denyer, N., "Pure second-order logic", Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 220--24. MR1167978 0760.03001 10.1305/ndjfl/1093636099 euclid.ndjfl/1093636099