David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Notre Dame Journal of Formal Logic 51 (3):351-360 (2010)
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 n th-order pure logics.
|Keywords||second-order logic nth-order logic elimination of quantifiers compactness decidability of validity model completeness|
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