Notre Dame Journal of Formal Logic

Boolean-Valued Second-Order Logic

Daisuke Ikegami and Jouko Väänänen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the range of the second-order variables. In our Boolean-valued second-order logic, the second-order variables range over all Boolean-valued subsets and relations on the domain. We show that under large cardinal assumptions Boolean-valued second-order logic is more robust than full second-order logic. Its validity is absolute under forcing, and its Hanf and Löwenheim numbers are smaller than those of full second-order logic.

Article information

Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 167-190.

First available in Project Euclid: 24 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03-06: Proceedings, conferences, collections, etc.
Secondary: 03C95: Abstract model theory 03E40: Other aspects of forcing and Boolean-valued models 03E57: Generic absoluteness and forcing axioms [See also 03E50]

Boolean-valued second-order logic full second-order logic $\Omega$-logic Boolean validity


Ikegami, Daisuke; Väänänen, Jouko. Boolean-Valued Second-Order Logic. Notre Dame J. Formal Logic 56 (2015), no. 1, 167--190. doi:10.1215/00294527-2835065.

Export citation


  • [1] Bagaria, J., N. Castells, and P. Larson, “An $\Omega$-logic primer,” pp. 1–28 in Set Theory, Trends in Mathematics, Birkhäuser, Basel, 2006.
  • [2] Bell, J. L., Boolean-valued Models and Independence Proofs in Set Theory, 2nd edition, vol. 12 of Oxford Logic Guides, Oxford University Press, New York, 1985.
  • [3] Henkin, L., “Completeness in the theory of types,” Journal of Symbolic Logic, vol. 15 (1950), pp. 81–91.
  • [4] Larson, P. B., The Stationary Tower: Notes on a Course by W. Hugh Woodin, vol. 32 of University Lecture Series, American Mathematical Society, Providence, 2004.
  • [5] Larson, P. B., “Three days of $\Omega$-logic,” Annals of the Japan Association for Philosophy of Science, vol. 19 (2011), pp. 57–86.
  • [6] Magidor, M., “On the role of supercompact and extendible cardinals in logic,” Israel Journal of Mathematics, vol. 10 (1971), pp. 147–57.
  • [7] Martin, D. A., and J. R. Steel, “Iteration trees,” Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73.
  • [8] Mitchell, W. J., and J. R. Steel, Fine Structure and Iteration Trees, vol. 3 of Lecture Notes in Logic, Springer, Berlin, 1994.
  • [9] Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, 3rd edition, vol. 41 of Monografie Matematyczne, PWN—Polish Scientific, Warsaw, 1970.
  • [10] Schindler, R., and J. R. Steel, “The self-iterability of L[E],” Journal of Symbolic Logic, vol. 74 (2009), pp. 751–79.
  • [11] Schlutzenberg, F. S., Measures in mice, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2007.
  • [12] Steel, J. R., “Local $K^{c}$ constructions,” Journal of Symbolic Logic, vol. 72 (2007), pp. 721–37.
  • [13] Steel, J., “An outline of inner model theory,” pp. 1595–1684 in Handbook of Set Theory, Vol. 3, Springer, Dordrecht, 2010.
  • [14] Väänänen, J., “Abstract logic and set theory, I: Definability,” pp. 391–421 in Logic Colloquium ’78 (Mons, 1978), vol. 97 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1979.
  • [15] Väänänen, J., “Second-order logic and foundations of mathematics,” Bulletin of Symbolic Logic, vol. 7 (2001), pp. 504–20.
  • [16] Väänänen, J., “Second-order logic or set theory?,” Bulletin of Symbolic Logic, vol. 18 (2012), pp. 91–121.
  • [17] Woodin, W. H., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, vol. 1 of de Gruyter Series in Logic and its Applications, Walter de Gruyter, Berlin, 1999.
  • [18] Woodin, W. H., “Suitable extender models I,” Journal of Mathematical Logic, vol. 10 (2010), pp. 101–339.