Second-order logic still wild
Journal of Philosophy 85 (2):75-87 (1988)
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Peter Koellner (2010). Strong Logics of First and Second Order. Bulletin of Symbolic Logic 16 (1):1-36.
Jouko Vaananen (2001). Second-Order Logic and Foundations of Mathematics. Bulletin of Symbolic Logic 7 (4):504-520.
Stewart Shapiro (1999). Do Not Claim Too Much: Second-Order Logic and First-Order Logic. Philosophia Mathematica 7 (1):42-64.
G. Aldo Antonelli & Richmond H. Thomason (2002). Representability in Second-Order Propositional Poly-Modal Logic. Journal of Symbolic Logic 67 (3):1039-1054.
Matti Eklund & Daniel Kolak (2002). Is Hintikka's Logic First-Order? Synthese 131 (3):371 - 388.
Alexander Paseau (2010). Pure Second-Order Logic with Second-Order Identity. Notre Dame Journal of Formal Logic 51 (3):351-360.
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