ACCURACY, COHERENCE AND EVIDENCE. 1. Setting the Stage

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1 2 KENNY EASWARAN AND BRANDEN FITELSON ACCURACY, COHERENCE AND EVIDENCE KENNY EASWARAN AND BRANDEN FITELSON Abstract. Takig Joyce s (1998; 2009) recet argumet(s) for probabilism as our poit of departure, we propose a ew way of groudig formal, sychroic, epistemic coherece requiremets for (opiioated) full belief. Our approach yields pricipled alteratives to deductive cosistecy, sheds ew light o the preface ad lottery paradoxes, ad reveals ovel coceptual coectios betwee alethic ad evidetial epistemic orms. 1. Settig the Stage This essay is about formal, sychroic, epistemic, coherece requiremets. We begi by explaiig how we will be usig each of these (five) key terms. Formal epistemic coherece requiremets ivolve properties of judgmet sets that are logical (ad, i priciple, determiable a priori). These are to be distiguished from other less formal ad more substative otios of coherece that oe ecouters i the epistemological literature. For istace, so-called coheretists like BoJour (1985) use the term i a less formal sese which implies (e.g.) that coherece is truth-coducive. While there will be coceptual coectios betwee the accuracy of a doxastic state ad its coherece (i the sese we have i mid), these coectios will be quite weak (too weak to merit the covetioal hoorific truth-coducive ). All of the varieties of coherece to be discussed below will be itimately related to deductive cosistecy. Cosequetly, whether a set of judgmets is coheret will be determied by (i.e., will supervee o) logical properties of the set of propositios that are the objects of the judgmets i questio. Sychroic epistemic coherece requiremets apply to the doxastic states of agets at idividual times. These are to be distiguished from diachroic requiremets (e.g., coditioalizatio, reflectio, etc.), which apply to sequeces of doxastic states across times. Presetly, we will be cocered oly with the former. 1 Date: 11/30/13. Peultimate versio: Fial versio to appear i Oxford Studies i Epistemology (Volume 5), T. Szabo Gedler & J. Hawthore (eds.), Oxford Uiversity Press. This material has bee preseted i various places over the past several years, ad the list of people with whom we ve had useful relevat coversatios is too log to eumerate here. But, i additio to three aoymous referees who provided very useful writte commets, we must sigle out (i alphabetical order) the followig people who have bee especially geerous with valuable feedback regardig this project: Rachael Briggs, Fabrizio Cariai, Jim Joyce, Ole Hjortlad, Haes Leitgeb, Be Levistei, Richard Pettigrew, Floria Steiberger, ad Joatha Weisberg. Brade Fitelso would also like to thak the Alexader vo Humboldt (AvH) Foudatio for their geerous support durig his summer (2011, 2012) visits to the Muich Ceter for Mathematical Philosophy at Ludwig-Maximilias-Uiversität Müche, where he preseted much of this material i semiars. 1 See Titelbaum (2013) for a excellet recet survey of the cotemporary literature o (Bayesia) diachroic epistemic coherece requiremets. Some, e.g., Moss (2013) ad Hedde (2013), have argued that there are o diachroic epistemic ratioal requiremets (i.e., that there are oly sychroic epistemic ratioal requiremets). We take o stad o this issue here. But, we will assume that there are (some) sychroic epistemic ratioal requiremets of the sort we aim to explicate (see f. 7). 1 Epistemic requiremets are to be distiguished from, e.g., pragmatic requiremets. Startig with Ramsey (1928), the most well-kow argumets for probabilism as a formal, sychroic, coherece requiremet for credeces have depeded o the pragmatic coectio of belief to actio. For istace, Dutch Book argumets ad Represetatio Theorem argumets (Hájek 2008) aim to show that a aget with o-probabilistic credeces (at a give time t) must (thereby) exhibit some sort of pragmatic defect (at t). 2 Followig Joyce (1998; 2009), we will be focusig o o-pragmatic (viz., epistemic) defects implied by the sychroic icoherece (i a sese to be explicated below) of a aget s doxastic state. To be more precise, we will be cocered with two aspects of doxastic states that we take to be distictively epistemic: (a) how accurate a doxastic state is, ad (b) how much evidetial support a doxastic state has. We will call these (a) alethic ad (b) evidetial aspects of doxastic states, respectively. 3 Coherece requiremets are global ad wide-scope. Coherece is a global property of a judgmet set i the sese that it depeds o properties of etire set i a way that is ot (i geeral) reducible to properties of idividual members of the set. Coherece requiremets are wide-scope i Broome s (2007) sese. They will be expressible usig shoulds (or oughts ) that take wide-scope over some logical combiatio(s) of judgmets. As a result, coherece requiremets will ot (i geeral 4 ) require specific attitudes toward specific idividual propositios. Istead, coherece requiremets will require the avoidace of certai combiatios of judgmets. We use the term coherece rather tha cosistecy because (a) the latter is typically associated with classical deductive cosistecy (which, as we ll see shortly, we do ot accept as a ecessary requiremet of epistemic ratioality), ad (b) the former is used by probabilists whe they discuss aalogous requiremets for degrees of belief (viz., probabilism as a coherece requiremet for credeces). Because our geeral approach (which was ispired by Joycea argumets for probabilism) ca be applied to may types of judgmet icludig both full belief ad partial belief 5 we prefer to maitai a commo parlace for the saliet requiremets i all of these settigs. Fially, ad most importatly, whe we use the term requiremets, we are talkig about ecessary requiremets of ideal epistemic ratioality. 6 The hallmark of a ecessary requiremet of epistemic ratioality N is that if a doxastic state 2 We realize that there are depragmatized versios of these argumets (Christese, 1996). But, eve these versios of the argumets trade essetially o the pragmatic role of doxastic attitudes (i sactioig bets, etc.). I cotrast, we will oly be appealig to epistemic coectios of belief to truth ad evidece. Our argumets do ot explicitly rely upo coectios betwee belief ad actio. 3 The alethic/evidetial distictio is cetral to the pre-ramseya debate betwee James (1896) ad Clifford (1877). Roughly speakig, alethic cosideratios are Jamesia, ad evidetial cosideratios are Cliffordia. We will be assumig for the purposes of this article that alethic ad evidetial aspects exhaust the distictively epistemic properties of doxastic states. But, our approach could be geeralized to accommodate additioal dimesios of epistemic evaluatio. 4 There are two otable exceptios to this rule. It will follow from our approach that (a) ratioal agets should ever believe idividual propositios ( ) that are logically self-cotradictory, ad (b) that ratioal agets should ever disbelieve idividual propositios ( ) that are logically true. 5 I fact, the framework ca be applied fruitfully to other types of judgmet as well. See (Fitelso & McCarthy 2013) for a applicatio to comparative cofidece, which leads to a ew foudatio for comparative probability. For a survey of applicatios of the geeral framework, see (Fitelso 2014). 6 Here, we adopt Titelbaum s (2013, chapter 2) locutio ecessary requiremet of (ideal) ratioality as well as (roughly) his usage of that locutio (as applied to formal, sychroic requiremets).

2 ACCURACY, COHERENCE AND EVIDENCE 3 4 KENNY EASWARAN AND BRANDEN FITELSON S violates N, the S is (thereby) epistemically irratioal. However, just because a doxastic state S satisfies a ecessary requiremet N, this does ot imply that S is (thereby) ratioal. For istace, just because a doxastic state S is coheret (i.e., just because S satisfies some formal, epistemic coherece requiremet), this does ot mea that S is (thereby) ratioal (as S may violate some other ecessary requiremet of epistemic ratioality). Thus, coherece requiremets i the preset sese are (formal, sychroic) ecessary coditios for the epistemic ratioality of a doxastic state. 7 Our talk of the epistemic (ir)ratioality of doxastic states is meat to be evaluative (rather tha ormative 8 ) i ature. To be more precise, we will (for the most part) be cocered with the evaluatio of doxastic states, relative to a idealized 9 stadard of epistemic ratioality. Sometimes we will speak (loosely) of what agets should do but this will (typically) be a evaluative sese of should (e.g., should o pai of occupyig a doxastic state that is ot ideally epistemically ratioal ). If a differet sese of should is iteded, we will flag this by cotrastig it with the idealized/evaluative should that features i our ratioal requiremets. Now that the stage is set, it will be istructive to look at the most well-kow coherece requiremet i the iteded sese. 2. Deductive Cosistecy, The Truth Norm, ad The Evidetial Norm The most well-kow example of a formal, sychroic, epistemic coherece requiremet for full belief is the (putative) requiremet of deductive cosistecy. (CB) All agets S should (at ay give time t) have sets of full beliefs (i.e., sets of full belief-cotets) that are (classically) deductively cosistet. May philosophers have assumed that (CB) is a ecessary requiremet of ideal epistemic ratioality. That is, may philosophers have assumed that (CB) is true, if its should is iterpreted as should o pai of occupyig a doxastic state that 7 For simplicity, we will assume that there exist some (sychroic, epistemic) ratioal requiremets i the first place. We are well aware of the curret debates about the very existece of ratioal requiremets (e.g., coherece requiremets). Specifically, we are cogizat of the saliet debates betwee Kolody (2007) ad others, e.g., Broome (2007). Here, we will simply adopt the o-elimiativist stace of Broome et al. who accept the existece of (ielimiable) ratioal requiremets (e.g., coherece requiremets). We will ot try to justify our o-elimiativist stace here, as this would take us too far afield. However, as we will explai below, eve coherece elimiativists like Kolody should be able to beefit from our approach ad discussio (see f. 45). As such, we (ultimately) see our adoptio of a o-elimiativist stace i the preset cotext as a simplifyig assumptio. 8 Normative priciples support attributios of blame or praise of agets, ad are (i some sese) actio guidig. Evaluative priciples support classificatios of states (occupied by agets) as defective vs o-defective ( bad vs good ), relative to some evaluative stadard (Smith, 2005, 3). 9 Deductive cosistecy ad the other formal coherece requiremets we ll be discussig are highly idealized ratioal epistemic requiremets. They all presuppose a stadard of ideal ratioality which is isesitive to sematic ad computatioal (ad other) limitatios of (actual) agets who occupy the doxastic states uder evaluatio. While this is, of course, a strog idealizatio (Harma 1986), it costitutes o sigificat loss of geerality i the preset cotext. This is because our aims here are rather modest. We aim (maily) to do two thigs i this paper: (a) preset the simplest, most idealized versio of our framework ad the (aïve) coherece requiremets to which it gives rise, ad (b) cotrast these ew requiremets with the (equally simple ad aïve) requiremet of deductive cosistecy. Owig to the idealized/evaluative ature of our discussio, we will typically speak of the (ir)ratioality of states, ad ot the (ir)ratioality of agets who occupy them. Fially, we will sometimes speak simply of ratioal requiremets or just requiremets. It is to be uderstood that these are shorthad for the full locutio ecessary requiremets of ideal epistemic ratioality. is ot ideally epistemically ratioal. Iterestigly, i our perusal of the literature, we have t bee able to fid may (geeral) argumets i favor of the claim that (CB) is a ratioal requiremet. Oe potetial argumet alog these lies takes as its poit of departure the (so-called) Truth Norm for full belief. 10 (TB) All agets S should (at ay give time t) have full beliefs that are true. 11 Presumably, there is some sese of should for which (TB) comes out true, e.g., should o pai of occupyig a doxastic state that is ot perfectly accurate (see f. 11). But, we thik most philosophers would ot accept (TB) as a ratioal requiremet. 12 Noetheless, (TB) clearly implies (CB) i the sese that all agets who satisfy (TB) must also satisfy (CB). So, if oe violates (CB), the oe must also violate (TB). Moreover, violatios of (CB) are the sorts of thigs that oe ca (ideally, i priciple) be i a positio to detect a priori. Thus, oe might try to argue that (CB) is a ecessary requiremet of ideal epistemic ratioality, as follows. If oe is (ideally, i priciple) i a positio to kow a priori that oe violates (TB), the oe s doxastic state is ot (ideally) epistemically ratioal. Therefore, (CB) is a ratioal requiremet. While this (TB)-based argumet for (CB) may have some prima facie plausibility, we ll argue that (CB) itself seems to be i tesio with aother plausible epistemic orm, which we call the Evidetial Norm for full belief. (EB) All agets S should (at ay give time t) have full beliefs that are supported by the total evidece. For ow, we re beig itetioally vague about what supported ad the total evidece mea i (EB), but we ll precisify these locutios i due course We will use the term orm (as opposed to requiremet ) to refer to local/arrow-scope epistemic costraits o belief. The Truth Norm (as well as the Evidetial Norm, to be discussed below) is local i the sese that it costrais each idividual belief it requires that each propositio believed by a aget be true. This differs from the ratioal requiremets we ll be focusig o here (viz., coherece requiremets), which are global/wide-scope costraits o sets of beliefs. Moreover, the sese of should i orms will geerally differ from the evaluative/global sese of should that we are associatig with ratioal requiremets (see f. 13). 11 Our statemet of (TB) is (itetioally) somewhat vague here. Various precisificatios of (TB) have bee discussed i the cotemporary literature. See Thomso (2008), Wedgwood (2002), Shah (2003), Gibbard (2005) ad Boghossia (2003) for some recet examples. The subtle distictios betwee these various reditios of (TB) will ot be crucial for our purposes. For us, (TB) plays the role of determiig the correctess/accuracy coditios for belief (i.e., it determies the alethic ideal for belief states). I other words, the should i our (TB) is iteded to mea somethig like should o pai of occupyig a doxastic state that is ot etirely/perfectly correct/accurate. I this sese, the versio of (TB) we have i mid here is perhaps most similar to Thomso s (2008, Ch. 7). 12 Some philosophers maitai that justificatio/warrat is factive (Littlejoh 2012; Merricks 1995). I light of the Gettier problem, factivity seems plausible as a costrait o the type of justificatio required for kowledge (Zagzebski 1994; Dretske 2013). However, factivity is implausible as a costrait o (the type of justificatio required for) ratioal belief. As such, we assume that is supported by the total evidece (i.e., is justified/warrated ) is ot factive. This assumptio is kosher here, sice it cross-cuts the preset debate regardig (CB). For istace, Pollock s defese of (CB) as a coherece requiremet does ot trade o the factivity of evidetial support (see f. 15). 13 The evidetial orm (EB) is [like (TB)] a local/arrow-scope priciple. It costrais each idividual belief, so as to require that it be supported by the evidece. We will ot take a stad o the precise cotet of (EB) here, sice we will (ultimately) oly eed to make use of certai (weak) cosequeces of (EB). However, the should of (EB) is ot to be cofused with the should of (TB). It may be useful (heuristically) to read the should of (EB) as should o pai of fallig short of the Cliffordia ideal ad the should of (TB) as should o pai of fallig short of the Jamesia ideal (see fs. 3 & 10).

3 ACCURACY, COHERENCE AND EVIDENCE 5 6 KENNY EASWARAN AND BRANDEN FITELSON Versios of (EB) have bee edorsed by various evidetialists (Clifford 1877; Coee & Feldma 2004). Iterestigly, the variats of (EB) we have i mid coflict with (CB) i some ( paradoxical ) cotexts. For istace, cosider the followig example, which is a global versio of the Preface Paradox. Preface Paradox. Let B be the set cotaiig all of S s justified first-order beliefs. Assumig S is a suitably iterestig iquirer, this set B will be a very rich ad complex set of judgmets. Ad, because S is fallible, it is reasoable to believe that some of S s first-order evidece will (ievitably) be misleadig. As a result, it seems reasoable to believe that some beliefs i B are false. Ideed, we thik S herself could be justified i believig this very secod-order claim. But, of course, addig this secod-order belief to B reders S s overall doxastic (full belief) state deductively icosistet. We take it that, i (some) such preface cases, a aget s doxastic state may satisfy (EB) while violatig (CB). Moreover, we thik that (some) such states eed ot be (ideally) epistemically irratioal. That is, we thik our Preface Paradox (ad other similar examples) establish the followig key claim: ( ) (EB) does ot etail (CB). [i.e., the Evidetial Norm does ot etail that deductive cosistecy is a requiremet of ideal epistemic ratioality.] We do ot have space here to provide a thorough defese of ( ). 14 Foley (1992) sketches the followig, geeral master argumet i support of ( ).... if the avoidace of recogizable icosistecy were a absolute prerequisite of ratioal belief, we could ot ratioally believe each member of a set of propositios ad also ratioally believe of this set that at least oe of its members is false. But this i tur pressures us to be uduly cautious. It pressures us to believe oly those propositios that are certai or at least close to certai for us, sice otherwise we are likely to have reasos to believe that at least oe of these propositios is false. At first glace, the requiremet that we avoid recogizable icosistecy seems little eough to ask i the ame of ratioality. It asks oly that we avoid certai error. It turs out, however, that this is far too much to ask. We thik Foley is oto somethig importat here. As we ll see, Foley s argumet dovetails icely with our approach to groudig coherece requiremets for belief. So far, we ve bee assumig that agets facig Prefaces (ad similar paradoxes of deductive cosistecy) may be opiioated regardig the (icosistet) sets of propositios i questio (i.e., that the agets i questio either believe or disbelieve each propositio i the set). I the ext sectio, we cosider the possibility that the appropriate respose to the Preface Paradox (ad other similar paradoxes) is to susped judgmet o (some or all) propositios implicated i the icosistecy. 14 Presetly, we are cotet to take ( ) as a datum. However, defiitively establishig ( ) requires oly the presetatio of oe example (preface or otherwise) i which (CB) is violated, (EB) is satisfied, ad the doxastic state i questio is ot (ideally) epistemically irratioal. We thik our Preface Paradoxes suffice. Be that as it may, we thik Christese (2004), Foley (1992), ad Klei (1985) have give compellig reasos to accept ( ). Ad, we ll briefly parry some recet philosophical resistace to ( ) below. Oe might eve wat to stregthe ( ) so as to imply that satisfyig (EB) sometimes requires the violatio of (CB). Ideed, this stroger claim is arguably established by our Preface Paradox cases. I ay evet, we will, i the iterest of simplicity, stick with our weaker reditio of ( ). 3. Suspesio of Judgmet to the Rescue? Some authors maitai that opiioatio is to blame for the discomfort of the Preface Paradox (ad should be abadoed i respose to it). We are ot moved by this lie of respose to the Preface Paradox. We will ow briefly critique two types of suspesio strategies that we have ecoutered. It would seem that Joh Pollock (1983) was the pioeer of the suspesio strategy. Accordig to Pollock, wheever oe recogizes that oe s beliefs are icosistet, this leads to the collective defeat of (some or all of) the judgmets comprisig the icosistet set. That is, the evidetial support that oe has for (some or all of) the beliefs i a icosistet set is defeated as a result of the recogitio of said icosistecy. If Pollock were right about this (i full geerality), it would follow that if the total evidece supports each of oe s beliefs, the oe s belief set must be deductively cosistet. I other words, Pollock s geeral theory of evidetial support (or warrat 15 ) must etail that ( ) is false. Ufortuately, however, Pollock does ot offer much i the way of a geeral argumet agaist ( ). His geeral remarks ted to be alog the followig lies (Pollock 1990, p. 231). 16 The set of warrated propositios must be deductively cosistet.... If a cotradictio could be derived from it, the reasoig from some warrated propositios would lead to the deial (ad hece defeat) of other warrated propositios, i which case they would ot be warrated. The basic idea here seems to be that, if oe (kowigly) has a icosistet set of (justified) beliefs, the oe ca deduce a cotradictio from this set, ad the use this cotradictio to perform a reductio of (some of) oe s (justified) beliefs. 17 Needless to say, ayoe who is already coviced that ( ) is true will fid this geeral argumet agaist ( ) ucovicig. Presumably, ayoe who fids themselves i the midst of a situatio that they take to be a couterexample to (EB) (CB) should be reluctat to perform reductios of the sort Pollock seems to have i mid, sice it appears that cosistecy is ot required by their evidece. Here, Pollock seems to be assumig a closure coditio (e.g., that is supported by the total evidece is closed uder logical cosequece/competet deductio) to provide a reductio of ( ). It seems clear to us that those who accept ( ) would/should reject 15 Pollock uses the term warrated rather tha supported by the total evidece. But, for the purposes of our discussio of Pollock s views, we will assume that these are equivalet. This is kosher, sice, for Pollock, S is warrated i believig p meas S could become justified i believig p through (ideal) reasoig proceedig exclusively from the propositios he is objectively justified i believig (Pollock 1990, p. 87). Our agets, like Pollock s, are idealized reasoers, so we may stipulate (for the purposes of our discussio of Pollock s views) that whe we say supported by the total evidece, we just mea whatever Pollock meas by warrated. Some (Merricks 1995) have argued that Pollock s otio of warrat is factive (see f. 12). This seems wrog to us (i the preset cotext). If warrat (i the relevat sese) were factive, the Pollock would t eed such complicated resposes to the paradoxes of cosistecy they would be trivially ruled out, a fortiori. This is why, for preset purposes, we iterpret Pollock as claimig oly that (EB) etails the cosistecy of (warrated) belief sets [(CB)], but ot ecessarily the truth of each (warrated) belief [(TB)]. 16 The elipsis i our quotatio cotais the followig parethetical remark: It is assumed here that a epistemic basis must be cosistet. That is, Pollock gives o argumet(s) for the claim that epistemic bases (which, for Pollock, are sets of iput propositios of agets) must be cosistet. 17 Rya (1991; 1996) gives a similar argumet agaist ( ). Ad, Nelki (2000) edorses Rya s argumet, as applied to defusig the lottery paradox as a couterexample to ( ) [i.e., (EB) (CB)].

4 ACCURACY, COHERENCE AND EVIDENCE 7 8 KENNY EASWARAN AND BRANDEN FITELSON closure coditios of this sort. We view (some) Preface cases as couterexamples to both cosistecy ad closure of ratioal belief. 18 While Pollock does t offer much of a geeral argumet for ( ), he does address two apparet couterexamples to ( ): the lottery paradox ad the preface paradox. Pollock (1983) first applied this collective defeat strategy to the lottery paradox. He later recogized (Pollock 1986) that the collective defeat strategy is far more difficult to (plausibly) apply i the case of the Preface Paradox. Ideed, we fid it implausible o its face that the propositios of the (global) Preface joitly defeat oe aother i ay probative sese. More geerally, we fid Pollock s treatmet of the Preface Paradox quite puzzlig ad upersuasive. 19 Be that as it may, it s difficult to see how this sort of collective defeat argumet could serve to justify ( ) i full geerality. What would it take for a theory of evidetial support to etail ( ) i full geerality via a Pollock-style collective defeat argumet? We re ot sure. But, we are cofidet that ay explicatio of supported by the total evidece (or warrated ) which embraces a pheomeo of collective defeat that is robust eough to etail the falsity of ( ) will also have some udesirable (eve uacceptable) epistemological cosequeces. 20 We ofte hear aother lie of respose to the Preface that is similar to (but somewhat less ambitious tha) Pollock s collective defeat approach. This lie of respose claims that there is somethig heterogeous about the evidece i the Preface Paradox, ad that this evidetial heterogeeity somehow udermies the claim that oe should believe all of the propositios that comprise the Preface Paradox. The idea seems to be 21 that the evidece oe has for the first-order beliefs (i B) is a (radically) differet kid of evidece tha the evidece oe has for the secod-order belief (i.e., the belief that reders B icosistet i the ed). Ad, because these bodies of first-order ad secod-order evidece are so heterogeous, there is o sigle body of evidece that supports both the first-order beliefs ad the secod-order belief i the Preface case. So, believig all the propositios of the Preface is ot, i fact, the epistemically ratioal thig to do. 22 Hece, the apparet tesio betwee (EB) ad (CB) is merely apparet. We thik this lie of respose is usuccessful, for three reasos. First, ca t we just gather up the first-order ad secod-order evidetial propositios, ad put them all ito oe big collectio of total Preface evidece? Ad, if we do so, why 18 See (Steiberger 2013) for a icisive aalysis of the cosequeces of the preface paradox for various priciples of deductive reasoig [i.e., bridge priciples i the sese of (MacFarlae 2004)]. 19 We do t have the space here to aalyze Pollock s (rather byzatie) approach to the Preface Paradox. Fortuately, Christese (2004) has already doe a very good job of explaiig why suspesio strategies like Pollock s ca ot, ultimately, furish compellig resposes to the Preface. 20 For istace, it seems to us that ay such approach will have to imply that supported by the total evidece is (geerally) closed uder logical cosequece (or competet deductio), eve uder complicated etailmets with may premises. See (Korb 1992) for discussio regardig (this ad other) upalatable cosequeces of Pollockia collective defeat. 21 We ve actually ot bee able to fid this exact lie of respose to the Preface aywhere i prit, but we have heard this kid of lie defeded i various discussios ad Q&A s. The closest lie of respose we ve see i prit is Leitgeb s (2013) approach, which appeals to the heterogeeity of the subject matter of the claims ivolved i the Preface. This does t exactly fall uder our evidetial heterogeeity rubric, but it is similar eough to be udermied by our Homogeeous Preface case. 22 Presumably, the, the ratioal thig to do is susped judgmet o some of the Preface propositios. But, which oes? As i the case of Pollock s suspesio strategy, it remais uclear (to us) precisely which propositios fail to be supported by the evidece i the Preface Paradox (ad why). would t the total Preface evidece support both the first-order beliefs ad the secod-order belief i the Preface case? Secod, we oly eed oe Preface case i which (EB) ad (CB) do geuiely come ito coflict i order to establish ( ). Ad, it seems to us that there are homogeeous versios of the Preface which do ot exhibit this (alleged) kid of evidetial heterogeeity. Here s oe such example. Homogeeous Preface Paradox. Joh is a excellet empirical scietist. He has devoted his etire (log ad esteemed) scietific career to gatherig ad assessig the evidece that is relevat to the followig first-order, empirical hypothesis: (H) all scietific/empirical books of sufficiet complexity cotai at least oe false claim. By the ed of his career, Joh is ready to publish his masterpiece, which is a exhaustive, ecyclopedic, 15-volume (scietific/empirical) book which aims to summarize (all) the evidece that cotemporary empirical sciece takes to be relevat to H. Joh sits dow to write the Preface to his masterpiece. Rather tha reflectig o his ow fallibility, Joh simply reflects o the cotets of (the mai text of) his book, which costitutes very strog iductive evidece i favor of H. O this basis, Joh (iductively) ifers H. But, Joh also believes each of the idividual claims asserted i the mai text of the book. Thus, because Joh believes (ideed, kows) that his masterpiece istatiates the atecedet of H, the (total) set of Joh s (ratioal/justified) beliefs is icosistet. I our Homogeeous Preface, there seems to be o evidetial heterogeeity available to udermie the evidetial support of Joh s ultimate doxastic state. Moreover, there seems to be o collective defeat loomig here either. Joh is simply beig a good empirical scietist (ad a good iductive o-skeptic) here, by (ratioally) iferrig H from the total, H-relevat iductive scietific/empirical evidece. It is true that it was Joh himself who gathered (ad aalyzed, etc.) all of this iductive evidece ad icluded it i oe hugely complex scietific/empirical book. But, we fail to see how this fact does aythig to udermie the (ideal) epistemic ratioality of Joh s (ultimate) doxastic state. So, we coclude that the heterogeeity strategy is ot a adequate respose to the Preface. 23 More geerally, we thik our Homogeeous Preface case udermies ay strategy that maitais oe should ever believe all the propositios i ay Preface. 24 We maitai that (adequate) resposes to the Preface Paradox eed ot require suspesio of judgmet o (ay of) the Preface propositios. Cosequetly, we 23 We said we rejected the heterogeous evidece lie of respose to the Preface for three reasos. Our third reaso is similar to the fial worry we expressed above regardig Pollock s collective defeat strategy. We do t see how a heterogeeity strategy could serve to establish ( ) i full geerality, without presupposig somethig very implausible about the geeral ature of evidetial support, e.g., that evidetial support is preserved by competet deductio (see f. 20). 24 This icludes Kapla s (2013) lie o the Preface, which appeals to the orms of what we are willig to say i the cotext of iquiry. Accordig to Kapla, what we are willig to say i the cotext of iquiry is govered by a requiremet of deductive cogecy, which is stroger tha (CB). Cogecy implies (CB) plus closure (uder competet deductio). Joh (the protagoist of our Homogeeous Preface Paradox) does ot seem to us to be violatig ay orms of what we are willig to say i the cotext of iquiry. It seems to us that othig prevets Joh from beig a perfectly ratioal scietific iquirer eve if he says everythig we ascribe to him i the Homogeeous Preface.

5 ACCURACY, COHERENCE AND EVIDENCE 9 10 KENNY EASWARAN AND BRANDEN FITELSON would like to see a (pricipled) respose to the Preface Paradox (ad other paradoxes of cosistecy) that allows for (full) opiioatio with respect to the propositios i the Preface ageda. Ideed, we will provide just such a respose (to all paradoxes of cosistecy) below. Before presetig our framework (ad respose), we will compare ad cotrast our ow view regardig the Preface Paradox (ad other paradoxes of cosistecy) with the views recetly expressed by a pair of philosophers who share our commitmet to ( ) i.e., to the claim that Preface cases (ad other similar cases) show that deductive cosistecy is ot a ecessary requiremet of ideal epistemic ratioality. 4. Christese ad Kolody o Coherece Requiremets We are ot aloe i our view that Prefaces (ad other paradoxes of deductive cosistecy) suffice to establish ( ). 25 For istace, David Christese ad Niko Kolody agree with us about Prefaces ad ( ). But, Christese ad Kolody react to the paradoxes of deductive cosistecy i a more radical way. They edorse ( ) There are o coherece requiremets (i the relevat sese) for full belief. That is to say, both Christese ad Kolody edorse elimiativism regardig all (formal, sychroic, epistemic) coherece requiremets for full belief. It is illumiatig to compare ad cotrast the views of Christese ad Kolody with our ow views about paradoxes of cosistecy ad proper resposes to them. Christese (2004) accepts the followig package of pertiet views. 26 (C 1 ) Partial beliefs (viz., credeces) are subject to a formal, sychroic, epistemic coherece requiremet (of ideal ratioality): probabilism. ( ) Full beliefs are ot subject to ay formal, sychroic, epistemic coherece requiremets (of ideal ratioality). (C 2 ) Epistemic pheomea that appear to be adequately explaiable oly by appeal to coherece requiremets for full belief (ad facts about a aget s full beliefs) ca be adequately explaied etirely by appeal to probabilism (ad facts about a aget s credeces). We agree with Christese about (C 1 ). I fact, our framework for groudig coherece requiremets for full belief is ispired by aalogous argumets for probabilism as a coherece requiremet for partial belief. We will retur to this importat parallel below. Christese s (C 2 ) is part of a error theory regardig epistemological explaatios that appear to ivolve coherece requiremets for full belief as (essetial) explaas. Some such error theory is eeded give ( ) sice epistemologists ofte seem to make essetial use of such coherece-explaas. 25 Other authors besides Christese (2004), Kolody (2007), Foley (1992) ad Klei (1985) have claimed that paradoxes of cosistecy place pressure o the claim that (EB) etails (CB). For istace, Kyburg (1970) maitais that the lottery paradox supports ( ). We are focusig o preface cases here, sice we thik they are, ultimately, more compellig tha lottery cases (see f. 38). 26 Strictly speakig, Christese (2004) ever explicitly edorses ( ) or (C2 ) i their full geerality. He focuses o deductive cosistecy as a coherece-explaas, ad he argues that it ca be elimiated from such explaatios, i favor of appeals oly to probabilism (ad facts about the agets credeces). So, our ( ) ad (C 2 ) may be stroger tha the priciples Christese actually accepts. I recet persoal commuicatio, Christese has voiced some sympathy with the (existece ad explaatory power of) the coherece requiremets for full belief developed here. Havig said that, our straw ma Christese allows for a more perspicuous cotrast i the preset cotext. Kolody (2007), o the other had, accepts the followig pair: (K 1 ) No attitudes (full belief, partial belief, or otherwise) are subject to ay formal, sychroic, epistemic coherece requiremets (of ideal ratioality). (K 2 ) Epistemic pheomea that appear to be adequately explaiable oly by appeal to coherece requiremets for full belief (together with facts about a aget s full beliefs) ca be adequately explaied etirely by appeal to the Evidetial Norm (EB), together with facts about a aget s full beliefs. Kolody s (K 1 ) is far more radical tha aythig Christese accepts. Of course, (K 1 ) etails ( ), but it also etails uiversal elimiativism about coherece requiremets i epistemology. Kolody does t thik there are ay (ielimiable) coherece requiremets (or ay ielimiable requiremets of ideal ratioality, for that matter), period. He does t eve recogize probabilism as a coherece requiremet for credeces. As a result, Kolody eeds a differet error theory to explai away the various epistemological explaatios that seem to appeal essetially to coherece requiremets for full belief. His error theory [(K 2 )] uses the Evidetial Norm for full belief (EB), alog with facts about the aget s full beliefs, to explai away such appeals to coherece requiremets. So, Kolody s error theory differs from Christese s i a crucial respect: Kolody appeals to local/arrow-scope orms for full belief to explai away apparet uses of coherece requiremets for full belief; whereas, Christese appeals to global/wide-scope requiremets of partial belief to explai away apparet uses of coherece requiremets for full belief. This is (partly) because Kolody is committed to the followig geeral claim: (K 3 ) Full beliefs are a essetial (ad ielimiable) part of epistemology (i.e., the full belief cocept is ielimiable from some epistemological explaatios). We agree with Kolody about (K 3 ). We, too, thik that full belief is a crucial (ad ielimiable) epistemological cocept. (Ideed, this is oe of the reasos we are offerig a ew framework for groudig coherece requiremets for full belief!) Christese, o the other had (at least o our readig, see f. 26), seems to be usympathetic to (K 3 ). Oe last epistemological priciple will be useful for the purposes of comparig ad cotrastig our views with the views of Christese ad Kolody. 27 ( ) If there are ay coherece requiremets for full belief, the deductive cosistecy [(CB)] is oe of them. [i.e., If ( ), the (CB).] Christese ad Kolody both accept ( ), albeit i a trivial way. They both reject the atecedet of ( ) [i.e., they both accept ( )]. We, o the other had, aim to provide a pricipled way of rejectig ( ). That is to say, we aim to groud ew coherece requiremets for full belief, which are distict from deductive cosistecy. We thik this is the proper respose to the paradoxes of cosistecy [ad ( )]. I the ext sectio, we preset our formal framework for groudig coherece requiremets for (opiioated) full belief. But, first, we propose a desideratum for such coherece requiremets, ispired by the cosideratios adduced so far. (D) Coherece requiremets for (opiioated) full belief should ever come ito coflict with either alethic or evidetial orms for (opiioated) full 27 The If..., the... i ( ) is a material coditioal. That is, ( ) asserts: either ( ) or (CB).

6 ACCURACY, COHERENCE AND EVIDENCE KENNY EASWARAN AND BRANDEN FITELSON belief. That is, coherece requiremets for (opiioated) full belief should be etailed by both the Truth Norm (TB) ad the Evidetial Norm (EB). I light of ( ), deductive cosistecy [(CB)] violates desideratum (D). If a coherece requiremet satisfies desideratum (D), we will say that it is coflict-proof. Next, we explai how to groud coflict-proof coherece requiremets for (opiioated) full belief. 5. Our (Naïve) Framework ad (Some of) its Coherece Requiremets As it happes, our preferred alterative(s) to (CB) were ot iitially motivated by thikig about paradoxes of cosistecy. They were ispired by some recet argumets for probabilism as a (sychroic, epistemic) coherece requiremet for credeces. James Joyce (1998; 2009) has offered argumets for probabilism that are rooted i cosideratios of accuracy (i.e., i alethic cosideratios). We wo t get ito the details of Joyce s argumets here. 28 Istead, we preset a geeral framework for groudig coherece requiremets for sets of judgmets of various types, icludig both credeces ad full beliefs. Our uified framework costitutes a geeralizatio of Joyce s argumet for probabilism. Moreover, whe our approach is applied to full belief, it yields coherece requiremets that are superior to (CB), i light of preface cases (ad other similar paradoxes of cosistecy). Applyig our framework to judgmet sets J of type J oly requires completig three steps. The three steps are as follows: Step 1. Say what it meas for a set J of type J to be perfectly accurate (at a possible world w). We use the term vidicated to describe the perfectly accurate set of judgmets of type J, at w, ad we use J w to deote this vidicated set. 29 Step 2. Defie a measure of distace betwee judgmet sets, d(j, J ). We use d to gauge a set J s distace from vidicatio at w [viz., d(j, J w )]. Step 3. Adopt a fudametal epistemic priciple, which uses d(j, J w ) to groud a (sychroic, epistemic) coherece requiremet for judgmet sets J of type J. This is all very abstract. To make thigs more cocrete, let s look at the simplest applicatio of our framework to the case of (opiioated) full belief. Let: B(p) S believes that p D(p) S disbelieves that p. Our agets will be formig (opiioated) judgmets o some saliet ageda A, which is a (possibly proper) subset of some fiite boolea algebra of propositios. That is, for each p A, S either believes p or S disbelieves p, ad ot both. 30 I 28 There are some importat disaalogies betwee Joyce s argumet for probabilism ad our aalogous argumets regardig coherece requiremets for full belief. Happily, the worry (articulated i Easwara & Fitelso 2012) that Joyce s argumet for probabilism may violate the credal aalogue of (D) does ot apply to our preset argumets (see f. 42). 29 As a heuristic, you ca thik of Jw as the set of judgmets of type J that a omisciet aget (i.e., a aget who is omisciet about the facts at world w) would have. 30 Our assumptio of opiioatio, relative to a saliet ageda A, results i o sigificat loss of geerality for preset purposes. As we have explaied above, we do ot thik suspesio of belief (o the Preface ageda there are may propositios outside this ageda o which it may be reasoable to susped) is a evidetially plausible way of respodig to the Preface Paradox. Cosequetly, oe of our preset aims is to provide a respose to paradoxes of cosistecy that this way, a aget ca be represeted by her belief set B, which is just the set of her beliefs (B) ad disbeliefs (D) over some saliet ageda A. Similarly, we thik of propositios as sets of (classical) possible worlds, so that a propositio is true at ay world that it cotais, ad false at ay world it does t cotai. 31 With our (aïve) setup i place, we re ready for the three steps. Step 1 is straightforward. It is clear what it meas for a set B of this type to be perfectly accurate/vidicated at a world w. The vidicated set B w is give by: B w cotais B(p) [D(p)] just i case p is true [false] at w. This is clearly the best explicatio of B w, sice B(p) [D(p)] is accurate just i case p is true [false]. Give the accuracy coditios for B/D, Step 1 is ucotroversial. Step 2 is less straightforward, because there are a great may ways oe could measure distace betwee opiioated sets of beliefs/disbeliefs. For simplicity, we adopt perhaps the most aïve distace measure, which is give by: d(b, B ) the umber of judgmets o which B ad B disagree. 32 I particular, if you wat to kow how far your judgmet set B is from vidicatio at w [i.e., if you wat to kow the value of d(b, B w )] just cout the umber of mistakes you have made at w. To be sure, this is a very aïve measure of distace from vidicatio. As it turs out, however, we (ultimately) wo t eed to rely o such a strog (or aïve) assumptio about d(b, B w ). I the ed, we ll oly eed a much weaker assumptio about d(b, B w ). But, for ow, let s ru with our aïve, coutig of mistakes at w defiitio of d(b, B w ). We ll retur to this issue later. Step 3 is the philosophically most importat step. Before we get to our favored fudametal epistemic priciple(s), we will digress briefly to discuss a stroger fudametal epistemic priciple that oe might fid (prima facie) plausible. Give our aïve setup, it turs out that there is a choice of fudametal epistemic priciple that yields deductive cosistecy [(CB)] as a coherece requiremet for opiioated full belief. Specifically, cosider the followig priciple: allows for full opiioatio (o the saliet agedas). Moreover, there are other applicatios of the preset framework for which opiioatio is required. Briggs et al. (2014) show how to apply the preset framework to the paradoxes of judgmet aggregatio, which presuppose opiioatio o the saliet agedas. Fially, we wat to preset the simplest ad clearest versio of our framework here. The aïve framework we preset here ca be geeralized i various ways. Specifically, geeralizig the preset framework to allow for suspesio of judgmet (o the saliet agedas) is, of course, desirable (Sturgeo 2008; Friedma 2013). See (Easwara 2013) for a geeralizatio of the preset framework which allows for suspesio of judgmet o the saliet agedas (see f. 39). Ad, see (Fitelso 2014) for several other iterestig geeralizatios of the preset framework. 31 It is implicit i our (highly idealized) framework that agets satisfy a weak sort of logical omisciece, i the sese that if two propositios are logically equivalet, the they may be treated as the same propositio i all models of the preset framework. As such, we re assumig that agets caot have distict attitudes toward logically equivalet (classical, possible-worlds) propositios. We have already explaied why such idealizatios are okay i the preset cotext (see f. 9). However, it is importat to ote that we are ot assumig agets satisfy a stroger sort of omisciece a aget may believe some propositios while disbelievig some other propositio etailed by them (i.e., our logical omisciece presuppositio does ot imply closure for ratioal belief). I other words, our agets are aware of all logical relatios, but their judgmet sets may ot be closed uder them. 32 This is called the Hammig distace betwee the biary vectors B ad B (Deza ad Deza 2009).

7 ACCURACY, COHERENCE AND EVIDENCE KENNY EASWARAN AND BRANDEN FITELSON Possible Vidicatio (PV). There exists some possible world w at which all of the judgmets i B are accurate. Or, to put this more formally, i terms of our distace measure d: ( w)[d(b, B w ) = 0]. Give our aïve setup, it is easy to show that (PV) is equivalet to (CB). 33 As such, a defeder of (CB) would presumably fid (PV) attractive as a fudametal epistemic priciple. However, as we have see i previous sectios, preface cases (ad other paradoxes of cosistecy) have led may philosophers (icludig us) to reject (CB) as a ratioal requiremet. This motivates the adoptio of fudametal priciples that are weaker tha (PV). Iterestigly, as we metioed above, our rejectio of (PV) was ot (iitially) motivated by Prefaces ad the like. Rather, our adoptio of fudametal priciples weaker tha (PV) was motivated (iitially) by aalogy with Joyce s argumet(s) for probabilism as a coherece requiremet for credeces. I the case of credeces, the aalogue of (PV) is clearly too strog. The vidicated set of credeces (i.e., the credeces a omisciet aget would have) are such that they assig maximal cofidece to all truths ad miimal cofidece to all falsehoods (Joyce, 1998). As a result, i the credal case, (PV) would require that all of oe s credeces be extremal. Oe does t eed Preface cases (or ay other subtle or paradoxical cases) to see that this would be a ureasoably strog (ratioal) requiremet. It is for this reaso that Joyce (ad all others who argue i this way for probabilism) back away from the aalogue of (PV) to strictly weaker epistemic priciples specifically, to accuracy-domiace avoidace priciples, which are credal aalogues of the followig fudametal epistemic priciple. Weak Accuracy-Domiace Avoidace (WADA). B is ot weakly 34 domiated i distace from vidicatio. Or, to put this more formally (i terms of d), there does ot exist a alterative belief set B such that: (i) ( w)[d(b, B w ) d(b, B w )], ad (ii) ( w)[d(b, B w ) < d(b, B w )]. (WADA) is a very atural priciple to adopt, if oe is ot goig to isist that as a requiremet of ratioality it must be possible for a aget to achieve perfect accuracy i her doxastic state. I the credal case, the aalogous requiremet was clearly too strog to cout as a ratioal requiremet. I the case of full belief, oe eeds to thik about Preface cases (ad the like) to see why (PV) is too strog. Retreatig from (PV) to (WADA) is aalogous to what oe does i decisio theory, whe oe backs off a priciple of maximizig (actual) utility to some less demadig requiremet of ratioality (e.g., domiace avoidace, maximizatio 33 Here, we re assumig a slight geeralizatio of the stadard otio of cosistecy. Stadardly, cosistecy applies oly to beliefs (ot disbeliefs), ad it requires that there be a possible world i which all the aget s beliefs are true. More geerally, we may defie cosistecy as the existece of a possible world i which all the aget s judgmets (both beliefs ad disbeliefs) are accurate. Give this more geeral otio of cosistecy, (PV) ad (CB) are equivalet i the preset framework. 34 Strictly speakig, Joyce et al. opt for the apparetly weaker priciple of avoidig strict domiace. However, i the credal case (assumig cotiuous, strictly proper scorig rules), there is o differece betwee weak ad strict domiace (Schervish et al. 2009). I this sese, there is o serious disaalogy. Havig said that, it is worth otig that, i the case of full belief, there is a sigificat differece betwee weak domiace ad strict domiace. This differece will be discussed i some detail i 6 below. I the meatime, wheever we say domiated what we mea is weakly domiated i the sese of (WADA). of expected utility, miimax, etc.). 35 Of course, there is a sese i which the best actio is the oe that maximizes actual utility; but, surely, maximizatio of actual utility is ot a ratioal requiremet. Similarly, there is clearly a sese i which the best doxastic state is the perfectly accurate [(TB)], or possibly perfectly accurate [(CB)/(PV)], doxastic state. But, i light of the paradoxes of cosistecy, (TB) ad (CB) tur out ot to be ratioal requiremets either. Oe of the mai problems with the existig literature o the paradoxes of cosistecy is that o pricipled alterative(s) to deductive cosistecy have bee offered as coherece requiremets for full belief. Such alteratives are just what our Joyce-style argumets provide. If a belief set B satisfies (WADA), the we say B is o-domiated. This leads to the followig, ew coherece requiremet for (opiioated) full belief: (NDB) All (opiioated) agets S should (at ay give time t) have sets of full beliefs (ad disbeliefs) that are o-domiated. Iterestigly, (NDB) is strictly weaker tha (CB). Moreover, (NDB) is weaker tha (CB) i a appropriate way, i light of our Preface Paradoxes (ad other similar paradoxes of cosistecy). Our first two theorems (each with a accompayig defiitio) help to explai why. The first theorem states a ecessary ad sufficiet coditio for (i.e., a characterizatio of) o-domiace: we call it Negative because it idetifies certai objects, the o-existece of which is ecessary ad sufficiet for o-domiace. The secod theorem states a sufficiet coditio for o-domiace: we call it Positive because it states that i order to show that a certai belief set B is o-domiated, it s eough to costruct a certai type of object. Defiitio 1 (Witessig Sets). S is a witessig set iff (a) at every world w, at least half of the judgmets 36 i S are iaccurate; ad, (b) at some world more tha half of the judgmets i S are iaccurate. Theorem 1 (Negative). B is o-domiated iff B cotais o witessig set. [We will use (NWS) to abbreviate the claim that o subset of B is a witessig set. Thus, Theorem 1 ca be stated equivaletly as: B is o-domiated iff (NWS).] It is a immediate corollary of this first theorem that if B is deductively cosistet [i.e, if B satisfies (PV)], the B is o-domiated. After all, if B is deductively cosistet, the there is a world w such that o judgmets i B are iaccurate at w (f. 33). However, while deductive cosistecy guaratees o-domiace, the coverse is ot the case, i.e., o-domiace does ot esure deductive cosistecy. This will be most perspicuous as a cosequece of our secod theorem. 35 The aalogy to decisio theory could be made eve tighter. We could say that beig accuracydomiated reveals that you are i a positio to recogize a priori that aother optio is guarateed to do better at achievig the epistemic aim of gettig as close to the truth as possible. This decisiotheoretic stace dovetails icely with the setimets expressed by Foley (op. cit.). See 8 for further discussio of (ad elaboratio o) this epistemic decisio-theoretic stace. 36 Here, we rely o aïve coutig. This is uproblematic, sice all of our agedas are fiite. The coherece orm we ll propose i the ed (see 7) will ot be based o coutig ad (as a result) will be applicable to both fiite ad ifiite agedas. All Theorems are proved i the Appedix.