The	Epistemology	of	Disagreement:	Why	Not	Bayesianism? (penultimate	draft-forthcoming	in	Episteme) Thomas	Mulligan Georgetown	University Put	people	together	and	before	long	you	will	find	them	at	odds. It	appears	to	be	a	fact	of our	species	that	we	have	disagreed,	do	disagree,	and	will	disagree-and	about	almost anything. We	disagree	about	important	issues	of	morality	and	politics;	we	disagree	about sports	and	other	banalities. Our	disagreements	are	sometimes	silly,	but	they	are sometimes	sober	and	reasonable,	at	least	prima	facie:	None	of	us	is	obviously	irrational,	no argument	beyond	the	pale. Philosophers	have	dutifully	attended	to	the	phenomenon	of	disagreement, considering,	in	particular,	its	epistemology.1 The	central	question	in	the	epistemology	of disagreement	debate	is	how	a	rational	person	modifies	her	belief	in	a	proposition	X	(if	at all)	when	she	learns	that	someone	else	disagrees	about	X. The	ultimate	reason	for	such revision,	if	indeed	it	is	warranted,	may	be	purely	intellectual-one	wishes	to	have	a maximally	justified	belief	for	its	own	sake. More	frequently,	though,	it	is	to	motivate	action: "The	goal	of	maximally	justified	belief	.	.	.	is	primarily	a	goal	of	individuals	who	need	to	act" 1	Philosophers	have	also	worked	on	(1)	disagreement's	ramifications	for	political	authority (e.g.	Rawls	1993);	(2)	social	choice	theory-the	aggregation	of	individual	preferences	into a	group	preference	(which	Arrow	(1950)	showed	to	be	intractable);	and	(3)	collective decision-making-how	individuals	who	share	a	common	goal	but	disagree	about	how	to pursue	that	goal	ought	to	comport	themselves	(the	most	famous	result	in	collective decision-making	is	the	Marquis	de	Condorcet's	jury	theorem	(1785)). 2 (Everett	2015:	278). One	wishes	to	form	a	maximally	justified	belief	about	what	20%	of the	restaurant	bill	is	(Christensen	2007)	to	leave	a	proper	tip. Although	an	area	of	active	research	for	only	about	a	decade,2	the	epistemology	of disagreement	has	attracted	intense	interest-a	result,	perhaps,	of	our	hyper-partisan political	climate,	and	worry	about	its	social	effects.3 The	method	used	by	philosophers	investigating	disagreement	today,	and	the method	which	underpins	the	foundational	work	(n.	4),	is	to	argue	for	a	disagreement	norm a	priori. For	example,	this	is	how	David	Christensen	makes	his	case	for	conciliationism:4 Disagreement	gives	one	evidence	that	one	has	made	a	mistake	in	interpreting	the original	evidence	.	.	.	Thus	the	persistence	of	the	degree	of	disagreement	on important	issues	.	.	.	indicates	that,	in	general,	practitioners	in	the	field	do	not	form beliefs	reliably. If	one	is	a	practitioner	in	such	a	field,	then,	absent	some	reason	to think	oneself	special,	one	should	not	have	confident	opinions	on	the	field's controversial	questions.	(2007:	757) On	the	other	side	of	the	debate,	we	have	Thomas	Kelly's	reasoning	in	favor	of	steadfastness: The	question	of	how	well	someone	has	evaluated	the	evidence	with	respect	to	a given	question	is	certainly	the	kind	of	consideration	that	is	relevant	to	deciding whether	his	or	her	judgement	ought	to	be	credited	with	respect	to	that	question. That	is,	it	is	exactly	the	sort	of	consideration	that	is	capable	of	producing	the	kind	of asymmetry	that	would	justify	privileging	one	of	the	two	parties	to	the	dispute	over the	other	party. And	from	my	vantage	point-as	one	of	the	parties	within	the dispute,	as	opposed	to	some	on-looking	third	party-it	is	just	this	undeniably relevant	difference	that	divides	us	on	this	particular	occasion.	(2005:	179) 2	One	can	find	harbingers	of	the	contemporary	disagreement	debate	in	Lehrer	and	Wagner 1981	and	Loewer	and	Laddaga	1985. 3	Seminal	work	in	the	epistemology	of	disagreement	includes	Christensen	2007,	Elga	2007, Kelly	2005,	Lackey	2010a,	and	van	Inwagen	1996. The	best	introduction	to	disagreement is	Feldman	2007	(see	also	Christensen	2009). 4	Conciliationists	hold	that	one	ought	to	revise	one's	belief	in	the	face	of	disagreement	from a	suitable	epistemic	interlocutor. See,	e.g.,	Christensen	2007,	Elga	2007,	and	Feldman 2007. Adherents	to	steadfastness	believe,	to	the	contrary,	that	one	may	"stick	to	one's guns"	epistemically,	maintaining	one's	original	confidence	in	the	proposition	at	dispute. For	defenses	of	steadfastness,	see,	e.g.,	Bergmann	2009,	Kelly	2005,	and	van	Inwagen	2010. 3 Christensen	and	Kelly	go	on	to	adduce	examples	from	real	life	which	are	supposed	to	show that	their	preferred	norms	are	correct. I	am	convinced	that	this	approach	is	misguided. We	should	not	endorse	a disagreement	norm	on	a	priori	grounds	and	a	handful	of	intuitive	cases,	and	then	impose	it on	those	cases	for	which	intuitions	go	the	opposite	way. Rather,	we	should	aspire	to	a principled	approach	to	belief	revision	that	yields	steadfast	norms	in	appropriate	cases	and conciliatory	norms	in	appropriate	cases. The	purpose	of	this	paper	is	twofold. First,	I	argue	that	such	a	principled	approach is	possible-if	we	apply	neglected	tools	from	Bayesian	analysis. Second,	I	show	that	the	Bayesian	modelling	strategy	I	commend	satisfies	three	vital desiderata	which	mainstream	approaches	to	disagreement,	conciliatory	and	steadfast,	do not. To	wit,	Bayesian	modelling	can	(1)	deal	with	multiple	epistemic	interlocutors;	(2) allow	one	to	suitably	modify	one's	beliefs	in	the	face	of	disagreement	from	epistemic superiors	and	inferiors	(i.e.	not	just	epistemic	peers);	and	(3)	accommodate	the	possibility of	dependence	between	interlocutors. I	have	organized	this	paper	as	follows. In	§1,	I	define	the	term	epistemic	peer	and provide	necessary	notation. §2	introduces	the	Bayesian	approach	to	belief	revision. §3 argues	that	dependence	among	epistemic	interlocutors	is	not	just	ubiquitous	in	the	realworld	but	critically	important	from	a	formal	point-of-view. Our	norms	must	be	capable	of accommodating	it. I	also	critique	a	recent	effort	of	Easwaran	et	al.	(2016)	to	provide	a disagreement	norm	in	the	Bayesian	spirit. §4	provides	what	I	believe	to	be	a	better	model for	disagreement. I	conclude	in	§5. 4 1. The	epistemology	of	disagreement:	concepts	and	modelling	assumptions I	shall	not	give	an	overview	of	the	epistemology	of	disagreement	debate	(for	that,	see	the references	listed	in	n.	4). In	this	section,	I	only	want	to	define	a	term	and	provide	some notation. The	term	is	"epistemic	peer",	first	coined	by	Gary	Gutting	(1982). Intuitively,	if	I believe	that	Elizabeth	Woodville	was	the	wife	of	Henry	VI,	and	my	8-year-old	cousin disagrees	with	me	about	this,	I	need	not	lose	confidence	in	my	belief-for	I	am	much	more likely	than	he	is	to	be	correct	about	this	historical	fact. On	the	other	hand,	when	a professor	of	British	history	tells	me	that	I	am	wrong,	I	certainly	must	lose	confidence. But what	is	the	rational	response	when	someone	as	likely	as	I	am	to	be	correct	about	Elizabeth Woodville	was	the	wife	of	Henry	VI	disagrees	with	me?5 Most	epistemologists	believe	that this	the	interesting	case-disagreement	with	an	epistemic	peer-and	it	has	been	the	focus of	the	debate. Kelly	defines	the	term	thus: Let	us	say	that	two	people	are	epistemic	peers	with	respect	to	some	question	if	and only	if	they	satisfy	the	following	two	conditions: (i)	they	are	equals	with	respect	to	their	familiarity	with	the	evidence	and arguments	which	bear	on	that	question,	and (ii)	they	are	equals	with	respect	to	general	epistemic	virtues	such	as intelligence,	thoughtfulness,	and	freedom	from	bias.	(2005:	174-75) This	is	a	typical	definition	(cf.	Gelfert	2011	and	Matheson	2015),	and	it	will	suffice	for	our purposes. Of	course,	that	disagreement	with	peers	is	of	epistemic	interest	does	not	imply	that disagreement	with	non-peers	is	not. Yet	little	attention	has	been	paid	to	how	one	should 5	Woodville	was	in	fact	the	wife	of	Edward	IV-not	Henry	VI. 5 revise	one's	beliefs	in	light	of	disagreement	from	an	epistemic	superior	or	an	epistemic inferior. And	the	attention	that	has	been	paid	to	those	cases	(e.g.	Zagzebski	2012)	tends	to focus	on	special	contexts,	like	morality. The	implicit	assumption	is	that	belief	revision	is warranted	when	one	comes	into	contact	with	an	epistemic	superior,	and	unwarranted when	one	comes	into	contact	with	an	epistemic	inferior. This	will	not	do. Set	aside	that	is	a	rare	thing	to	interact	with	people	whom	we	can honestly	say	are	pure	epistemic	peers. All	our	epistemic	interactions	take	place	within	a complicated	nexus	of	inferiors,	peers,	and	superiors. The	"smartest"	person	we	know	may believe	X;	two	marginally	less	smart	people	may	believe	~X;	two	peers	may	believe	X	while one	believes	~X;	and	all	the	while	four	inferiors	believe	~X. What	to	do,	epistemically? As things	stand	now,	no	guidance	is	forthcoming. What	we,	epistemologists	interested	in	disagreement,	should	like	to	have	is	an approach	to	disagreement	that	incorporates	not	only	gradations	in	confidence,	which	our norms	already	do,	but	also	gradations	in	competence,	which	our	norms	do	not. I	define	some	terms. The	goal	of	the	epistemology	of	disagreement	debate	is	to identify	the	correct	disagreement	norm. We	shall	be	considering	a	finite	set	of	people,	V	= {v1,	v2,	.	.	.,	vn},	who	have	opinions	about	some	proposition	X. Although	philosophers typically	consider	only	the	special	case	of	n	=	2,6	we	shall	not	so	limit	ourselves. Associated	with	each	vi	is	a	confidence,	ci	∈	(0,	1). We	interpret	a	person's confidence	in	X	as	follows:	As	ci	approaches	1	(0),	vi	approaches	certainty	that	X	is	true (false). Although	within	the	philosophical	literature	one	more	commonly	sees	ci	∈	[0,	1],	it is	better	to	use	the	open	interval. This	becomes	relevant	for	technical	reasons	later	on,	but 6	Exceptions	include	Gardiner	(2014)	and	Mulligan	(2015). 6 it	also	makes	more	sense	from	the	Bayesian	point-of-view. To	say	that	one	has	a confidence	of	1	(0)	is	to	say	that	no	future	evidence	could	shake	one's	belief	that	the proposition	at	issue	is	true	(false). That	is	wrong	even	in	the	strongest	real-world	contexts. Even	our	beliefs	about	putative	necessary	truths	might	one	day	be	undermined. Sometimes we	discover	that	a	mathematical	"proof"	we	thought	was	rigorous	is	in	fact	subtly	flawed. Let	us	suppose,	without	loss	of	generality,	that	v1	is	the	one	trying	to	decide	whether or	not	to	modify	her	belief	in	X	in	light	of	disagreement. 2. Bayesian	belief	revision It	is	surprising	to	me	that	little	effort	has	been	devoted	to	tackling	the	core	problem	of	the epistemology	of	disagreement-how	to	revise	one's	belief	given	the	beliefs	of	others-with tools	from	Bayesian	analysis. I	am	aware	of	only	two	broad	attempts	along	these	lines	in the	philosophical	literature. Typically,	the	relevant	papers	argue,	often	convincingly,	that some	disagreement	norm	is	incompatible	with	a	Bayesian	principle	or	its	overarching philosophy. The	goal	of	this	paper,	in	contrast,	is	to	provide	a	general	Bayesian	solution	to the	disagreement	problem. First,	there	are	a	number	of	persuasive	arguments	that	concilationism,	usually understood	in	its	"equal	weight"	form	(the	idea,	roughly,	that	when	two	epistemic	peers disagree,	the	rational	thing	for	them	to	do	is	to	"meet	in	the	middle")	is	incompatible	with Bayesianism-because,	for	example,	it	violates	conditionalization. (Cf.	Isaacs	2019,	Jehle and	Fitelson	2009,	Lasonen-Aarnio	2013,	Levinstein	2015,	and	Shogenji	Manuscript.) Second,	there	is	Easwaran	et	al.'s	(2016)	derivation	of	a	disagreement	norm	they call	"Upco"	("Updating	on	the	credences	of	others"). This	is	the	best-developed	Bayesian 7 approach	to	disagreement	in	the	epistemological	literature,	and	I	shall	consider	it	in	some detail	in	the	next	section. As	we	shall	see,	not	only	does	a	proper	Bayesian	strategy	provide	a	means	for updating	confidences	given	others'	confidences	(indeed,	this	is	Bayesianism's	raison d'être),	it	satisfies	the	three	desiderata	mentioned	in	the	introduction:	It	deals	with multiple	epistemic	interlocutors;	provides	guidance	for	updating	beliefs	given disagreement	from	interlocutors	of	whatever	competences;	and	it	ensures	that	facts	about epistemic	dependence	influence	beliefs	appropriately. Moreover,	the	Bayesian	strategy	provides	steadfast	norms	in	those	cases	in	which, intuitively,	it	is	rational	to	"stick	to	your	guns". And	it	provides	conciliatory	norms	for those	cases	in	which	belief	revision	seems	right. For	a	good	Bayesian,	sometimes steadfasters	like	Kelly	are	correct;	other	times,	conciliationists	like	Christensen	are	on	the right	side	of	things. We	should	not	impose	either	norm	on	scenarios	for	which	it	is inappropriate-even	though	philosophers	often	try	to	do	just	that. Now,	some	epistemologists,	like	Richard	Feldman	(2009),	have	argued	informally against	a	"one	size	fits	all"	approach	to	disagreement,	often	as	part	of	a	"total	evidence" approach: I	am	not	endorsing	universal	principles	asserting	that	it	is	never	reasonable	to maintain	one's	belief,	I	am	arguing	that	evidence	of	peer	disagreement	is	evidence against	one's	original	belief. It	is	consistent	with	this	that,	in	many	cases,	it	is	strong evidence	against	one's	original	belief,	strong	enough	to	render	that	belief	no	longer justified.	(Feldman	2009:	304) And	Kelly	(2010)	discusses	coming	to	terms	with	multiple	epistemic	interlocutors	and dependence	through	a	total	evidence	approach. For	adherents	of	this	approach,	perhaps 8 this	paper,	and	the	Bayesian	paradigm	more	broadly,	can	provide	a	useful	formal framework	for	determining	how	one's	total	evidence	should	bear	on	a	given	hypothesis. Our	key	move	will	be	for	v1	to	regard	her	interlocutors'	judgments	as	random variables,	the	values	of	which-namely,	c2,	.	.	.,	cn-are	revealed	to	her	by	v2,	.	.	.,vn. Then,	v1 treats	c2,	.	.	.,	cn	as	data	relevant	to	X	on	which	she	can	update	c1. Obviously	this	is	a	very different	methodology	than	the	typical,	a	priori	approach	described	in	the	introduction. But	it	is	also	different	than	Easwaran	et	al.'s	Upco,	which	involves	no	probabilistic modelling	at	all,	but	is,	rather,	a	function	which	falls	out	as	a	special	case	of	Bayes's	Law (under	the	assumption	that	epistemic	interlocutors	are	independent	conditional	on	X	).7 We	begin	by	noting	that	confidence	is	typically	interpreted	as	subjective	probability (cf.	§1): cw = Prw(X) (2.1) Note	that	since	X	and	~X	are	mutually	exclusive	and	jointly	exhaustive	of	the	sample	space (the	proposition	is	either	true	or	it	is	false,	and	not	both),	(1	–	ci)	is	vi's	subjective probability	that	~X. Next,	consider	the	special	case	of	n	=	2.8 v1	and	her	interlocutor,	v2,	are	considering X	(e.g.	the	defendant	is	guilty). v2	reports	a	confidence	of	c2	in	X. Denoting	v1's	post- 7	See	Morris	1974	for	a	derivation	and	discussion	of	Upco. 8	The	foundational	work	here,	underlying	what	follows,	was	done	by	Morris	(1974). 9 disagreement	confidence	by	c′1,	Bayes's	Law	provides	unambiguous	guidance	to	v1	about how,	rationally,	she	should	proceed:9 c{| = Pr(X	|	c~) = c| × Pr(c~	|	X) c| × Pr(c~	|	X) + (1 − c|) × Pr(c~	|	~X) (2.2) Notice	how	the	disagreement	problem	reduces	to	specification	of	the	likelihoods Pr(c2	|	X)	and	Pr(c2	|	~X). That	is,	to	reach	a	maximally	justified	belief,	v1	must	answer	two questions:	(1)	"What	is	the	probability	that	my	interlocutor	would	say	what	he	did	(viz.	c2) if	the	state	of	the	world	is	such	that	X	(e.g.	the	defendant	is	in	fact	guilty)?" And	(2)	"What is	the	probability	that	my	interlocutor	would	say	what	he	did	if	the	state	of	the	world	is such	that	~X	(the	defendant	is	innocent)?" One	can	see	immediately	how	the	Bayesian	approach	satisfies	the	desideratum	of gradations	in	competence	(which,	again,	dominant	epistemological	approaches	do	not); these	are	incorporated	into	the	likelihood	functions	themselves. For	example,	take	the	special	case	in	which	v1's	interlocutor	is	not	only	her epistemic	superior	but	is	epistemically	infallible	(and	v1	knows	this):	v2	reports	c2	=	0	if	X is	false,	c2	=	1	if	X	is	true,	and	nothing	else. Then	if	v1	hears	"0"	from	v2,	c′1	=	0. (Because Pr(c2	=	0	|	X)	=	0.) If	v1	hears	"1"	from	v2,	then	c′1	=	1. (Because	the	second	term	on	the RHS	of	equation	(2.2)	becomes |  .) Of	course,	in	general	v1	will	have	to	specify	likelihood functions	that	cover	the	entire	domain	of	c2-from	0	to	1. But	the	principle	is	the	same. 9	I	suppress	the	subscript	on	the	probability	function	here	on	out. I	am	also	going	to	abuse notation	a	little,	using	ci	to	refer	both	to	the	realization	of	a	random	variable	and	to	the random	variable	itself. 10 Another	alluring	feature	of	this	approach	is	that	it	satisfies	the	second	desideratum: It	easily	generalizes	to	n	of	arbitrary	size. Again,	by	Bayes's	Law: c{| = Pr(X	|	c~, . . . , c) = c| × Pr(c~, . . . , c	|	X) c| × Pr(c~, . . . , c	|	X) + (1 − c|) × Pr(c~, . . . , c	|	~X) (2.3) The	likelihoods	can	be	put	into	more	manageable	form. By	the	definition	of	conditional probability,	Pr(A,	B	|	C)	=	Pr(A	|	B,	C)	×	Pr(B	|	C). Thus, Pr(c~, . . . , c	|	X) = Pr(c	|	c~, . . . , c|, X) × Pr(c~, . . . , c|	|	X) (2.4) and Pr(c~, . . . , c	|	~X) = Pr(c	|	c~, . . . , c|, ~X) × Pr(c~, . . . , c|	|	~X) (2.5) The	second	terms	on	the	right-hand-sides	of	equations	(2.4)	and	(2.5)	can	be	expanded	in	a similar	way. Doing	that,	and	substituting	into	equation	(2.3),	yields: c{| = c| × ∏ Pr(cw	|	c~, . . . , cw|, X)w~ c| × ∏ Pr(cw	|	c~, . . . , cw|, X)w~ + (1 − c|) × ∏ Pr(cw	|	c~, . . . , cw|, ~X)w~ (2.6) Again	the	disagreement	problem	is	one	of	specifying	likelihoods. Now,	especially when	it	comes	to	multiple	interlocutors,	this	may	be	an	onerous	task. It	requires	that	v1 detail	her	interlocutors'	intelligence,	susceptibility	to	bias,	and	other	epistemic	features. As 11 a	result,	in	the	1970s	and	1980s,	decision	theorists	proposed	a	number	of	coarse-grained models	for	real-world	use.10 It	seems	that	philosophers	are	unaware	of	this	body	of	work, despite	the	relevance	for	the	epistemology	of	disagreement	suggested	by	some	of	its	titles (e.g. French's	"Updating	of	Belief	in	the	Light	of	Someone	Else's	Opinion"). I	want	to describe	one	such	model	here,	to	illustrate	the	applicability	of	the	Bayesian	modelling approach	to	our	contemporary	disagreement	debate.11 The	first	model	was	developed	by	Peter	Morris	(1983)	and	Robert	Winkler	(1968). I	have	chosen	it	because	it	is	simple	and	because	it	incorporates	the	two	desiderata	just described. The	idea	underlying	the	model	is	to	treat	individuals'	beliefs	about	some	event,	like a	defendant's	guilt,	as	Beta-distributed	random	variables. The	reported	cis	are	regarded	as the	means	of	those	distributions. The	beta	distribution	is	appropriate	because	we	are seeking	to	represent	a	distribution	of	probabilities. And	it	yields	a	new,	post-disagreement distribution	by	summing	over	parameters	that	define	the	individual	distributions. The mean	of	that	new	distribution	may	then	be	adopted	as	the	post-disagreement	confidence. This	provides	the	following	disagreement	norm	(I	omit	the	derivation	here,	it	may be	found	in	the	cited	work): c{| = wwcw  w| (2.7) 10	Good	summaries	of	this	literature	may	be	found	in	Clemen	and	Winkler	1990	and	1999 and	French	1985. 11	Other	models,	not	described	here,	include	those	of	Clemen	and	Winkler	(1987),	French (1981),	and	Lindley	(1985). 12 where ww  w| = 1 (2.8) This	is	a	simple	weighted	average,	where	the	opinions	of	the	vis	are	granted	influence	in accordance	with	v1's	view	of	their	relative	competence. In	the	special	case	in	which	v1 regards	them	all	as	epistemic	peers,	weights	are	set	to	|  . Note	two	things. First,	our	two	desiderata	are	incorporated-the	wis	provide	for differences	in	competence,	and	as	many	interlocutors	as	v1	likes	may	offer	their	opinions on	X	for	v1's	consideration. Second,	this	norm	is	essentially	the	same	as	conciliationism's equal	weight	view,	albeit	more	general.12 An	example	of	the	norm	in	action	may	be	helpful. Consider	the	"complicated	nexus" problem	described	in	§1:	v1	is	trying	to	form	a	maximally	justified	belief	about	X	in	light	of disagreement	from	10	epistemic	interlocutors-some	peers,	some	superiors,	and	some inferiors. Under	Morris	and	Winkler's	model,	v1	ought	to	do	two	things:	(1)	obtain	reports from	v2,	.	.	.,	v11	regarding	their	confidences	in	X;	and	(2)	assess	the	relative	competences	of v1,	.	.	.,	v11. Suppose	that	this	yields: 12	It	is	also	essentially	the	same	as	the	(non-Bayesian)	"linear	opinion	pool"	(see,	e.g.,	Stone 1961). 13 Epistemic	agent Reported	confidence Relative	competence v1 0.5 0.1 v2 0.7 0.16 v3,	v4 0.3 0.14 v5,	v6 0.8 0.1 v7 0.4 0.1 v8,	v9,	v10,	v11 0.2 0.04 Then	c′1	=	0.48. The	computation	is	straightforward,	as	is	the	solicitation	of	confidence information	from	v2,	.	.	.,	v11. The	only	challenge	for	v1	is	assessing	relative	competence. Now	this	norm	has	a	serious	drawback,	a	drawback	which	plagues	theories	in	the epistemology	of	disagreement	but	which	has	hardly	been	attended	to	in	the	disagreement literature.13 Namely,	it	implicitly	endorses	the	idea	that	if	there	is	no	disagreement	to	begin with	(i.e.	if	all	epistemic	interlocutors	share	the	same	confidence),	then	the	postdisagreement	confidence	should	simply	equal	the	shared,	pre-disagreement	confidence. This	is	sometimes	known	as	the	"unanimity	condition",	but	it	is	a	bug,	not	a	feature,	of	a theory,	even	in	the	special	case	of	agreement	with	epistemic	peers. Whether	the	unanimity	condition	should	hold	or	not	turns	on	whether	there	exists dependence	between	(1)	our	epistemic	agent	and	her	interlocutors,	and	(2)	the interlocutors	themselves. For	reasons	I	shall	now	give,	any	viable	disagreement	norm	must be	capable	of	modelling	such	dependence. 13 But	see	Barnett	Forthcoming,	Dietrich	2010,	and	Easwaran	et	al.	2016. 14 3. Disagreement	and	dependence v1,	v2,	and	v3	are	professional	horseplayers,	trying	to	form	a	maximally	justified	belief	in Judy's	Lightning	will	win	the	race. They	regard	each	other	as	epistemic	peers,	and	they have	good	evidence	that	they	are	in	fact	peers:	They've	been	betting	on	races	for	a	long time,	and	have	had	the	same	success	in	picking	winners. But	v1,	v2,	and	v3	are	not	identical. In	particular,	v1	and	v2	share	the	same handicapping	methodology:	They	rely	on	how	horses	appear	the	morning	of	the	race. v3, on	the	other	hand,	has	developed	a	mathematical	system	for	predicting	winners	on	the basis	of	diverse	historical	data. Nevertheless,	these	two	methodologies	appear	equally good;	v1,	v2,	and	v3	win	with	equal	frequency. Naturally,	v1	and	v2	tend	to	win	together, because	they	share	the	same	methodology. In	contrast,	v3	sometimes	wins	when	v1	and	v2 lose	(and	vice	versa). Suppose	that,	before	this	race,	each	reports	the	same	confidence,	g,	in	Judy's Lightning	will	win	the	race. According	to	standard	disagreement	theory,	v1	should	not change	her	confidence	in	this	proposition	(obviously	g	is	unmodified	under	steadfastness, and	it	implicitly	stays	the	same	under	most	variants	of	conciliationism,	too-the	arithmetic average	of	{g,	g,	g}	is	g).14 Two	questions	to	consider:	(1)	Should	v1's	confidence	in	Judy's	Lightning	will	win the	race	be	unchanged	by	her	interaction	with	v2	and	v3,	given	that	the	three	do	not disagree	about	the	probability	of	this	event? (2)	If	v1's	post-"disagreement"	confidence 14	Although	see	Elga	2013. 15 should	not	remain	the	same,	should	v2	and	v3	exert	the	same	epistemic	influence	on	v1 when	she	modifies	her	judgment? The	answer	to	(1)	is,	pace	current	theory,	"no". Even	though	v3	is	an	epistemic	peer, equally	good	at	getting	to	the	truth	of	Judy's	Lightning	will	win	the	race,	v3	is	different	than v1. And	that	difference	means	that	their	assessments	are	at	least	partially	independent. If she	is	rational,	v1	will	recognize	that	independence	and	use	it	appropriately	to	modify	her confidence. For	example,	if	g	=	0.8,	then	v1's	post-"disagreement"	confidence	will	be greater	than	0.8. The	knowledge	that	a	different	handicapping	methodology,	even	if	no better	than	your	own,	is	also	highly	confident	that	it	has	picked	a	winner	provides	you	with greater	reason	to	believe	that	you	have	got	things	right. As	far	as	(2)	is	concerned,	v2	and	v3	should	certainly	not	exert	the	same	epistemic influence	over	v1. Indeed,	because	v2	is	more-or-less	a	copy	of	v1,	c2	is	not	a	useful	datum when	c1	=	c2	=	0.8. That	is	precisely	what	v1	expects	to	hear	prior	to	her	interaction	with v2,	and	so	conditionalizing	upon	it	should	not	affect	her	prior	judgment. Because	there	is perfect	dependence	between	v1's	judgment	and	v2's	judgment,	and	v1	knows	this,	there	is nothing	to	be	gained	epistemically	through	interaction	with	v2	in	this	case. The	possibility	of	dependence	between	epistemic	agents	illuminates	the	limitations of	Easwaran	et	al.'s	Upco: c{| = c| × c~ c| ×	c~ + (1 − c|) × (1 − c~) (3.1) 16 While	this	norm	does	allow	for	violations	of	unanimity	(what	Easwaran	et	al.	call "synergy")-as	it	sometimes	should-it	fails	to	account	for	dependence,	as	in	the	example just	given. It	would	allow	v2	and	v3	to	exert	the	same	epistemic	influence	over	v1. And	that, as	we	have	seen,	would	be	a	mistake. Under	Upco,	if	v1	and	v2	interact	when	both	have	a	confidence	of	g	=	0.8,	then	the post-"disagreement"	confidence	is	0.94. The	same	result	is	reached	if	v1	and	v3	interact. But	such	a	high	post-"disagreement"	confidence	is	only	plausible	in	the	latter	case. Because v2	provides	no	independent	insight,	v1	should	maintain	a	confidence	of	0.8	after	interacting with	v2	alone. Upco	fails	to	account	for	this	important	difference. To	their	credit,	Easwaran	et	al.	recognize	that	Upco	is	limited	in	this	way. But	they do	not	grapple	with	"the	general	question	of	how	to	deal	with	peer	update	when	we	think there	are	correlations	between	one's	peers"	(p.	31),	suggesting,	instead,	that	Upco's	use	be restricted	to	scenarios	of	disagreement	involving	perfect	independence	between	epistemic agents. But	I	stress	that	this	is	not	a	minor	limitation;	it	is	a	loss	of	generality	which renders	the	norm	useless	for	real-world	use. It	is	a	struggle	to	imagine	a	real-world scenario	in	which	perfect	independence	holds. Two	philosophers	disagree	about	the morality	of	some	new	law? Consider	all	the	common	training	they	receive. Two	jurors disagree	about	a	defendant's	guilt? Think	of	the	common	evidence,	presented	at	trial,	on which	their	judgments	rely. Two	weathermen	disagree	about	whether	it	will	rain tomorrow? Note	that	both	base	their	judgments	on	the	same	radar	data. Easwaran	et	al.	do	offer	a	conjecture	about	how	Upco	might	handle	dependence- but	I	do	not	think	that	it	will	work. They	suggest	that	we	assign	to	each	term	in	Upco	an exponent	representing	the	weight	of	that	interlocutor's	opinion,	where	the	weight	assigned 17 to	the	c1	terms	is	set	to	1,	and	the	weight	of	"fully	independent	peers"	is	likewise	1. Then,	if peers'	opinions	are	correlated,	we	reduce	the	weights	assigned	to	those	peers. For example,	if	two	peers	are	perfectly	correlated,	then	they	should	be	treated	by	Upco	as	one "fully	independent	peer"	by	assigning	each	a	weight	of	one-half. At	the	same	time,	the	exponents	are	supposed	to	handle	gradations	in	competence; indeed,	this	is	why	they	are	introduced	in	the	first	place. (Again,	Easwaran	et	al.	explicitly avoid	in-depth	analysis	of	dependence	in	their	paper.) For	example,	"if	we	raise	[ci]	to	the power	of	2,	we	are	treating	[vi's]	report	.	.	.	as	equivalent	to	the	report	of	two	independent peers	with	weight	1	reporting	that	credence."	(Easwaran	et	al.	2016:	30). Here's	the	problem. Suppose	that	I	think	some	proposition	is	false. My	epistemic superior	thinks	that	it	is	true	with	confidence	c2. Whatever	else	we	might	want	to	say about	my	post-disagreement	confidence,	it	should	not	be	greater	than	c2. But	we	can choose	values	for	this	Upco	variant	that	delivers	such	a	result: c{| = 0.4| × 0.7~ 0.4| × 0.7~ + (1 − 0.4)| × (1 − 0.7)~ = 0.78 (3.2) To	deal	with	dependence,	I	suggest	we	look	elsewhere. 4. A	better	way We	ought	to	model	disagreement	in	a	way	that	explicitly	takes	into	account	correlation between	epistemic	interlocutors. Our	models	should	not	hew	to	unanimity;	they	should 18 yield	a	post-disagreement	confidence	which	turns	in	part	on	pre-disagreement dependence. An	excellent	starting	point	would	be	the	work	of	Christian	Genest	and	Mark Schervish	(1985),	who	recognize	that	even	though	ideally	rational	agents	should	employ equation	(2.6),	that	formula	is	unhelpful	for	real-world	use. This	is	because	it	requires	that v1	specify	Pr(c~, . . . , c	|	X)	and	Pr(c~, . . . , c	|	~X),	each	of	which	are	joint	distributions	over n	–	1	random	variables. Genest	and	Schervish	show	that	so	long	as	(1)	v1	can	specify	the	means	of	the marginal	distributions	and	(2)	plausible	consistency	conditions	hold	(more	on	this	below), then	a	maximally	justified	belief	is	given	by: c{| = c| +λw(cw − μw)  w~ (4.1) where	μi	is	the	mean	of	i's	confidence	distribution	(as	specified	by	v1),	and	the	λws	are	the coefficients	of	linear	regression	of	X	on	the	vector	(c2,	.	.	.,	cn). To	reiterate:	c1	is	v1's	pre-disagreement	confidence	in	the	proposition	at	issue. ci	is the	stated	confidence	of	vi	in	the	proposition. v1	says	to	herself,	for	each	of	her	epistemic interlocutors,	"My	interlocutor	might	report	many	possible	confidences-from	very	close to	0	to	very	close	to	1. Some	values	are	more	likely	than	others. What	is	the	mean confidence	that	I	expect	to	hear?" That	is	μi. 19 Note	that	if	v1's	interlocutor	reports	what	she	expects	him	to	(that	is,	if	cw = μw),	then her	confidence	is	unchanged	by	their	interaction. This	makes	sense	from	the	Bayesian point-of-view;	the	interlocutor's	judgment	was	already	baked	into	c1. Suppose,	for	example,	that	the	government	announces	a	tax	hike	on	the	rich. I	have a	confidence	of	c1	=	0.8	that	this	policy	is	just. My	colleague	down	the	hall	is	a	conservative, so	I	expect	he'll	find	the	policy	to	be	unjust. Perhaps	I	think	he's	most	likely	to	report	a confidence	of	0.2	(if	c2,	qua	random	variable,	is	Normal,	the	mean	equals	the	mode). Now	if I	ask	him	what	he	thinks	about	the	policy,	and	he	tells	me,	as	I	expect	he	will,	that	it	is unjust,	my	confidence	should	be	little	affected. I	already	knew	that. But	if	this	conservative agrees	with	me	that	the	tax	hike	is	just-well,	that	is	surprising. It	is	genuinely	new	and useful	information,	and	so	it	gives	me	grounds	to	increase	my	confidence	that	the	policy	is	a just	one. Of	course,	technically	c2	cannot	be	Normal-because	the	support	of	the	normal distribution	is	(−∞,∞)	and	c2	requires	a	support	of	(0,	1). One	can	truncate	the	normal distribution	to	(0,	1),	but	then	it	is	no	longer	generally	true	that	its	mean	will	equal	its mode. Nevertheless,	for	the	purpose	of	on-the-fly	belief	revision,	v1	would	not	go	far	wrong by	imagining	c2	distributed	"normalish"	on	(0,	1),	and	taking	its	peak	to	be	μ2. And,	if	v1 desires	a	precise	answer,	she	can	calculate	the	actual	mean	for	her	chosen	Truncated Normal	on	(0,	1). Note	that	it	will	frequently	be	sensible	for	v1	to	set	μi	=	c1. That	is,	v1	can	presume that	the	mean	realization	of	cw	(qua	random	variable)	is	what	she,	v1,	already	believes. This is	not	the	case	in	the	above	example,	because	there	I	know	that	my	opinion	about	the	policy and	my	colleague's	opinion	are	likely	to	be	opposed	given	our	political	differences. But 20 often	μi	=	c1	is	sensible. (Example:	I'm	walking	down	the	street,	the	sky	is	looking	gloomy, and	I	think	there's	a	60%	chance	of	rain. I	ask	a	passerby	what	he	thinks. It's	highly unlikely	that	he'll	say	"60%",	exactly,	but	he's	more	likely	to	say	"60%"	than	anything	else. But	if	there	were	some	evidence	of	bias-if	he	were	wearing	a	t-shirt	that	said,	"I	Hate Rain"-then	I	would	be	justified	in	taking	μ2	1	c1. The	λws	satisfy	certain	inequalities	to	ensure	that	c{|	is	a	bona	fide	probability measure,	and	can	be	interpreted	as	indicators	of	how	much	independent	insight	vi provides,	above	and	beyond	what	was	provided	by	{v1,	.	.	.,	vi-1}. Namely: max  λwμw c| ,  w~  λw(1 − μw) (1 − c|)  w~  ≤ 1 (4.2) (N.B.	here	we	assume	that	the	λws	are	positive. For	the	most	general	cases,	which	could include	negative	λws,	there	are	2n	inequalities.) The	intuition	is	that	as	some	λw	gets	close	to 1,	c1	must	be	kept	close	to	μw. It	does	not	make	sense	for	v1	to	believe,	pre-disagreement, that	both	(1)	X	is	very	improbable,	and	(2)	vi,	who	has	great	insight	into	the	truth	of	X, likely	believes	that	X	is	very	probable. To	illustrate	equation	(4.1)	in	action,	let	us	consider	the	horse	racing	case,	with	c| = c~ = c = 0.80,	as	above. We	have	three	epistemic	peers	who	report	the	same	confidence in	the	proposition. Here,	v1	should	certainly	take	μ~ = c| = 0.8	(they	are	copies). μ,	in contrast,	will	be	somewhat	less	than	this-say,	0.4. Of	course,	the	distribution	of	c,	in	v1's eyes,	will	depend	on	many	things. But	surely	its	mean	will	be	less	than	v|′s	0.80,	which	is 21 extraordinarily	high	in	the	context	of	horse	racing. (A	horse	that	goes	off	at	0.3	to	win	is considered	a	heavy	favorite.) v1	might	then	evaluate	λ~ = 0.01	and	λ = 0.30,	representing	little	possible epistemic	help	from	v2	and	significant	possible	help	from	v3. These	values	satisfy	the necessary	inequalities,	and	they	yield	a	disagreement	norm	under	which	(1)	v2	exerts	no epistemic	influence	over	v1,	and,	in	contrast,	(2)	v3's	judgment	does	provide	reason	for	v1	to become	more	confident	in	her	judgment-even	though,	I	stress,	v1	and	v3	do	not	disagree about	the	proposition	at	dispute. In	particular,	for	the	values	given,	v1's	confidence	rises from	0.80	to	0.92. This	model	also	prevents	the	bad	result	of	equation	(3.2). Again,	Upco	with dependence	fails	for	c| = 0.4	and	c~ = 0.7. But	here,	with	μ~ = c| = 0.4,	we	get: c{| = 0.4 + λ~(0.3) (4.3) One	may	see	that	the	troublesome	inequality,	c{| > c~,	would	arise	if λ~ > 1. But consistency	conditions	require	that max  0.4 0.4 − 1 , 0.4 − 1 0.4  ≤ λ~ ≤ min  0.4 0.4 , 1 − 0.4 1 − 0.4  (4.4) The	latter	inequality	ensures	that	c{| ≤ c~. 22 Notice	how	this	approach	satisfies	our	three	desiderata. First,	multiple	epistemic interlocutors	are	generally	admissible,	and	the	oft-considered	scenarios	in	which	n	=	2	are simply	dealt	with	as	special	cases. Second,	we	incorporate	differences	in	competence	via	the	λws. To	take	the	extreme cases,	suppose	v1	believes	that	v2	is	epistemically	infallible. Then	v1	will	set	λ~	=	1,	because v2's	judgment	is	perfectly	correlated	with	the	truth. Then	c{| = c| + (c~ − μ~). Since cw, μw ∈ (0, 1),15	consistency	conditions	require	that	c| = μ~. Therefore,	c{| = c~. v1	entirely abrogates	her	judgment	and	adopts	v2's	opinion,	as	one	would	expect. Next,	suppose	that	v1	believes	that	v2	is	epistemically	useless;	v2's	judgment	is utterly	uncorrelated	with	the	truth. Then	λ~ = 0	and	c{| = c|. v1	maintains	her	judgment in	the	face	of	disagreement	from	this	interlocutor. And	less	extreme	cases	of	epistemic superiority	and	inferiority	are	dealt	with	accordingly.16 The	λws	also	allow	us	to	incorporate	the	third	desideratum:	the	possibility	of dependence. We	have	already	seen	examples	of	this,	but	I	would	like	to	point	out	here	two possible	sources	of	dependence	and	how	each	gets	handled. First,	there	may	be	dependence	between	v1	and	her	interlocutor(s). One	imagines two	weathermen	(§3),	each	of	whom	makes	a	prediction	about	rain	on	the	basis	of,	and only	on	the	basis	of,	weather	data	which	they	both	possess. Recall	the	interpretation	of	λw as	how	much	independent	insight	vi	provides,	above	and	beyond	what	was	provided	by	{v1, .	.	.,	vi-1}. In	this	case,	even	if	v2	is	v1's	epistemic	superior	(being	better	when	it	comes	to 15 Note that Genest and Schervish derive equation (4.1) under the assumption that the support of the cis is [0, 1]. However, the posterior is the same in either case. 16	West	and	Crosse	(1992)	provide	a	useful	discussion	of	how	to	select	the	λws. 23 "general	epistemic	virtues	such	as	intelligence,	thoughtfulness,	and	freedom	from	bias"- §1),	v2	provides	no	independent	insight,	and	so	λ~	is	set	equal	to	0,	and	so	c{| = c|.17 Second,	there	may	be	dependence	between	interlocutors. Suppose,	as	a	variant	on the	horse	racing	case,	that	it	is	v2	and	v3,	not	v1	and	v2,	who	use	the	same	handicapping methodology. Then	λ~	will	be	positive,	because	it	is	useful	for	v1	to	know	that	one	of	v2	and v3	agree	with	her	about	the	winner. But	it	is	not	useful	(given	that	knowledge)	to	know that	both	v2	and	v3	agree	with	her. So	λ	will	be	set	to	0. v1	will	modify	her	belief	only	in light	of	one	interlocutor's	opinion. 5. Conclusion Since	the	beginning	of	the	disagreement	debate,	epistemologists	have	presented	a	number of	real-world	cases	of	disagreement	which	yield,	variously,	conciliatory	and	steadfast intuitions. The	typical	reaction	has	been	to	endorse	one	set	of	intuitions	over	the	other	and then	force	a	theoretical	structure,	conforming	to	that	set,	onto	the	other	scenarios. For	a	Bayesian	like	me,	this	is	misguided. When	a	person	faces	disagreement	from others-whether	they	be	peers,	superiors,	inferiors,	or	some	combination	thereof-she should	specify	likelihood	functions	that,	in	her	best	judgment,	accurately	model	the circumstance	she	finds	herself	in. There	is	no	"one	size	fits	all"	disagreement	norm, 17 This	is	an	idealized	example. We	are	assuming	that	there	is	no	possibility	of,	say, misreading	the	radar	data. If	there	were,	λ~	>	0	would	be	appropriate. We	are	also ignoring	that	the	move	from	raw	radar	data	to	weather	prediction	requires	judgment	and experience-and	thus	v2's	assent	carries	useful	information	to	v1. Generally,	any	time	that v2	can	serve	as	a	check	on	v1's	work,	v1	will	wish	to	incorporate	v2's	judgment	to	some degree. 24 because	each	real-world	case	of	disagreement	displays	different	features:	competence, confidence,	bias,	dependence,	and	all	the	rest. I	am,	therefore,	convinced	that	a	Bayesian	modelling	approach	to	the epistemological	problem	of	disagreement,	as	expressed	in	equations	(2.2)	(for	the	single interlocutor	case)	and	(2.6)	(for	the	multiple	interlocutor	case)	is	the	only	promising	one. As	we	have	seen,	this	approach	incorporates	features	which	are	not	only	alluring but	apparently	necessary. The	toy	examples	in	the	literature,	involving	disagreement	with a	single	epistemic	peer,	do	not	get	at	what	is	the	real	import	of	disagreement	research: Helping	human	beings,	in	all	their	diversity,	work	together	and	overcome-indeed harness-differences	of	opinion. Under	the	approach	I	recommend,	we	are	no	longer restricted	to	disagreement	with	a	single	person,	nor	to	disagreement	with	epistemic	peers. The	ubiquitous	fact	of	dependence	between	judgments	is	handled. Bias	may	be	explicitly modelled. And	so	on. Indeed,	even	further	generality	can	be	incorporated. If,	for	example,	one's	epistemic interlocutors	specify	not	just	a	single	confidence	but	an	entire	distribution	over	(0,	1),	that can	be	handled	as	well.18 And,	as	we	have	seen,	steadfast	and	conciliatory	disagreement	norms	fall	out naturally	as	special	cases	of	equations	(2.2)	and	(2.6). When	a	real-world	scenario	evokes a	conciliatory	intuition,	our	model	can	provide	a	"conciliatory"	belief	revision	function;	and when	a	scenario	evokes	a	steadfast	intuition,	it	can	provide	a	"steadfast"	function. To	make this	plain,	let	us	apply	the	model	of	§4	to	two	prominent	scenarios	from	the	epistemological 18	See	Clemen	and	Winkler	1999,	Genest	and	Zidek	1986,	and	Winkler	1981. 25 literature,	one	of	which	is	supposed	to	support	conciliationism,	and	the	other, steadfastness. First,	consider	"Restaurant	Tip": Suppose	that	five	of	us	go	out	to	dinner. It's	time	to	pay	the	check,	so	the	question we're	interested	in	is	how	much	we	each	owe. We	can	all	see	the	bill	total	clearly, we	all	agree	to	give	a	20	percent	tip,	and	we	further	agree	to	split	the	whole	cost evenly,	not	worrying	over	who	asked	for	imported	water,	or	skipped	desert	[sic],	or drank	more	of	the	wine. I	do	the	math	in	my	head	and	become	highly	confident	that our	shares	are	$43	each. Meanwhile,	my	friend	does	the	math	in	her	head	and becomes	highly	confident	that	our	shares	are	$45	each. How	should	I	react,	upon learning	of	her	belief?	(Christensen	2007:	193) According	to	Christensen,	"it	seems	quite	clear	that	I	should	lower	my	confidence	that	my share	is	$43"	(2007:	193). And	surely	that	intuition-that	a	rational	person	will	lose confidence	in	my	share	is	$43-is	widely	shared. Let	us	model	this	scenario. First,	c1	will	be	something	like,	say,	0.8. v1	must	round off	the	bill	to	the	nearest	whole	number	and	divide	that	by	five. These	are	not	difficult operations	for	an	educated	adult,	but	it	is	certainly	possible	to	make	a	mistake. Second,	it makes	sense	for	v1	to	take	μ~ ≈ c|	(see	§4). If	these	are	five	philosophers	we're	talking about,	they'll	be	aware	of	the	possibility	of	making	an	arithmetic	error,	and	so	they	are likely	to	report	a	high,	but	not	perfect,	confidence	in	the	proposition	at	issue. Third,	and finally,	there	is	the	question	of	specifying	λ~. By	the	consistency	conditions	(and	assuming non-negative	correlation),	0 ≤ λ~ ≤ 1. v1	is	free	to	select	the	amount	of	epistemic	weight she	wishes	to	assign	to	v2's	judgment. If	v1	wishes	to	treat	v2	as	a	pure	epistemic	peer,	as that	term	is	typically	defined	and	as	the	case	is	typically	interpreted,	then	she	sets	λ~ = 0.5. This	yields	the	following	disagreement	norm: c{| ≈ 0.5c| + 0.5c~ (5.1) 26 which	is	conciliationism's	equal	weight	view-precisely	the	norm	that	has	been	regarded as	appropriate	for	"Restaurant	Tip". Next,	consider	Jennifer	Lackey's	"Elementary	Math": Harry	and	I,	who	have	been	colleagues	for	the	past	six	years,	were	drinking	coffee	at Starbucks	and	trying	to	determine	how	many	people	from	our	department	will	be attending	the	upcoming	APA. I,	reasoning	aloud,	say:	"Well,	Mark	and	Mary	are going	on	Wednesday,	and	Sam	and	Stacey	are	going	on	Thursday,	and,	since	2+2=4, there	will	be	four	other	members	of	our	department	at	that	conference." In response,	Harry	asserts:	"But	2+2	does	not	equal	4." Prior	to	this	disagreement, neither	Harry	nor	I	had	any	reason	to	think	that	the	other	is	evidentially	or cognitively	deficient	in	any	way,	and	we	both	sincerely	avowed	our	respective conflicting	beliefs.	(Lackey	2010b:	283) The	modelling	is	straightforward. v1	must	choose	λ~,	and	so	she	asks	herself	what independent	epistemic	insight	v2	(Harry)	has	into	the	truth	of	2+2=4. And	the	answer	is, of	course,	almost	none. The	proposition	is	so	simple,	and	so	ubiquitous,	and	accessible	to v1	via	so	many	means,	that	there	is	nothing	novel	that	v2	might	say	about	it	(though	of course	he	might	be	unhelpful	in	myriad	ways,	if,	e.g.,	he	says	crazy	things-as	appears	to	be the	case	here). So	λ~ ≈ 0,	and	we	have	the	steadfast	disagreement	norm: c{| ≈ c| (5.2) There	is	the	potential	for	flexibility	here. Modify	the	case	so	that	v2	is	a	renowned philosopher,	working	on	the	foundations	of	mathematics. He	is	known	as	the	smartest	man who	has	ever	lived. v1	has	the	results	of	a	recent	psychiatric	evaluation	of	v2	which	attests to	his	competence. 27 Now	it	is	no	longer	obvious	that	v2	has	nothing	useful	to	say	about	2+2=4. Maybe he's	really	discovered	something	profound	about	arithmetic. Certainly,	a	lesson	of intellectual	history	is	that	notions	long	thought	false,	even	bizarre	("time	is	relative")	may come	to	be	acknowledged	as	absolutely	right. And	so	v1	may	take	λ~ > 0. Then,	when	v2 offers	his	opinion,	it	will	affect	v1's	confidence	in	2+2=4,	lowering	it	(if	c~ < μ~,	as	in "Elementary	Math")	or	raising	it	(if	c~ > μ~),	as	appropriate. One	final	point. I	stress	that	there	is	no	sense	in	which	the	model	of	equations	(2.78),	or	(4.1),	or	any	other	for	that	matter,	is	correct	simpliciter. Rather,	we	should	give careful	thought	to	the	epistemic	features	of	any	given	circumstance	of	disagreement (features	like	dependence	and	bias)	and	then	choose	an	appropriate	Bayesian	model. 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