# Higher-Order Contingentism, Part 3: Expressive Limitations

## Abstract

Two expressive limitations of an infinitary higher-order modal language interpreted on models for higher-order contingentism – the thesis that it is contingent what propositions, properties and relations there are – are established: First, the inexpressibility of certain relations, which leads to the fact that certain model-theoretic existence conditions for relations cannot equivalently be reformulated in terms of being expressible in such a language. Second, the inexpressibility of certain modalized cardinality claims, which shows that in such a language, higher-order contingentists cannot express what is communicated using various instances of talk of ‘possible things’, such as ‘there are uncountably many possible stars’.

## Keywords

Contingentism Higher-order modal logic Expressivity## Notes

### Acknowledgments

In addition to those thanked in the acknowledgements of Part 1, I would like to thank a reviewer for comments on Part 3, and the editor, Frank Veltman, for all his help with the publication of the three parts.

## References

- 1.Adams, R.M. (1979). Primitive thisness and primitive identity.
*The Journal of Philosophy*,*76*(1), 5–26.CrossRefGoogle Scholar - 2.Blackburn, P., de Rijke, M., & Venema, Y. (2001).
*Modal logic*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - 3.Fine, K. (1977).
*Postscript to worlds, times and selves (with A. N. Prior)*. London: Duckworth.Google Scholar - 4.Fine, K. (1977). Properties, propositions and sets.
*Journal of Philosophical Logic*,*6*(1), 135–191.CrossRefGoogle Scholar - 5.Fine, K. (2003). The problem of possibilia. In Loux, M.J., & Zimmerman, D.W. (Eds.)
*The Oxford handbook of metaphysics*(pp. 161–179). Oxford: Oxford University Press.Google Scholar - 6.Fraïssé, R. (1958). Sur une extension de la polyrelation et des parentés tirant son origine du calcul logiques du k-ème échelon. In
*Le raisonnement en mathématiques et en sciences expérimentales, volume 70 of Colloques Internationaux du CNRS, pages 45–50. Paris: Editions du Centre National de la Recherche Scientifique*.Google Scholar - 7.Fritz, P. (2013). Modal ontology and generalized quantifiers.
*Journal of Philosophical Logic*,*42*(4), 643–678.Google Scholar - 8.Fritz, P. Higher-order contingentism, part 2: Patterns of indistinguishability.
*Journal of Philosophical Logic*, forthcoming.Google Scholar - 9.Fritz, P., & Goodman, J. (2016). Higher-order contingentism part 1: Closure and generation.
*Journal of Philosophical Logic*,*45*(6), 645–695.CrossRefGoogle Scholar - 10.Fritz, P., & Goodman, J. Counting incompossibles.
*Mind*, forthcoming.Google Scholar - 11.Hintikka, J., & Rantala, V. (1976). A new approach to infinitary languages.
*Annals of Mathematical Logic*,*10*(1), 95–115.CrossRefGoogle Scholar - 12.Hodges, W. (1997).
*A shorter model theory*. Cambridge: Cambridge University Press.Google Scholar - 13.Leuenberger, S. (2006). A new problem of descriptive power.
*The Journal of Philosophy*,*103*(3), 145–162.CrossRefGoogle Scholar - 14.Lewis, D. (1986).
*On the plurality of worlds*. Oxford: Basil Blackwell.Google Scholar - 15.Stalnaker, R. (2012).
*Mere possibilities*. Princeton: Princeton University Press.Google Scholar - 16.Williamson, T. (2013).
*Modal logic as metaphysics*. Oxford: Oxford University Press.CrossRefGoogle Scholar