Journal of Philosophical Logic

, Volume 47, Issue 1, pp 123–142

# Second-order Logic and the Power Set

• Ethan Brauer
Article

## Abstract

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments.

## Keywords

Second-order logic Set theory Power set Axiom of separation Jane Entanglement of logic and mathematics

## References

1. 1.
Boolos, G. (1971). The Iterative Conception of Set. The Journal of Philosophy, 68(8), 215–231.
2. 2.
Boolos, G. (1984). To be is to be a Value of a Variable (Or Some Values of Some Variables). The Journal of Philosophy, 81(8), 430–449.
3. 3.
Boolos, G. (1985). Nominalist Platonism. The Philosophical Review, 94(3), 327–344.
4. 4.
Chang, C. C., & Keisler, H. J. (2012). Model Theory, 3rd edn. Mineola, NY: Dover.Google Scholar
5. 5.
Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University Press.Google Scholar
6. 6.
Dummett, M. (1976) In Evans, G., & McDowell, J. (Eds.), What is a Theory of Meaning?(II). Oxford: Clarendon Press.Google Scholar
7. 7.
Dummett, M. (1993). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press.Google Scholar
8. 8.
Dummett, M. (2000). Elements of Intuitionism, 2nd edn. Oxford: Oxford University Press.Google Scholar
9. 9.
Jané, I. (1993). A critical appraisal of second-order logic. History and Philosophy of Logic, 14(1), 67–68.
10. 10.
Jech, T. (2002). Set Theory, 3rd edn. New York: Springer.Google Scholar
11. 11.
Koellner, P. (2010). Strong Logics of the First and Second Order. Bulletin of Symbolic Logic, 16(1), 1–36.
12. 12.
McCarty, D. C. (1994). On Theorems of Gödel and Kreisel: Completeness and Markov’s Principle. Notre Dame Journal of Formal Logic, 35(1), 99–107.
13. 13.
Priest, G. (1979). The Logic of Paradox. Journal Philosophical Logic, 8, 219–241.
14. 14.
Quine, W.V. (1970). Philosophy of Logic. Englewood Cliff, NJ: Prentice Hall.Google Scholar
15. 15.
Rayo, A., & Uzquiano, G. (1999). Toward a Theory of Second-Order Consquence. Notre Dame Journal of Formal Logic, 40(3), 1–11.Google Scholar
16. 16.
Shapiro, S. (1991). Foundations Without Foundationalism. New York: Oxford University Press.Google Scholar
17. 17.
Shapiro, S. (1999). Do not claim too much: Second-order logic and first-order logic. Philosophia Mathematica, 3(7), 42–64.
18. 18.
Shapiro, S. (2012). Higher-order Logic or Set Theory: A False Dilemma. Philosophia Mathematica, 3(20), 305–323.
19. 19.
de Swart, H. (1976). Another Intuitionistic Completeness Proof. The Journal of Symbolic Logic, 41(3), 644–662.
20. 20.
Väänänen, J. (2001). Second Order Logic and Foundations of Mathematics. Bulletin of Symbolic Logic, 7(4), 504–520.
21. 21.
Väänänen, J. (2015). Categoricity and Consistency in Second Order Logic. Inquiry, 58(1), 20–27.
22. 22.
Veldman, W. (1976). An Intuitionistic Completeness Theorem for Intuitionistic Predicate Logic. The Journal of Symbolic Logic, 41(1), 159–166.Google Scholar
23. 23.
Weston, T. (1976). Kreisel, The Continuum Hypothesis, and Second Order Set Theory. The Journal of Philosophical Logic, 5, 281–298.
24. 24.
Williamson, T. (1994). Vagueness. New York: Routledge.Google Scholar
25. 25.
Williamson, T. (2014). Logic, Metalogic, and Neutrality. Erkenntnis, 79, 211–231.