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An Accuracy Argument in Favor of Ranking Theory Journal of Philosophical Logic manuscript No. (will be inserted by the editor) An Accuracy Argument in Favor of Ranking Theory Eric Raidl · Wolfgang Spohn Received: date / Accepted: date Abstract Fitelson and McCarthy (2014) have proposed an accuracy measure for con- fidence orders which favors probability measures and Dempster-Shafer belief func- tions as accounts of degrees of belief and excludes ranking functions. Their accuracy measure only penalizes mistakes in confidence comparisons. We propose an alter- native accuracy measure that also rewards correct confidence comparisons. Thus we conform to both of William James’ maxims: “Believe truth! Shun error!” We com- bine the two maxims, penalties and rewards, into one criterion that we call prioritized accuracy optimization (PAO). That is, PAO punishes wrong comparisons (preferring the false to the true) and rewards right comparisons (preferring the true to the false). And it requires to prioritize being right und avoiding to be wrong in a specific way. Thus PAO is both, a scoring rule and a decision rule. It turns out that precisely con- fidence orders representable by two-sided ranking functions satisfy PAO. The point is not to argue that PAO is the better accuracy goal. The point is only that ranking theory can also be supported by accuracy considerations. Thus, those considerations by themselves cannot decide about rational formats for degrees of belief, but are part and parcel of an overall normative assessment of those formats. Eric Raidl’s work was supported by the Deutsche Forschungsgemeinschaft (DFG) Research Unit FOR 1614. Eric Raidl Cluster of Excellence “Machine Learning: New Perspectives for Science” Eberhard Karls University Tubingen¨ Tubingen¨ Germany E-mail: [email protected] Wolfgang Spohn (corresponding author) Department of Philosophy, University Konstanz, 78457 Konstanz, Germany E-mail: [email protected] 2 Eric Raidl, Wolfgang Spohn Keywords Comparative Belief · Accuracy · Accuracy-first Epistemology · Degrees of Belief · Probability Theory · Ranking Theory · Representation Theorems Mathematics Subject Classification (2000) 00A30 · 03A05 · 06A05 · 60A05 · 62C05 1 Introduction Joyce (1998, 2009) has proposed a novel justification of probability theory in terms of accuracy dominance: A possible epistemic state, i.e., an assignment of degrees of belief, is an assignment f of numbers between 0 and 1 to propositions as a possible epistemic state. For each possible world w, truth in w (truth = 1, falsity = 0) is also such an assignment iw, the perfectly accurate one. The given assignment f deviates from each iw to some extent d( f ;iw). Now, one may argue that this inaccuracy mea- sure d has to satisfy some plausible conditions; the quality of that argument is crucial, of course. Joyce then shows that if f violates the axioms of probability theory, then there is a probability measure p that weakly dominates f in accuracy, i.e., for which d( f ;iw) ≥ d(p;iw) for all w and d( f ;iw) > d(p;iw) for some w. This is an exciting kind of argument. This argument and the broader philosophical position defended by that argument is now referred to as accuracy-first epistemology. It may be summa- rized as claiming that we must accept probability theory for purely epistemic reasons, i.e., only in order to approximate the truth as well as possible. There are variations and extensions of that argument; the discussion is vigorous and ramified. This the- oretical field is most comprehensively presented by Pettigrew (2016). It should be noted, though, that it is always about justifying probability. Easwaran and Fitelson (2015) have developed a more general kind of accuracy dominance argument applicable to belief sets. Fitelson and McCarthy (2014) did the same regarding comparative judgments. For them, the basic notion is that of a con- fidence order : B A iff A is at least as plausible or credible as B. They go on defining an inaccuracy measure for such orders. They argue that this measure must have certain properties. And then they can run the same kind of argument. That is, such an order is required not to be weakly dominated in accuracy by some other or- der, i.e., to satisfy weak accuracy dominance avoidance (WADA). And so they study which properties those orders must have in order to satisfy WADA and which prop- erties guarantee WADA to be satisfied. One significant result is, e.g., that WADA is satisfied by qualitative probability. However, they widen the perspective and discuss also other epistemic formats. Spohn (1988) has proposed another representation of degrees of belief, which he claims also to represent belief (in contrast to probability theory, which is prima facie plagued by the lottery paradox). This is called ranking theory nowadays and widely developed and applied in Spohn (2012). Spohn (2012, ch. 5 and 8) presents his own justifications. But it is an obvious question how ranking theory fares under this type of justificatory argument. This is the topic of this paper. At first, it seems that Joyce’s kind of argument is inapplicable to ranking theory, simply because ranks measure degrees of belief not by real numbers between 0 and 1, but by (non-negative) integers. However, the generalized argument is applicable. An Accuracy Argument in Favor of Ranking Theory 3 Ranking functions also induce confidence orders, and then one may check how they relate to WADA. Indeed, Fitelson (2017) has done so and shown that rank-induced confidence orders violate WADA. For Fitelson this throws doubt, to say the least, on the normative foundations of ranking theory. One may evade this criticism by pointing to all the other normative arguments in favor of ranking theory. And then one may dispute which kind of justification weighs more. This is not what we want to do here; that would just mean repeating the old justifications. Rather we want to take the bull by the horns. Again, the crucial issue is which kind of inaccuracy measure we should accept. And we would like to propose a novel kind of inaccuracy measure that confirms rank-induced confidence orders and disconfirms, e.g., qualitative probability. Moreover, we want to argue that this novel inaccuracy measure is at least plau- sible. We don’t claim that it is superior to Fitelson’s and McCarthy’s measure; this would be presumptuous. But we do claim that the discussion is symmetric. Each plausible epistemic format can be confirmed by its own plausible accuracy measure. Accuracy dominance arguments are not decisive. Thus, other normative arguments need not fear to be dominated by those accuracy arguments. That’s the moral we would like to draw in this paper. Although our paper provides an accuracy argument for ranking theory and could thus be seen as extending the accuracy-first epistemology from probability theory to ranking theory, we conceive of this rather as a criticism of the accuracy-first strategy: Each epistemic format will require its particular notion of accuracy, so that in the end, accuracy-first arguments are part of a holistic picture and non-detachable from the epistemic format they are supposed to ground. For this purpose, section 2 introduces Fitelson’s and McCarthy’s accuracy dom- inance argument concerning confidence orders. Section 3 briefly introduces ranking theory, shows which kind of confidence orders it entails, and explains Fitelson’s crit- icism, the bad behavior of rank-induced confidence orders according to his criteria. Thus the problem we want to deal with is exposed in detail. Section 4 then develops a different kind of accuracy measure. Of course, it is tailored to ranking theory, but we will argue that it has an independent kind of plausibility. Section 5 proves our main result: namely that weak accuracy dominance avoidance with respect to this measure entails and is in fact equivalent to the properties of rank-induced confidence orders. Our case is summarized in the final section 6. Longer proofs can be found in the appendix A. 2 Confidence Orders and Their Accuracy Fitelson and McCarthy (2014) take comparative orders to be the basic epistemic level, with good reason. Beliefs seem to be an all-or-nothing affair, providing too coarse- grained a description, and quantitative degrees of belief seem to derive from epistemic orders by some measurement procedure. So, let’s accept their starting point. Let W be a non-empty finite set of possibilities or possible worlds. The subsequent consid- erations don’t work for infinitely many possibilities. Throughout the paper let A be the power set of W, that is, the set of all subsets of W. We interpret A as the set of 4 Eric Raidl, Wolfgang Spohn all propositions over W and consider these to be the objects of epistemic attitudes. We could take sentences instead, which have those propositions as truth conditions. However, speaking directly about propositions simplifies our business. Then we say: Definition 1. is a confidence order over A iff (1) is a weak order (transitive and total relation) over A ,1 (2) /0 ≺ W (non-triviality), and (3) if A ⊆ B, then A B (monotonicity). Moreover, is regular iff (4) /0 ≺ A ≺ W for all A 6= /0;W. Transitivity of confidence orders has never been under dispute. Totality, however, is a strong requirement, which we do not want to defend, but assume here for the sake of theorizing. One could try to do without it, as Halpern (2003, sect. 2.8) does with his plausibility measures. (2) seems indispensable. The tautology W can’t be just as credible as the contradiction /0.And (3) seems indispensable, too. If A entails B, B cannot be less credible than A. These (together with transitivity) are also the two basic properties of Halpern’s plausibility measures. Let’s not further argue about them.
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