We develop an account of logical consequence for the second-order language of set theory in the spirit of Boolos's plural interpretation of monadic second-order logic.
References
[1] Boolos, G., ``To be is to be a value of a variable (or to be some values of some variables),'' The Journal of Philosophy, vol. 81 (1984), pp. 430--49.
Mathematical Reviews (MathSciNet):
MR755478
[2] Boolos, G., ``Nominalist platonism,'' Philosophical Review, vol. 94 (1985), pp. 327--44.
[3] Boolos, G., ``Reading the Begriffsschrift,'' Mind, vol. 94 (1985b), pp. 331--34.
Mathematical Reviews (MathSciNet):
MR796077
[4] Cartwright, R., ``Speaking of everything,'' Noûs, vol. 28 (1994), pp. 1--20.
[5] Etchemendy, J., The Concept of Logical Consequence, Harvard University Press, Cambridge, 1990.
[6] Kreisel, G., ``Informal rigour and completeness proofs,'' pp. 138--71 in Problems in the Philosophy of Mathematics, edited by I. Lakatos, North-Holland, Amsterdam, 1967.
[7] McGee, V., ``Two problems with Tarski's theory of consequence,'' Proceedings of the Aristotelian Society, vol. 92 (1992), pp. 273--92.
[8] McGee, V., ``How we learn mathematical language,'' Philosophical Review, vol. 106 (1997), pp. 35--68.
[9] Rayo, A., and S. Yablo, ``Nominalism through de-nominalization,'' Noûs, vol. 35 (2001), pp. 74--92.
[10] Shapiro, S., ``Principles of reflection and second-order logic,'' Journal of Philosophical Logic, vol. 16 (1987), pp. 309--33.
[11] Shapiro, S., Foundations Without Foundationalism, A Case for Second-Order Logic, Clarendon Press, Oxford, 1991.
[12] Weir, A., ``Naive set theory is innocent!,'' Mind, vol. 107 (1998), pp. 763--98.
[13] Yi, B.-U., ``Is two a property?,'' Journal of Philosophy, vol. 96 (1999), pp. 163--90.