Skip to main content
Log in

Political ignorance is both rational and radical

  • Original Research
  • Published:
Synthese Aims and scope Submit manuscript

“If there is one thing in this world

that you can depend on,

that you can bet your last dollar on,

it’s the ignorance of the American people.”

-Pase Rock

Abstract

It is commonly held that political ignorance is rational, a response to the high costs and low benefits of acquiring political information. But many recent critics of the claim that political ignorance is rational instead urge that it is a simple consequence of agents not concerning themselves with the acquisition of political information whatsoever. According to such critics, political ignorance is inadvertent radical ignorance rather than a rational response to the incentives faced by agents in democracies. And since political ignorance is not a response to incentives, these critics urge, it cannot be ameliorated by incentivizing the acquisition of political information. This paper has two goals. First, I show that these seemingly competing accounts of political ignorance are in fact complementary, together explaining much political ignorance. Indeed, there is a sense in which political ignorance can be both rational and radical at the same time. Second, I more closely examine the relationship between incentives, kinds of political ignorance, and the acquisition of political information. On the one hand, from the fact that political ignorance is rational it does not follow that it can be overcome by incentivizing the acquisition of information. On the other hand, from the fact that political ignorance is radical it does not follow that it cannot be overcome by incentivizing the acquisition of information. Lastly, the complexity of the information in question is more relevant to determining whether ignorance can be overcome than whether such ignorance is rational or radical.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. Of course, this is consistent with there being some disagreement concerning the extent of such ignorance. See Lepoutre (2022) for relevant discussion.

  2. This shouldn’t be taken to exclude the possibility that other factors play a role in exacerbating levels of political ignorance. For example, perhaps the prevalence of political misinformation also explains some public political ignorance. See Benkler, Faris, and Roberts (2018), Brown (2018), O’Connor and Weatherall (2019), Fritts and Cabrera (2022), Lynch (2022), and Gibbons (2023a; 2023b) for relevant discussion.

  3. With that said, in Sect. 3 I suggest that even the models with the relevant assumptions capture the behavior of some agents.

  4. See Somin (2021) for a helpful overview of many issues related to models of rational political ignorance.

  5. For an overview of the epistemic demands placed upon agents by competing theories of democratic participation, see Somin (2013: pp. 38–61). Notably, on some theories, the sort of information mentioned above understates the amount of information required. If so, then the costs of information acquisition are even steeper than the preceding discussion suggests. For related discussion of the epistemic responsibilities of agents in democracies, see Boult (2021) and Giavazzi (2022).

  6. There may even be straightforward monetary costs related to acquisition in some cases (for example, if the relevant information is behind a paywall, requires a subscription to access, and so forth), though the opportunity costs are more heavily emphasized in standard treatments of rational ignorance.

  7. Precisely how unlikely voters are to cast a decisive vote, though an interesting question, is less important for our purposes than the fact that the relevant likelihood is assumed to be low and, more significantly, known—at least intuitively—by rational agents. On the probability of casting a decisive vote, see Ferejohn and Fiorina (1974), Owen and Grofman (1984), Aldrich (1993), and Gelman et al. (2012). See Somin (2013: pp. 62–4) for relevant discussion.

  8. A closely related theory—rational irrationality—purports to explain the prevalence of seemingly widespread political irrationality in more-or-less the same way that a theory of rational ignorance explains the pervasiveness of political ignorance (Caplan, 2007). Broadly put, it holds that the costs of political rationality outweigh its benefits (and that the benefits of political irrationality outweigh its costs). And one might think that this theory ought to be considered alongside models of rational ignorance and radical ignorance as a possible explanation for high levels of political ignorance. However, I opt not to focus on it for two reasons. First, given the clear parallels between theories of rational ignorance and rational irrationality, it is plausible to assume that whatever applies to theories of rational ignorance also applies, at least in broad outline, to theories of rational irrationality. Moving forward, then, I assume that whatever is true of the comparative merits of theories of rational ignorance and radical ignorance (together with an assessment of their respective practical implications) is true of theories of rational irrationality and what we could call radical irrationality. Second, rational irrationality is of course best viewed as an account of political irrationality rather than political ignorance per se. Notably, not all politically irrational agents are politically ignorant, though (rational) political irrationality may explain the manner in which some political agents are selectively ignorant (Gibbons, 2022; Hannon, 2022). Thanks to an anonymous referee for discussion on this point.

  9. Of course, if people find the process of acquiring information fun or entertaining, this also suggests that the costs of information acquisition are lower than they might initially seem, at least for some people.

  10. Naturally, the political information acquired from entertaining sources is likely often to be of dubious quality.

  11. For related discussion, see Gunn (2015: pp. 280–82).

  12. Throughout this paper I write of political ignorance being “a choice”, being “chosen”, and so on. To clarify, when I say that political ignorance is chosen, I primarily have in mind cases where an agent is confronted with a range of alternatives and makes a conscious and deliberate choice of one alternative over the others. Specifically, these are cases where agents deliberately choose to forgo the acquisition of political information instead of some other alternative that involves the acquisition of information, and where the agent is aware of―and considering―the various alternatives. This contrasts with cases where an agent chooses to refrain from acquiring political information but without at the same time considering other alternatives involving the acquisition of information (as when I choose to listen to Thin Lizzy but without considering the possibility of listening to Motörhead instead). Thanks to an anonymous referee for pushing for clarity on this matter.

  13. It is worth noting that not all proponents of rational ignorance models intend for these models to be psychologically realistic. Downs, for instance, writes that “[theoretical] models should be tested primarily by the accuracy of their predictions rather than by the reality of their assumptions” (Downs, 1957: p. 21). Others, most notably Milton Friedman (1953) and Satz and Ferejohn (1994), share similar methodological views. For such proponents of rational choice models, the challenge raised by proponents of radical ignorance misfires, since the relevant models are not intended to accurately capture the psychological profile of actual agents. However, other proponents of these models—such as Somin—do intend for their models to be empirically realistic (Somin, 2013: p. 89; though see also Somin, 2015: p. 382). For these proponents, the potential falsity of the relevant assumptions is an issue. See Hoffman (2015: p. 306) for further discussion.

  14. It is notable―and admittedly controversial―that this way of characterizing rationality effectively makes rational choice a “theory of preference maximization” (Kogelmann and Gaus, 2017). It takes for granted an agent’s preferences, construing rationality as about, inter alia, always choosing the most preferred option from among a set of alternatives, and so on. For some, this way of viewing rationality may be somewhat unsatisfactory. Perhaps such people are drawn to the claim that rationality requires beliefs (and preferences) that are, in some sense, well-founded, rather than simply taking an agent’s preferences for granted (Gaus, 2008: pp. 9–12). However, in what follows, I adopt the approach to rationality utilized by scholars such as Downs and others who have argued that political ignorance is rational.

  15. Indeed, some evidence indicates that people willingly give up a chance to win money in order to avoid being exposed to ideologically incongruent information (Frimer, Skitka, and Motyl, 2017). Relatedly, some people have belief-based preferences which lead them to seek out and process information in highly selective ways (Bénabou and Tirole, 2016). See Williams (2020a: pp. 5–7) for further discussion.

  16. For a fascinating discussion of socially adaptive belief more generally, see Williams (2020b).

  17. See Hertwig and Engels (2016), Gigerenzer and Garcia-Retamero (2017), and Brown and Walasek (2020) for sustained discussion of reasons people may have for remaining deliberately ignorant. For a more detailed investigation of deliberate ignorance, see the collection of articles in Hertwig and Engels (2020).

  18. In a recent paper, Somin makes a similar claim about rational political ignorance, writing that “the prediction of rational behavior in this sphere is not dependent on the assumption that voters are hyper-logical or capable of making complex calculations about odds” (Somin, 2023: p. 289). Indeed, he elsewhere writes that precisely calculating overall costs and benefits is often irrational insofar as it takes more effort than it is worth to most agents (Somin, 2021: p. 242).

  19. Tversky (1969), for instance, famously argues that, under certain conditions, intransitivity of preferences arises.

  20. Cf. Bovens (2022) on the intransitivity of preferences.

  21. Such people may have been influenced by these models to deliberately refrain from the acquisition of political information in ways they didn’t previously. Their political ignorance may even have been radical up until the point they first encountered models of rational ignorance, whereupon the option to acquire political information became salient for them, only to be rejected in favor of options involving deliberate ignorance.

  22. In addition to the sources already mentioned earlier, see also Schwartz et al. (2020).

  23. I assume here that the agents in question are typical adults. Children, for instance, especially young children, are likely to be radically ignorant of the overwhelming majority of politically relevant information.

  24. For some recent examples of this literature, see Kogelmann and Ogden (2018) and Van der Vossen (2021).

  25. See for instance Kunda (1990), Von Hippel and Trivers (2011), Kurzban (2012), Kahan (2017), Simler and Hanson (2017), and Williams (2021).

  26. See Lupia and Prior (2008) for some relevant experimental evidence in support of this claim.

  27. Somin discusses some of these issues, concluding that “[at] the very least, we should devote more sustained analysis” to this sort of proposal (Somin, 2023: p. 300).

  28. This is consistent with granting that such agents could be incentivized to acquire political information in some other fashion. However, it remains the case that institutionally realistic (i.e., legal, affordable, and practically implementable) ways of incentivizing the acquisition of political information may be ineffective for some people.

  29. Of course, moving from one jurisdiction to another is costly. But political systems which enable a greater degree of foot voting make it less costly than it would be otherwise.

  30. See also Friedman (2006; 2019). On complexity and politics more generally, see Hayek (1945), Zolo (1992), Jervis (1998), and Little (2008).

  31. Cf. Guerrero (2021: p. 420). See also Reiss (2021). But see Somin (2015) for some criticism.

  32. Thanks to Daniel Williams and two anonymous referees for their insightful feedback on different versions of this paper.

References

  • Aldrich, J. (1993). Rational choice and turnout. American Journal of Political Science, 37, 246–278.

    Google Scholar 

  • Barnett, Z. (2020). Why you should vote to change the outcome. Philosophy and Public Affairs, 48, 422–446.

    Google Scholar 

  • Bénabou, R., & Tirole, J. (2016). Mindful economics: the production, consumption, and value of beliefs. Journal of Economic Perspectives, 30(3), 141–164.

    Google Scholar 

  • Benkler, Y., Faris, R., & Roberts, H. (2018). Network propaganda: Manipulation, disinformation, and radicalization in american politics. Oxford University Press.

    Google Scholar 

  • Bennett, S. E., & Friedman, J. (2008). The irrelevance of economic theory to understanding economic ignorance. Critical Review: A Journal of Politics and Society, 20(1), 195–258.

    Google Scholar 

  • Boult, C. (2021). The epistemic responsibilities of citizens in a democracy. In M. Hannon & J. de Ridder (Eds.), The routledge handbook of political epistemology. Routledge.

    Google Scholar 

  • Bovens, L. (2022). Four structures of intransitive preferences. In C. M. Melenovsky (Ed.), The routledge handbook of philosophy, politics, and economics. Routledge.

    Google Scholar 

  • Brennan, J. (2016). Against democracy. Princeton University Press.

    Google Scholar 

  • Brennan, J., & Freiman, C. (2023). Why swing-state voting is not effective altruism: the bad news about the good news about voting. Journal of Political Philosophy, 31(1), 60–79.

    Google Scholar 

  • Brown, E. (2018). Propaganda, misinformation, and the epistemic value of democracy. Critical Review: A Journal of Politics and Society, 30(3–4), 194–218.

    Google Scholar 

  • Brown, G. D. A., & Walasek, L. (2020). Models of deliberate ignorance in individual choice. In R. Hertwig & C. Engels (Eds.), Deliberate Ignorance: choosing not to know. Cambridge: The MIT Press.

    Google Scholar 

  • Caplan, B. (2007). The myth of the rational voter. Princeton University Press.

    Google Scholar 

  • Converse, P. E. (1964). The nature of belief systems in mass publics. In D. E. Apter (Ed.), Ideology and discontent. Free Press of Glencoe.

    Google Scholar 

  • DelliCarpini, M. X., & Keeter, S. (1996). What Americans know about politics and why it matters. Yale University Press.

    Google Scholar 

  • Downs, A. (1957). An economic theory of democracy. Harper and Row.

    Google Scholar 

  • Edlin, A., Gelman, A., & Kaplan, N. (2008). Vote for charity’s sake. The Economists’ Voice, 5, 1–4.

    Google Scholar 

  • Evans, A. J., & Friedman, J. (2011). search versus browse: a theory of error grounded in radical (Not Rational) ignorance. Critical Review: A Journal of Politics and Society, 23(1–2), 73–102.

    Google Scholar 

  • Ferejohn, J., & Fiorina, M. (1974). The paradox of not voting: A decision theoretic analysis. American Political Science Review, 68, 525–536.

    Google Scholar 

  • Friedman, J. (2005). Popper, weber, and hayek: The epistemology and politics of ignorance. Critical Review: A Journal of Politics and Society, 17(1–2), 1–58.

    Google Scholar 

  • Friedman, J. (2006). Taking ignorance seriously: Rejoinder to critics. Critical Review: A Journal of Politics and Society, 18(4), 467–532.

    Google Scholar 

  • Friedman, J. (2019). Power without knowledge: A critique of technocracy. Oxford University Press.

    Google Scholar 

  • Friedman, M. (1953). The methodology of positive economics’, in essays in positive economics. University of Chicago Press.

    Google Scholar 

  • Frimer, J. A., Skitka, L. A., & Motyl, M. (2017). Liberals and conservatives are similarly motivated to avoid exposure to one another’s opinions. Journal of Experimental Social Psychology, 72, 1–12.

    Google Scholar 

  • Fritts, M., & Cabrera, F. (2022). Fake news and epistemic vice: Combating a uniquely noxious market. Journal of the American Philosophical Association. https://doi.org/10.1017/apa.2021.11

    Article  Google Scholar 

  • Gaus, G. F. (2008). On philosophy, politics, and economics. Wadsworth Cengage Learning.

    Google Scholar 

  • Gelman, A., Silver, N., & Edlin, A. (2012). What is the probability your vote will make a difference. Economic Inquiry, 50(2), 321–326.

    Google Scholar 

  • Giavazzi, M. (2022). The epistemic responsibilities of voters: towards an assertion-based account. Journal of Moral Philosophy, 20(1–2), 111–131.

    Google Scholar 

  • Gibbons, A. F. (2021). Political disagreement and minimal epistocracy. Journal of Ethics and Social Philosophy, 19(2), 192–201.

    Google Scholar 

  • Gibbons, A. F. (2022). Is epistocracy irrational? Journal of Ethics and Social Philosophy, 21(2), 251–281. https://doi.org/10.26556/jesp.v21i2.1581

    Article  Google Scholar 

  • Gibbons, A. F. (2023a). Bullshit in politics pays. Episteme. https://doi.org/10.1017/epi.2023.3

    Article  Google Scholar 

  • Gibbons, A. F. (2023b). Bad language makes good politics. Inquiry: an Interdisciplinary Journal of Philosophy. https://doi.org/10.1080/0020174X.2023.2203164

    Article  Google Scholar 

  • Gigerenzer, G., & Garcia-Retamero, R. (2017). Cassandra’s regret: The psychology of not wanting to know. Psychological Review, 124, 179–196.

    Google Scholar 

  • Gilens, M. (2001). Political Ignorance and collective policy preferences. American Political Science Review, 95(2), 379–396.

    Google Scholar 

  • Guerrero, A. (2021). The epistemic case for non-electoral forms of democracy. In M. Hannon & J. de Ridder (Eds.), The routledge handbook of political epistemology. Routledge.

    Google Scholar 

  • Gunn, P. (2015). Looking but not seeing: The (Ir)relevance of incentives to political ignorance. Critical Review: A Journal of Politics and Society, 27(3–4), 270–298.

    Google Scholar 

  • Hannon, M. (2022). Are knowledgeable voters better voters? Politics, Philosophy & Economics, 21(1), 29–54. https://doi.org/10.1177/1470594X211065080

    Article  Google Scholar 

  • Hardin, R. (2009). How do you know? The economics of ordinary knowledge. Princeton University Press.

    Google Scholar 

  • Hayek, F. A. (1945). The use of knowledge in society. American Economic Review, 35(4), 519–530.

    Google Scholar 

  • Hertwig, R., & Engels, C. (2016). Homo ignorans: deliberately choosing not to know. Perspectives on Psychological Science, 11(3), 359–372.

    Google Scholar 

  • Hertwig, R., & Engels, C. (2020). Deliberate ignorance: Choosing not to know. The MIT Press.

    Google Scholar 

  • Hoffman, T. (2015). Rational choice and political irrationality in the new millennium. Critical Review: A Journal of Politics and Society, 27(3–4), 299–315.

    Google Scholar 

  • Ikeda, S. (2003). How compatible are public choice and austrian political economy? Review of Austrian Economics, 16, 63–75.

    Google Scholar 

  • Jervis, R. (1998). System effects: Complexity in political and social life. Princeton University Press.

    Google Scholar 

  • Kahan, D. (2017). The expressive rationality of inaccurate perceptions. Behavioral and Brain Sciences., 40, 26–28.

    Google Scholar 

  • Kogelmann, B., & Gaus, G. (2017). Rational choice theory. In A. Blau (Ed.), Research methods in analytic political theory. Cambridge: Cambridge University Press.

    Google Scholar 

  • Kogelmann, B., & Ogden, B. G. (2018). Enough and as good: A formal model of lockean first appropriation. American Journal of Political Science., 62(3), 682–694.

    Google Scholar 

  • Kolodny, N. & Brunero, B. (2020) Instrumental Rationality, The Stanford Encyclopedia of Philosophy (Spring 2020 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2020/entries/rationality-instrumental/

  • Kunda, Z. (1990). The case for motivated reasoning. Psychological Bulletin., 108, 480–498.

    Google Scholar 

  • Kurzban, R. (2012). Why everyone (Else) is a hypocrite. Princeton University Press.

    Google Scholar 

  • Lau, R. R., & Redlawsk, D. P. (2001). Advantages and disadvantages of cognitive heuristics in political decision making. American Journal of Political Science, 45(4), 951–971.

    Google Scholar 

  • Lepoutre, M. (2022). Political understanding. British Journal of Political Science. https://doi.org/10.1017/S0007123422000023

    Article  Google Scholar 

  • Little, A. (2008). Democratic piety: Complexity, Conflict, and violence. Edinburgh University Press.

    Google Scholar 

  • Lupia, A. (2016). Uninformed: Why people know so little about politics and what we can do about it. Oxford University Press.

    Google Scholar 

  • Lupia, A., & McCubbins, M. (1998). The Democratic dilemma: Can citizens learn what they need to know? Cambridge University Press.

    Google Scholar 

  • Lupia, A., & Prior, M. (2008). Money, time, and political knowledge: Distinguishing quick recall and political learning skills. American Political Science Review, 52(1), 169–183.

    Google Scholar 

  • Lynch, M. (2022). Memes, misinformation, and political meaning. Southern Journal of Philosophy, 60(1), 38–56.

    Google Scholar 

  • MacAskill, W. (2015). Doing good better: How effective altruism can help you make a difference. Penguin.

    Google Scholar 

  • O’Connor, C., & Weatherall, B. (2019). The misinformation age: How false beliefs spread. Yale University Press.

    Google Scholar 

  • Oppenheimer, D., & Edwards, M. (2012). Democracy despite itself: Why a system that shouldn’t work at all works so well. MIT Press.

    Google Scholar 

  • Owen, G., & Grofman, B. (1984). To vote or not to vote: The paradox of nonvoting. Public Choice, 42, 311–325.

    Google Scholar 

  • Page, S. E. (2010). Diversity and complexity. Princeton University Press.

    Google Scholar 

  • Popkin, S. L. (1991). The reasoning voter: communication and persuasion in presidential campaigns. University of Chicago Press.

    Google Scholar 

  • Reiss, J. (2021). Why do experts disagree? Critical Review: A Journal of Politics and Society, 32(1–3), 218–241.

    Google Scholar 

  • Riker, W. H., & Ordeshook, P. (1968). A theory of the calculus of voting. American Political Science Review, 62(1), 24–42.

    Google Scholar 

  • Satz, D., & Ferejohn, J. (1994). Rational choice and social theory. The Journal of Philosophy, 91(2), 71–87.

    Google Scholar 

  • Schumpeter, J. A. (1942). Capitalism, socialism, and democracy. Harper and Brothers.

    Google Scholar 

  • Schwartz, B., Richerson, P. J., Berkman, B. E., Frankenreiter, J., Hagmann, D., Isaacowitz, D. M., Pachur, T., Schooler, L. J., & Wehling, P. (2020). The deep structure of deliberate ignorance: mapping the terrain. In R. Hertwig & C. Engels (Eds.), Deliberate ignorance: choosing not to know. The MIT Press.

    Google Scholar 

  • Simler, K., & Hanson, R. (2017). The elephant in the brain. Oxford University Press.

    Google Scholar 

  • Sinhababu, N. (2016). In defense of partisanship. In D. Killoren, E. Crookston, & J. Trerise (Eds.), Ethics in politics: new papers on the rights and obligations of political agents. Routledge.

    Google Scholar 

  • Somin, I. (2008). Knowledge about ignorance: New directions in the study of political information. Critical Review: A Journal of Politics and Society, 18(1–3), 255–278.

    Google Scholar 

  • Somin, I. (2013). Democracy and political ignorance: why smaller government is better. Stanford University Press.

    Google Scholar 

  • Somin, I. (2015). The ongoing debate over political ignorance: reply to my critics. Critical Review: A Journal of Politics and Society, 27(3–4), 380–414.

    Google Scholar 

  • Somin, I. (2020). Free to move: Foot voting, migration, and political freedom. Oxford University Press.

    Google Scholar 

  • Somin, I. (2021). Is political ignorance rational? In M. Hannon & J. de Ridder (Eds.), The routledge handbook of political epistemology. Routledge.

    Google Scholar 

  • Somin, I. (2023). Top-down and bottom-up solutions to the problem of political ignorance. In H. Samaržija & Q. Cassam (Eds.), The epistemology of democracy. Routledge.

    Google Scholar 

  • Stigler, G. J. (1976). The Xistence of X-Efficiency. American Economic Review, 66(1), 213–216.

    Google Scholar 

  • Sweeny, K., Melnyk, D., Miller, W., & Shepperd, J. (2010). Information avoidance: Who, What, When, and Why. Review of General Psychology, 14(4), 340–353.

    Google Scholar 

  • Tullock, G. (1972). Toward a mathematics of politics. University of Michigan Press.

    Google Scholar 

  • Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76(1), 31–48.

    Google Scholar 

  • Van der Vossen, B. (2021). Property, the environment, and the lockean proviso. Economics & Philosophy, 37(3), 395–412. https://doi.org/10.1017/S0266267120000401

    Article  Google Scholar 

  • Von Hippel, W., & Trivers, R. (2011). The evolution and psychology of self-deception. Behavioral and Brain Sciences., 34(1), 1–16.

    Google Scholar 

  • Williams, D. (2020a). Motivated Ignorance, rationality, and democratic politics. Synthese, 198, 7807–7827.

    Google Scholar 

  • Williams, D. (2020b). Socially adaptive belief. Mind & Language, 36(3), 333–354.

    Google Scholar 

  • Williams, D. (2021). Epistemic irrationality and the Bayesian brain. British Journal for the Philosophy of Science., 72(4), 913–938.

    Google Scholar 

  • Wittman, D. (1995). The myth of democratic failure. University of Chicago Press.

    Google Scholar 

  • Zaller, J. R. (1992). The nature and origins of mass opinion. Cambridge University Press.

    Google Scholar 

  • Zolo, D. (1992). Democracy and complexity: a realist approach. The Pennsylvania State University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Adam F. Gibbons.

Ethics declarations

Conflict of interest

There is no conflict of interest to declare.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gibbons, A.F. Political ignorance is both rational and radical. Synthese 202, 61 (2023). https://doi.org/10.1007/s11229-023-04293-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11229-023-04293-1

Keywords

Navigation