Toward an Epistemic Foundation for Comparative Confidence

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1 Toard an Epistemic Foundation for Comparative Confidence Branden Fitelson 1 David McCarthy 2 1 Philosophy & Rutgers & LMU branden@fitelson.org 2 HKU mccarthy@hku.hk Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 1 The contemporary literature focuses mainly on to types of non-comparative judgment: belief and credence. Not much attention is paid to comparative judgment (but see [16]). It asn t alays thus. Keynes [21], de Finetti [3, 4] and Savage [24] all emphasized the importance (and perhaps even fundamentality) of comparative confidence. Comparative confidence is a three-place relation beteen an agent S (at a time t) and a pair of propositions hp, qi. We ll use [p q\ to express this relation, viz., [S is at least as confident in the truth of p as she is in the truth of q\. It is difficult to articulate the meaning of ithout someho implicating that it essentially involves some non-comparative judgments [e.g., b(p) b(q)]. But, it s important to think of as autonomous and irreducibly comparative i.e., as a kind of comparative judgment that may not reduce to anything non-comparative. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 2 Aim: give epistemic justifications of coherence requirements for that have appeared in the contemporary literature. Means: exploit a generalization of Joyce s non-pragmatic argument for probabilism [18, 19]. Note: something similar has already been done for full belief [10, 1, 8, 13]. Joyce as inspired by an elegant geometrical argument of de Finetti [5] (see Extras). Hoever, unlike de Finetti, Savage, et. al. [24, 15, 17] Joyce s approach is epistemic in nature. Abstracting aay from Joyce s argument, e have developed a frameork [13] for grounding epistemic coherence requirements for judgment sets J ={j 1,...,j n } (of type J) over agendas of propositions A={p 1,...,p n }. Applying our frameork involves three steps. Step 1: Identify a precise sense in hich individual judgments j of type J can be (qualitatively) inaccurate (or alethically defective/imperfect) at a possible orld. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 3 Step 2: Define an inaccuracy score i(j, ) for individual judgments j of type J. This is a numerical measure of ho inaccurate (in the sense of Step 1) j is (at ). For each set J ={j 1,...,j n }, e define its total inaccuracy at as the sum of the i-scores of its members: I(J,) P i i(j i,). Step 3: Adopt a fundamental epistemic principle, hich uses I(J,) to ground a (formal, synchronic, epistemic) coherence requirement for judgment sets J of type J. In the case of Joyce s argument for probabilism, e have: Step 1: [b(p) = r \ is inaccurate at just in case r differs from the value assigned to p by the indicator function v (p), hich is 1 (0) if p is true (false) at. Step 2: i(b(p), ) is (squared) Euclidean distance (or Brier score) beteen b(p) and v (p). I(b, ) = P i i(b(p i ), ). Step 3: The fundamental epistemic principle: b shouldn t be eakly dominated (by any b 0 ), according to I(,). Today: e apply the frameork to comparative confidence. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 4

2 We begin ith some background assumptions about. Our first assumption is that our agents S form comparative confidence judgments regarding all pairs of propositions on some m-proposition agenda A, dran from some n-proposition Boolean algebra B n (m apple n, viz., A B n ). Our second assumption is that is a total preorder on A, i.e., satisfies the folloing conditions, for all p, q, r 2 A. Totality. (p q) _ (q p). Transitivity. If p q and q r, then p r. Global versions of these are controversial [14, 12, 23]. We re only assuming local versions of them (for some agendas A). Once e ve got a total preorder on A, e can then define a strictly more confident than relation on A, as follos. p q p q and q Ë p. Because is a total preorder on A, it ill follo that is an asymmetric, transitive, irreflexive relation on A. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 5 We can also define an equally confident in (or epistemically indifferent beteen ) relation on A, as: p q p q and q p. Since is a total preorder, is an equivalence relation. Next, e ll assume our agents S are logically omniscient. (LO) S respects all logical equivalencies. ê If p, q are logically equivalent, then S judges p q. And, if S judges p q, then p, q are not logically equivalent. Finally, e ll assume our agents S have regular -orderings. Regularity. If p is contingent, then p?and > p. We can represent -relations on agendas A via their 0/1 adjacency matrices A, here A ij = 1 iff p i p j. Toy example: let A=B 4 be the smallest sentential BA, ith four propositions h>,p, P,?i, for some contingent P. Specifically, interpret P as a tossed coin lands heads. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 6 > P P? > P P ? P > P 0 > P P? > P P ? P > P?? The above figure shos the adjacency matrix and graphical representation of a relation ( ) on B 4. This relation is supported by S s evidence E, if E says that the coin is fair. Consider an alternative relation ( 0 ) on B 4, hich agrees ith on all judgments, except for P P. That is, P 0 P; hereas, P P. [ 0 is depicted on the next slide.] Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 7 This alternative relation 0 on B 4 is supported by S s evidence E, if E says that the coin is biased toard heads. Intuitively, neither nor 0 should be deemed (formally) incoherent. After all, either could be supported by an agent s evidence. We ll return to evidential requirements for comparative confidence relations belo. Meanhile, Step 1. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 8

3 Step 1 involves articulating a precise sense in hich an individual comparative confidence judgment p q is inaccurate at. Here, e follo Joyce s [18, 19] extensionality assumption, hich requires inaccuracy to supervene on the truth-values of the propositions in A at. An individual comparative confidence judgment p q is inaccurate at iff p q entails that the ordering fails to rank all truths strictly above all falsehoods at. 1 On this conception, there are to facts about the inaccuracy of individual comparative confidence judgments p q. Fact 1. If q & p is true at, then p q is inaccurate at. Fact 2. If p. q is true at, then p q is inaccurate at. 1 One might be tempted by a eaker (and more Joycean ) definition of inaccuracy, according to hich p q is inaccurate iff it contradicts the comparison p q induced by the indicator function v. This eaker definition (hich also deems p q inaccurate if p q is true at ) is untenable for us. This ill follo from our Fundamental Theorem, belo. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 9 Step 2 requires a point-ise inaccuracy measure i(p q, ). There are to kinds of inaccurate -judgments (Facts 1 and 2). Intuitively, these to should kinds of inaccuracies should not receive equal i-scores. Mistaken judgments should receive greater i-scores than mistaken judgments. Ho much more inaccurate than mistakes are mistakes? Tice as inaccurate! Suppose (by convention) that e assign an i-score of 1 to mistaken judgments. We must (!) assign an i-score of 2 to mistaken judgments. 8 2 if q & p is true at, and p q, >< i(p q, ) 1 if p. q is true at, and p q, >: 0 otherise. s total inaccuracy (on A at ) is the sum of s i-scores. I(,) i(p q, ). p,q2a Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 10 Step 3 involves the adoption of a fundamental epistemic principle. Here, e ill follo Joyce and adopt: Weak Accuracy-Dominance Avoidance (WADA). should not be eakly dominated in inaccuracy (according to I). More formally, there should not exist a 0 (on A) such that (i) (8)[I( 0,)apple I(,)], and (ii) (9)[I( 0,)<I(,)]. Recall our toy relations and 0 over B 4. Neither of these relations should be ruled-out as incoherent, as each could be supported by some body of evidence [19, pp ]. Theorem. Neither nor 0 is eakly dominated in I-inaccuracy by any binary relation on B 4. This result is a corollary of our Fundamental Theorem, hich ill also explain hy e ere forced to assign an inaccuracy score of exactly 2 to inaccurate judgments. More on that later. Meanhile, a historical interlude. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 11 Various coherence requirements for have been discussed [15, 2, 26]. We ll focus on a particular family of these. We begin ith the fundamental requirement (C), hich has (near) universal acceptance. We ill state (C) in to ays: axiomatically, and in terms of numerical representability. (C) S s -relation (assumed to be a total preorder on B n ) should satisfy the folloing to axiomatic constraints: (A 1 ) >?. (A 2 ) For all p, q 2 B n, if p entails q then q p. A plausibility measure (a.k.a., a capacity) on a Boolean algebra B n is real-valued function Pl : B n, [0, 1] hich satisfies the folloing three conditions [15, p. 51]: (Pl 1 ) Pl(?) = 0. (Pl 2 ) Pl(>) = 1. (Pl 3 ) For all p, q 2 B n, if p entails q then Pl(q) Pl(p). Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 12

4 To kinds of representability of, by a real-valued f. is fully represented by f for all p, q 2 B n p q () f (p) f (q). is partially represented by f for all p, q 2 B n p q =) f (p) > f (q). No, (C) can be expressed equivalently, as follos: (C) S s -relation (assumed to be a total preorder on B n ) should be fully representable by some plausibility measure. Theorem 1. (WADA) entails (C). [See Extras for a proof.] There are several other coherence requirements for that can be expressed both axiomatically, and in terms of numerical representability by some real-valued f. We ll state these, and say hether or not they follo from (WADA). The next requirements involve belief functions. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 13 A mass function on a Boolean algebra B n is a function m : B n, [0, 1] that satisfies the folloing to conditions: (M 1 ) m(?) = 0. (M 2 ) m(p) = 1. p 2 B n A belief function Bel : B n, [0, 1] is generated by an underlying mass function m on B n in the folloing ay: Bel m (p) m(q). q 2 B n q entails p No, consider the folloing coherence requirement: (C 0 ) S s -relation (assumed to be a total preorder on B n ) should be partially representable by some belief function. A total preorder satisfies (C 0 )iff satisfies (A 2 ) [26]. So, Theorem 1 has a Corollary: [ Thm 2 ] (WADA) entails (C 0 ). What about full representability of a belief function? To it: Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 14 (C 1 ) S s -relation (assumed to be a total preorder on B n ) should be fully representable by a belief function. As it turns out [26], a relation is fully representable by some belief function if and only if satisfies (A 1 ), (A 2 ), and (A 3 ) If p entails q and hq, ri are mutually exclusive, then: q p =) q _ r p _ r. (WADA) also entails (A 3 ). That is, e have the folloing: Theorem 3. (WADA) entails (C 1 ). [See Extras.] Moving beyond (C 1 ) takes us into comparative probability. A t.p. is a comparative probability iff satisfies (A 1 ), (A 2 ), & (A 5 ) If hp, qi and hp, ri are mutually exclusive, then: q r () p _ q p _ r (C 2 ) S s -relation (assumed to be a total preorder on B n ) should be a comparative probability relation. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 15 Theorem 4. (WADA) does not entail (C 2 ). [See Extras.] The folloing axiomatic constraint is a eakening of (A 5 ). (A? 5 ) If hp, qi and hp, ri are mutually exclusive, then: q r =) p _ r Ê p _ q And, the folloing coherence requirement is a (corresponding) eakening of coherence requirement (C 2 ). (C? 2 ) should (be a total preorder and) satisfy (A 1), (A 2 ) and (A? 5 ). Theorem 5. (WADA) does not entail (C? 2 ). [See Extras.] Our final pair of coherence requirements for involve representability by some probability function. I m sure everyone knos hat a Pr-function is, but... Probability functions are special kinds of belief functions (just as belief functions ere special kinds of Pl-measures). Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 16

5 A probability mass function is a function m hich maps states of B n to [0, 1], and hich satisfies these to axioms. (M 1 ) m(?) = 0. (M 2 ) m(s) = 1. s 2 B n A probability function Pr : B n, [0, 1] is generated by an underlying probability mass function m in the folloing ay Pr m (p) m(s). s 2 B n s entails p That brings us to our final pair of requirements for. (C 3 ) should be be partially representable by some Pr-function. (C 4 ) should be be fully representable by some Pr-function. de Finetti [3, 4] famously conjectured that (C 2 ) entails (C 4 ). But, Kraft et. al. [22] shoed that (C 2 ) h (C 3 ). [See Extras.] Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 17 We have the folloing logical relations beteen the C s. full rep. by Pr (C 4 ) 7 ========) (C 3 ) 7 partial rep. by Pr qualitative prob. (C 2 ) 7 =======) (C? 2 )7 full rep. by Bel (C 1 ) full rep. by Pl (C) partial rep. by Bel (C 0 ) (================ (A 1 ) (A 2 ) (A? 5 ) If a requirement follos from (WADA), it gets a. If a requirement does not to follo from (WADA), it gets an 7. We conclude ith our final (and most important) Fundamental Theorem(s). [See Extras for proofs.] Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 18 We assume that numerical probabilities reflect evidence, i.e., e adopt the folloing evidential requirement. (R) is representable by some regular probability function. Fundamental Theorem. If a comparative confidence relation satisfies (R), then satisfies (WADA). The proof of our Fundamental Theorem (see Extras) reveals that I(,) is evidentially proper, in this sense [13]. Definition (Evidential Propriety). Suppose a judgment set J of type J is supported by the evidence. That is, suppose there exists some evidential probability function Pr( ) hich represents J (in the appropriate sense of represents for judgment sets of type J). If this is sufficient to ensure that J minimizes expected inaccuracy (relative to Pr), according to the measure of inaccuracy I(J,), then e ill say that the measure I is evidentially proper. Note: the decision to eight -mistakes tice as heavily as -mistakes is forced by evidential propriety (see Extras). Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 19 Proof. Theorem 1. (WADA) entails (C), viz., (WADA) ) (A 1 ) & (A 2 ). Suppose violates (A 1 ). Because is total, this means is such that? >. Consider the relation 0 hich agrees ith on all comparisons outside the h?, >i-fragment, but hich is such that > 0?. We have: (8) i(> 0?,)= 0 < 1 apple i(? >,). Suppose violates (A 2 ). Because is total, this means there is a pair of propositions p and q in A such that (a) p entails q but (b) p q. Consider the relation 0 hich agrees ith outside of the hp, qi-fragment, but hich is such that q 0 p. The table on the next slide depicts the hp, qi-fragments of the relations and 0 in the three salient possible orlds 1 3 not ruled out by (a) p Ó q. By (b) & (LO), p and q are not logically equivalent. So, orld 2 is a live possibility, and 0 eakly I-dominates. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 20

6 Theorem 3. (WADA) entails (C 1 ). i p q 0 I(, i ) I( 0, i ) 1 T T p q q 0 p 0 0 T F 2 F T p q q 0 p F F p q q 0 p 0 0 Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 21 Proof. Having already proved Theorem 1, e just need to sho that (WADA) entails (A 3 ). Suppose violates (A 3 ). Because is total, this means there must exist p, q, r 2 A such that (a) p Ó q, (b) hq, ri are mutually exclusive, (c) q p, but (d) p _ r q _ r. Let 0 agree ith on every judgment, except (d). That is, let 0 be such that (e) q 0 p and (f) q _ r 0 p _ r. There are only four orlds (or hp, q, ri state descriptions) compatible ith the precondition of (A 3 ). These are the folloing (state descriptions). 1 = p & q & r 2 = p & q & r 3 = p & q & r 4 = p & q & r By (c) & (LO), p and q are not logically equivalent. As a result, orld 2 is a live possibility. Moreover, (f) ill not be inaccurate in any of these four orlds. But, (d) must be inaccurate in orld 2. This suffices to sho that 0 eakly I-dominates. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 22 Proof. Theorem 4. (WADA) does not entail (C 2 ). Having already proved Theorem 1, e just need to sho that (WADA) does not entail (A 5 ). Suppose (a) hp, qi and hp, ri are mutually exclusive, (b) q r, and (c) p _ r p _ q. It can be shon (by exhaustive search) that there is no binary relation 0 on the agenda hp, q, ri such that (i) 0 agrees ith on all judgments except (b) and (c), and (ii) 0 eakly I-dominates. There are only four alternative judgment sets that need to be compared ith {(b), (c)}, in terms of their I-values across the five possible orlds ( 1 5 ) compatible ith the precondition of (A 5 ): (1) {q r,p_ r p _ q}, (2) {r q, p _ r p _ q}, (3) {q r,p_ r p _ q}, and (4) {q r,p_ r p _ q}. It is easy to verify that none of these alternative judgment sets eakly I-dominates the set {(b), (c)}, across the five salient possible orlds. Note: this argument actually establishes the stronger claim (Theorem 5) that (WADA) does not entail (A? 5 )/(C? 2 ). Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 23 Proof. Fundamental Theorem. If a comparative confidence relation satisfies (R), then satisfies (WADA). That is, (R) ) (WADA). Suppose Pr( ) fully represents. Consider the expected I-inaccuracy, as calculated by Pr( ), of : EIPr P Pr() I(,). Since I(,) is a sum of the i(p q, ) for each hp, qi 2A, and since E is linear: EIPr = E Pr i(p q, ) p,q2a (1) Suppose Pr(p) > Pr(q). Then e have: E Pr i(p q, ) = 2 Pr(q & p) < E Pr i(p q, ) = Pr(p. q), and E Pr i(p q, ) = 2 Pr(q & p) < E Pr i(q p, ) = 2 Pr(p & q). (2) Suppose Pr(p) = Pr(q). Then e have: E Pr i(p q, ) = Pr(p. q) = E Pr i(p q, ) = 2 Pr(q & p). As a result, if is fully representable by any Pr( ), then cannot be strictly I-dominated, i.e., (C 4 ) ) (SADA). Moreover, if e assume Pr( ) to be regular, then must satisfy (WADA) [13]. ê (R) ) (WADA). Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 24

7 Theorem. a := 2; b := 0 is the only assignment to a, b that ensures the folloing definition of i is evidentially proper. 8 a if q & p is true in, and p q, >< b if q p is true in, and p q, i(p q, ) 1 if p. q is true in, and p q, >: 0 otherise. Let m 4 = Pr(p & q), m 3 = Pr( p & q), and m 2 = Pr(p & q). Then, the propriety of i is equivalent to the folloing (universal) claim. And, the only assignment that makes this (universal) claim true is a := 2; b := 0. 0 B m 2 m 4 > m 3 m 4 & 0 B m 2 m 4 = m 3 m 4 a m 3 b (1 (m 2 m 3 )) apple a m 2 b (1 (m 2 m 3 )) & a m 3 b (1 (m 2 m 3 )) apple m 2 m 3 m 2 m 3 apple a m 2 b (1 (m 2 m 3 )) & m 2 m 3 apple a m 3 b (1 (m 2 m 3 )) Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 25 1 C A 1 C A Our ordering presuppositions (Totality & Transitivity) are not universally accepted as rational requirements [14, 12, 23]. In our book [13], e analyze both of the ordering presuppositions in more detail. Specifically, e sho that: (1) Totality does not follo from eak accuracy dominance avoidance. That is, (WADA) does not entail Totality. (2) Transitivity does not from eak accuracy dominance avoidance. That is, (WADA) does not entail Transitivity. These to negative results [especially (1)] are probably not very surprising. But, it is somehat interesting that none of the three instances of Transitivity is entailed by (WADA). Transitivity 1. If p q and q r, then r Ê p. Transitivity 2. If p q and q r, then r Ê p. Transitivity 3. If p q and q r, then p r. The first instance of Transitivity is the least controversial of the three. And, the last (transitivity of ) is the most [23]. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 26 In their seminal paper, Kraft et. al. [22] refute de Finetti s [3, 4] conjecture: (C 2 ) ) (C 4 ). In fact, they sho (C 2 ) h (C 3 ). Their counterexample involves a linear order on an algebra B 32 generated by five states: {s 1,...,s 5 }. We on t rite don the entire linear order as this involves a complete ranking of 32 propositions. Instead, e focus only the folloing, salient 8-proposition fragment. s 1 s 2 _ s 4 s 3 _ s 4 s 1 _ s 2 s 2 _ s 5 s 1 _ s 4 s 1 _ s 2 _ s 4 s 3 _ s s s 2 _ s s 3 _ s s 1 _ s s 2 _ s s 1 _ s s 1 _ s 2 _ s s 3 _ s Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence Simplest case of df s Theorem [5]: b(p) = x; b( P) = y. The diagonal lines are the probabilistic b s (on hp, P i). The to directions of de Finetti s theorem (for hp, Pi) can be established via these to figures. And, this simplest (hp, P i) version of the Theorem generalizes from the simplest propositional Boolean algebra B 4 to B n, for any n. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 28

8 There are to, eaker I-dominance requirements that e discuss in the book [13]. These are as follos. Strict Accuracy-Dominance Avoidance (SADA). should not be strictly dominated in inaccuracy (according to I). More formally, there should not exist a 0 (on A) such that (8)[I( 0,)<I(,)]. Of course, (SADA) is strictly eaker than (WADA). And, here is a requirement that is even eaker than (SADA). Let M(,) the set of s inaccurate judgments at. Strong Strict Accuracy-Dominance Avoidance (SSADA). There should not exist a 0 on A such that: (8)[M( 0,) M(,)]. Some of our (WADA) results also go through for (SADA) and/or (SSADA). Finally, e give a complete, big picture of all the logical relations among all the requirements. Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 29 (WADA) (SADA) (SSADA) (============== ====================) (================= =======) (A 1 ) (======= (R) (C 4 ) =============) (C 3 ) (C 2 ) =============) (C? 2 ) (C 1 ) (C) (C 0 ) (=================== Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 30 [1] R. Briggs, F. Cariani, K. Easaran and B. Fitelson Individual Coherence and Group Coherence, to appear in Essays in Collective Epistemology, J. Lackey (ed.), OUP. [2] A. Capotorti and B. Vantaggi, Axiomatic characterization of partial ordinal relations, International Journal of Approximate Reasoning, [3] B. de Finetti, Foresight: Its Logical Las, Its Subjective Sources (1935), in H. Kyburg and H. Smokler (eds.), Studies in Subjective Probability, Wiley, [4], La logica del plausibile secondo la concezione di Polya, Societa Italiana per il Progresso delle Scienze, [5] Theory of probability, Wiley, [6] M. Deza and E. Deza, Encyclopedia of Distances, Springer, [7] C. Duddy and A. Piggins, A measure of distance beteen judgment sets, Social Choice and Welfare, [8] K. Easaran, Dr. Truthlove or: Ho I Learned to Stop Worrying and Love Bayesian Probability, 2013, manuscript. [9] K. Easaran and B. Fitelson, An Evidentialist Worry about Joyce s Argument for Probabilism, Dialectica, [10], Accuracy, Coherence & Evidence, to appear in Oxford Studies in Epistemology (Volume 5), T. Szabo Gendler & J. Hathorne (eds.), Oxford University Press, [11] P. Fishburn, Weak qualitative probability on finite sets. Annals of Mathematical Statistics, Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 31 [12], The Axioms of Subjective Probability, Statistical Science, [13] B. Fitelson, Coherence, book manuscript, [14] P. Forrest, The problem of representing incompletely ordered doxastic systems, Synthese, [15] J. Halpern, Reasoning about uncertainty, MIT Press, [16] J. Hathorne, The lockean thesis and the logic of belief, in F. Huber and C. Schmidt-Petri (eds.), Degrees of Belief, Springer, [17] T. Icard, Pragmatic Considerations on Comparative Confidence, 2014, manuscript. [18] J. Joyce, A Nonpragmatic Vindication of Probabilism, Philosophy of Science, [19], Accuracy and Coherence: Prospects for an Alethic Epistemology of Partial Belief, in F. Huber and C. Schmidt-Petri (eds.), Degrees of Belief, Springer, [20] J. Kemeny and J. Snell, Mathematical models in the social sciences, Ginn, [21] J.M. Keynes, A Treatise on Probability, MacMillan, [22] C. Kraft, J. Pratt and A. Seidenberg, Intuitive Probability on Finite Sets, The Annals of Mathematical Statistics, [23] K. Lehrer and C. Wagner, Intransitive Indifference: The Semi-Order Problem, Synthese, [24] L. Savage, The Foundations of Statistics, Dover, [25] D. Scott, Measurement Structures and Linear Inequalities, Journal of Mathematical Psych., [26] S. Wong, Y. Yao, P. Bollmann and H. Burger, Axiomatization of qualitative belief structure, IEEE Transactions on Systems, Man and Cybernetics, [27] P. Young, Optimal voting rules, The Journal of Economic Perspectives, Fitelson & McCarthy Toard an Epistemic Foundation for Comparative Confidence 32