## Notre Dame Journal of Formal Logic

### Actualism, Serious Actualism, and Quantified Modal Logic

William H. Hanson

#### Abstract

This article studies seriously actualistic quantified modal logics. A key component of the language is an abstraction operator by means of which predicates can be created out of complex formulas. This facilitates proof of a uniform substitution theorem: if a sentence is logically true, then any sentence that results from substituting a (perhaps complex) predicate abstract for each occurrence of a simple predicate abstract is also logically true. This solves a problem identified by Kripke early in the modern semantic study of quantified modal logic. A tableau proof system is presented and proved sound and complete with respect to logical truth. The main focus is on seriously actualistic T (SAT), an extension of T, but the results established hold also for systems based on other propositional modal logics (e.g., K, B, S4, and S5). Following Menzel it is shown that the formal language studied also supports an actualistic account of truth simpliciter.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 2 (2018), 233-284.

Dates
Accepted: 5 February 2015
First available in Project Euclid: 17 November 2017

https://projecteuclid.org/euclid.ndjfl/1510888080

Digital Object Identifier
doi:10.1215/00294527-2017-0022

Mathematical Reviews number (MathSciNet)
MR3778310

Zentralblatt MATH identifier
06870291

#### Citation

Hanson, William H. Actualism, Serious Actualism, and Quantified Modal Logic. Notre Dame J. Formal Logic 59 (2018), no. 2, 233--284. doi:10.1215/00294527-2017-0022. https://projecteuclid.org/euclid.ndjfl/1510888080

#### References

• [1] Burge, T., “Truth and singular terms,” Noûs, vol. 8 (1974), pp. 309–25.
• [2] Chihara, C. S., The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic, Oxford University Press, New York, 1998.
• [3] Davies, M., and L. Humberstone, “Two notions of necessity,” Philosophical Studies, vol. 38 (1980), pp. 1–30.
• [4] Fine, K., “Model theory for modal logic, III: Existence and predication,” Journal of Philosophical Logic, vol. 10 (1981), pp. 293–307.
• [5] Fine, K., Modality and Tense: Philosophical Papers, Oxford University Press, Oxford, 2005.
• [6] Fitting, M., Types, Tableaus, and Gödel’s God, vol. 12 of Studia Logica Library, Kluwer, Dordrecht, 2002.
• [7] Fitting, M., and R. L. Mendelsohn, First-Order Modal Logic, vol. 277 of Synthese Library, Kluwer, Dordrecht, 1998.
• [8] Garson, J. W., “Quantification in modal logic,” pp. 267–323 in Handbook of Philosophical Logic, Vol. 3, edited by D. M. Gabbay and F. Guenther, Kluwer, Dordrecht, 2001.
• [9] Gibbard, A., “Contingent identity,” Journal of Philosophical Logic, vol. 4 (1975), pp. 187–221.
• [10] Gilbert, D. R., and E. D. Mares, “Completeness results for some two-dimensional logics of actuality,” Review of Symbolic Logic, vol. 5 (2012), pp. 239–58.
• [11] Hanson, W. H., “Actuality, necessity, and logical truth,” Philosophical Studies, vol. 130 (2006), pp. 437–59.
• [12] Hanson, W. H., “Logical truth in modal languages: Reply to Nelson and Zalta,” Philosophical Studies, vol. 167 (2014), pp. 327–39.
• [13] Hodes, H. T., “Axioms for actuality,” Journal of Philosophical Logic, vol. 13(1984), pp. 27–34.
• [14] Hodes, H. T., “On modal logics which enrich first-order $\mathrm{S}5$,” Journal of Philosophical Logic, vol. 13 (1984), pp. 423–54.
• [15] Hodes, H. T., “Some theorems on the expressive limitations of modal languages,” Journal of Philosophical Logic, vol. 13 (1984), pp. 13–26.
• [16] Jager, T., “An actualistic semantics for quantified modal logic,” Notre Dame Journal of Formal Logic, vol. 23 (1982), pp. 335–49.
• [17] Jager, T., “De re and de dicto,” Notre Dame Journal of Formal Logic, vol. 29 (1988), pp. 81–90.
• [18] Kaplan, D., “On the logic of demonstratives,” Journal of Philosophical Logic, vol. 8 (1979), pp. 81–98.
• [19] Kripke, S. A., “Semantical considerations on modal logic,” Acta Philosophica Fennica, vol. 16 (1963), pp. 83–94.
• [20] Lambert, K., Free Logic: Selected Essays, Cambridge University Press, Cambridge, 2003.
• [21] Lambert, K., and E. Bencivenga, “A free logic with simple and complex predicates,” Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 247–56.
• [22] Mates, B., Elementary Logic, 2nd edition, Oxford University Press, New York, 1972.
• [23] Menzel, C., “Actualism, ontological commitment, and possible world semantics,” Synthese, vol. 85 (1990), pp. 355–89.
• [24] Menzel, C., “The true modal logic,” Journal of Philosophical Logic, vol. 20(1991), pp. 331–74.
• [25] Percival, P., “Predicate abstraction, the limits of quantification, and the modality of existence,” Philosophical Studies, vol. 156 (2011), pp. 389–416.
• [26] Plantinga, A., The Nature of Necessity, Oxford University Press, Oxford, 1974.
• [27] Plantinga, A., “Replies to my colleagues,” pp. 313–96 in Alvin Plantinga, edited by J. Tomberlin and P. van Inwagen, Springer, Dordrecht, 1985.
• [28] Plantinga, A., “Self-profile,” pp. 3–97 in Alvin Plantinga, edited by J. Tomberlin and P. van Inwagen, Springer, Dordrecht, 1985.
• [29] Prior, A. N., and K. Fine, Worlds, Times and Selves, Duckworth, London, 1977.
• [30] Ray, G., “Ontology-free modal semantics,” Journal of Philosophical Logic, vol. 25 (1996), pp. 333–61.
• [31] Stalnaker, R. C., “Complex predicates,” The Monist, vol. 60 (1977), pp. 327–39.
• [32] Stalnaker, R. C., “The interaction of modality with quantification and identity,” pp. 12–28 in Modality, Morality, and Belief: Essays in Honor of Ruth Barcan Marcus, edited by W. Sinnott-Armstrong, D. Raffman, and N. Asher, Cambridge University Press, Cambridge, 1995.
• [33] Stalnaker, R. C., Ways a World Might Be, Oxford University Press, Oxford, 2003.
• [34] Stalnaker, R. C., Mere Possibilities: Metaphysical Foundations of Modal Semantics, Princeton University Press, Princeton, 2012.
• [35] Stalnaker, R. C., and R. H. Thomason, “Abstraction in first-order modal logic,” Theoria, vol. 34 (1968), pp. 203–7.
• [36] Stephanou, Y., “Investigations into quantified modal logic,” Notre Dame Journal of Formal Logic, vol. 43 (2002), pp. 193–220.
• [37] Stephanou, Y., “First-order modal logic with an ‘actually’ operator,” Notre Dame Journal of Formal Logic, vol. 46 (2005), pp. 381–405.
• [38] Stephanou, Y., “Serious actualism,” Philosophical Review, vol. 116 (2007), pp. 219–50.
• [39] Thomason, R. H., and R. C. Stalnaker, “Modality and reference,” Noûs, vol. 2 (1968), pp. 359–72.
• [40] Thomson, J., and A. Byrne, Content and Modality: Themes from the Philosophy of Robert Stalnaker, Oxford University Press, Oxford, 2006.
• [41] Vlach, F., “`Now’ and ‘then’: A formal study in the logic of tense and anaphora,” Ph.D. dissertation, University of California, Los Angeles, 1973.
• [42] Williamson, T., Modal Logic as Metaphysics, Oxford University Press, Oxford, 2013.