Skip to main content
SpringerLink
Log in

Observations on the Trivial World

  • Published:
Erkenntnis Aims and scope Submit manuscript

Abstract

A world is trivial if it makes every proposition true all at once. Such a world is impossible, an absurdity. Our world, we hope, is not an absurdity. It is important, nevertheless, for semantic and metaphysical theories that we be able to reason cogently about absurdities—if only to see that they are absurd. In this note we describe methods for ‘observing’ absurd objects like the trivial world without falling in to incoherence, using some basic techniques from modal logic. The goal is to begin to locate the trivial world’s relative position in modal space; the outcome is that the less we assume about relative possibility, the more detail we can discern at the edge of reason.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Trivialism is distinct from dialetheism, the thesis that some (not all!) contradictions are true. But it is in the context of dialetheism that trivialism often ends up being discussed, and, as we will see, truth value ‘gluts’ may have a role to play. See Priest (2000) (reprinted in chapter 3 in Priest (2006a)), where the name ‘trivialism’ seems to have originated. For some sympathetic discussions, see Kabay (2011), Azzouni (2013), Bueno (2007), Estrada-González (2012) and Plebani (2015). To avert any confusion: our paper is not a defense of trivialism.

  2. Thanks to a referee for clarifying this point.

  3. It is not the purpose of this paper to determine the right modal logic(s) for reasoning about the trivial world (although some are putatively ruled out), nor to isolate some class of models that might establish the consistency of a theory of the trivial world. Or to provide a ‘theory of the trivial world’ at all. To start on any of those projects, one needs to have some intuitions for the subject matter—minimally, some ‘perspective’ on the target, like how its appearance changes when you get closer or farther away. That is what we aim to provide here.

  4. Lewis argued that there are no impossible worlds (Lewis 1986, p. 7, ftnt 3), of which the trivial world would be the most extreme (non) example; his reasoning concerns inconsistent worlds. Others have disagreed. See, for example, Mares (1997), Nolan (1997, 2013), Berto (2013), Jago (2013, 2014), Krakauer (2013) and cf. Bjerring (2013). In light of this ongoing discussion, the purpose of this section is only to motivate and contextualize the study of one particular impossible world.

  5. See, for example, Stalnaker (1968, p. 103), Cresswell (1975), Veldman (1976) and Fine (2014). A trivial world is explicitly considered in Routley et al. (1982, p. 380), along with an empty world (one making nothing true); cf. Priest (1998).

  6. For discussion of the first edition of Priest (2016b), see Kroon (2008), Nolan (2008) and Woodward (2013). For elaborations and further details, see Berto (2012). A recent objection to modal Meinongianism is Bueno and Zalta (2017).

  7. Related to ‘American style’ impossible worlds, as glossed in Berto (2013, \(\S \)4.1), although ‘Australasian style’ impossible worlds can also satisfy unrestricted comprehension, given a suitable background logic. Cf. Nolan (1997).

  8. Thanks to a referee for finding it strange.

  9. There is the question of ‘how far away’ the nearest impossible worlds are—for instance, the strangeness of impossibility condition (Nolan 1997; Mares 1997), whereby the nearest impossible world is farther away from the actual world than even the most bizarre of possible worlds. Without weighing in on this (dubious) condition, on our way of setting things up, the trivial world (when closed under logic, as it (trivially) is) may still be closer to the actual world than the nearest open world. But similarity ordering on impossible worlds must wait for another day; see Berto (2014b, p. 115).

  10. As argued in Priest (1990), Sylvan (1992) and Weber et al. (2016). So, conjunction, disjunction, and implication could all potentially be non-classical. On this, it is worth mentioning a potential way for open worlds (dismissed above) to re-enter the picture, while still assuming that worlds are closed under the semantic clauses just given: pick some bizarre enough non-classical logic to govern the connectives, e.g. one where conjunction elimination fails, and one could get back some ‘wild’ behaviour. [Thanks to a referee for suggesting this point.] That said, our intended class of logics, while undefined, would be expected to have connectives within a more narrow range of familiar properties. For example, we will assume that ‘implies’ is at least reflexive and transitive, and obeys modus ponens.

  11. It is standard in non-classical logics (Priest 2008, p. 65) to partition the space of worlds into possible and impossible worlds, and then to relativize the semantics accordingly, e.g. an argument is valid iff for any possible world that makes the premises true, it also makes the conclusion true. For reasons to question this approach, see Girard and Weber (2015). Here we are simply working with worlds, without categorizing them.

  12. Insofar as \(\bot \) is itself a sentence, this clause is circular—or impredicative, as is said more politely. (Thanks to a referee for saying this politely.) In nearby frameworks, like unrestricted naive property theory, (1) these sort of definitions are allowed as ‘fixed points’ (e.g. Cantini 2003, p. 357), and (2) the proposition \(\bot \) can be defined as the property that all things have at the intersection of the universe (Priest 2006b, p. 253). But to be careful, the clause here is taken as a substantive assumption, following from Meinongian characterization intuitions, rather than a recursively specified condition on when a world satisfies \(\bot \).

  13. Correspondence theorems are conditionals, but we have not specified the properties of our implication connective, \(\Rightarrow \), beyond a few basic constraints (see footnote 10 above). It is an open question as to which and to what extent non-classical logics can support meta-theoretic reasoning about Kripke frames. A start at answering this question is in Girard and Weber (unpublished), which uses positive linear logic to prove basic correspondence theorems.

  14. One may wonder what exactly will break the standard proof of equivalence between two presentations. Here is a brief sketch. The standard proof, e.g. as spelled out by Priest (2008, Theorem 3.7.5), argues contrapositively: assume that \(\psi \) is invalid in an equivalence model, and show it is also invalid in a universal model. The idea is to consider a world in the equivalence model that is a counterexample to \(\psi \), and take the cluster of worlds connected to it to be the ‘universe’ for a universal model, whereby \(\psi \) is invalidated there too. This works perfectly well if the cluster does not include the trivial world—but not so if the cluster includes \(\bullet \), since the valuation will no longer behave for the required induction.

  15. The results from here onward are connected to hybrid logic, where nominals uniquely denote worlds (Areces and ten Cate 2006); so \(\bot \) would label the trivial world. We are not aware of work in hybrid logic along the lines we are pursing.

  16. There are troubles in the infinite case to do with modal languages’ inability to distinguish bisilimilar structures; see Blackburn et al. (2002, p. 68). Thanks to Marta Bilková for discussion on this point.

  17. The theorem does not hold for the apparently similar claim that, e.g. \(\Diamond \Diamond \Diamond \varphi \) is true at w, for arbitrary\(\varphi \), if w is three steps from the trivial world. As it happens, w may be one step away from some other world at which \(\varphi \) holds.

  18. Proof. Let \(w \,\Vdash\, \Diamond \bot \). Then w accesses some u at which \(\bot \) (so in fact \(u = \bullet \), but we don’t need this). At all worlds, \(\bot \) implies \(\varphi \) for any \(\varphi \). So u also makes \(\varphi \) true, and therefore \(w \,\Vdash\, \Diamond \varphi \), since w accesses u.

References

  • Alban, M. (1943). Independence of the primitive symbols of Lewis’s calculi of propositions. Journal of Symbolic Logic, 8, 25–26.

    Google Scholar 

  • Areces, C., & ten Cate, B. (2006). Hybrid logics. In B. van Benthem & F. Wolter (Eds.), Handbook of modal logic. Amsterdam: Elsevier.

    Google Scholar 

  • Azzouni, J. (2013). Inconsistency in natural languages. Synthese, 190(15), 3175–3184.

    Google Scholar 

  • Baron, S., Colyvan, M., & Ripley, D. (2017). How mathematics can make a difference. Philosopher’s Imprint, 17(3), 1–19.

    Google Scholar 

  • Berto, F. (2012). Existence as a real property. New York: Springer.

    Google Scholar 

  • Berto, F. (2013). Impossible worlds. In E. Zalta (Ed.), Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/impossible-worlds/. Accessed 2 Mar 2018.

  • Berto, F. (2014a). Modal noneism: Transworld identity, identification, and individuation. Australasian Journal of Logic, 11(2), 61–89.

    Google Scholar 

  • Berto, F. (2014b). On conceiving the inconsistent. Proceedings of the Aristotelian Society, 114(1), 103–121.

    Google Scholar 

  • Bjerring, J. C. (2013). On counterpossibles. Philosophical Studies, 168(2), 327–353.

    Google Scholar 

  • Blackburn, P., de Rijke, M., & Venema, Y. (2002). Cambridge tracts in theoretical computer science. In Modal logic (Vol. 53). Cambridge: Cambridge University Press.

    Google Scholar 

  • Bueno, O. (2007). Troubles with trivialism. Inquiry, 50(6), 655–667.

    Google Scholar 

  • Bueno, O., & Zalta, E. N. (2017). Object theory and modal meinongianism. Australasian Journal of Philosophy, 95(4), 761–778.

    Google Scholar 

  • Cantini, A. (2003). The undecidability of Grisin’s set theory. Studia Logica, 74(3), 345–368.

    Google Scholar 

  • Cresswell, M. (1975). Hyperintensional logic. Studia Logica, 34, 25–38.

    Google Scholar 

  • Estrada-González, L. (2012). Models of possibilism and trivialism. Logic and Logical Philosophy, 21, 175–205.

    Google Scholar 

  • Fine, K. (2014). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43(2), 549–577.

    Google Scholar 

  • Girard, P., & Weber, Z. (2015). Bad worlds. Thought, 4, 93–101.

    Google Scholar 

  • Girard, P., & Weber, Z. (unpublished). Modal logic without contraction in a metatheory without contraction. (manuscript).

  • Halldén, S. (1950). Results concerning the decision problem of Lewis’s calculi S3 and S6. Journal of Symbolic Logic, 14, 230–236.

    Google Scholar 

  • Humberstone, L. (2011). Variations on a trivialist argument of Paul Kabay. Journal of Logic, Language, and Information, 20, 115–132.

    Google Scholar 

  • Jago, M. (2013). Impossible worlds. Noûs, 47(3), 713–728.

    Google Scholar 

  • Jago, M. (2014). The impossible: An essay on hyperintensionality. Oxford: Oxford University Press.

    Google Scholar 

  • Kabay, P. D. (2011). A defense of trivialism. Ph.D. thesis, University of Melbourne, Australia.

  • Kielkopf, C. (1975). The logic of Nihilism. The New Scholasticism, 49, 162–167.

    Google Scholar 

  • Krakauer, B. (2013). What are impossible worlds? Philosophical Studies, 165(3), 989–1007.

    Google Scholar 

  • Kroon, F. (2008). Much ado about nothing: Priest and the reinvention of noneism. Philosophy and Phenomenological Research, 76(1), 199–207.

    Google Scholar 

  • Lamb, E. (2014). The saddest thing i know about the integers. Scientific American. Roots of unity blog. http://blogs.scientificamerican.com/roots-of-unity/the-saddest-thing-i-know-about-the-integers/.

  • Lewis, D. (1990). Noneism or allism? Mind, 99(393), 23–31.

    Google Scholar 

  • Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell.

    Google Scholar 

  • Mares, E. D. (1997). Who’s afraid of impossible worlds? Notre Dame Journal of Formal Logic, 38(4), 516–526.

    Google Scholar 

  • Martin, B. (2014). Dialetheism and the impossibility of the world. Australasian Journal of Philosophy, 93(1), 61–75.

    Google Scholar 

  • Meinong, A. (1910). On assumptions. (Trans: J. Heanue) University of California Press, 1983.

  • Mortensen, C. (1989). Anything is possible. Erkenntnis, 30(3), 319–337.

    Google Scholar 

  • Mortensen, C. (2005). It isn’t so, but could it be? Logique et Analyse, 48(189/192), 351–360.

    Google Scholar 

  • Nolan, D. (1997). Impossible worlds: A modest approach. Notre Dame Journal of Formal Logic, 38(4), 535–572.

    Google Scholar 

  • Nolan, D. (2008). Properties and paradox in Graham Priest’s towards non-being. Philosophy and Phenomenological Research, 76(1), 191–198.

    Google Scholar 

  • Nolan, D. (2013). Impossible worlds. Philosophy Compass, 8(4), 360–372.

    Google Scholar 

  • Omori, H., & Weber, Z. (unpublished). Just true? On the metatheory for paraconsistent truth (to appear).

  • Plebani, M. (2015). Could everything be true? Probably not. Philosophia, 43(2), 499–504.

    Google Scholar 

  • Priest, G. (1990). Boolean negation and all that. Journal of Philosophical Logic, 19, 201–215.

    Google Scholar 

  • Priest, G. (1998). The trivial object and the non-triviality of a semantically closed theory with descriptions. Journal of Applied Non-Classical Logics, 8(1–2), 171–183.

    Google Scholar 

  • Priest, G. (2000). Could everything be true? Australasian Journal of Philosophy, 78(2), 189–195.

    Google Scholar 

  • Priest, G. (2006a). Doubt truth to be a liar. Oxford: Oxford University Press.

    Google Scholar 

  • Priest, G. (2006b). In contradiction: A study of the transconsistent (2nd ed.). Oxford: Oxford University Press.

    Google Scholar 

  • Priest, G. (2008). An introduction to non-classical logic (2nd ed.). Cambridge: Cambridge University Press.

    Google Scholar 

  • Priest, G. (2014). One. Oxford: Oxford University Press.

    Google Scholar 

  • Priest, G. (2016a). Thinking the impossible. Philosophical Studies, 173, 2649–2662.

    Google Scholar 

  • Priest, G. (2016b). Towards non-being (2nd ed.). Oxford: Oxford University Press.

    Google Scholar 

  • Routley, R. (1980). Exploring Meinong’s jungle and beyond. Philosophy Department, RSSS, Australian National University, Canberra. (New four volume edition from Synthese Library forthcoming).

  • Routley, R. (1983). Nihilism and Nihilist Logics. In Discussion Papers in Environmental Philosophy (Vol. 4). Canberra: Australian National University.

  • Routley, R., Plumwood, V., Meyer, R. K., & Brady, R. T. (1982). Relevant Logics and their Rivals. Atascardero, CA: Ridgeview.

    Google Scholar 

  • Slaney, J. K. (1989). RWX is not curry-paraconsistent. In G. Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: Essays on the inconsistent (pp. 472–480). Munchen: Philosophia Verlag.

    Google Scholar 

  • Stalnaker, R. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory (pp. 98–112)., American philosophical quarterly monograph series 2 Oxford: Blackwell.

    Google Scholar 

  • Sylvan, R. (1992). On interpreting truth tables and relevant truth table logic. Notre Dame Journal of Formal Logic, 33(2), 207–215.

    Google Scholar 

  • Veldman, W. (1976). An intuitionistic completeness theorem for intuitionistic predicate logic. Journal of Symbolic Logic, 41, 159–166.

    Google Scholar 

  • Weber, Z. (2012). Transfinite cardinals in paraconsistent set theory. Review of Symbolic Logic, 5(2), 269–293.

    Google Scholar 

  • Weber, Z., Badia, G., & Girard, P. (2016). What is an inconsistent truth table? Australasian Journal of Philosophy, 94(3), 533–548.

    Google Scholar 

  • Williams, B. (1981). Persons, character, and morality. In Moral luck (pp. 1–19). Cambridge: Cambridge University Press.

  • Wittgenstein, L. (1961). Notebooks 1914–1916. New York: Harper. (Translated by G.E.M. Anscombe).

    Google Scholar 

  • Woodward, R. (2013). Towards being. Philosophy and Phenomenological Research, 86(1), 183–193.

    Google Scholar 

Download references

Acknowledgements

Thanks to Patrick Girard and several anonymous referees for very helpful comments. Thanks to audiences at: the University of Otago, Kyoto University, the Buenos Aires Logic Group, and the University of Otago. This was in part written while both authors were visiting the Institute for Computer Science, Czech Academy of Science, with our gratitude (especially to Petr Cintula). This work was supported by the Marsden Fund, Royal Society of New Zealand, and the Japan Society for the Promotion of Science JSPS KAKENHI Grant Number JP16K16684.

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Otago, 117 Union St E, Dunedin, 9054, New Zealand

    Zach Weber

  2. Department of Philosophy, Graduate School of Letters, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Hitoshi Omori

Authors
  1. Zach Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hitoshi Omori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zach Weber.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Weber, Z., Omori, H. Observations on the Trivial World. Erkenn 84, 975–994 (2019). https://doi.org/10.1007/s10670-018-9990-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10670-018-9990-y

Search

Navigation