Conditionalization	Does	Not	(in	general)	Maximize	Expected	Accuracy MIRIAM	SCHOENFIELD The	University	of	Texas	at	Austin	/	New	York	University mschoenfield@austin.utexas.edu Abstract: Greaves and Wallace argue that conditionalization maximizes expected accuracy. In this paper I show that their result only applies to a restricted range of cases. I then show that the update procedure that maximizes	expected	accuracy	in	general	is	one	in	which,	upon	learning	P,	we conditionalize, not on P, but on the proposition that we learned P. After proving	this	result,	I	provide	further	generalizations	and	show	that	much	of the accuracy-first epistemology program is committed to KK-like iteration principles	and	to	the	existence	of	a	class	of	propositions	that	rational	agents will	be	certain	of	if	and	only	if	they	are	true. 1.	Introduction Rational	agents	revise	their	beliefs in light	of	new	information	they	receive. But	how	should	agents	revise	their	beliefs	in	response	to	new	information?	To	state this	question	more	precisely,	it	will	be	helpful	to	think	of	information	processing	as occurring	in	two	(not	necessarily	temporal)	stages:1	First,	there	is	a	non-inferential stage at which an agent, through some non-inferential means, gains some information. We'll	call	this	exogenous	information	gaining. Metaphorically,	we	can think	of	this	stage	as	involving	the	world	'flinging'	some	information	at	the	agent. In the second stage, the agent revises her beliefs in response to the exogenous information gaining (the flinging) that took place. These are the revisions that we are interested in evaluating. Sometimes, as a result of such revisions,	the	agent	may	come	to	possess	additional	information,	in	which	case	we'll say that this information came to be possessed endogenously. For example, I	may gain the information that Gabe is at the party exogenously, and, as a result of revising	my	beliefs in response to this information, also come to (endogenously) possess	the	information	that	his	partner	Henry	is	at	the	party. 1	The	two-stage	model	is	discussed	(or	implicit)	in	much	of	the	literature	on	Bayesian	updating. See, for example,	Howson and	Urbach (1989, p.285), Jeffrey (1992, p.38), Bronfman (2014, p.872) and Miller	(forthcoming). More precisely, then, the question	we're interested in is this: how	does an ideally rational agent revise her opinions in light of the information she receives exogenously? According to	Bayesian	epistemology, rational	agents2	revise their	credences by conditionalization. Informally, conditionalizing on E involves setting your new credence in every propositions, P, to what your old credence in P was on the supposition	that	E.	Formally,	you	conditionalize	on	E	if pnew(⋅)	=	pold(	⋅	|	E) where p(A|B)	=	p(A&B)	/	p(B). Since	conditionalizing is	an	operation	performed	on	a	proposition,	thinking of conditionalizing as a way of responding to new information requires characterizing each possible body of information an agent might receive as a proposition.	Since	one	of the	aims	of this	paper is to	evaluate	an	argument for	the claim that conditionalizing is the rational response to gaining information, I will assume	for	now	(as	is	standard)	that	any	body	of	information	that	an	agent	receives exogenously can be uniquely characterized as a proposition (one that is often a conjunction	of	many	other	propositions).3 Later	we'll	see	what	happens	if	we	relax this	assumption. The	proposition that uniquely characterizes the entire body	of information the agent exogenously receives is sometimes referred to in the literature as 'the strongest	proposition	one learns'. To	emphasize	the	exogenous	aspect,	however, I will sometimes call this proposition 'the strongest proposition one exogenously learns'. For short, I will sometimes just call it 'the proposition one exogenously learns'	or	'the	proposition	one	learns'. Note	that	what	I	am	taking	as	primitive	is	the	notion	of	exogenously	gaining information.	I	am	using	the	term	'the	strongest	proposition	one	exogenously	learns' 2	Unless	stated	otherwise,	when	I	talk	about	rational	agents,	I	mean	ideally	rational	agents. I	discuss non-ideal	agents	in	§4. 3	Why	uniquely? Because if there	were	more than one	proposition that characterized the body of information	the	agent	receives,	then	the	claim	that	one	should	conditionalize	on	the	proposition	that characterizes one's new information wouldn't make sense. If one claimed that one should conditionalize on a	proposition characterizing this information, then conditionalization would no longer output a unique credence function given an agent's priors and the new information she received. Conditionalization,	then,	would	no	longer	count	as	an	update	procedure	in	the	sense	that	is necessary	for	the	arguments	under	discussion. as a technical term, which presupposes that any body of information can be uniquely characterized as the sort of thing (a proposition) that one can conditionalize	on. Conditionalization is the process of revising one's credences by conditionalizing on the strongest proposition one exogenously learns. Why think that conditionalization is a rational	way	of revising	one's credences? There are a variety	of	arguments	that	have	been	offered,4	but	the	focus	of	this	paper	will	be	an argument by Hilary Greaves and David Wallace (2006) for the claim that conditionalization	maximizes	expected	accuracy. The	Greaves	and	Wallace	argument	is	part	of	a	larger	philosophical	program that	has	been	of	increasing	interest:	accuracy-first	epistemology. The	basic	tenet	of accuracy-first	epistemology	is	that	accuracy	is	the	fundamental	epistemic	value,	and the	central	project	that	accuracy-firsters	pursue	involves	the	derivation	of	rational requirements from accuracy based considerations.5 A cluster of accuracy based arguments for rational requirements, including arguments for the requirement to conditionalize,	rely	on	the	following	claim: RATACC: The rational update procedures are those that	maximize expected accuracy	according	to	a	strictly	proper	scoring	rule. (The	terms	used	in	this	principle	will	be	defined	precisely	in	what	follows). I will argue that Greaves and	Wallace's result applies only to a restricted range	of	cases. Thus,	even	if	RATACC	is	true,	Greaves	and	Wallace's	argument	does not show that, in general, conditionalizing on the proposition one learns is the update	procedure	that	is	rational. So the question then arises: which update procedure	maximizes expected accuracy in general? I show that, in fact, what maximizes expected accuracy in general is not conditionalization, but a rule that I will call 'conditionalization*'. Conditionalization*	has	us	conditionalize	on	the	proposition	that	we	learn	P,	when	P is	the	proposition	we	learn.6	I	will	show	that	conditionalization*	happens	to	coincide 4	See,	for	example,	Teller	(1976),	Williams	(1980)	and	van	Fraassen	(1989,	p.331-7)	and	(1999). 5	For	an	overview,	see	Pettigrew	(2016). 6	I	borrow	the	term	'conditionalization*'	from	Hutchison	(1999).	Hutchison	describes	a	class	of	cases that	have	been	thought	to	pose	problems	for	conditionalization.	One	proposal	he	describes	(though does	not	commit	to)	for	how	to	deal	with	these	cases	is	to	deny	that	conditionalization	is	the	rational update	procedure. Rather,	he	proposes,	perhaps	what's	rational,	upon	learning	P,	is	conditionalizing on	the	proposition	that	we	learn	P. Defenders	of	conditionalization	have	offered	alternative	ways	of with	conditionalization	in	the	special	cases	that	Greaves	and	Wallace	consider,	but	it yields	different	results	in	all	other	cases.	So	my	central	thesis	is	the	following: Central Thesis: If RATACC is true, then the rational update procedure is conditionalization*,	and	not	conditionalization. I	will	not,	in	this	paper,	evaluate	the	merits	of	RATACC	or	the	accuracy-first	program. This	is	why	my	central	thesis	is	a	conditional	claim. After	arguing	for	this	thesis,	I	discuss	some	of	the	interesting	implications	of my results for iteration principles in epistemology. In particular, I show that if RATACC	is	true,	it	follows	that, if	we	learn	P,	we're	rationally	required	to	be	certain that	we	learned	P. I	then	show	that,	regardless	of	how	we	think	about	exogenously gaining	information,	it	follows	from	RATACC	that	there	is	a	class	of	propositions	that rational agents will be certain of if and only if they are true. Since many of the results	of the	accuracy-first	program	rely	on	RATACC, those	who	deny	these	claims cannot	accept	much	of	what	accuracy-first	epistemology	has	to	offer. 2.	Setup What does it mean to say that an update procedure maximizes expected accuracy? In	this	section	I	lay	out	the	formal	framework	that	I	will	use	to	prove	the main	result. 2.1	Accuracy	and	expected	accuracy Accuracy	is	measured	by	a	scoring	rule,	A,	which	takes	a	state	of	the	world,	s, from	a	partition	of	states, S,	and	a	credence	function	c	defined	over	S,	from	the	set	of such	credence	functions,	CS,	and maps	the	credence	function/state	pair	to	a	number between	0	and	1	that	represents	how	accurate	the	credence	function	is	in	that	state. A:	CS X S à	[0,1] Intuitively,	we	can	think	of	the	accuracy	of	some	credence	function	as	its	'closeness to the truth'. c is maximally accurate if it assigns 1 to all truths and 0 to all falsehoods. It is minimally accurate if it assigns 1 to all falsehoods and 0 to all truths. treating	the	cases	that	Hutchison	describes,	though	Hutchison	raises	worries	for	these	proposals.	My paper provides an independent argument for Hutchison's proposal that doesn't appeal to the controversial	cases	discussed	in	his	paper. If	an	agent	does	not	know	which	state	obtains	she	will	not	able	to	calculate the	accuracy	of	a	credence	function	c.	However, if	she	is	probabilistically	coherent, she will be able to calculate the expected accuracy of c. (Throughout, I will be assuming that rational agents are probabilistically coherent). The expected accuracy	of	credence	function	c	∈	CS	relative	to	a	probability	function	p	∈	CS	is: EAp(c)	=	∑	p(s)	A(c,	s) s∈S That	is,	the	expected	accuracy	of	a	credence	function	c	relative	to	p	is	the	average	of the	accuracy	scores	c	would	get in	the	different	states	that	might	obtain,	weighted by	the	probability	that	p	assigns	to	those	states	obtaining. A	strictly	proper	scoring	rule	is	a	scoring	rule	with	the	feature	that	every probability	function	maximizes	expected	accuracy	relative	to	itself. In	other	words, if	A	is	strictly	proper,	then	the	quantity: EAp(c)	=∑	p(s)	A(c,	s) s∈S is maximized when c = p. I will not argue here for the claim that our accuracy measures should be strictly proper. I	will simply assume this to be true in	what follows since the accuracy based argument for the claim that we should conditionalize (in addition to other arguments in accuracy-first epistemology7) requires strict propriety.	8	See	Greaves and	Wallace (2006), Gibbard (2008), Joyce (2009),	Moss	(2011),	Horowitz	(2013)	and	Pettigrew	(2016)	for	a	discussion	of	the motivation	for	using	strictly	proper	scoring	rules. 7	For	example,	the	argument	for	probabilism. See	Pettigrew	(2016). 8	Although	the	accuracy	based	argument for the	claim	that	conditionalization	is the	rational	update procedure	requires	strict	propriety,	it's	worth	noting	that	Greaves	and	Wallace	state	their	main	result slightly	more	generally:	rather	than	assuming	RATACC	and	that	the	scoring	rule	is	strictly	proper,	they remain	neutral	on	propriety	and	assume	that	the	rational	update	procedures	will	be	those	in	which one adopts a credence function that is recommended by a credence function yielded by an update procedure that	maximizes	expected	accuracy.	As	a result, their	main	argument	does	not show	that conditionalization	is	always	rational,	but	rather,	that	what	they	call	quasi-conditionalization	is	always rational. In their	Corollary	2, they	point	out that that if	we	assume that the	scoring	rule is strictly proper,	conditionalization	always	maximizes	expected	accuracy,	and	so	is	always	rational. It	is	also true that if	we assume that the scoring rule is strictly proper, their constraint on rational update procedures is equivalent to RATACC. In this paper, I'm interested in arguments for the claim that conditionalizing (rather than quasi-conditionalizing) is always rationally required and, for these purposes,	RATACC	and	strict	propriety	must	be	assumed. 2.2	Learning	experiences	and	update	procedures We're trying to figure out how to revise our credences in light of the exogenous information	we gain.	What exactly is involved in gaining information? Greaves	and	Wallace	don't	say	much	about	this,	and	I	too	will	remain	as	neutral	as possible. All	that	is	being	assumed	(by	Greaves	and	Wallace	and	myself)	is	that	the body of information one exogenously receives can be uniquely characterized as a proposition. Suppose	you	know	that	you're	going	to	undergo	some	experience,	E. E	might be	waking	up tomorrow,	or	arriving	at the	office. Assuming	you	are	probabilistic, for any proposition P, the set {P, ~P} is a partition of your possibility space. (A partition	of	a	probabilistic	agent's	possibility	space	is	a	set	of	propositions	that	the agent regards as	mutually exclusive and jointly exhaustive.) So the following is a partition	of	your	possibility	space:	{I	gain	some	new	information	upon	undergoing E, I don't gain any new information upon undergoing E}. We can represent this partition	as	follows: I	gain	some	new	information	upon undergoing	E. I	don't	gain	new	information	upon undergoing	E. Now consider all of the possibilities in which you gain new information upon undergoing	E. Call	these	bodies	of	information:	i1,	i2...	You	can	further	subdivide	the region	in	which	you	gain	new	information	as	follows: I gain i1 I gain i2 I	gain i3 I	gain i4 ... I	don't	gain	new	information	upon undergoing	E. Since we are assuming for now that we can uniquely characterize each possible body of information that you gain as a proposition, and we are describing the possibility	in	which	you	gain	a	body	of	information	as	a	case	in	which	you	learn	that proposition,	we	can	redescribe	the	partition	above	as	follows: I learn X1 I learn X2 I learn X3 I	learn X4 ... I	don't	gain	new	information	upon undergoing	E. (Recall that 'I learn	Xi' is short for:	Xi is the strongest proposition I exogenously learn.) We'll let L(P)	name the	proposition that	P is the strongest	proposition	you exogenously learn upon undergoing E. For ease of notation, we'll describe the possibility in	which	you	gain	no	new	information	as	a	case in	which	you learn	the tautology	(T). So	yet	another	redescription	of	the	partition	above	is: L(X1) L(X2) L(X3) L(X4) ... L(T) We'll	call	an	event in	which	an	agent	exogenously learns	a	proposition	a learning experience	(and	note	that,	given	our	terminology,	it	is	consistent	with	this	that	the agent	'learns'	the	tautology	and	so	gains	no	new	information).	Now	suppose	that	an agent is considering some learning experience that she will undergo. She can represent	her	future	learning	experience	by	the	set	of	propositions	that	she	assigns non-zero credence to exogenously learning. So we'll say that an agent whose possibility	space	is	as	depicted	above	represents	her	future	learning	experience	by the	set: X:	{X1, X2, X3..., T} I will sometimes use the name of the set that represents an agent's learning experience	as	a	name	for	the	learning	experience	itself. It	will	be	useful for	what follows	to	note	that, in	general, if	X	represents	an agent's	future	learning	experience,	and	L(X)	is	the	set	of	propositions	L(Xi)	for	each Xi	∈X,	then	L(X)	is	a	partition	of	the	agent's	possibility	space. Here's	why:	First, imagine	a	case in	which the	agent is	certain that	she	will gain	some	new	information	upon	undergoing	the	learning	experience.	Then	she	will be	certain	that	there	will	be	exactly	one	proposition	in	X	that	uniquely	characterizes the	new	information	that	she	will	exogenously	receive.	Thus,	she	will	be	certain	that exactly	one	member	of	L(X)	is	true. So	if	the	agent	is	certain	that	she	will	gain	some new information,	L(X) is	a	partition	of	her	possibility	space. If,	on the	other	hand, the	agent	leaves	open	the	possibility	of	gaining	no	new	information,	then	T	will	be	a member	of	X. Since	our	agent	is	certain	that	she	will	gain	no	new	information	(learn T)	or	gain	some	new	information	(learn	exactly	one	of	the	Xi	that	is	not	T),	but	not both,	she	too	is	certain	that	exactly	one	proposition	in	L(X)	is	true. Thus,	whether the	agent	leaves	open	the	possibility	of	gaining	no	new	information	or	not,	L(X)	is	a partition	of	the	agent's	possibility	space. An update procedure, U, in response to a learning experience, X, is a function that assigns a probability distribution to each member of X, with the intended	interpretation	that	an	agent	conforming	to	U	adopts	U(Xi)	as	her	credence function if and only if the proposition she learns upon undergoing the learning experience is Xi. In other words, on the intended interpretation, an agent conforming to	U adopts	U(Xi) if and	only if L(Xi) is true. The fact that an	update procedure	is	a	mapping	from	the	propositions	the	agent	might	learn	to	probability functions	guarantees	that	update	procedures	satisfy	what	Greaves	and	Wallace	call 'availability':	In	any	two	worlds	in	which	the	agent	learns	the	same	information,	the update procedure recommends the same credence function. Conceiving of update procedures	in	this	way	is	motivated	by	the	thought	that	what	an	agent	is	rationally required	to	do	in	response	to	learning	a	proposition	must	be	determined	completely by	which	proposition	she	learns. Later	in	the	paper	we'll	consider	generalizations	of the	notion	that	don't	take	this	assumption	for	granted. It	will sometimes	be convenient to think	of	U as assigning to	each	possible state	a	credence	function. So	we	can	let	U(s)	be	U(Xi),	where	Xi is	the	proposition that	the	agent	learns	in	state	s. U(s)	=	U(Xi)	where	s∈L(Xi) As	we'll see in a	moment,	what	Greaves and	Wallace call 'an experiment' is just a special	kind	of	learning	experience,	and	what	Greaves	and	Wallace	call	'an	available act' is just	an	update	procedure in response to	an	experiment. So	my	notions	are generalizations	of	the	notions	that	Greaves	and	Wallace	use. 2.3	Experiments	and	available	acts Greaves	and	Wallace's	discussion	assumes	that	the	agent	contemplating	her future	learning	experience	satisfies	the	following	two	conditions: PARTITIONALITY:	The	propositions	that	the	agent	assigns	non-zero	credence	to exogenously	learning	form	a	partition	of	the	agent's	possibility	space. FACTIVITY:	The	agent	is	certain	that	if	she	learns	P,	P	is	true.	9 In cases in which PARTITIONALITY and FACTIVITY hold, we will say that the agent's future	learning	experience	is	representable	as	an	experiment. Greaves and Wallace's definition of an available epistemic act A is: 'an assignment	of	a	probability	distribution	to	each	piece	of	possible	information	Ej∈E [where E is	a	partition] with	the	intended	interpretation	that	if	A(Ej)	=	pj	then	pj	is the	probability	function	that	an	agent	performing	act	a	would	adopt	as	his	credence distribution if he received the new information that the actual state was some member	of	Ej' (p.	611-612). Thus,	an	available	act is just	an	update	procedure	in response	to	an	experiment. Now, if every rational agent satisfied PARTITIONALITY and FACTIVITY, then perhaps it	wouldn't	matter that	Greaves and	Wallace's result only applies to such agents (for their	account	could	still	be	a	general	account	of	how	to	revise	rational credence	functions). So	it's	worth	thinking	about	whether	a	rational	agent	may	fail to	satisfy	these	conditions. To begin, note that,	prima facie, it would be quite surprising if all rational agents satisfied	PARTITIONALITY.	To return to	our flinging	analogy, imagine that the world	has	a	'bucket'	of	propositions	{X1,X2	...}	that	you	think	it	might	fling	at	you.	If you	know	that	the	world	will	fling	exactly	one	proposition	in	the	bucket	at	you,	then the	set:	{the	world	flings	X1,	the	world	flings	X2,	the	world	flings X3...}	is,	indeed,	a partition	of	your	possibility	space. But	so	far	we've	been	given	no	reason	to	think that the	propositions in	the	bucket	itself	form	a	partition	of your	possibility space. After	all,	what	if	the	bucket	contains	both	P	and	P&Q? Since	P&Q	entails	P,	any	set that	contains	P&Q	and	P	is	not	a	partition. This	means	that	if	an	agent	leaves	open the	possibility that	P is the	strongest	proposition	she	exogenously learns,	and	also leaves	open the	possibility that	P&Q is the strongest	proposition she exogenously 9	Greaves	and	Wallace	are	explicit	about	PARTITIONALITY,	but	not	FACTIVITY. However,	as	we'll	see, FACTIVITY	must	be	assumed	for	their	arguments	to	work. learns, then the agent doesn't satisfy PARTITIONALITY. But it's hard to see why it would be irrational for an agent to leave open the possibility that the strongest proposition she learns is P, and also leave open the possibility that the strongest proposition	she	learns	is	P&Q. To illustrate the strength of the claim that all rational agents satisfy PARTITIONALITY	and	FACTIVITY,	it	will	be	helpful	to	prove	the	following	lemma	(I	call	it a	'lemma'	because	it	will	play	an	important	role	in	a	proof	that	comes	later): Lemma	1 An	agent	satisfies	PARTITIONALITY	and	FACTIVITY	if	and	only	if,	for	each	Xi such that	she	assigns	non-zero	credence	to	Xi	being	the	strongest	proposition	she exogenously	learns,	the	agent	assigns	credence	1	to: L(Xi)	↔	Xi Proof Suppose	that	PARTITIONALITY	and	FACTIVITY	are	satisfied.	FACTIVITY	entails	that the	agent	assigns	credence	1	to	the	left-to-right	direction	of	the	biconditional: L(Xi)	à	Xi for	any	Xi.	For	FACTIVITY	says	that, for	all	Xi, the	agent is	certain that if she learns Xi, Xi is true. What about the right-to-left direction? If PARTITIONALITY	holds,	then	the	agent	is	certain	that	exactly	one	proposition	in X is true. Since, by assumption, the agent is certain that she	will learn	one proposition	in	X,	and	that	(due	to	FACTIVITY)	it	will	be	a	true	proposition,	she will	have	to	learn	the	one	true	proposition	in	X. So	if	X	forms	a	partition,	she is	certain that the	Xi that is true is the	proposition that	she	will learn.	This gives us: Xi à L(Xi). Thus, any agent that satisfies PARTITIONALITY and FACTIVITY	will,	for	each	Xi	∈	X,	assign	credence	1	to	L(Xi)	↔	Xi. Conversely,	suppose	that	for	every	proposition	Xi	that	an	agent	assigns	nonzero	credence	to learning,	she	assigns	credence	1 to:	L(Xi)	↔	Xi.	And	recall that the L(Xi) form a partition of the agent's possibility space.10 It follows 10	See §2.2 for the detailed argument for this, but here's the gist: L(Xi) is the proposition that the strongest	proposition	an	agent	exogenously	learns	is	Xi. So	an	agent	can't	leave	open	the	following possibility: For distinct X1 and X2, the strongest proposition I exogenously learn is X1 and the strongest	proposition	I	exogenously	learn	is	X2. This	is	because,	assuming	X1	and	X2	are	distinct,	if the	agent	exogenously	learns	X2,	then	it's	false	that	the	strongest	proposition	she	exogenously	learns is	X1. Since	the	agent	can't leave	open	the	possibility	that	there	are	two	propositions	that	are	each that	an	agent	who	regards	the	Xi	as	equivalent	to	the	L(Xi)	will	be	such	that the	Xi	also	form	a partition	of	the	agent's	possibility	space.	So	any	agent	who is	certain	that,	for	each	Xi,	L(Xi)	↔	Xi,	satisfies	PARTITIONALITY.	And	under	the assumption	that	the	agent	is	certain	that,	for	each	Xi,	L(Xi)	à	Xi	(which	is	the just the left-to-right	direction	of the	biconditional), it follows	that the	agent satisfies	FACTIVITY	as	well:	she	is	certain	that	if	she	learns	some	proposition, Xi, that	proposition is true.	Thus,	any	agent that is	certain that, for	each	Xi, L(Xi)	↔	Xi	satisfies	PARTITIONALITY	and	FACTIVITY. So the question of	whether a rational agent could fail to satisfy PARTITIONALITY or FACTIVITY amounts to the following question:	Might there be some proposition, P, such that a rational agent assigns non-zero credence to exogenously learning P, while	leaving	open	the	possibility	that	P	is	true,	though	she	won't	learn	it,	OR	leaving open	the	possibility	that	she	will	learn	P,	but	P	isn't	true? Let's begin by considering the first type of case: a case in which an agent leaves	open	the	possibility	that	P,	but	she	doesn't	learn	that	P. P	but	not	L(P) Seemingly, there are	many cases in	which, for some	P that I	might learn, I leave	open	the	possibility	that	P	is	true	though	I	don't	learn	it. Suppose,	for	example, that I am about to turn on my radio and am considering the possible bodies of information	I	might	receive. I	think	that	one	possibility	is	that	I	learn: R:	It	is	raining	in	Singapore. and	nothing	else. I	also	think,	however,	that	it	might	be	raining	in	Singapore	even	if I	don't	learn	that	it	is	when	I	turn	on	the	radio.	This	seems	perfectly	rational,	but	if so, then it is rational to leave open the possibility that R (a proposition I might learn)	is	true	but	I	don't	learn	that	it	is. In	response,	one	might	claim	that	it	is,	in	fact,	irrational	for	me	to	leave	open the	possibility	that	I	exogenously	learn	R	and	nothing	else.	For	perhaps	one	thinks the	strongest	proposition	she	exogenously	learns,	the	agent	must	think	that	at	most	one	member	of the set L(Xi) is true. She	will also think that at least one	member of the set is true since	we are assuming	that	she	is	certain	that	she	will	undergo	a	learning	experience	represented	by	X:	that	is,	she is	certain	that	she	will	learn	one	member	of	X.	(Recall	that	this	is	consistent	with	her	leaving	open	the possibility of gaining no new information and merely 'learning' the tautology.) Thus, she will be certain	that	at	least	one	member	of	L(X)	is	true	and	that	at	most	one	member	of	L(X)	is	true. that I should	be	certain that	any	case in	which I come to	exogenously	possess the information	that	R	as	a	result	of	turning	on	the	radio	is	a	case	in	which	the	strongest proposition	that	I	exogenously	learn	is	something	like: R(R):	It	is	being	reported	on	the	radio	that	it	is	raining	in	Singapore. So,	one	might	claim,	if	I	am	certain	that	I	will	turn	on	the	radio,	I	should	be	certain that	if	R(R)	is	true,	I	will	learn	that	it	is. But should I?	What if I leave	open the	possibility that upon turning	on the radio	all	I	will	hear	is	static? In	that	case	I	might	leave	open	the	possibility	that	it	is being	reported	on	the	radio	that	it	is	raining	in	Singapore,	even	if	I	don't	learn	that	it is being reported on the radio that it is raining in Singapore. Surely it is not irrational	to	leave	such	a	possibility	open. In	response	to	this,	one	might	claim	that	it	is	also	irrational	for	me	to	think	of R(R)	as	a	proposition	in	the	bucket	of	propositions	that	the	world	might	fling	at	me (that	is,	as	a	potential	strongest	proposition	I	exogenously	learn).	Rather,	one	might claim, the proposition in the vicinity that I should assign non-zero credence to exogenously	learning	is: E(R(R)): I	have	an	experience	as	of it	being reported	on the radio that it is raining	in	Singapore. And	perhaps,	one	thinks,	I	am	rationally	required	to	be	certain	that	if	E(R(R))	is	true, I	will	learn	that	it	is. Note, however, that for this this strategy to generalize the following two claims	must	be	true: (a) If P is a proposition about one's experience (that one could, in principle, learn about), then a rational agent should regard it as impossible	for	P	to	be	true	without	her	learning	that	P. (b) Every	agent	should	assign	credence	zero	to	P	being	the	strongest proposition she exogenously learns, unless P is a proposition about	her	own	experience. Why is (b)	necessary?	Because it's	plausible that for any	proposition	P that is	not about	an	agent's	experiences,	an	agent	can	rationally	leave	open	the	possibility	that P	is	true	though	the	agent	doesn't	learn	that	it	is. So	if	agents	are	to	be	certain	that all	propositions they	might learn	will	be true	only	if	they	learn	them, they	must	be certain that the only kinds of propositions they will exogenously learn are propositions	about their	experience.	Why is (a)	necessary? Because	claiming that the	only	propositions	I	learn	are	about	my	experience	will	be	of	no	help	if	I	can	leave open	the	possibility	that	some	proposition	about	my	experience is true	but	I	don't learn	that	it	is. But (a) and (b) are far from obvious. Let's begin with (a). Consider, for example,	the	following	proposition: Detailed-E(R(R)):	I	have	an	experience	as	of	a	reporter	with	a	British	accent saying that it is raining in Singapore with a slight emphasis on the word 'raining'	and	a	pause	between	'raining'	and	'Singapore'. This seems like a proposition I could learn. But it also seems possible that my experience	could	have	the	described	features	and	yet	I	don't	exogenously	learn	that it	does. I	may	not	notice	the	accent,	or	the	pauses,	or	the	emphases,	despite	the	fact that	these	features	are	present	in	my	experience. So	why	couldn't	a	rational	agent leave open the possibility that a proposition like this is true, though she doesn't learn	that	it	is? (b) is also a very substantive assumption. Why should every agent be antecedently certain that propositions about her experience are the	only	kinds of propositions she will exogenously learn? Presumably small children exogenously learn things: the world flings bodies of information at them. But small children might not even have the conceptual apparatus that	makes it possible for them to exogenously	learn	propositions	about	their	own	experience. So	one	might	want	to claim	that	children,	at	least,	can	exogenously	learn	propositions	that	are	not	of	this sort. But	if	the	world	can	fling	propositions	like	R,	or	R(R),	into	a	child's	belief	box, what should make me antecedently certain that the world won't fling such a proposition at me? In other words, if propositions that aren't about one's experience can, in principle, be exogenously learned, why should every agent be certain	that	she	won't	undergo	this	sort	of	learning? In sum, while there is nothing incoherent about the view that, for any proposition	P one	might learn, one is rationally required to be certain that if P is true,	one	will learn it,	such	a	view	requires	some	rather	hefty	commitments	about the kinds of propositions that can be exogenously learned. The resulting commitments are stronger than even the kinds of luminosity commitments that (some) internalists are happy to sign up for and that Timothy	Williamson (2000) and	others	have	argued	against. For	it's	not	just	that	one	can't	be	wrong	about	one's own	experiences. And it's not just that, for some class of experiences, having the experience	always	puts	one	in	a	position	to	know	that	one	is	having	it. It's	not	even that,	whenever some	proposition is true of one's experience, one in fact	comes to know	that	proposition. It is that	every rational	agent	must	antecedently	be	certain that	any	proposition	P	that	could	be	true	of	her	experience	(and	which	it	is	possible to	learn	about)	is	a	proposition	that	she	will learn	exogenously	whenever	P	is	true and	that	there	are	no	other	propositions	that	she	could	exogenously	learn. L(P)	but	not	P If	you	think	that	the	word	'learn'	is	factive,	and	that	any	rational	agent	should be certain of this, you	might think that a rational agent can never leave open the possibility	of	learning	a	proposition	that	is	false.	But	let's	set	aside	the	semantics	of 'learn'. For	various reasons, some	philosophers	have thought that	an	agent	might have	a	false	proposition	as	part	of	her	evidence.11	So	if	we	redescribed	the	project	as an investigation into how an agent should revise her credences in light of the evidence	she	receives	(instead	of	'in	light	of	what	she	exogenously	learns'),	we	might want	an	account	that	allows	a	rational	agent	to	leave	open	the	possibility	of	gaining a	false	proposition	as	part	of	her	evidence. In	this	case,	we	would	want	an	account that	would	apply	to	agents	that	fail	to	satisfy	FACTIVITY. Given	the	considerations	above,	I	think	that	it	should	remain	a	live	possibility that	a	rational	agent	may	fail	to	satisfy	one	of	PARTITIONALITY	or	FACTIVITY. So	if	we want	a	fully	general	account	of	credal	revision,	we	should	consider	how	such	agents should	revise	their	credences	in	light	of	what	they	learn. This	forces	us	to	consider learning	experiences	that	aren't	representable	as	experiments. 2.4	The	expected	accuracy	of	update	procedures 11	See, for example, Rizzierie (2011), Arnold (2013), Comesaña and McGrath (2014) and Drake (forthcoming). So	far,	we	have	defined	the	expected	accuracy	of	a	credence	function. But	we don't yet have a definition of 'the expected accuracy of an update procedure in response to a future learning	experience'.	Greaves and	Wallace	do	provide such a definition.	However,	Greaves	and	Wallace's	definition	can	only	be	used	to	describe the expected accuracy of an update procedure for an agent that satisfies PARTITIONALITY and	FACTIVITY. Since, in this	paper, I am interested in	which	update procedures	maximize expected accuracy in general, I	will have to generalize their notion. So	what	do	we	mean	by	the	expected	accuracy	of	an	update	procedure	U in response	to	a	future	learning	experience	X? On	an	intuitive	level,	what	we're	trying to capture is how accurate we expect to be upon learning a member of	X if we conform	to	U. And	recall	that,	on	the	intended	interpretation,	an	agent	conforms	to U if she adopts	U(Xi) whenever the proposition she learns upon undergoing the learning	experience	is	Xi. Suppose that an agent knows that she will undergo a learning experience represented	by	X.	Let	A(U(s),s)	represent	the	accuracy	score	of	an	agent	conforming to	U	in	s.	It	is	natural	to	think	of	the	expected	accuracy	that	an	agent	assigns	to	U	as the	weighted	average	of the	accuracy	scores that	an	agent	conforming to	U	would adopt	in	each	state	in	which	she	learns	a	member	of	X. This	gives	us	the	following understanding of the expected accuracy of an update procedure: The expected accuracy	of	an	update	procedure	U	in	response	to	a	future	learning	experience	X, relative	to	an	agent's	probability	function	p	is:12 EAp(U)	= ∑ p(s)	A(U(s),	s) s∈L(X) = ∑ ∑ p(s)*	A(U(X	i)),	s) L(Xi)∈L(X) s∈	L(Xi) 12	My	definition	of	expected	accuracy	is	inspired	by	the	definition	provided	by	Greaves	and	Wallace (though there is one important difference, the reason for which will become clear shortly). A limitation of defining expected accuracy using summations is that if the number of things being summed over is infinite, the sum may not be defined. Kenny Easwaran (2013) provides an alternative	way	of	understanding the	notion	of expected	accuracy that coincides	with	Greaves	and Wallace's	definition	when	finite	quantities	are	involved,	but	also	applies	to	cases	when	the	quantities are infinite. The results that follow can be carried out in Easwaran's framework (see note 15). However, since the crucial points in this	paper are	most easily	brought out	using the	Greaves and Wallace	inspired	definition,	I	will	continue	using	summations	in	the	main	text. I	will	now	prove	a	second	lemma: Lemma	2 If	an	agent's	future	learning	experience	is	representable	as	an	experiment,	E, and	U	is	an	update	procedure	in	response	to	E, then: EAp(U) = ∑ ∑ p(s)*	A(U(Ei)),	s) = ∑ ∑ p(s)*	A(U(Ei)),	s) L(Ei)∈L(E) s∈	L(Ei) Ei∈E s∈	Ei Proof Note	that	the	first	(leftmost)	double	sum	is	just	the	definition	of	the	expected accuracy	of	an	update	procedure. The	second	double	sum	is	just	like	the	first except	that,	rather	than	summing	over	the	L(Ei),	we're	summing	over	the	Ei. We know from Lemma 1 that if an agent's future learning experience is representable as an experiment – that is, the agent satisfies PARTITIONALITY and	FACTIVITY	–	then	the	agent	is	certain	that	for	all	propositions	Ei	∈E	: Ei	↔	L(Ei) Given this, there is no harm in replacing the 'L(Ei)' that features in the definition	of	the	expected	accuracy	of	an	update	procedure	with	'Ei'. Since Greaves and Wallace assume PARTITIONALITY and FACTIVITY, they can simply define the expected accuracy of an update procedure (which they call 'an act') in response to an	experiment as the	average	accuracy scores that	would result from adopting U(Ei) whenever Ei is true. And this, indeed, is what they do. Their definition	of	the	expected	accuracy	of	an	act	corresponds	to	the	double	sum	on	the right-hand	side	of	the	lemma.	But	it's	important	to	realize	that	they	wouldn't	define expected	accuracy this	way if they	weren't assuming	PARTITIONALITY	and	FACTIVITY. This is	because,	without these	assumptions, the	double sum	on the right	does	not represent a weighted average of the scores that would result from an agent performing the	act. For	Greaves	and	Wallace, in	defining	an	act, say that	an	agent performs	act	U	in	response	to	X	if	she	adopts	U(Xi)	as	her	credence	function	if	and only if she learns	Xi (p.612). But if an agent leaves open the	possibility that	Xi is true, though	she	doesn't learn	it	(PARTITIONALITY	fails),	or	that	she learns it, though it's	not true (FACTIVITY fails), then	an	agent	performing	U	would	not adopt	U(Xi) if and	only	if	Xi	is	true. Thus,	it	is	only	if	PARTITIONALITY	and	FACTIVITY	are	assumed	that the	double	sum	on	the	right	represents	the	expected	accuracy	of	the	credences	that result	from	an	agent	performing	U. 2.5	Summing	up The	purpose	of	this	section	was	to	develop	a	precise	definition	of	the	notion of the expected accuracy of an update procedure in response to a learning experience. Although Greaves and	Wallace provide a definition for the expected accuracy	of	an	act	in	response	to	an	experiment,	this	definition	won't	apply	to	cases in	which	PARTITIONALITY	or	FACTIVITY	fail. I defined the expected accuracy of an update procedure as the weighted average	of the	accuracy	scores that	would	result form	an	agent conforming to the update	procedure	(adopting	U(Xi)	whenever	she	learns	Xi).	I	then	showed	that	if	the agent	can	represent	her	future	learning	experience	as	an	experiment,	this	quantity will	equal the	weighted	average	of the	accuracy	scores that	would	result from	her adopting	U(Xi)	whenever	Xi is	true. This	gives	us	Greaves	and	Wallace's	definition of the expected accuracy of an act. Thus, my framework, in terms of update procedures and learning experiences, is a generalization of the framework developed	by	Greaves	and	Wallace. In the next section I	will use the generalized framework to derive	Greaves and Wallace's result: the claim that, for an agent who can represent her future learning	experience	as	an	experiment,	conditionalizing	on	the	proposition	she	learns maximizes expected accuracy. I then prove a more general result: for any agent contemplating a future learning experience, the	update procedure that	maximizes expected	accuracy	is	one	in	which,	upon	learning	Xi,	the	agent	conditionalizes	on	the proposition that she learned Xi. In cases in which the learning experience is representable	as	an	experiment	(and	only	in	such	cases),	this	amounts	to	the	same thing	as	conditionalizing	on	Xi. 3.	The	Greaves	and	Wallace	Result	and	its	Generalization Greaves and Wallace argue that (given a strictly proper scoring rule) conditionalizing on the proposition one learns is the update procedure that maximizes	expected	accuracy	in	response	to	an	experiment. We can think of the argument for this claim as involving two steps. First, there	is	a	purely	formal	result	that	demonstrates	that	plugging	in	certain	values	in certain	quantities	maximizes those	quantities. Second, there is an	argument from this	formal	result	to	the	claim	that,	given	our	understanding	of	update	procedures, expected accuracy of update procedures, learning, and experiments, the update procedure (or available act) that	maximizes expected accuracy in response to an experiment is the one that has the agent conditionalize on the proposition she learns. It	will	be	important	to	keep	these	two	steps	separate. I	will	call	the	purely formal	result	that	can	be	extracted	from	Greaves	and	Wallace's	paper	'G&W'. G&W:	For	any	partition	of	states	P:	{P1...Pn},	consider	the	set	of	functions,	F, that	assign	members	of	P	to	probability	functions. The	member	of	F,	F,	that maximizes	this	quantity: ∑ ∑ p(s)*	A(F(Pi),	s) Pi∈P s∈Pi is: F(Pi)	=	Cond(Pi)	=	p(⋅	|	Pi) when	A	is	strictly	proper. G&W	can	be	used	to	derive	Greaves	and	Wallace's	claim	about	experiments: CONDMAX:	Suppose	you	know	that	you	are	going to	perform	an	experiment, E. The	update	procedure	that	maximizes	expected	accuracy	in	response	to	E, relative to probability function p, is the update-procedure that assigns, to each	Ei,	p(⋅	|	Ei). The	argument	from	G&W	to	CONDMAX,	using	our	generalized	framework,	is	simple. Proof	of	CONDMAX: (1)	The	expected	accuracy	of	an	update	procedure	U	in	response	to	an experiment	E,	relative	to	a	probability	function	p	is: (*) ∑ ∑ p(s)*	A(U(Ei),	s) Ei∈E s∈	Ei (from	Lemma	2). (2)	The	value	of	U	that	maximizes	(*)	is	U=Cond(Ei). (This	follows	from	G&W	and	the	fact	that	E	is	a	partition) (3)	The	update	procedure	U	that	maximizes	expected	accuracy	in	response	to an experiment E is U=Cond(Ei). That is, the update procedure that maximizes	expected	accuracy	is	the	one	that	has	the	agent	conditionalize	on the	member	of	E	that	she	learns. (This	follows	from	(1)	and	(2)). But what about cases in which our future learning experiences aren't representable as experiments? Which update procedure maximizes expected accuracy	in	those	cases? Here	is	the	answer: GENERALIZED	CONDMAX: Suppose you know that you are going to undergo a learning experience, X. The update procedure that maximizes expected accuracy in response to	X, relative to probability function	p, is the update procedure	that	assigns, to	each	Xi,	p(⋅|L(Xi))	where	L(Xi) is the	proposition that Xi is the strongest proposition the agent exogenously learns upon undergoing	the	learning	experience. Proof	of	GENERALIZED	CONDMAX: Recall	that	the	expected	accuracy	of	an	update	procedure,	U,	in	response	to	a learning	experience	X	is	defined	as: (#) ∑ ∑ p(s)*	A(U(Xi)),	s) L(Xi)∈L(X) s∈	L(Xi) We	are	aiming	to	show	that	(#)	is	maximized	when	U(Xi)	=	Cond(L(Xi)). So suppose	for	reductio	that	this	is	false:	that	is,	that	there	exists	a	function,	U*, such	that: ∑ ∑ p(s)*	A(U*(Xi)),	s) > ∑ ∑ p(s)*	A(Cond(L(Xi)),	s) L(Xi)∈L(X) s∈	L(Xi) L(Xi)∈L(X) s∈	L(Xi) Now,	define	μ(L(Xi))	as	U*(Xi).13 It	follows	that: ∑ ∑ p(s)*	A(μ(L(Xi)),	s) > ∑ ∑ p(s)*	A(Cond(L(Xi)),	s) L(Xi)∈L(X) s∈	L(Xi) L(Xi)∈L(X) s∈	L(Xi) But	this	is	impossible,	because	it	follows	from	G&W	that	the	quantity: (##) ∑ ∑ p(s)*	A(F(L(Xi)), s) L(Xi)∈L(X) s∈	L(Xi) is	maximized	when	F(L(Xi))=	Cond(L(Xi)). Thus,	there	cannot	exist	a	μ	that satisfies	the	inequality	above. Contradiction. Here is the lesson to be learned from	CONDMAX and its generalization: the	update procedure	that	maximizes	expected	accuracy	in	response	to	any	learning	experience is	one in	which	an	agent	who learns	Xi conditionalizes	on	the	proposition	that	she learns Xi upon undergoing the learning experience. 14 The reason that conditionalizing	on	the	proposition	that	one	learns	maximizes	expected	accuracy	in response	to	an	experiment is that, in these special cases, the agent knows that she will	learn	Xi	if	and	only	if	Xi	is	true. In	these	cases,	conditionalizing	on	Xi	amounts to	the	very	same	thing	as	conditionalizing	on	L(Xi).15 13	How	do	we	know	that	there	is	such	a	μ? Since	there	is	a	bijection	between	the	Xi	and	the	L(Xi), there exists an inverse of L(Xi),	which	we'll call 'L-(Xi)', such that L-(L(Xi)) =	Xi. We can then let μ(L(Xi))	be	U*	composed	with	L-. Thus:	μ(L(Xi))	=	U*(L-(L(Xi))	=	U*(Xi). 14	Note	that	this	is	true	for	any	proposition	that	is	the	strongest	proposition	one	exogenously	learns, including propositions that are, themselves, about gaining information. So if, say, in a	Monty Hall case,	one	thinks	that	the	strongest	proposition	learned	is	something	along	the	lines	of: 'I	gained	the information that there is a goat behind door 2', the update procedure that maximizes expected accuracy	will	have	you	conditionalize	on:	'I	learned	that	I	gained	the	information	that	there	is	a	goat behind	door	2'. 15	The result can be generalized further to cases in which the possible number of propositions learned	is	infinite. However,	to	perform	this	generalization,	we	need	a	notion	of	expected	accuracy that doesn't rely on summation. Easwaran (2013) provides such a notion and argues, using this notion, that conditionalization	maximizes expected accuracy. Like Greaves and	Wallace, however, Easwaran relies on both PARTITIONALITY and FACTIVITY. So some	modifications need to be	made to derive GENERALIZED CONDMAX using Easwaran's framework. Since Easwaran's notion of expected accuracy is quite complex, I cannot, in this note, explain in general terms how the proof	must be modified. But for those readers familiar with Easwaran's argument, here are the relevant details: First, Easwaran's claim that 'V and	V' are identical on	~E' (p.136) relies on	FACTIVITY. For suppose FACTIVITY	is	violated. Then	it's	possible	that,	for	some	s,	the	agent	learns	E	in	s	but	~E	is	true	in	s. In such	a	state	V(s)	= I(A,	x, s)	and	V'(s)	= I(A,	x', s). Since it	has	not	been	assumed that	x	and	x'	are identical, it cannot be assumed that	V	and	V'	are identical on	~E.	What can	be assumed, however, 4.	Iteration	Principles The update procedure that	maximizes expected accuracy in general is not conditionalization. It	is	conditionalization*:	conditionalizing	on	the	proposition	that one	learned	P,	when	P	is	the	proposition	learned. Recall	that	we	are	interested	in	the	expected	accuracy	of	update	procedures like conditionalization or conditionalization* because of the possibility that expected	accuracy	considerations	can	be	used	to	support	claims	about	which	update procedures	are	rational.	And	recall	that	underlying	the	arguments	under	discussion for	the	rationality	of	various	update	procedures	is	the	following	assumption: RATACC: The rational update procedures are those that	maximize expected accuracy	according	to	a	strictly	proper	scoring	rule. Together,	RATACC	and	GENERALIZED	CONDMAX	entail: COND*:	The	rational	update	procedure	is	conditionalization*. In	other	words, upon learning P, an ideally rational agent will conditionalize on the proposition	that	she	learned	P.16 Since conditionalizing on any proposition involves assigning credence 1 to that proposition, and conditionalization* has us conditionalize on the proposition that we	learned	P,	when	P	is	learned,	it	follows	from	COND*	that: LL:	If	one	learns	P,	one	is	rationally	required	to	be	certain	that	one	learned	P. I	suspect	that	people	who	deny	KK	–	the	principle	that	whenever	one	knows	P	one	is in	a	position	to	know	that	one	knows	P17	–	or	related	iteration	principles,	will find without	relying	on	FACTIVITY,	is	that	V	and	V'	are	identical	on	~L(E). Second,	Easwaran's	claim	that	'on E, V(s) = I(A, x, s) and V'(s) = I(A, x', s)' (p.136) relies on PARTITIONALITY. For suppose that PARTITIONALITY	is	violated. Then	it's	possible	that	there	is	some	state	s	in	which	E	is	true	but	the	agent doesn't	learn	E	–	rather,	she	learns	some	other	proposition	E*. In	such	a	case,	V(s)	=	I(A,	f(E*),	s)	and V'(s)	=	I(A,	f'(E*),	s). Since	it	is	not	assumed	that	f(E*)	is	x,	or	that	f'(E*)	is	x',	we	cannot	assume	that, on E, V(s) = I(A, x, s) and V'(s) = I(A, x', s). What can be assumed, however, without relying on PARTITIONALITY,	is	that,	on	L(E),	V(s)	=	I(A,	x,	s)	and	V'(s)	=	I(A,	x',	s). Plugging	in	these	substitutions throughout the remainder of the proof yields the result that, in general, conditionalizing on L(E) (rather than E), where E is the proposition learned, is the update procedures that maximizes expected	accuracy. 16	Recall	that	the	proposition	one	'learns'	refers	to	the	strongest	proposition	one	exogenously	learns. LL	unattractive.18 But	if	LL	is	rejected,	COND*	must	also	be	rejected. In	this	section,	I explore	a	number	of	ways	of	resisting	the	conclusion	that	conditionalization*	is	the rational update procedure, and the resulting commitment to LL. The most straightforward way to do this is to simply reject RATACC – the claim that the rational	update	procedures	are	those	that	maximize	expected	accuracy.	Ultimately,	I think that this is the most promising route for those who wish to reject COND* and/or	LL.	But	first	I'd	like	to	describe	two	alternatives. The	first	involves	claiming that	all rational	agents	do, in fact, satisfy	PARTITIONALITY and	FACTIVITY.	The	second involves	a	modification	of	RATACC. 4.1	Endorsing	the	Requirements	of	PARTITIONALITY	and	FACTIVITY The	argument	against the	claim	that	conditionalization	maximizes	expected accuracy in general relied on the thought that rational agents may fail to satisfy PARTITIONALITY or FACTIVITY. I offered considerations that tell against the requirement	that	rational	agents	satisfy	both	of	these	conditions. But	perhaps,	upon realizing	that	endorsing	conditionalization*	as	the	rational	update	procedure	brings with	it	a	commitment	to	LL,	one	may	want	to	revisit	this	issue. However, even if a case can be made that all rational agents satisfy PARTITIONALITY and FACTIVITY, this	won't help the LL-denier. For CONDMAX tells us that if all rational agents satisfy PARTITIONALITY and FACTIVITY, ordinary conditionalization	will	be	the	update	procedure	that	maximizes	expected	accuracy. However,	by	Lemma	1,	all	rational	agents	who	satisfy	PARTITIONALITY	and	FACTIVITY will	regard	L(P)	and	P	as	equivalent. So,	if	rational	agents	conditionalize	on	P,	upon learning	P,	they	will	assign	credence	1	to	P. But,	since	these	agents	assign	credence 1	to	P	↔	L(P),	conditionalizing	on	P	will	result	in	the	agent	assigning	credence	1	to L(P) as	well. Thus, if PARTITIONALITY and	FACTIVITY	are satisfied, conditionalization yields	the	result	that	an	agent	that	learns	P	will	be	certain	that	she	learned	P.19 17	See,	for	example,	Williamson	(2000). 18	Note,	however, that	at least some	objections to	KK	don't	extend to	LL. KK	has the	consequence that	if	an	agent	knows	P,	she	knows	that	she	knows	P,	she	knows	that	she	knows	that	she	knows	P, and	so	on. However,	recall that	by 'learn'	we	mean	exogenously learn.	Thus,	LL just	says	that if	an agent	exogenously learns	P	she	must	become	certain	that	she	exogenously	learned	P. It	doesn't	say that	if	she	exogenously	learns	P,	she	exogenously	learns	that	she	exogenously	learns	P.	The	certainty in	learning	P	need	not, itself,	be	the	result	of	exogenous	learning.	Thus,	unlike	KK,	LL	'iterates'	only once. 19	Bronfman	(2014)	gives	a	related	argument for	the	claim	that	agents	that	satisfy	these	conditions will	conform	to	KK. This brings out an important point: conditionalization and conditionalization*	only	yield	different	results	when	an	agent	doesn't	satisfy	at	least one	of	PARTITIONALITY	or	FACTIVITY. I	suggested	that, in	many	ordinary	cases, these requirements are not both satisfied. In such cases, conditionalization*, and not conditionalization,	maximizes	expected	accuracy. But	even	if	one	disagrees	with	me about whether rational agents always satisfy PARTITIONALITY and FACTIVITY, one shouldn't	reject	the	claim	that	conditionalization*	maximizes	expected	accuracy.	For conditionalization and conditionalization* amount to the very same thing when PARTITIONALITY and	FACTIVITY are satisfied.	Thus, COND*	and	LL follow from	RATACC even	if	agents	are	rationally	required	to	satisfy	PARTITIONALITY	and	FACTIVITY. 4.2	Modifying	RATACC Aaron	Bronfman	(2014,	p.	887-8)	considers	and	rejects	a	rule	that	is	similar to	conditionalization*. His	reason	for	rejecting	the	rule	is	based	on	the	thought	that when	we're	considering	which	update	procedures	maximize	expected	accuracy,	we should	only	consider	those	procedures	that the	agent in	question	can	competently execute. On this view, the rational update procedure isn't the update procedure from the pool of possible update procedures that maximizes expected accuracy. Rather, the rational update procedure is the procedure from the pool of update procedures that the agent can competently execute that maximizes expected accuracy. As an example, suppose that Al fails to satisfy one of PARTITIONALITY or FACTIVITY. I have shown that the update procedure that maximizes expected accuracy	for	Al	from	the	pool	of	possible	update	procedures	is	conditionalization*.	But now	suppose	that	Al	sometimes	exogenously learns	P,	but is	unable	to	realize	that he	learned	P.	Arguably,	Al	can't	competently	execute	conditionalization*.20	If	this	is right, then according to modified RATACC, which has us consider only update procedures	that	Al	can	competently	execute,	Al	is	not	required	to	conditionalize*. I think that this is an interesting suggestion, but it is worth noting a few things: First, when we calculate the expected accuracy of update procedures, we always do so from the perspective of the agent prior to undergoing the learning experience. Bronfman's suggestion is that	we remove from the	pool of candidate update	procedures those	that the	agent	cannot	execute. But	what if the	agent	has 20	How plausible this claim is depends on the modal scope of 'can'. I will simply assume that someone	who	is	sympathetic	to	this	line	of	thought	will	have	a	way	of	making	sense	of	the	modal	that yields	the	desired	result. false	(but	rational)	beliefs	about	which	update	procedures	she	can	execute? Then the update procedure that maximizes expected accuracy from the pool of procedures that she thinks she can	execute	may	differ from the	update	procedure that	maximizes	expected	accuracy	from	the	pool	of	procedures	that	she	can	execute. But it seems against the spirit of Bronfman's proposal to demand that the agent update in accord with the update procedure that maximizes expected accuracy relative to her actual abilities when she has no way of knowing which update procedure	this	is. One might modify Bronfman's proposal so that what's relevant is not the agent's	actual	abilities,	but	the	agent's	opinions	concerning	her	abilities. But	if	the only	update	procedures	in	the	pool	that	she	should	be	choosing	from	are	those	that she	is	certain	that	she	will	be	able	to	execute,	the	pool	may	well	be	empty. Perhaps, then,	the	pool	shouldn't	only	contain	procedures	that	she	is	certain	she	will	be	able to execute. Maybe it should contain those procedures that she believes she can execute,	or	those	that	she	is	sufficiently	confident	that	she	can	execute.21 But	there are	additional	complications. For	suppose I	now	rationally	believe	that I	won't	be able to refrain from being certain that my child is the best player on the team, whatever	evidence	I	receive.	But	I	am	wrong	about	this. In	fact,	I	will	perfectly	well be able to evaluate the evidence concerning the relative abilities of	my child. The view under consideration entails that even if, when the time comes, all of my evidence suggests that	my child is	mediocre, and I am	capable	of recognizing this fact,	in	virtue	of	the	fact	that,	at	an	earlier	time,	I	believed	that	I	couldn't	help	but	be certain that she is	best, I am	rationally	required to	be certain that she is the	best! This	seems	highly	implausible. I don't	mean to claim that these complexities are insurmountable, but it is worth	noting	that	nothing	that	looks	like	ordinary	conditionalization	will	emerge	as a result of Bronfman's modification. If we modify RATACC in the way Bronfman suggests	and thereby	avoid	a	commitment to	COND*	and	LL,	what	we	are left	with isn't	good	old-fashioned	conditionalization. Rather, the	rational	update	procedure will	be	something	very	messy	and	agent-relative	that	can't	be	neatly	characterized in	a	formal	framework. If	we	want	to	account	for	the	limitations	of	non-ideal	agents 21	If	we	included	only	those	procedures	that	the	agent	knows	she	can	execute,	then,	since	'knows'	is factive,	we	will	run	into	the	earlier	problem. If	the	agent	rationally	believes	that	she	can	execute	all of	the	procedures	in	set	S,	but	she	only	knows	that	she	can	execute	the	procedures	in	S',	then	the	view would	imply	that	it's	rational	for	her	to	accord	with	the	update	procedure	that	maximizes	expected accuracy	relative	to	S'. this is to be expected, but	we are now quite far from the project as Greaves and Wallace,	and	others	involved	in	accuracy-first	epistemology,	originally	conceived	of it. In describing the idealized agents under discussion Greaves and Wallace say: 'Real	epistemic	agents	are	not	(at	least	not	quite)	like	this. Bayesian	epistemology	is a normative theory rather than a purely descriptive one' (p. 608). Greaves and Wallace are interested in a notion of ideal	rationality that doesn't take an agent's cognitive	limitations	into	account. One	might	have	qualms	about	such	idealizations, but	these	qualms	will	extend	to	Bayesian	epistemology	more	generally	and	are	not ones	that	I	will	address	here. Still, one	might claim, even the idealized notion of rationality that Greaves and	Wallace are	working	with takes into account some of the agent's limitations. After	all, if	any	update	procedure	were	allowed	in	the	pool,	then	surely	the	update procedure that	maximizes expected accuracy would be one that requires that, in every	state, the	agent	assign	credence	1 to	all the truths	and	credence	0 to	all the falsehoods! Now,	as	a	matter	of	fact,	given	the	way	we	have	defined	'update	procedure', the	rule	'assign	credence	1	to	all	truths	and	0	to	all	falsehoods'	(let's	call	it	'the	truth rule')	simply	isn't	an	update	procedure.	For	recall	that	an	update	procedure	is	just	a function from the propositions one	might learn to credence functions. Since the truth	values	of	some	propositions	may	vary	amongst	the	worlds	in	which	the	agent learns	the	same	information,	but	the	recommended	credence	function	cannot	vary amongst these	worlds, a function from the propositions the agent	might learn to credence	functions	will	not,	in	general,	be	one	that,	when	conformed	to,	results	in	an agent	assigning	credence	1	to	all	truths	in	every	state. Nonetheless, one	might think that the reason	we defined the notion of an update	procedure	in	a	way	that	rules	out	the	truth	rule	is	that	we	are	only	interested in	procedures that	are, in some	sense,	available	to the	agent	upon	undergoing the learning	experience.	We	don't	want	to	require	that	the	agent	be	certain	that it	will rain	tomorrow,	in	virtue	of	the	fact	that	it	will	rain	tomorrow,	if	all	she	learns	is,	say, that a coin landed Heads. Similarly, you	might think, we	must find some way of ruling	out	update	procedures	that	require	an	agent	to	be	certain	that	she	learned	P, in virtue of that fact that she did learn P, even though the only information she exogenously	received	was	that	P. Perhaps	so. But	the	issue	here	will	be:	available	in what	sense? It will be helpful to make use of Ned Hall's (2004) distinction between analyst	experts	and	database	experts. We	defer to	database	experts	because they possess a great deal of information.	We defer to analyst experts because of their superior information	processing	abilities.	Thus,	we	can	distinguish	agents	who	are idealized	along	the	database	dimension	(they	are	certain	of	all	and	only	the	truths), and	agents	who	are idealized	along the	analyst	dimension. It is the latter	kind	of idealization	that	Greaves	and	Wallace	are	interested	in. They	want	to	know	how	an idealized	analyst	will	revise	her	beliefs in light	of	new	information. Since	they	are interested in idealized information processing, and not idealized information possession, it is clear why they require that update procedures issue the same recommendations	in	any	two	states	in	which	the	agent	gains	the	same	information. It will, however, be difficult to come up with a principled way of ruling out conditionalization* as the ideal update procedure if the ideal in question is ideal information	processing.	This is	because, like	conditionalization,	conditionalization* is simply an operation performed on the proposition exogenously learned. The operation is the following: If	P is the	proposition learned, take	P,	attach	an	L to it, and	conditionalize	on	the	resulting	proposition:	L(P). If	we	were	happy	with	ordinary	conditionalization,	then	we	were	happy	with requiring that (ideal!) agents	be certain that	Q,	upon learning	P, if	P entails	Q. In endorsing this commitment we needn't suppose that any event in which one exogenously	learns	P	constitutively	involves	a	learning	of	Q. Rather,	the	Q-learning may be a kind of endogenous learning that idealized agents will undergo upon exogenously learning	P. The requirement that agents be certain that they learned what	they	learned	is, in	the	relevant	sense,	no	different	from	the	requirement	that agents be certain in the propositions that their evidence entails. Here too, we needn't suppose that any event in which one learns P constitutively involves a learning that	one learned	P. The claim is rather that ideal agents	will come to	be certain	that	they	learned	P	upon	appropriate	processing	of	the	information	that	P. In sum, Bronfman's modification of RATACC may well be worth serious consideration, but it does not engage with the project as Greaves and Wallace conceived of it: figuring out the ideally rational way to revise beliefs, where the idealization in question is along the information processing dimension. Conditionalization*, just like	conditionalization, is	an	operation	on the	proposition an	agent	learns.	If	we	are	interested	in	ideal	information	processing,	there	shouldn't be	any	restrictions	on	what	operations	can	be	performed	on	this	proposition. 4.3	Giving	up	RATACC Rather than trying to modify RATACC, one may simply reject the idea that anything in the vicinity of RATACC is true. On this view, there simply is no straightforward connection between the rational way of revising one's credences and	considerations	of	expected	accuracy. There	is	plenty	of	literature	devoted	to	evaluating	the	merits	of	accuracy-first epistemology22	and	entering into this	debate	will take	us	beyond the	scope	of this paper. But	it	is	important	to	realize	that	RATACC	plays	an	important	role	in	much	of the	accuracy-first	project.23	So	I	will	simply	note	that	this	is	one	way	that	someone who wants to reject COND* and LL might go. If this turns out to be the only acceptable way to reject these claims, then we will have learned the following interesting fact:	Many accuracy-first epistemologists (those	who endorse RATACC) are	committed	to	some	substantive	iteration	principles,	and,	conversely,	those	who reject	such	principles	are	committed	to	rejecting	large	portions	of	the	accuracy-first project. 5.	Further	Generalizations	and	Further	Consequences I	have	shown	that	conditionalizing	on	the	propositions	we	learn	does	not,	in general,	maximize expected accuracy. Rather, conditionalizing on the	proposition that	we	learned	P,	when	P	is	the	proposition	learned, is	the	update	procedure	that maximizes expected accuracy. In this section I provide further generalizations of this result and show that, no matter which features of an agent's situation the RATACC-er	thinks	the	rationality	of	an	agent's	credence	function	depends	on,	she	is committed	to: LUMINOUS	INFALLIBILITY:	There	is	a	class	of	propositions	concerning	an	agent's situation,	such	that,	for	any	subject	S,	if	S	is	rational,	these	propositions	will be	true	of	S	if	and	only	if	she	is	certain	of	them. To begin, I will give an argument for the following generalization of GENERALIZED	CONDMAX: 22 See, for example, Caie (2013), Greaves (2013), Pettigrew (2016), Konek and Levinstein (forthcoming),	and	Carr	(ms.). 23	Though	see	Schoenfield	(forthcoming),	section	4,	for	an	alternative	conception	of	how	rationality and accuracy considerations interrelate, which takes accuracy as fundamental, but gives up on RATACC. SUPER	GENERALIZED	CONDMAX:	Let	U	be	a	function	from	a	set	of	propositions	X to credence functions with the intended interpretation that an agent conforming	to	U	adopts	U(Xi)	whenever	the	agent	bears	relation	R	to	Xi.	Let R(Xi)	be	the	proposition	'The	agent	bears	relation	R	to	Xi'. If	the	R(Xi)	form	a partition,	then	the	function,	U,	such	that	conforming	to	U	maximizes	expected accuracy,	is	the	one	that	has	the	agent	conditionalize	on	the	proposition	'the agent	bears	relation	R	to	Xi'	whenever	the	agent	bears	relation	R	to	Xi. Why is this	principle true? Here's the intuitive idea: Suppose	you could choose a credence function that you knew an agent would adopt whenever she bears the relation	R to	some	proposition	P. Even	without	knowing	anything	else	about this relation,	if	you	wanted	her	to	be	as	accurate	as	possible,	the	following	seems	like	a sensible	first	step:	have	her	assign	credence	1	to	the	proposition:	she	bears	relation R to P	whenever she bears relation R to P. For this will guarantee that if she conforms to the procedure, she will assign credence 1 to a truth! What conditionalizing on 'she bears relation R to P' adds to this is just that she'll renormalize	the	rest	of	her	credences	in	response	to	her	newfound	certainty. More formally, note that SUPER GENERALIZED CONDMAX differs from GENERALIZED	CONDMAX	only in that	we	are talking	about	bearing the	R	relation to	a proposition,	rather	than	the	learning	relation,	and	we	are	understanding	what	it is for	an	agent	to	conform	to	U	as	the	agent	adopting	U(Xi)	whenever	she	bears	R	to	Xi instead	of	whenever	she	learns	Xi	. But	the	proof	of	GENERALIZED	CONDMAX	didn't	rely on	any	special feature	of	L.	So	if, instead	of	asking	how	an	agent	should	revise	her credences	as	a	function	of	which	proposition	she	learns,	we	ask	how	an	agent	should revise	her	credences	as	a function	of	which	proposition	she	bears	R	to, it	will turn out	that	the	update	procedure	that	maximizes	expected	accuracy	is	the	one	that	has the	agent	conditionalize	on	R(Xi)	whenever	she	bears	R	to	Xi	. SUPER GENERALIZED CONDMAX explains why the results in this paper are completely	neutral	with	respect	to	one's	understanding	of	'learning'. A	theorist	can take	any	notion	of	learning	that	she's	interested	in	(coming	to	assign	credence	1	to P,	coming	to	know	P,	coming	to	believe	P),	and	partition	an	agent's	possibility	space in accord with the different 'learnings' she might undergo (perhaps including a trivial	instance	of	learning	to	capture	a	case	in	which	no	new	information	is	gained). Then,	SUPER	GENERALIZED	CONDMAX	will	say	that	the	update	procedure	that	maximizes expected accuracy in response to whatever kind of learning takes place is conditionalizing	on	the	proposition	that	the	relevant	kind	of	learning	has	taken	place. We	can	now	generalize	the	result	even	further.	For	it	also	follows	from	G&W that: SUPER-DUPER (SD) GENERALIZED CONDMAX: Consider any partition of propositions	Pi	over	a	set	of	states	Ω.	Let	U	be	a	function	from	Pi	to	credence functions with the intended interpretation that an agent adopts U(Pi) whenever	Pi	obtains.	The	U	that	maximizes	expected	accuracy	is	the	one	that assigns	to	each	Pi	the	credence	function	that	results	from	conditionalizing	on Pi. Proof Let	Pi	be	a	set	of	propositions	that	partition	Ω. It	follows	directly	from	G&W that	the	function	U	that	maximizes	this	quantity: ∑ ∑ p(s)*	A(U(Pi),	s)) Pi	∈Ω s∈	Pi is:	U=Cond(Pi). Note	that	the	quantity	above	represents	the	expected	accuracy	of	an	agent's credences	if	that	agent	adopts	U(Pi),	whenever	Pi	obtains.	It	follows	that	if	we understand conforming to	U	as adopting	U(Pi) whenever Pi obtains, the	U which	is	such	that	conforming	to it	maximizes	expected	accuracy	is	the	one that	has	the	agent	adopt	Cond(Pi)	whenever	Pi	obtains. This result allows us even greater flexibility in terms of how we think about exogenously gaining information because we are no longer required to think of information	gaining	as	an	agent	coming	to	bear	a	relation	to	a	proposition.	Suppose, for example, that one thought that the world doesn't fling single propositions at agents, but sets of	propositions. We then	might ask:	how	should	one revise	one's credences in response to learning a set of propositions? SD-GENERALIZED	CONDMAX entails	that	the	update	procedure	that	maximizes	expected	accuracy	in	response	to learning sets of propositions is the one that has the agent conditionalize on the proposition 'S is the	set	of	propositions that	was learned'	whenever	S is the	set	of propositions learned.	Or suppose that, like	Richard Jeffrey (1992),	one thinks that exogenous learning involves the world shifting around some of an agent's credences.	We	then	might	ask:	how	should	one	revise	one's	credences	in	response	to certain credal	changes taking place? If	we're looking for the update procedure in response	to	credal	changes	that	maximizes	expected	accuracy, the	answer	will	not be	to	Jeffrey-conditionalize. The	answer	will	be	to	regular-old-conditionalize	on	the proposition	'such	and	such	credal	changes	have	occurred'	whenever	such	and	such credal	changes	have	occurred. There is a sense, then, in	which a defender of RATACC can't help but adopt some	version	of	the	truth	rule. For	whatever	one's	theory	of	rationality	is,	one	can partition	the	space	of	possible	situations	an	agent	might	find	herself	in	in	such	a	way that the same	doxastic state is rational in each cell of the	partition. Perhaps, for example,	a	theorist	partitions	the	space	based	on	what	the	agent's	phenomenology is: {She	has	phenomenology	P1, she	has	phenomenology	P2...}	or	what she learns {She learns	X1, she learns	X2...} or what her evidence is: {She possesses E1, She possess	E2...}. Call this	partition,	whatever it is,	P. There	is, then,	a function	from the Pi	∈ P to credence functions that (for this theorist) represents the credence function	that	is	rational	for	an	agent	to	adopt	in	any	given	cell	of	the	partition. We'll call the propositions in P the propositions whose truth determines what credence function	it	is	rational	for	an	agent	to	adopt. Now,	suppose	that	our	theorist	is	a	RATACC-er.	It	follows	from	SD-GENERALIZED CONDMAX that conditionalizing on Pi whenever Pi obtains is the way to assign credence	functions	to	the	members	of	P	that	maximizes	expected	accuracy. So	the RATACC-er	will	think	that	if	Pi	is	true,	a	rational	agent	will	conditionalize	on	Pi	and	so become	certain	that	it	is	true.	The	RATACC-er	must	also	think	that	if	a	rational	agent is certain that	Pi, then	Pi is true. This is	because the	Pi form	a	partition, and so	a rational	agent	will	be	certain	of	at	most	one	Pi. (If	she	were	certain	of	more	than	one Pi,	then	she	would	be	certain	of	two	incompatible	propositions).	We	also	know	that she	will	be	certain	of	at	least	one	Pi,	since	at	least	one	Pi	will	be	true,	and	we	already established that if Pi is true she will be certain that it is (since she will have conditionalized	on	it). It follows	that	she	will	be	certain	of	exactly	one	Pi: the	true one. Thus, for	any	Pi, the	agent	will	be certain that	Pi, if and	only if	Pi is true. In other	words: If RATACC is true, then the propositions whose truth determines what credence	function	it	is	rational	for	an	agent	to	adopt	are	propositions	that	a rational agent is luminously infallible	about	– that is, they	are	propositions that	she	will	be	certain	of	if	and	only	if	they	are	true. We	are	now	in	a	better	position	to	recognize	the	awkwardness	that	arises	in the Greaves and	Wallace framework – an awkwardness that, I believe, reflects a tension in our thinking about these issues	more generally. In defining an update procedure, Greaves and Wallace committed themselves to the view that which credence	function	it	is	rational	for	an	agent	to	adopt	depends	on	which	proposition the agent learns. In other words, for Greaves and Wallace, the Pi – those propositions	whose truth determines	which credence function it is rational for an agent to adopt – are the L(Xi): the propositions describing	which proposition an agent	learns. While	this	seems	like	a	perfectly	plausible	choice	for	one's	Pi,	it	would not	be	plausible	to	suppose	that	the	credence	function	that	it	is	rational	for	an	agent to	adopt	depends	on	whether, for example, it's raining in	Singapore, regardless	of what evidence the agent	has to that effect. Thus, propositions about the	weather conditions in Singapore are	not a plausible choice of Pi. But here's the problem: Intuitively, we think that propositions about the weather in Singapore are propositions	we	might	learn,	and	we	all	grew	up	liking	the	idea	that	conditionalizing on	what	we learn is rational. But	what follows from SD-GENERALIZED	CONDMAX	and RATACC is that	whichever	propositions	are such that their truth determines	which credence function is rational are the propositions that a rational agent will conditionalize on. So a RATACC-er can't think that which of the L(Xi) is true, determines	which	credence	function	is	rational,	and	think	that	the	propositions	that the	agent	will	conditionalize	on	are	propositions	about	the	weather,	unless	she	also thinks	that	the	propositions	about	the	weather	are	equivalent	to	the	propositions	in L(X). This	is	why,	to	get	the	Greaves	and	Wallace	result	–	that	conditionalizing	on the	content	of	what	one	learns	maximizes	expected	accuracy	–	we	must	impose	such severe	restrictions	on	the	possible	contents	of	learning.	For	whatever	these	contents are,	they	must	be	equivalent	from	our	perspective	to	the	proposition	that	we	learn them. 6.	Conclusion I have argued that conditionalization is not the update procedure that, in general, maximizes expected accuracy. The update procedure that maximizes expected accuracy is conditionalization*: conditionalizing on the proposition that one learned P when P is the strongest proposition one exogenously learned. Conditionalizing on P, it turns out, only	maximizes expected accuracy in cases in which	the	agent	is	antecedently	certain	that,	for	all	P	she	might	learn,	if	P	is	true	she will	learn	it,	and	if	she	learns	P,	it	is	true. If	the	rational	update	procedures	are	those	that	maximize	expected	accuracy (that is, if RATACC is true), the fact that conditionalization* maximizes expected accuracy entails that conditionalization* is rational, and if conditionalization* is rational,	then	one	is	rationally	required	to	be	certain	that	one	learned	P	whenever	it is	true	that	one	learned	P. These results are instances	of a yet	deeper	phenomenon.	Anyone	who	accepts RATACC	is	committed	to	the	existence	of	a	class	of	propositions	that	rational	agents will	be luminously infallible	about:	a	class	of	propositions	that	rational	agents	will be	certain	of	if	and	only	if	they	are	true. The results in this paper can thus be summarized as follows: It follows from RATACC	that: (1) If which credence function it is rational to adopt is determined by which proposition one learns, then conditionalizing on the proposition that one learned	Xi,	when	Xi	is	the	proposition	learned,	is	the	rational	way	of	revising one's credences. The class of propositions L(Xi) will be the class of propositions	that	a	rational	agent	is	luminously	infallible	about. (2) If a rational agent regards any proposition Xi that she might learn as equivalent to L(Xi), the claim that she learned it, then the rational update procedure (conditionalizing on L(Xi)) will amount to the same thing as conditionalizing	on	Xi. In	this	case,	the	class	of	propositions	one	might	learn (the	Xi) are also propositions that a rational agent is luminously infallible about. (3) If which credence function it is rational to adopt is determined by some other feature	of	an	agent's	situation,	such	that, for	some	partition	P,	which credence function it is rational for an agent to adopt depends on which member of P is true, then a rational agent will conditionalize on Pi (a member of P) whenever Pi is true. The propositions Pi	∈ P will be the propositions	that	a	rational	agent	is	luminously	infallible	about. Committed	Rat-Accers	might	take	these	results	as	favoring	a	kind	of	foundationalist epistemology	on	which there is	some	privileged	class	of	propositions that	rational agents	will be certain of if and only if they are true. Adamant deniers of such an epistemology	might	take	these	arguments	as	a	reason	to	abandon	the	idea	that	the rational update procedures are those that maximize expected accuracy. But however	we	proceed,	it	is	important	to	be	aware	of	the	extent	to	which	the	thought that	rationality	involves	maximizing	expected	accuracy	and	such	claims	as	LUMINOUS INFALLIBILITY	are	intertwined. They	will,	I	believe,	stand	or	fall	together.24 References Arnold,	A.	(2013).	'Some	Evidence	is	False',	Australasian	Journal	of	Philosophy	91 (1):	165-72 Bronfman,	A.	(2014).	'Conditionalization	and	not	Knowing	that	one	Knows', Erkenntnis	79(4):	871-92 Caie,	M.	(2013).	'Rational	Probabilistic	Incoherence',	Philosophical	Review	122(4): 527-75 Carr,	J.	(ms.).	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