Abstract
In situations where we ignore everything but the space of possibilities, we ought to suspend judgment—that is, remain agnostic—about which of these possibilities is the case. This means that we cannot sum our degrees of belief in different possibilities, something that has been formalised as an axiom of non-additivity. Consistent with this way of representing our ignorance, I defend a doxastic norm that recommends that we should nevertheless follow a certain additivity of possibilities: even if we cannot sum degrees of belief in different possibilities, we should be more confident in larger groups of possibilities. It is thus shown that, in the type of situation considered (in so-called “classical ignorance”, i.e. “behind a thin veil of ignorance”), it is epistemically rational for advocates of suspending judgment to endorse this comparative confidence, while on the other hand it is shown that, even in classical ignorance, no stronger belief—such as a precise uniform probability distribution—is warranted.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For recent philosophical discussions on other aspects of suspension of judgment, see, e.g., Friedman (2013a, 2013b, 2015); Tang (2015); Staffel (2019); Rosa (2021); McGrath (2023). Its origins date back to the ancient scepticism of Pyrrho (Empiricus, 1994) A number of formal representational frameworks in inductive logic have been elaborated to properly represent a suspension of judgment. The different approaches include non-probabilistic non-numerical calculi such as Norton (2008), and imprecise probabilities such as de Cooman and Miranda (2007). For an overview of approaches, see Halpern (2003 Ch. 2) and Dubois (2007).
As we will see, the resulting preference order verifies what is known as monotonicity in cardinality. For details on comparative confidence judgments, see Titelbaum (2022 ch. 3). This norm must not be confused with the usual axiom of additivity of comparative orders (in turn different from Kolmogorov’s axiom of additivity stated above), which can be stated as follows: for disjoint events A, B, and C, \( A \succ B \Leftrightarrow A \cup C \succ B \cup C \) (de Finetti, 1937). Unless stated otherwise, throughout this paper the term “preference” will thus refer to an epistemic preference, i.e., a comparative confidence. It will be only in Section 6 when we will explore the connections of our epistemic norm with decision-theoretic and pragmatic norms for action.
Other representations of ignorance include Dempster-Shafer’s theory of belief functions (Shafer, 1976) or plausibility measures and entrenchment rankings from AGM theory (Rott, 2001; Halpern, 2003) (which do not satisfy additivity, and are only super-additive). For more details, see (Genin and Huber, 2022 esp.§3).
To further frame the present research, let me underline that there has been an orthogonal long discussion between qualitative and quantitative approaches, mostly focused on whether there is an agreement between them. This is an orthogonal debate. The agnostic view, which I assume and explain in this paper and that has been studied in epistemology, decision theory, and philosophy of science, is weaker than any qualitative probability. (Hence, discussions like Kraft et al. (1959) and Scott (1964) are here irrelevant; that is why in Section 3, I explore and show that the lack of agreement between the two frameworks under analysis.) The result is that a specific comparative order is warranted on top of the agnostic view.
The lack of justification for a uniform probability distribution has been widely stressed in the foundations of statistical mechanics (see, e.g., Albert (2000 Ch. 3.2)).
In our example, the objective chance distribution describes the chances of the treasure being inside any of the chests. The physical significance of this distribution can be considered modeling an unknown underlying stochastic process delivering the treasure to one of the chests. Similarly, this chance distribution can be also considered modeling a process without genuine objective chances, such as a deterministic process instead of a genuinely stochastic process (corresponding, for instance, to someone putting the treasure in one chest).
The Principal Principle states that, if p is a certain proposition about the outcome of some chancy event and E is our background evidence at t, which must be admissible evidence, then \(Cr(p|Ch(p) = x \wedge E) = x\). (Evidence E is admissible relative to p if it contains no information relevant to whether p will be true, except perhaps information bearing on the chance of p.) See Lewis (1980 86) and Hoefer (2018 Ch. 3).
Some details about our setting may help to prevent the temptation of thinking that the existence of “counterexamples” would undermine the rationality of endorsing our norm. Given our ignorance, we must not make conjectures about the chances and allow that any chance distribution could be the case. Obviously, there are plenty of “counterexamples” in which our recommendation leads to a wrong confidence (i.e., prefer the path where there is no treasure)! Still, the recommendation claims that, given our ignorance regarding the presence or lack of “counterexamples” and therefore acknowledging the obvious fallibility of the norm, it is nevertheless epistemically irrational not to be more confident in the larger group of possibilities. So, strictly speaking, such cases are not counterexamples.
A way to justify the assumption of \((**)\) is to maintain that we began our discussion in an epistemic situation of ignorance in which we were suspending judgment; thus, we were assuming that there was no reason that could lead us to any comparative order. Hence, the only reason is (AddPoss); therefore, \( A \succ B\) only if \(|A| > |B| \). This justification, however, seems to conflate our epistemological ignorance of reasons with the objective lack of reasons. Yet, this justification might be acceptable insofar as no other reason has been provided in the literature; that is, in an epistemic situation of suspension of judgment, we assume that there is no other reason to endorse such comparative order.
One might be tempted to conclude \((*)\) due to the following invalid proof by reductio. By switching the names of the variables in (AddPoss*), we have the logically equivalent version with the signs changed (AddPoss**): \( A \prec B \longleftrightarrow | A| < | B | \). Let us show that it is impossible that \((*): | A | = | B | \rightarrow A \sim B \) is false and both (AddPoss*) and (AddPoss**) true. Let A and B be such that \(| A | = | B |\) is true and \( A \sim B \) false. Then, \((*)\) is false, \( | A| > | B | \) and \( | A| < | B | \) are false, and either \( A \succ B\) or \( A \prec B\) is true. But then either (AddPoss*) or (AddPoss**) is false. Therefore, \((*)\). \(\blacksquare \) (Note that if we use (AddPoss) instead of the stronger (AddPoss*), the proof does not follow, since \((*)\) would be false and the premises (AddPoss) and the analogous of (AddPoss**) vacuously true.) This proof is, however, incorrect. Again, due to an intuitive but wrong disregard of the agnostic position, the proof is wrongly supposing that if \( A \sim B \), then it must be the case that either \( A \succ B\) or \( A \prec B\). This is wrong, for in fact one can suspend judgment and have no confidence whatsoever about the relation between A and B.
Proof. Consider a generic non-uniform quantitative probability P. Then, for some atomic possibilities \(a,b \in \Omega \) (where \(\Omega \) is the finite possibility space), \(P(a)>P(b)\). Since a and b are atomic possibilities, \( | \{a\} |= | \{b\} | \). By \((*)\), we have that \(a \sim b\). Then, it is not the case that \(a \succ b\). Therefore, it is false that \(P(a)>P(b)\) iff \(a \succ b\). Hence, no non-uniform P can represent the order determined by (AddPoss*) \(\blacksquare \)
Similar but more sophisticated arguments based on the notion of accuracy have been proposed to defend a stronger claim: the assigning of equiprobability in situations of ignorance. Pettigrew (2016a, 2016b) directly appeals to accuracy, while Williamson (2010 §3.4) and Landes and Williamson (2013) appeal to a pragmatic caution understood as the minimisation of loss, although Williamson (2018) adapts the argument to a purely non-pragmatic epistemic version appealing to accuracy. I do not assume these stronger results, which would imply our weaker claim (for a numerical uniform, distribution implies that the larger set is more likely). See also Joyce (1998) argument for probabilism from accuracy, and Pettigrew (2016a) and Titelbaum (2022 ch.10) for a clear treatment of the notion of accuracy.
As is usually understood in the literature, “risk” just means the possibility that some undesirable event may occur (Hansson 2018 §1), in this case related to the inaccuracy of our beliefs.
If you say \(A \succ B\) and it is in A, then you are accurate, but not maximally accurate (“+1”) because your claim was weak, so your accuracy scores some positive value “\(+k\)”, \(0<k<1\). If, instead, you say \(A \succ B\) and it is in B, then you are wrong, and so inaccurate to a certain degree. The score would not be maximally inaccurate (“\(-1\)”) for the same reason (your claim was merely a weak comparative statement, not an assignment of a precise credence distribution). It would instead be some negative accuracy value: let us say “\(-k\)”. For more on this, see, along the same lines, Konek (2015) p.2 and Sections 3 and 4. In any case, potential variations in the specific values would not affect the argument below, since the decision rules would yield the same results.
Maximax is a rule associated not with caution but with taking risks, since what it recommends is to prefer the option with the greatest utility, even if its worst possible outcome is worse than those of its alternatives. In contrast, the other rules are cautious in that they prioritise avoiding the option with the worst utility. We will discuss these rules later, when I apply them to our scenario. In any case, for a discussion of whether these are the most cautious rules, see Williamson (2007) for synchronic caution and Radzvilas et al. (forthcoming) for diachronic caution.
In more detail, these rules determine the following (in both tables): there is neither strong nor weak Dominance for neither A nor B are in all worlds as good or better than the other. Minimax makes no choice either, for it would choose the path with the best worst-case inaccuracy, but that is “\(-1\)” for both A and B. Minimax-regret makes no choice either, for the worst-case regret is the same (the worst-case regret of A is +1- -1=+2; idem for B). Finally, Hurwicz takes into account the worst and the best values of each option, conferring a relative importance to each through a parameter \(\alpha \): \( A \succ B \) if and only if \(\alpha \cdot \max (A) + (1-\alpha ) \cdot \min (A) > \alpha \cdot \max (B) + (1-\alpha ) \cdot \min (B).\) Also, Hurwicz recommends equally both A and B, no matter the value of \(\alpha \), for in the formula \(\alpha V_{MAX} + (1-\alpha ) V_{\min }\), the maximum and minimum accuracy values \(V_{MAX}\) and \(V_{\min }\) are the same for A and B. We will now see that the tie is broken when we recur to the remaining rule Leximin.
More specifically, the axiom states: “8. Column duplication. The ordering is not changed if a new column, identical with some old column, is adjoined to the matrix. (Thus we are only interested in what states of Nature are possible, and not in how often each state may have been counted in the formation of the matrix.)” (Milnor, 1954 52).
Not all possibilities are real, they can be just merely conceptual (or logical): they cannot occur; that is, they are not physically possible; in other words, they are not an outcome of an objective chance distribution. Now, we do not know a priori what is physically possible, and in many scenarios, we never know which is the space of real, physical possibilities.
Some differences between James’s norm and mine: James defended belief on insufficient evidence without specifying which belief we could hold (albeit having in mind religious belief). Here, instead, I defend holding a specific belief on insufficient evidence, and one that is much less ambitious than religious belief (as I clarify in §5.3): a partial order of confidence among sets of possibilities, ordered according to their size.
In Savage’s sense, this kind of situation can be described as those in which the agents do not have the so-called null-act in their current set of possible actions.
References
Albert, D.Z. (2000). Time and chance. Harvard University Press.
Benétreau-Dupin, Y. (2015). The Bayesian who knew too much. Synthese, 192(5), 1527–1542.
Binmore, K. (2008). Rational decisions. Princeton University Press .
Chignell, A. (2016). The ethics of belief. In Zalta, E.N. (Ed.), Stanford Encyclopedia of Philosophy, Stanford University, Winter 2016 edition.
Clifford, W.K. (1877). The ethics of belief and other essays. Prometheus Books, 1999 edition.
de Cooman, G., & Miranda, E. (2007). Symmetry of models versus models of symmetry. In W. Harper, & Wheeler G. (Eds.), Probability and Inference: Essays in Honour of Henry E. Kyburg, College Publications, 67–149.
de Finetti, B. (1937). La Prévision: Ses Lois Logiques, Ses Sources Subjectives. Annales de l’Institut Henri Poincaré, 17, 1–68.
Dubois, D. (2007). Uncertainty theories: A unified view. In IEEE Cybernetic Systems Conference, Dublin, Ireland , 4–9.
Dubois, D., Prade, H., & Smets, P. (1996). Representing partial ignorance. Transaction System Man Cybernetics Part A, 26(3), 361–377.
Ellsworth, L. (1978). Decision-theoretic analysis of Rawls’ original position. In Hooker, A. Leach, J.J. E. F. McClennen, & D. Reidel (Eds.), Foundations and Applications of Decision Theory, 29–45.
Empiricus, S. (1994). Outlines of Pyrrhonism. Harvard University Press, I c. A.C., edition.
Fine, T. (1973). Theories of probability: An examination of foundations. Academic Press.
Friedman, J. (2015). Why suspend judging? Noûs 50, 3.
Friedman, J. (2013). Rational agnosticism and degrees of belief. Oxford Studies in Epistemology, 4, 57.
Friedman, J. (2013). Suspended judgment. Philosophical Studies, 162(2), 165–181.
Genin, K., & Huber, F. (2022). Formal representations of belief. In The Stanford Encyclopedia of Philosophy. Fall 2022 edition.
Hacking, I. (2001). An introduction to probability and inductive logic. Cambridge University Press.
Halpern, J.Y. (2003). Reasoning about uncertainty. MIT Press.
Hansson, S. O. (1994). Decision theory. A brief introduction. Department of Philosophy and the History of Technology. Royal Institute of Technology. Stockholm.
Hansson, S.O. (2018). Risk. In E.N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab: Stanford University.
Hoefer, C. (2019). Chance in the world: A skeptic’s guide to objective chance. Oxford University Press.
James, W. (1979). The will to believe and other essays in popular philosophy. Harvard University Press.
Joyce, J. M. (1998). A nonpragmatic vindication of probabilism. Philosophy of Science, 65(4), 575–603.
Konek, J. (2015). Epistemic conservativity and imprecise credence. Philosophy and Phenomenological Research.
Kraft, C. H., Pratt, J. W., & Seidenberg, A. (1959). Intuitive probability on finite sets. The Annals of Mathematical Statistics, 30(2), 408–419.
Landes, J., & Williamson, J. (2013). Objective Bayesianism and the maximum entropy principle. Entropy, 15(9), 3528–3591.
Lewis, D. K. (1980). A subjectivist’s guide to objective chance. In R.C. Jeffrey (Ed.), Studies in Inductive Logic and Probability, University of California Press, 83–132.
Luce, R. D., & Raiffa, H. (1957). Games and decisions: Introduction and critical survey. Dover Publications: Dover books on advanced mathematics.
McGrath, M. (2023). Being neutral: Suspension of judgement, agnosticism and inquiry. Noûs.
Milnor, J. (1954). Games against nature. In Coombs, C.H. Thrall, R.M. & R.L Davis (eds.), Decision Processes, Wiley, New York.
Myrvold, W.C. (2014). Probabilities in statistical mechanics. In C. Hitchcock, & A. Hajek (Eds.), The Oxford Handbook of Probability and Philosophy, Oxford University Press.
Norton, J. D. (2021). The material theory of induction. Calgary, Alberta, Canada: University of Calgary Press.
Norton, J.D. (2023). Eternal inflation: When probabilities fail. Synthese, special edition Reasoning in Physics.
Norton, J. D. (2008). Ignorance and indifference. Philosophy of Science, 75(1), 45–68.
Norton, J.D. (2023). The material theory of induction. in progress.
Norton, J. D., & Confusions, C. (2010). Not supporting versus supporting not. Philosophy of Science, 77(4), 501–523.
Norton, J. D. (2007). Disassembled, probability. British Journal for the Philosophy of Science, 58(2), 141–171.
Peterson, M. (2009). An introduction to decision theory. Cambridge Introductions to Philosophy: Cambridge University Press.
Pettigrew, R. (2016a). Accuracy and the laws of credence. Oxford University Press UK
Pettigrew, R. (2016). Accuracy, risk, and the principle of indifference. Philosophy and Phenomenological Research, 92(1), 35–59.
Radzvilas, M., William P., & Francesco, D.P. (forthcoming). The Ambiguity Dilemma for Imprecise Bayesians. The British Journal for the Philosophy of Science.
Rosa, L. (2021). Rational requirements for suspended judgment. Philos Stud 178, 385–406.
Rott, H. (2001). Change, choice and inference: A study of belief revision and nonmonotonic reasoning. Oxford University Press.
Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1(2), 233–247.
Shafer, G. (1976). A mathematical theory of evidence. Princeton University Press.
Staffel, J. (2019). Credences and suspended judgments as transitional attitudes. Philosophical Issues, 29(1), 281–294.
Tang, W. H. (2015). Reliabilism and the suspension of belief. Australasian Journal of Philosophy, 94(2), 362–377.
Titelbaum, M.G. (2022). Fundamentals of Bayesian Epistemology 1: Introducing Credences. Oxford University Press
Urbach, P. (1993). and Colin Howson. Scientific reasoning: The Bayesian approach. Open Court.
White, R. (2009). Evidential symmetry and mushy credence. In T. Szabo Gendler & J. Hawthorne (Eds.), Oxford Studies in Epistemology, Oxford University Press, 161–186.
White, R. (2005). Epistemic permissiveness. Philosophical Perspectives, 19(1), 445–459.
Williamson, J. (2010). In defence of objective Bayesianism. OUP Oxford.
Williamson, J. (2018). Justifying the principle of indifference. European Journal for Philosophy of Science, 8, 1–28.
Williamson, J. (2007). Motivating objective Bayesianism: from empirical constraints to objective probabilities. In: Probability and Inference: Essays in Honour of Henry E. Kyburg Jr, William L. Harper & Gregory R. Wheeler (eds), p- 151–179, College Publications.
Acknowledgements
For their helpful comments, I would like to thank Brice Bantegnie, Carl Hoefer, John Horden, Kine Josefine Aurland-Bredesen, John Norton, Elías Okon, Nils-Hennes Stear, Carlos Romero, Marta Sznaider, David Teira, Alessandro Torza, an anonymous reviewer of “Philosophy of science”, two anonymous reviewers of “Acta Analytica”, and audiences at the ALFAN conference and the LOGOS seminar.
Funding
This work was supported by the grant “Formal Epistemology - the Future Synthesis”, in the framework of the programme Praemium Academicum realised at the Institute of Philosophy of the Czech Academy of Sciences, by the Pontificia Universidad Católica de Valparaíso through the grants DI-emergente 039.346/2021 and DI-consolidado 039.310/2022, and by the grant FONDECYT iniciación 11240113.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The author declares no competing interests.
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
Filomeno, A. Suspension of Judgment, Non-additivity, and Additivity of Possibilities. Acta Anal (2024). https://doi.org/10.1007/s12136-024-00590-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12136-024-00590-7