Abstract
If possible worlds are conjunctions of states of affairs, as in David Armstrong’s combinatorial theory, then is the empty world to be thought of as the null conjunction of states of affairs? The proposal seems plausible, and has received support from David Efird, Tom Stoneham, and Armstrong himself. However, in this paper, it is argued that the proposal faces a trilemma: either it leads to the absurd conclusion that the actual world is empty; or it reduces to a familiar representation of the empty world in which the concept of a null conjunction plays no role; or it needs to make room for the null individual of certain non-classical mereologies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This description of Armstrong’s view is perfectly compatible with the fact that he was a fictionalist about possible worlds other than the actual world. After all, one can also be a fictionalist about nonactual human beings and, at the same time, hold that all possible human beings are animals.
Cf. “there is no way to combine elements and make nothing at all” (Lewis 2001 [1986], 74n53). Armstrong also thought that, if an empty world were possible, then no other world would be accessible from it (Armstrong 1989, 64). It is not clear, however, why he thought this an unwelcome consequence. For example, Armstrong did not (at the time) think that the accessibility relation is symmetric (“at the time”, because he seems to have changed his mind in Armstrong 2004, 84–5).
Jan Heylen (2017) would probably disagree, since he rules out as “question-begging” any such deductive answer to the question of why there is something rather than nothing. There is an issue, however, of whether the notion of a question-begging answer can be captured in purely logical terms (as in Heylen 2017, 544–545).
Brauer’s proof requires another assumption (PWS), which I am inclined to reject, but which others may be willing to accept: “for any world w and sentence φ, it is true at w that ◊φ just in case it is true at w that there is a possible world v such that φ is true at v” (Brauer, 2022, 2754).
In an analogous way, one can explain why the null disjunction is identified with the False.
Tom Stoneham (personal conversation) suggested that the null conjunction can be understood on this model. The suggestion is plausible, as some philosophers (e.g., Lewis 2001[1986]) have identified propositions with sets of possible worlds. However, since we are modelling propositions here we can regard the identification as an idealization.
If necessary truths are to be modelled as sets of possible worlds, then it is hard to avoid the assumption that there is a set of all possible worlds. We can sidestep the issue of whether this assumption is true. Armstrong himself thought that it is incompatible with the principles of recombination accepted by Lewis 2001[1986] and himself (Armstrong 1989, 25–30). However, the point of modelling propositions as sets of possible worlds in the main text is not to provide a picture of propositions that is accurate by Armstrong’s standards, but to provide an alternative way of understanding the identification of the null conjunction with the True. To the extent that the identification holds up under different assumptions, it can be regarded as robust.
The “side pockets” mentioned here are somewhat akin to the “local absolute absences” discussed by Roy Sorensen in his history of nothingness (Sorensen 2022, 102–115).
In this case, the quantifier cannot range over all states of affairs, since, if it did, then it would deny the existence of the state of affairs that it itself is supposed to represent. According to Efird and Stoneham, quantifier restrictions do not automatically make the issue of whether an empty world is possible uninteresting (2009, 226).
The null individual can be identified with the null fusion. See Cotnoir and Varzi 2021 (p. 141) for a proof. It is commonly thought that the null individual, if it exists, is part of everything and, therefore (by the antisymmetry of parthood), unique. One referee for this journal asks if the null individual in the empty world is, then, both the null fusion of first-order states of affairs and the null fusion of apples. This may seem strange because the null individual seems to bear the totalling relation to the property of being a first-order state of affairs only under the first description. Friends of the null individual who find this an unwelcome consequence may consider giving up the assumption that the null individual is part of everything and, therefore, unique (as in Bunge 1966, where a different null individual is proposed for each kind of thing).
Filippo Casati and Naoya Fujikawa point out a problem for Priest’s identification of nothingness with the empty fusion (Casati and Fujikawa 2019, 3752). However, the problem does not arise if one replaces Priest’s definition of fusion with a different, “algebraic” one (that is, if one defines fusions in terms of least upper bounds). Casati and Fujikawa (following Weber and Cotnoir 2015) themselves prefer such a definition. There may be other reasons for Priest to replace his definition of fusion (Cotnoir 2018, 644). For a general discussion of definitions of fusion and their connection with the empty fusion, see Cotnoir and Varzi 2021, 160–174, and, especially, 165.
References
Andrews, P. B. (2002). An introduction to mathematical logic and type theory: To truth through proof (2nd ed.). Springer.
Armstrong, D. M. (1989). A combinatorial theory of possibility. Cambridge University Press.
Armstrong, D. M. (1997). A world of states of affairs. Cambridge University Press.
Armstrong, D. M. (2004). Truth and truthmakers. Cambridge University Press.
Armstrong, D. M. (2006). Reply to Efird and Stoneham. Australasian Journal of Philosophy, 84(2), 281–283.
Brauer, E. (2022). Metaphysical Nihilism and modal logic. Philosophical Studies, 179(9), 2751–2763.
Bunge, M. (1966). On null individuals. Journal of Philosophy, 63(24), 776–778.
Casati, F., & Fujikawa, N. (2019). Nothingness, Meinongianism and inconsistent mereology. Synthese, 196(4), 3739–3772.
Comtet, L. (1974). Advanced Combinatorics: The Art of Fine and Infinite Expansions (Revised and enlarged). D. Reidel Publishing Company.
Cotnoir, A. J. (2018). A note on priest’s mereology. Australasian Journal of Logic, 15(4), 642–645.
Cotnoir, A. J., & Varzi, A. Z. (2021). Mereology. Oxford University Press.
Dugundji, J. (1978[1966]). Topology. Allyn and Bacon, Inc.
Efird, D., & Stoneham, T. (2005). Genuine modal realism and the empty world. European Journal of Analytic Philosophy, 1(1), 21–37.
Efird, D., & Stoneham, T. (2006). Combinatorialism and the possibility of nothing. Australasian Journal of Philosophy, 84(2), 269–280.
Efird, D., & Stoneham, T. (2009). Is metaphysical nihilism interesting? Pacific Philosophical Quarterly, 90, 210–231.
Heylen, J. (2017). Why is there something rather than nothing? A Logical Investigation. Erkenntnis, 82(3), 531–559.
Hodges, W. (1997). A shorter model theory. Cambridge University Press.
Hudson, H. (2006). Confining composition. Journal of Philosophy, 103(12), 631–651.
Lewis, D. (1991). Parts of classes. Blackwell Publishers.
Lewis, D. (2001[1986]) On the Plurality of Worlds. Blackwell Publishers.
Martin, R. M. (1965). Of time and the null individual. Journal of Philosophy, 62(24), 723–736.
Nozick, R. (1981). Philosophical explanations. Belknap Press of Harvard University Press.
Priest, G. (2014). Much ado about nothing. Australasian Journal of Logic, 11(2), 146–158.
Rodriguez-Pereyra, G. (1997). There might be nothing: The subtraction argument improved. Analysis, 57(3), 159–166.
Sider, T. (2005). Another Look at Armstrong’s Combinatorialism. Noûs, 39(4), 679–695.
Simons, P. (2003[1987]). Parts: A study in ontology. Clarendon Press.
Sorensen, R. (2022). Nothingness: A philosophical history. Oxford University Press.
Varzi, A. (2019). Mereology. The Stanford Encyclopedia of Philosophy (Spring 2019 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2019/entries/mereology/>
Weber, Z., & Cotnoir, A. J. (2015). Inconsistent boundaries. Synthese, 192(5), 1267–1294.
Acknowledgements
This paper was read by four anonymous referees for this journal, three of whom provided comments that were helpful in revising earlier drafts. At an earlier stage, I also received helpful comments from Ethan Brauer, Leon Horsten, and Dan Marshall.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
De Clercq, R. The Empty World as the Null Conjunction of States of Affairs. Erkenn (2023). https://doi.org/10.1007/s10670-023-00740-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10670-023-00740-0