Jennifer Nagel, University of Toronto November 23, 2018 The Psychological Dimension of the Lottery Paradox Penultimate version; final version forthcoming in The Lottery Paradox, Igor Douven, ed., Cambridge University Press. 1. Shifting	and	controversial	intuitions The	lottery	paradox	exposes	some	tensions	in	our	natural	ways	of	thinking	about	probabilities,	and in	how	we	think	about	belief	itself. This	chapter	explores	the	paradox	from	a	psychological	angle, arguing	that	it	arises	from	the	flexibility	of	our	cognitive	capacities	to	represent	(and	reason	about) the	empirical	realm.	A	better	understanding	of	these	capacities	can	give	us	a	clearer	sense	of	our theoretical	options.	Ultimately,	I	take	a	broad	view	of	the	paradox:	in	my	view,	it	can	be	triggered not	only	by	discussion	of	games	with	stipulated	odds,	but	by	topics	of	all	sorts.	However,	it	will	be simplest	to	start	with	an	example	inspired	by	Kyburg's	classic	(1961)	discussion,	in	which	you	hold one	ticket	in	a	fair	lottery,	with	odds	of	(let	us	say)	a	million	to	one,	in	which	the	draw	has	been	held but	the	single	winner	not	yet	announced.	It	is	very	likely	that	your	ticket	has	lost,	but	what	is	the significance	of	this	high	likelihood	for	the	rationality	of	believing	that	your	ticket	has	lost?	If	we insist	that	a	threshold	of	.999999	is	not	high	enough	for	rational	belief,	it	may	seem	we	are	trapping ourselves	in	skepticism:	surely	many	of	the	ordinary	things	we	rationally	believe	about	the	world are	less	certain	than	logical	truths.	On	the	other	hand,	if	we	do	believe	that	this	ticket	has	lost,	by symmetry	we	should	say	the	same	for	any	of	the	other	tickets	in	the	lottery,	and	as	long	as	conjuncttion	of	rational	beliefs	is	a	rational	operation,	it	seems	we	would	be	rational	to	deduce	that	all	the tickets	have	lost,	in	contradiction	to	our	other	beliefs	about	this	fair	lottery. As	a	rational	person	in	this	situation,	do	you	believe	that	your	ticket	has	lost?	Do	you instead	restrict	yourself	to	believing	only	that	it	is	very	likely	to	have	lost?	Do	you	even	sense	a clear	contrast	between	believing	a	proposition,	and	believing	that	this	proposition	is	very	likely? These	are	genuine	questions:	we	will	see	that	there	is	a	strange	controversy	about	what	is	intuitive here.	Whatever	one's	own	intuitive	inclinations	might	be-whether	to	believe	that	the	ticket	has lost,	or	to	believe	only	the	weaker	proposition,	or	to	shift	between	these	stances	in	different contexts-one	may	wonder	about	what	explains	these	inclinations,	why	others	might	feel differently,	and	then	also	how	it	would	be	most	rational	for	us	to	think. 2 One	of	the	many	notable	features	of	the	lottery	paradox	is	that	experts	disagree	not	only	in	their final	theoretical	analyses	of	the	paradox,	but	also	in	their	initial	intuitive	reactions	to	it,	and	their sense	of	how	broadly	these	intuitions	are	shared.	Is	it	intuitively	justified	or	rational	to	believe	that one's	ticket	in	a	large	fair	lottery	has	lost,	given	its	high	chances	of	losing?	According	to	some	surveys	of	the	literature,	prima	facie	intuition	is	widely	taken	to	speak	against	outright	belief.	Julia Staffel	summarizes	the	literature	as	follows,	using	'lottery	proposition'	to	denote	a	proposition	such as	'my	ticket	in	this	fair	million-to-one	lottery	will	lose': The	following	two	constraints	on	what	constitutes	rational	outright	beliefs	have	been	proposed in	the	literature.	They	are	meant	to	capture	our	intuitions	about	the	ordinary	notion	of	belief. (1)	It	is	irrational	to	have	outright	beliefs	in	lottery	propositions;	(2)	It	is	irrational	to	hold	outright	beliefs	based	on	purely	statistical	evidence.	(Staffel	2016,	1725) Staffel	goes	on	to	explain	the	reasoning	behind	rejecting	the	rationality	of	outright	belief	in	lottery propositions,	notwithstanding	their	high	probability.	According	to	Staffel,	"This	response	seems quite	intuitive:	if	I	can	rationally	believe	that	my	ticket	will	lose,	why	buy	it	in	the	first	place?	Why not	throw	it	away?	Moreover,	I	can	rationally	think	I	might	win,	which	is	not	consistent	with believing	I	will	lose"	(Staffel	2016,	1725-6). Not	everyone	shares	Staffel's	perception	that	our	"ordinary	notion	of	belief"	makes	it intuitively	irrational	to	believe	a	lottery	proposition.	John	Hawthorne,	Daniel	Rothschild	and	Levi Spectre	contend	that	"at	least	intuitively,	it	seems	reasonable	to	believe	that	one's	ticket	will	lose," adding	that	they	take	the	initial	interest	of	the	lottery	paradox	to	depend	on	precisely	this	intuition (Hawthorne,	Rothschild	et	al.	2016).	In	a	similar	spirit,	Philip	Ebert,	Martin	Smith	and	Ian	Durbach look	at	the	position	described	by	Staffel	as	the	standard	one	in	the	literature,	and	characterize	it quite	differently.	As	Ebert	and	colleagues	see	it,	there	is	widespread	acknowledgement	in	the literature	that	common	sense	speaks	in	favor	of	belief	in	lottery	propositions:	"As	well	as	denying that	one	can	know	a	lottery	proposition	based	purely	on	the	odds	against	it	winning,	some philosophers	have	gone	further	and	denied	that	one	can	even	justifiably	believe	a	lottery proposition	on	this	basis	[they	cite	Dana	Nelkin,	Jonathan	Sutton,	Martin	Smith,	and	Declan Smithies].	Those	who	endorse	this	claim,	however,	rarely	describe	it	as	'intuitive'	or	widely accepted,	and	almost	universally	regard	it	as	something	standing	in	need	of	substantial	argument" (Ebert,	Smith	et	al.	2018,	2).	The	contrast	with	Staffel's	presentation	of	our	pre-theoretical	starting point	invites	curiosity	about	what	really	is	intuitive	or	widely	accepted. 3 There	are	a	number	of	philosophers	in	each	of	these	camps,	including	for	example	Dylan Dodd	(2017)	on	Staffel's	side,	and	others,	back	to	Henry	Kyburg	(1961),	on	Hawthorne's.	Indeed, the	starting	point	of	Kyburg's	argument	for	the	rationality	of	accepting	lottery	propositions	is	that, despite	their	small	chance	of	being	false,	"everyone	simply	accepts"	statistical	hypotheses	of	just	the sort	that	Staffel	has	suggested	would	be	outlawed	by	"our	ordinary	notion	of	belief."	In	his	defense of	the	reasonableness	of	accepting	that	the	lottery	ticket	has	lost,	Kyburg	writes:	"If	it	is	objected that	this	is	not	reasonable,	on	the	grounds	that	there	is	a	finite	probability	that	ticket	j	will,	after	all, win	the	lottery,	we	can	answer	by	pointing	out	that	according	to	the	same	line	of	reasoning,	there	is a	finite	probability	that	any	statistical	hypothesis	of	the	sort	that	everyone	simply	accepts,	is	false" (Kyburg	1961,	197).	According	to	Kyburg,	there	are	many	propositions	which	are	highly	probable but	not	certain-having,	in	his	view,	"nothing	but	their	probability	to	recommend	them"	(Kyburg 1961,	87)	-where	these	propositions	are	nonetheless	simply	accepted,	or	believed	outright,	by	all rational	people.	His	core	example	of	this	type	of	proposition	is	that	the	sun	will	rise	tomorrow; surely,	he	urges,	it	is	an	unreasonable	form	of	skepticism	to	resist	outright	belief	in	such propositions. We	can	agree	with	Kyburg's	claim	that	only	a	skeptic	would	resist	the	notion	that	the	sun will	rise	tomorrow,	even	while	wondering	whether	this	notion	is	relevantly	similar	to	the	claim	that a	given	lottery	ticket	will	lose.	It	is	not	obvious	how	we	could	pull	apart	the	truths	about	sunrises that	we	want	to	embrace	from	the	truths	about	lotteries	that	will	get	us	into	trouble:	it	seems	there is	no	straightforward	way	to	carve	out	the	problematic	territory	in	terms	of	its	logical	or probabilistic	features	(Douven	and	Williamson	2006).	But	whatever	side	we	are	taking,	we	can	also grant	the	last	point	made	by	Ebert	and	colleagues:	substantial	argument	would	be	needed	to	defend the	rational	legitimacy	of	rejecting	outright	belief	in	a	lottery	proposition,	whether	or	not	rejection was	intuitive.	Any	attempt	to	resolve	this	paradox	will	require	substantial	argument.	But	there	is something	striking	about	the	mismatch	in	initial	perceptions	of	what	seems	at	the	outset	to	be intuitively	rational	(or	reasonable,	or	justified-for	present	purposes	I	take	these	notions	to	be equivalent).	The	divergence	here	marks	a	difference	between	this	paradox	and	the	knowledge version	of	the	lottery	paradox,	where	almost	all	find	it	intuitive	that	we	lack	knowledge	that	the ticket	has	lost,	despite	strong	odds.	Somehow,	the	belief	version	of	the	paradox	has	an	extra	layer	of difficulty	about	its	starting	point,	even	before	we	search	for	possible	solutions. One	might	imagine	that	the	best	way	forward	on	the	question	of	what	is	intuitive	would	be to	turn	to	formal	data	collection,	and	run	surveys	(on	laypeople	or	experts,	or	both)	on	whether 4 they	feel	it	is	rational	to	believe	lottery	propositions.	Indeed,	Ebert	and	colleagues	(2018)	have done	just	this,	finding	majority	support	for	answers	in	line	with	their	view	of	what	is	intuitive, although	interestingly	they	find	a	significantly	weaker	majority	for	the	positive	attribution	of rational	belief	than	for	the	negative	knowledge	attribution.	John	Turri	and	Ori	Friedman	report similar	findings,	with	roughly	80%	of	their	participants	taking	a	person	considering	purchasing	a lottery	ticket	to	be	justified	in	believing	that	it	will	lose,	while	90%	of	their	participants	took	this agent	not	to	know	the	ticket	will	lose	(Turri	and	Friedman	2014).	However,	it	might	be	premature to	take	the	question	as	settled	by	the	majority	response	to	these	studies,	not	least	because	it	is possible	that	naïve	experimental	participants	may	have	understood	the	justification	question	as asking	about	the	pragmatic	(as	opposed	to	the	epistemic)	rationality	of	believing	that	the	ticket	will lose. Another	possibility	is	that	both	sides	in	the	epistemological	debate	are	in	a	sense	right: perhaps	there	are	some	ways	of	speaking	or	thinking	(about	lotteries,	or	about	belief	itself)	that make	it	seem	intuitive	to	say	or	judge	that	a	rational	person	will	believe	that	the	ticket	will	lose,	and there	are	other	ways	of	speaking	or	thinking	that	make	this	seem	irrational.	A	particular	way	of setting	up	the	question	or	running	a	survey	may	tend	to	cue	one	of	these	ways	of	speaking	or thinking	at	the	expense	of	another.	If	this	is	right,	then	the	question	of	what	is	intuitive-either among	experts	or	among	laypeople-remains	an	empirical	question,	but	a	question	with	a	more complex	answer	than	we	might	at	first	have	supposed.	The	main	task	in	what	follows	will	be	to explain	and	defend	the	idea	that	we	have	diverse	ways	of	speaking	and	thinking	here,	giving	a descriptive	account	of	their	core	descriptive	features	before	turning	to	the	normative	question	of their	rational	merits. With	a	better	understanding	of	our	resources	for	detecting	and	speaking about	belief,	and	perhaps	also	a	deeper	understanding	of	belief	itself,	we	will	be	better	positioned	to find	a	safe	path	between	the	threat	of	skepticism	and	the	threat	of	odd	restrictions	on	logical reasoning. 2.	Speaking	of	believing,	and	thinking If	we	want	to	understand	our	intuitive	ways	of	thinking	about	belief,	one	possible	source	of	data could	be	our	spontaneous,	everyday	use	of	the	verb	'believe',	and	similar	verbs.	Some	researchers are	inclined	to	make	much	of	this	data.	Kevin	Dorst,	for	example,	argues	that	linguistic	evidence decisively	favors	the	Lockean	view,	according	to	which	there	is	nothing	more	to	belief	than	high 5 confidence	(Dorst	2018).	Following	Hawthorne,	Rothschild	and	Spectre	(HRS),	Dorst	takes	the proposition-embedding	sense	of	the	verb	'think'	to	be	the	most	natural	expression	of	belief	in English;	Dorst	also	agrees	with	HRS	that	this	sense	of	'think'	is	synonymous	with	'believe,'	on	the strength	of	phenomena	such	as	the	incongruity	of	saying	things	like,	"He	thinks	that	p	but	it's	not	as if	he	believes	that	p."	In	Dorst's	view,	although	the	relevant	senses	of	'think'	and	'believe'	express the	same	mental	state,	our	use	of	'think'	is	an	even	better	guide	to	our	ordinary	intuitive	understanding	of	belief	than	our	use	of	'believe',	both	because	'think'	is	"more	colloquial"	and	because philosophers	use	'believe'	in	perhaps	technical	expressions	such	as	'believe	outright'	and	'fully believe',	possibly	contaminating	our	intuitions	about	belief-talk	with	theory	(Dorst	2018,	3). When we	look	at	how	'think'	is	ordinarily	used,	Dorst	contends,	we	will	see	that	it	is	utterly	routine	to equate	high	confidence	with	belief.	For	example,	he	observes	that	having	heard	a	forecast	that	there is	a	70%	chance	of	rain	makes	it	seem	fine	to	answer	the	question	of	whether	it	will	rain	with,	"I think	it	will."	Equally,	one	can	say	of	a	lottery	ticket	holder,	"I	think	her	ticket	will	lose,"	without	any obvious	irrationality.	In	his	view,	"We	are	constantly	having	to	figure	out	what	to	think	based	on merely	statistical,	inconclusive	evidence-and	outside	the	philosophy	room	we	do	so	without	pause or	compunction"	(Dorst	2018,	5).	If	we	apply	the	T-schema	to	ordinary	language,	with	a	charitable presumption	of	common	rationality,	Dorst	concludes,	we	will	see	that	there	is	nothing	irrational about	what	he	takes	to	be	the	common	practice	of	believing	lottery	propositions. Dorst	is	certainly	right	that	'think'	is	more	common	than	'believe':	in	a	balanced	corpus	of written	and	spoken	English,	these	are	the	12th	and	50thmost	common	verbs,	respectively	(Davies and	Gardner	2010,	317),	and	in	spoken	language,	'think'	is	more	than	six	times	as	common	as 'believe'	(Davies	2008-,	accessed	November	13,	2018).	Focusing	on	the	first-person,	one	recent study	found	'I	think'	to	be	used	roughly	once	every	three	minutes	in	unplanned	American	English speech,	and	every	six	minutes	in	planned	speech;	it	is	taken	to	be	the	most	frequent	'epistemic phrase',	while	'I	believe'	does	not	figure	in	the	top	ten	(Kärkkäinen	2010).	Given	its	relative frequency,	'think'	has	a	good	claim	to	being	the	default	nonfactive	epistemic	mental	state	verb	in English,	and,	if	we	restrict	ourselves	to	the	sense	that	embeds	a	propositional	complement,	its interchangeability	with	'believe'	is	largely	regarded	as	unproblematic	in	the	mental	state attribution	literature.1 1	It	is	an	interesting	question	why	philosophical	discussions	focus	on	'believe';	I	suspect	that	this	choice	may be	driven	by	the	easier	nominalization	and	by	a	desire	to	keep	clear	of	any	ambiguities	arising	from	the	sense of	'think'	that	means	'ruminate'. 6 What	is	more	controversial	is	whether	the	nature	of	belief	can	be	read	directly	off	our everyday	use	of	these	verbs.	One	worry	about	Dorst's	first-person	examples	in	particular	is	that both	'I	think'	and	'I	believe'	are	most	frequently	used	not	to	self-attribute	a	mental	state,	but	to modulate	an	assertion.	For	example,	the	conspiracy	theorist	who	says,	"I	think	it	was	Bush	who planned	the	9/11	attacks"	is	ordinarily	heard	as	having	said	something	false	about	the	attacks,	not something	true	about	his	or	her	own	mental	state.	Note	that	the	'I	think'	here	could	easily	be	moved to	medial	or	final	position	in	the	sentence,	as	befits	an	adverbial	expression.	While	it	is	true	that initial	'I	believe'	and	'I	think'	can	have	main	clause	status,	this	status	is	rare,	comprising	less	than five	percent	of	the	instances	in	a	recent	study	of	spoken	usage	(Kaltenböck	2009).	Both	expressions much	more	commonly	appear	as	parentheticals	with	an	adverbial	use	modifying	the	speaker's commitment,	or	a	discourse-marking	use,	said	to	carry	little	of	the	phrase's	original	semantic meaning.	There	are	a	number	of	rival	theories	about	the	proper	taxonomy	here.	Gunther Kaltenböck	(2010)	identifies	four	different	functions	of	parenthetical	'I	think':	(1)	shield,	(2) approximator,	(3)	structural,	and	(4)	booster.	The	first	of	these	uses	is	by	far	the	most	common (over	80%	of	the	instances	in	his	2010	study	of	parenthetical	'I	think').	As	a	shield,	'I	think'	has	the function	of	a	hedge,	or	"an	epistemic	marker	expressing	the	speaker's	tentativeness,	lack	of certainty	or	commitment	with	regard	to	the	truth	value	of	its	host"	(Kaltenböck	2010,	248).	As	an approximator,	'I	think'	takes	phrasal	rather	than	clausal	scope,	inserted	before	precise	expressions such	as	dates	and	measurements-as	in,	"the	story	was	in	I	think	the	October	issue"-and	serves "to	trigger	the	need	for	greater	fuzziness	or	vagueness"	(2010,	254).	The	'structural'	or	discoursemarking	function	of	'I	think'	is	harder	to	pin	down,	not	least	because	its	scope	is	unclear.	Kaltenböck observes	that	'I	think'	disproportionately	occurs	amidst	disfluencies,	surrounded	by	pauses, backtracking	and	hesitation	sounds	(uh,	um);	in	his	view,	'I	think'	often	functions	as	some	type	of filler,	bleached	of	literal	meaning,	perhaps	signaling	the	speaker's	effort	to	maintain	their conversational	turn,	or	it	may	serve	some	linking	function	between	complex	topics	and	comments. The	booster	function	is	intriguing:	with	the	right	pitch	movement	and	emphasis	on	the	personal pronoun,	'I	think'	can	express	heightened	rather	than	diminished	commitment	to	the	host proposition.	In	Kaltenböck's	view,	the	booster	function	is	possible	for	subjective	(for	example, aesthetic	or	political)	evaluations	("I	think	Opera	North	is	always	inventive"),	where	the	hedging shield	function	is	invoked	for	'verifiable'	matters	("I	think	John	is	in	London")	(2010,	264). Nicole	Dehé	and	Anne	Wichmann	argue	that	these	function	types	are	distinguished	from	each other	by	prosody,	both	for	'think'	and	for	'believe'.	In	their	corpus	study	of	sentence-initial	I	think (that)	and	I	believe	(that),	they	argue	that	initial	'I	think'	(or	'I	believe')	has	main	clause	status	(they 7 locate	the	boost	function	here)	when	the	speaker	stresses	the	pronoun;	it	functions	as	a	hedging parenthetical	or	comment	clause	when	the	verb	is	stressed	(sometimes	the	pronoun	is	even dropped	in	this	usage),	and	as	a	discourse	marker	when	both	pronoun	and	verb	are	unstressed (Dehé	and	Wichmann	2010).	This	approach	suggests	one	possible	step	towards	an	explanation	for our	philosophical	divergence	over	what	is	initially	intuitive. With	a	difference	of	emphasis	of	the sort	that	is	not	usually	marked	in	writing,	the	following	sentences	can	be	read	as	making	quite different	claims: (1) I	think	this	lottery	ticket	will	lose. (2) I	think	this	lottery	ticket	will	lose. In	a	context	where	the	long	odds	are	public,	tentative-sounding	(1)	sounds	good.	But	because	the	'I think'	here	functions	as	a	parenthetical	hedge	indicating	that	what	follows	is	only	probable,	rather than	as	a	main	clause	self-attribution	of	a	mental	state,	there	is	limited	value	in	the	fact	that	(1) sounds	like	a	truth	spoken	by	a	rational	person.	We	cannot	simply	follow	Dorst's	advice	to	reason "according	to	the	T-schema"	(2018,	3),	and	conclude	that	a	rational	person	believes	the	ticket	will lose,	full	stop.	By	contrast,	the	bolder	declaration	(2)	does	make	a	mental	state	claim:	here	'I	think" has	main	clause	status.	However,	in	contrasting	the	speaker's	epistemic	position	with	the	hearer's, (2)	seems	more	questionable	if	the	speaker	does	not	have	private	information	warranting	a	deeper pessimism	than	the	publically	known	low	chance	of	winning.	If	Dorst	is	focusing	on	cases	like	(1) while	Staffel	is	focusing	on	cases	like	(2),	they	are	responding	to	different	stimuli,	although	it	remains	to	be	seen	what	the	significance	is	for	their	underlying	disagreement	about	rational	belief itself,	and	what	happens	when	we	switch	to	the	third-person	perspective. There	are	a	number	of	theories	about	how	and	why	the	phrases	'I	think'	and	'I	believe'	have evolved	to	serve	this	variety	of	functions.	It	is	widely	noted	that	there	is	something	oddly	redundant about	overtly	tagging	a	statement	with	'I	think':	as	Graham	Ranger	puts	it,	"the	operation	involved in	I	think	is	implied	in	the	act	of	assertion,	in	any	case,	as	part	of	the	enunciative	scenario.	The question	therefore	is	to	determine	what	the	explicit	presence	of	the	string	I	think	adds	to	an utterance"	(Ranger	2018,	282).	In	Ranger's	view,	typical	contexts	of	assertion-for	example, discussions	of	whether	John	is	in	London-involve	a	shared	default	orientation	towards	knowledge that	will	set	the	speaker's	overt	use	of	'think'	in	contrast	with	the	stronger	'know',	generating	a tentative	effect.	Meanwhile,	in	more	subjective	evaluative	contexts,	such	as	discussions	of	music	or politics,	overt	'I	think'	is	understood	in	contrast	to	'they	think'	or	'you	think',	boosting	the	speaker's commitment	to	what	is	said	(2018,	281-2).	This	reading	fits	with	Dehé	and	Wichmann's 8 observations	about	the	difference	between	prosodic	emphasis	on	the	verb	and	subject,	and Kaltenböck's	position	on	the	contexts	where	'I	think'	serves	to	boost	rather	than	hedge.	However, we	have	at	best	the	start	of	a	theory	here,	leaving	much	unexplained,	including	cases	like	our	(2),	a boosted	but	not	especially	subjective	or	aesthetic	evaluation.	In	summation,	it	remains controversial	exactly	what	range	of	functions	is	served	by	'I	think',	but	it	is	widely	agreed	that	these functions	are	various,	and	that	many	of	them	involve	bleaching	of	the	literal	mental	state	meaning of	'think'	(on	this	point,	see	also	Van	Bogaert	2011).	Where	Dorst	argued	that	ordinary	usage	of 'think'	could	be	a	direct	guide	to	the	nature	of	belief,	it	appears	that	considerable	caution	is	in	order. 3.	The	difficulty	of	belief	attributions,	and	beliefs	themselves It	might	seem	that	we	could	make	things	easy	by	switching	to	third-person	attributions.	Focusing on	a	third	party	who	knows	that	it	is	very	likely	that	the	ticket	has	lost,	we	can	ask	whether	this person	would	be	rational	to	believe	that	the	ticket	has	lost.	It	is	possible	we	could	still	suffer	some contamination	from	formulaic	uses	of	'I	think'	to	mark	weakened	commitment,	for	example	if	in evaluating	the	mindset	of	a	third	party	we	imagine	them	saying	"I	think	this	ticket	will	lose",	or	if	we imagine	ourselves	in	their	shoes,	and	ask	whether	we	could	reasonably	say	such	a	thing.	Furthermore,	there	is	also	an	evidential	use	of	third-person	mental	state	language.	Mandy	Simons	(2007) notes	that	one	might	answer	the	question,	"Who	was	Louise	with	last	night?"	with	something	like, "Henry	thinks	that	she	was	with	Bill." To	choose	this	answer	rather	than	the	simpler	"She	was	with Bill"	is	one	way	the	speaker	could	signal	diminished	commitment	to	the	embedded	clause,	Simons observes.	Given	that	it	is	the	embedded	clause	that	is	an	answer	to	the	question,	this	clause	has main	point	status:	if	we	are	concerned	with	Louise's	company,	not	Henry's	state	of	mind,	our	talk	of thinking	serves	chiefly	to	mark	a	tentative	attitude	to	the	key	content.	Simons	does	hold	that	the speaker	is	still	making	an	assertion	about	Henry's	mental	state	here,	but	it	is	an	open	question whether	this	evidential	use	of	the	mental	state	verb	clouds	the	significance	of	our	intuitions	about third-person	belief	attributions. But	even	if	we	do	not	suffer	this	kind	of	linguistic	interference,	explaining	how	we intuitively	judge	the	beliefs	of	third	parties	remains	a	tricky	business.	There	is	no	question	that belief	attribution	and	evaluation	forms	a	major	part	of	mature	human	social	intelligence;	indeed,	it is	argued	that	the	capacity	to	attribute	beliefs	(as	opposed	to	just	states	of	knowledge	and ignorance)	lies	at	the	very	core	of	what	sets	us	apart	from	other	socially	intelligent	animals 9 (Tomasello	2018).	Spontaneously,	and	often	without	any	deliberate	effort,	we	see	other	agents	as having	beliefs,	sometimes	misconceptions	or	accidentally	correct	opinions,	sometimes	states	of mind	we	evaluate	more	positively.	We	rely	on	a	rich	set	of	cues	in	belief	attribution,	including	signs of	what	others	can	perceive	and	have	perceived	in	the	recent	past,	what	they	say,	and	how	they	act. This	is	a	complex	matter,	not	least	because	these	cues	can	point	in	different	directions	(for	some interesting	complications,	see	Nyarko	and	Schotter	2002,	Schlag,	Tremewan	et	al.	2015). Because	the	processes	that	support	spontaneous	belief	attributions	are	not	generally	open to	introspection,	it	takes	significant	work	to	develop	a	theory	of	how	we	are	attributing	and evaluating	the	beliefs	of	others.	In	fact,	there	are	even	questions	about	the	nature	and	extent	of	our access	to	our	own	beliefs.	According	to	one	prominent	theory	of	mental	state	attribution,	Peter Carruthers's	Interpretive	Sensory	Access	(ISA)	model,	even	in	the	first-person	case	we	gauge	belief indirectly,	from	behavioral	cues	(Carruthers	2011).	While	the	range	of	data	we	can	draw	on	in	the first-person	case	is	different,	because	it	includes	inner	as	well	as	outer	speech,	and	direct	access	to phenomenal	states	such	as	hunger	and	pain,	the	ISA	theory	maintains	that	we	apply	the	same mindreading	processes	to	ourselves	as	to	other	people	in	attributing	propositional	attitudes:	for example,	querying	(inner	or	outer)	speech,	and	interpreting	what	we	hear	or	find	ourselves	saying in	our	minds.	According	to	the	ISA,	mindreading	involves	rapid	processes	of	unconscious	inference. We	instinctively	model	ourselves	as	being	self-transparent	or	having	unmediated	access	to	our	own propositional	attitudes,	and	these	simplified	models	are	generally	unproblematic	because	we	are largely	accurate	in	our	self-attributions;	however,	the	ISA	model	maintains	that	we	can	be	mistaken about	what	we	believe	or	want.	Carruthers	supports	his	theory	by	reviewing	a	large	body	of empirical	literature	on	manipulations	and	special	circumstances	that	can	make	people misunderstand	their	own	beliefs	and	motivations,	separating	their	self-conception	from	the	real underpinnings	of	their	agency	(Carruthers	2011,	ch.	11).	If	the	ISA	model	is	right,	we	are	able	to self-attribute	mental	states	not	by	direct	conscious	awareness	of	the	propositional	attitudes themselves,	but	through	an	interpretive	process	anchored	in	our	awareness	of	the	effects	of	these attitudes	in	(inner)	speech	and	outer	behavior.	I	consider	the	ISA	model	to	be	the	most	plausible explanation	of	our	existing	empirical	evidence,	including	phenomenological	evidence,	and	I	also think	that	it	can	support	a	new	way	of	making	progress	on	the	lottery	paradox,	so	I	will	be	assuming it	in	what	follows.	Those	who	are	initially	unconvinced	by	Carruthers	might	see	some	progress	on the	lottery	paradox	as	a	reason	to	reconsider	the	merits	of	the	ISA;	failing	that,	there	may	still	be some	observations	of	interest	in	what	follows,	as	not	everything	in	my	approach	to	the	problem	will depend	directly	on	the	ISA. 10 Meanwhile,	there	is	a	delicate	relationship	between	the	nature	of	belief	and	the	evidence	we can	draw	from	the	behavior	of	agents.	If	someone	keeps	a	lottery	ticket,	rather	than	throwing	it away,	one	could	read	this	either	as	a	sign	that	they	do	not	believe	outright	that	it	will	lose,	or perhaps	instead	as	a	sign	that	their	practical	reasoning	is	guided	by	degrees	of	belief	rather	than binary	states,	and	they	have	rationally	calculated	a	positive	expected	value	in	holding	onto	the ticket	(on	the	significance	of	this	line	of	thought	to	the	knowledge	version	of	the	lottery	paradox, see	Douven	2008).	The	latter	way	of	reading	the	action	leaves	it	open	that	the	rational	agent	might believe	the	ticket	will	lose,	assuming	high	probability	is	sufficient	for	outright	belief. Because	the	relationship	between	belief,	credence	and	action	is	so	heavily	contested,	it seems	that	an	easier	place	to	start	investigating	the	attribution	and	evaluation	of	binary	and	graded mental	states	could	be	the	domain	of	explicit	reasoning,	where	it	is	easier	to	see	what	we	are working	with.	Does	the	rational	person	reason	explicitly	in	terms	of	binary	or	graded	states?	Here again,	the	answer	may	not	be	simple:	arguably,	we	can	do	both.	In	a	series	of	recent	papers, Jonathan	Weisberg	has	defended	the	view	that	both	binary	and	graded	representations	figure	in rational	thought,	and	neither	is	reducible	to	the	other	(Weisberg	2013,	Weisberg	forthcoming). Depending	on	the	circumstances,	we	can	engage	either	in	thinking	which	takes	categorical	claims	as input,	such	as	syllogistic	reasoning	or	processes	such	as	Elimination	By	Aspects,	or	we	can	engage in	probabilistic	calculations,	assigning	subtle	credal	weights	to	relevant	factors.	Indeed,	as	Weisberg has	pointed	out,	some	interesting	general	models	of	rational	judgment,	such	as	the	Evidence Accumulation	Model	(Newell	and	Lee	2011),	take	both	binary	and	weighted	input	at	different stages	of	processing. Explicit	reasoning	is	also	a	domain	in	which	intuitive	evaluations	of	the	rationality	of	belief can	separate	from	intuitive	evaluations	of	knowledge:	the	person	who	engages	in	valid	reasoning from	premises	she	could	not	have	known	were	false	might	still	be	evaluated	as	rational,	even	if	she fails	to	attain	knowledge	from	her	reasoning.	Special	attention	to	this	feature	of	inference	should help	us	understand	the	relationship	between	the	knowledge	version	of	the	lottery	paradox	and	the rational	belief	version.	Our	first	step	will	be	to	look	at	the	emergence	of	both	forms	of	the	paradox in	contexts	of	switching	from	thinking	unreflectively	to	engaging	in	explicit	reasoning,	and	here	we will	turn	to	an	example	involving	subject	matter	beyond	lotteries. 11 4.	Intuitive	and	reflective	cognition Who	is	the	current	pope?	For	an	easy	trivia	question	like	this,	an	answer	comes	to	mind	without conscious	effort	or	reflection.2	At	the	time	of	writing,	the	answer,	which	we	will	call	proposition	p,	is that	the	current	pope	is	Pope	Francis.	Notoriously,	however,	it	is	possible	for	me	to	induce	conscious reflection	in	the	vicinity	of	proposition	p	in	a	way	that	can	trigger	a	different	frame	of	mind	about	it. For	example,	we	could	review	mortality	statistics	for	the	relevant	demographic:	according	to	the World	Health	Organization's	most	recent	report,	men	of	Pope	Francis's	advanced	age	in	his	world region	have	a	background	annual	mortality	rate	of	6.9%	(WHO	2016).	Other	things	being	equal, one's	credence	that	any	such	man	has	not	died	in	the	last	hour	should	be	about	.9999918384.	On the	question	of	whether	other	things	are	equal,	one	might	also	reflect	on	the	risk	of	assassination and	airplane	crashes	for	a	very	prominent	and	somewhat	controversial	leader	who	travels	frequently.	It	is	extremely	unlikely	that	anything	untoward	has	happened	to	Pope	Francis	since	the last	time	our	news	feeds	were	updated,	but	he	is	a	man,	all	men	are	mortal,	and	few	of	us	know	the precise	hour	of	any	remote	individual's	passing.	Having	accepted	that	it	is	not	impossible	that Francis	has	died	in	the	last	hour,	we	might	then	attend	to	the	further	fact	that	if	he	has	died,	we have	entered	a	papal	interregnum,	making	it	no	longer	the	case	that	Francis	is	the	current	pope. With	these	facts	in	mind,	let	us	revisit	our	attitude	to	the	proposition	p:	the	current	pope	is Pope	Francis,	where	this	is	explicitly	recognized	to	entail	that	it	is	not	the	case	that	this	man	has died	in	the	last	hour.	How	are	we	inclined	to	think	about	this	proposition	now?	As	philosophers	at least	since	Jonathan	Vogel	(1990)	have	observed,	when	we	reflect	this	way,	the	proposition	no longer	seems	to	be	known.	Before	we	turn	to	the	question	of	whether	our	current	frame	of	mind allows	rational	belief	in	the	proposition,	we	can	examine	the	nature	of	this	frame	of	mind,	and	the reasons	why	we	tend	to	feel	knowledge	has	lapsed. If	my	example	worked	as	intended,	the	initial	presentation	of	a	question	about	the	identity of	the	Pope	triggered	an	answer	immediately,	without	conscious	reflection	on	any	supporting	(or potentially	undermining)	considerations.	The	second	time	the	question	was	posed,	it	was	after attention	had	been	directed	to	a	series	of	premises	presented	as	relevant	to	the	issue.	There	is	a psychologically	significant	contrast	between	the	two	ways	of	thinking	employed	in	these	successive moments,	ways	of	thinking	now	commonly	labelled	as	Type	1	(intuitive)	and	Type	2	(reflective) 2	Any	readers	who	encounter	resistance	with	this	example	should	construct	another	they	find	easier;	any readers	within	the	pope's	immediate	ambit	should	construct	an	example	within	the	realm	of	background trivia	rather	than	current	experience. 12 cognition.3	They	differ	in	a	variety	of	ways,	with	intuitive	processes	operating	in	a	swift	and autonomous	fashion,	where	reflective	thinking	is	generally	slower	and	often	controlled.	It	is	now widely	thought	that	the	ultimate	basis	of	the	contrast	concerns	the	involvement	of	our	limited capacity	central	working	memory	resource	(Evans	and	Stanovich	2013,	Carruthers	2015).	Working memory	keeps	contents	in	mind	or	consciously	available	during	complex	thought,	enabling 'cognitive	decoupling',	or	representations	of	alternatives,	or	hypothetical	states	of	affairs	potentially at	odds	with	what	the	subject	is	currently	experiencing	(Baddeley	2007,	Stanovich	2011).	To	take	a classic	example	(inspired	by	Sloman	1996),	when	intuitive	cognition	solves	a	slightly	shuffled anagram	like	"auotmonous",	the	answer	("autonomous")	springs	to	mind	with	no	consciousness	of intermediary	stages;	when	reflective	cognition	solves	a	more	complex	and	deeply	shuffled	anagram like	"tebardelie",	there	are	successive	conscious	presentations	of	possible	solutions	in	which	the	letters	are	re-ordered,	where	each	such	presentation	is	evaluated	intuitively	as	a	non-word,	and	further	permutations	are	brought	to	mind,	until	the	solution	is	found	("deliberate").	The	two	types	of thinking	are	deeply	integrated:	reflective	cognition	can	answer	new	questions	whose	answers	are not	automatic	for	us,	but	it	does	so	through	organizing	a	series	of	cycles	of	intuitive	cognition. Although	reflective	cognition	is	sometimes	characterized	as	"conscious",	both	types	of	thinking incorporate	a	substantial	amount	of	processing	below	the	radar	of	consciousness,	such	as	the	processes	that	govern	the	allocation	of	attention	and	the	retrieval	of	relevant	memories	(Evans	2009). Some	problems	can	be	answered	through	either	way	of	thinking,	but	partly	because	reflective thinking	is	effortful,	our	default	is	to	answer	intuitively	or	take	for	granted	the	most	plausible	state of	affairs,	expending	the	effort	to	consider	alternative	possibilities	only	when	prompted	(Evans 2007,	Stanovich	and	West	2008). The	distinction	between	categorical	and	probabilistic	judgments	is	orthogonal	to	the distinction	between	intuitive	and	reflective	cognition:	the	suggestion	here	will	not	be	simply	that intuition	is	always	categorical,	or	that	reflection	on	its	own	cues	probabilistic	thinking.	Either	type of	thinking	can	be	engaged	with	categorical	content,	for	example	in	solving	arithmetical	problems either	automatically	or	deliberately.	Although	very	easy	problems	like	single-digit	addition	are typically	solved	in	intuitive	cognition	by	the	automatic	production	of	an	overlearned	answer,	they can	be	solved	through	reflective	thinking,	for	example	by	application	of	deliberate	counting procedures.	Either	type	of	thinking	can	also	be	engaged	to	thinking	about	probabilities,	with	some differences	worth	watching.	Intuitive	thinking	is	more	common	with	verbal	scales	of	likelihood 3	I	am	following	the	recent	joint	work	of	Jonathan	Evans	and	Keith	Stanovich	(2013)	in	using	the	adjectives "intuitive"	and	"reflective"	to	denote	Type	1	and	Type	2	cognition,	respectively. 13 ('very	likely'),	and	switching	from	these	qualitative	scales	to	numerical	scales	('80%	chance')	tends to	shift	us	towards	reflective	thought	(Windschitl	and	Wells	1996).	However,	it	remains	possible	to engage	in	reflective	cognition	with	qualitative	probabilities,	for	example	when	the	premises	of	one's explicit	syllogistic	reasoning	include	modal	terms	such	as	'likely'. Still,	this	is	not	to	say	that	either type	of	cognition	can	be	used	for	any	problem:	reflective	cognition	becomes	mandatory	when	we need	to	engage	in	controlled	reasoning	about	precise	numerical	representations	of	likelihoods, given	novel	problems	to	which	no	answer	has	previously	been	learned.	Differences	in	our spontaneous	epistemic	evaluations	of	intuitive	and	reflective	cognition	can	play	a	role	in	explaining diverse	responses	to	the	lottery	paradox	if	subjects	can	differ	in	presenting	the	key	proposition	in the	paradox	as	generated	by	one	form	of	cognition	or	the	other. One	final	psychological	factor	that	seems	relevant	to	the	lottery	paradox	is	the	distinction between	'verbatim'	and	'gist'	representations	of	probabilities	(Reyna	2004).	Precise	verbatim representations	enable	detailed	calculations,	but	they	are	hard	to	retain	and	are	handled	through slow	serial	processing;	fuzzy	gist	representations	may	distort	fringe	cases	but	are	more	easily retained	and	can	be	handled	through	swifter	parallel	processing.	In	the	rougher	intuitive	format	of gist	representation,	"very	small	risks	are	edited	to	'essentially	nil	risk'"	(Reyna	2004,	63). Something	like	gist	representation	may	be	behind	the	phenomenon	of	insurance	resistance	for	lowprobability	harms:	for	example,	in	simulated	management	games,	people	are	(surprisingly)	more than	twice	as	likely	to	purchase	$500	insurance	for	a	.25	probability	of	a	$1,980	loss	than	for	a	.002 probability	of	a	$247,500	loss	(Slovic,	Fischhoff	et	al.	1977).	If	precise	and	controlled	reasoning were	in	effect	here,	one	would	not	expect	to	see	a	lowered	tendency	to	secure	coverage	for	the more	catastrophic	prospect	of	equal	expected	value.	The	intuitive	tendency	to	write	off	tiny probabilities	as	zero	also	means	that	we	respond	differently	to	presentations	of	equivalent	risks over	different	time	horizons.	In	one	classic	study	of	seat	belt	use,	participants	were	presented	with either	the	increased	risk	of	disabling	injury	for	travel	without	a	seat	belt	for	each	trip	taken (.00001)	or	the	equivalent	risk	over	a	lifetime	of	driving	(.33).	Only	10%	of	those	exposed	to	the single	trip	statistics	indicated	that	their	seat	belt	use	would	change	as	a	result,	versus	39%	of	those exposed	to	the	lifetime	statistics	(Slovic,	Fischhoff	et	al.	1978).	While	these	particular	examples	are generally	taken	as	evidence	that	gist	representations	are	normatively	problematic,	Reyna	argues that	reliance	on	gist	often	increases	the	accuracy	of	reasoning:	where	children	and	novices	can	be misled	by	incidental	features	of	verbatim	representations,	adults	and	experts	in	fields	such	as cardiology	develop	stronger	capacities	to	represent	risks	intuitively,	typically	increasing	the consistency	of	their	judgments	in	doing	so	(Reyna,	Lloyd	et	al.	2003).	This	observation	is	in	line 14 with	a	larger	program	concerning	the	ecological	validity	of	heuristic	thinking:	given	limitations	in time,	attention	and	computational	skill,	there	are	many	circumstances	in	which	human	beings reasoning	heuristically	on	the	basis	of	a	subset	of	available	information	will	be	more	accurate	than those	attempting	complex	procedures	to	weigh	all	available	evidence	in	full	precision	(Gigerenzer and	Brighton	2009).	The	moment	of	making	a	decision	about	the	purchase	of	insurance	is	not	the time	to	discard	a	remote	possibility	as	the	kind	of	thing	that	doesn't	happen,	but	in	daily	life	it	could be	taxing	to	the	point	of	paralysis	to	try	to	make	ordinary	judgments	with	maximally	precise weights	attached	to	all	conceivable	possibilities,	and	overall	higher	accuracy	in	judgment	could	very plausibly	be	secured	through	a	general	practice	of	reliance	on	gist	representation.	Indeed	in	many situations	we	have	no	choice	but	to	rely	on	gist:	verbatim	representations	(of	probabilities,	or	of situations	more	broadly)	are	hard	to	retain,	and	in	practice	as	time	elapses,	gist	representations tend	to	dominate	in	recall	and	assertion	(Goldsmith,	Koriat	et	al.	2002). 5.	The	knowledge	version	of	the	paradox We	are	now	in	position	to	return	to	our	question	about	the	pope.	In	an	earlier	paper	(Nagel	2011),	I argued	that	shifts	between	intuitive	and	reflective	cognition	help	to	explain	the	knowledge	version of	the	paradox:	someone	who	initially	knows	that	Francis	is	the	current	pope	might	be	unable	to come	to	know	that	Francis	has	not	died	in	the	last	hour,	even	when	she	recognizes	that	this	latter proposition	is	entailed	by	the	former.	Focusing	on	the	question	of	whether	Francis	might	have	died in	the	last	hour	can	undermine	one's	prior	knowledge	that	he	is	the	current	pope,	I	argued,	either by	destroying	this	knowledge	through	weakening	confidence,	or	simply	by	thwarting	access	to	it	in occurrent	judgment,	making	it	impossible	for	her	to	use	this	prior	knowledge	as	the	premise	of	an appropriate	deduction.	It	was	a	starting	point	of	this	earlier	paper	that	either	kind	of	cognition	can yield	knowledge:	paradigm	sources	of	knowledge	include	both	intuitive	processes	such	as	face recognition	and	reflective	processes	such	as	sound	deduction.	In	fact,	notwithstanding	the	psychological	differences	between	these	modes	of	thought,	what	makes	a	judgment	of	either	type	count	as knowledge	could	ultimately	be	the	same	across	the	board	(for	example,	that	one	is	arriving	safely	at the	truth).	Even	so,	it	is	sometimes	possible	for	a	person	to	know	a	proposition	only	when	she judges	that	proposition	in	one	manner	rather	than	the	other:	for	example,	if	she	has	unreliable intuitions	in	a	given	domain	but	is	capable	of	accurate	calculations	when	she	thinks	reflectively,	or sound	intuitions	but	unreliable	powers	of	reflection. 15 Pairs	of	judgments	like	those	about	the	pope-this	earlier	paper	labels	these	as	instances	of the	"Harman-Vogel	paradox"-exploit	a	special	kind	of	disparity	in	what	we	can	judge	intuitively and	reflectively.	What	is	special	about	the	propositions	that	feature	in	the	Harman-Vogel	paradox	is that	the	subject	is	asked	to	make	reflective	judgments	that	question	some	standing	presupposition behind	the	corresponding	intuitive	judgment.	Ordinary	intuitive	judgements	about	generally	stable environmental	facts	like	the	identity	of	world	leaders	are	made	on	the	basis	of	semantic	memory.	In answering	questions	about	these	facts,	we	typically	presuppose	that	what	we	recall	was	originally anchored	in	knowledgeable	testimony,	and	that	the	world	remains	relevantly	as	it	was	when	we came	to	know	what	we	now	recall.	When	the	appropriate	source	and	world	stability	conditions	are taken	by	an	evaluator	to	be	met,	intuitive	judgments	easily	register	as	instances	of	knowledge.	But these	are	presuppositions	as	opposed	to	conditions	consciously	available	to	the	subject:	as	subjects, we	are	typically	unable	to	recall	to	consciousness	which	original	source	furnished	us	with	any particular	trivia	fact	we	recall	intuitively,	let	alone	evaluate	the	quality	of	that	source	or	collect consciously	available	materials	to	support	non-question-begging	explicitly	reasoned	judgments about	the	relevant	stability	in	the	environment. The	question	introducing	our	example	("Who	is	the	current	pope?")	presupposed	that	we were	not	in	a	papal	interregnum.	When	we	were	invited	to	consider	the	answer	to	that	question ("The	current	pope	is	Pope	Francis")	a	second	time,	this	presupposition	was	no	longer	available, given	our	active	consideration	of	the	hypothetical	possibility	of	Pope	Francis's	recent	demise. Working	memory	had	been	charged	with	the	task	of	contemplating	this	hypothetical	possibility, and	we	were	switched	into	a	reflective	mode	of	thought,	in	which	we	were	obliged	to	search	for consciously	available	reasons	to	deny	the	problematic	focal	possibility.	Thinking	reflectively,	we become	self-conscious	about	the	timing	of	what	is	retrieved	from	memory,	and	we	can	recognize that	what	is	consciously	available	does	not	enable	the	construction	of	any	good	explicit	argument against	a	very	recent	change	in	the	world	that	would	undermine	our	claim	to	know	our	key proposition	p	(that	the	current	pope	is	Pope	Francis).	When	we	are	challenged	to	produce	an explicit	argument	in	support	of	the	judgment	about	the	current	pope	that	we	initially	made intuitively,	we	have	no	choice	but	to	survey	the	materials	consciously	available	to	us,	even	though the	contents	of	consciousness	played	no	role	in	producing	the	original	unreflective	judgment.	We cannot	put	forward	the	truth	of	that	judgment	as	a	persuasive	argument	in	its	own	support:	this would	be	manifestly	circular	reasoning.	It	remains	possible	for	a	person	to	judge	that	p	is	the	case	in such	circumstances,	but	not	by	valid	reasoning	from	premises	that	are	not	in	question.	One	could, for	example,	think	to	oneself	that	Pope	Francis	simply	must	be	alive,	because	one	is	very	much 16 hoping	to	see	him	in	Vatican	City	next	Sunday,	but	such	reasoning	has	no	claim	to	a	positive epistemic	evaluation. One	might	wonder	why	exactly	we	can	no	longer	make	(and	positively	evaluate	ourselves	as making)	our	original	unreflective	judgment,	once	worries	have	been	introduced.	The	distinction between	reflective	and	unreflective	judgment	is	a	matter	of	their	causation,	but	we	do	not	have direct	access	to	the	etiology	of	what	comes	to	mind	in	any	mode	of	thought. When	we	engage	in reflective	cognition,	we	have	conscious	access	to	a	sequence	of	contents,	and	the	global	broadcast	of these	contents	in	consciousness	triggers	the	activation	of	multiple	mental	modules	that	respond	to the	relevant	content,	producing	further	conscious	contents	such	as	visual	imagery	and	inner	speech. While	these	contents	are	conscious,	the	manner	in	which	we	react	to	them	involves	multiple unconscious	and	involuntary	processes,	including	processes	that	interpret	and	evaluate	inner speech,	and	processes	that	intuitively	construct	and	evaluate	arguments	(for	a	review	of	the literature	on	these	processes,	see	Mercier	and	Sperber	2017).	This	process	of	reasoning	can	also produce	new	beliefs,	as	intuitive	modules	for	reasoning	pick	up	logical	patterns	among	the	contents we	assert	in	inner	speech,	producing	conclusions	we	decide	to	accept.	Arguments	presented	in inner	speech	are	subject	to	the	same	type	of	spontaneous	evaluation	that	we	apply	when	hearing the	voice	of	another	person.	We	cannot	by	sheer	force	of	will	make	something	said	in	inner	speech (such	as,	"Francis	is	the	current	pope")	register	as	the	manifestation	of	an	unreflectively	formed belief.	Once	we	have	started	to	reason	explicitly	on	a	question,	as	we	will	be	triggered	to	do	by rehearsal	of	the	precisely	represented	mortality	statistics	and	the	question	about	the	hypothetical possibility	of	his	recent	death,	we	will	find	ourselves	evaluating	the	things	we	hear	ourselves	saying as	part	or	product	of	our	reasoning.	If	no	good	explicit	reasoning	can	support	a	claim	which answers	the	question	we	are	aiming	to	settle,	then	as	long	as	we	are	in	a	mode	of	thought	that	is weighing	the	epistemic	merits	of	explicit	reasoning,	it	will	seem	we	lack	knowledge	of	the	claim	in question.4	If	the	ISA	model	of	indirect	(or	non-transparent)	access	to	mental	states	is	right,	it	is possible	that	we	actually	retain	knowledge	of	the	key	proposition	throughout	this	manipulation,	but 4	Although	the	original	Harman-Vogel	cases	involved	a	switch	from	intuitive	to	reflective	cognition,	I	think	the paradox	can	be	extended	to	any	case	of	switching	upwards	to	a	more	self-conscious	mode	of	thought,	for example,	moving	within	the	domain	of	reflective	cognition	from	plain	syllogistic	reasoning	to	reasoning	about one's	syllogistic	reasoning. One	ordinarily	trusts	one's	rational	capacities	unselfconsciously,	although	these capacities	are	structured	in	a	way	which	is	largely	shielded	from	introspection:	we	are	typically	aware	of successive	contents	but	not	directly	of	the	nature	of	the	processing	that	alerts	us	to	inconsistencies	or	enables the	deduction	of	sound	conclusions	(on	this	point,	see	Evans	2009).	But	it	is	possible	to	construct	cases	in which	we	have	doubts	about	those	processes,	for	example	on	receipt	of	misleading	information	to	the	effect that	we	have	consumed	a	drug	that	impairs	logical	reasoning	(as	in	Christensen	2010).	Detailed	discussion	of these	cases	is	however	a	project	for	another	occasion. 17 are	unable	to	register	ourselves	as	knowing	when	we	switch	to	the	self-doubting	reflective	mode	of thought.	If	our	grasp	of	our	own	propositional	mental	states	works	through	indirect	processes,	such as	querying	inner	speech,	our	inability	to	construct	a	good	argument	will	bar	us	from	producing	an apparently	knowledgeable	occurrent	judgment	that	p	is	the	case.	We	cannot	intuitively	evaluate	the propositional	attitude	by	being	directly	conscious	of	the	attitude	itself,	assuming	that	the	ISA	is right.	However,	knowledge	itself	could	well	be	a	state	which	can	endure	even	when	its	typical manifestations	are	temporarily	blocked:	if	so,	the	intuitive	sense	of	knowledge	loss	is	misleading. 6.	The	belief	version	of	the	problem The	claim	that	Francis	is	the	current	pope	(p)	is	a	claim	that	could	be	made	on	the	basis	of	either intuitive	or	reflective	thought:	I	have	argued	that	it	will	typically	come	across	as	knowledge	if	made intuitively,	assuming	that	the	subject	learned	this	fact	from	a	knowledgeable	source,	and	the	world conditions	have	been	appropriately	stable.	In	contexts	where	these	presuppositions	have	been challenged,	efforts	to	reach	p	reflectively	by	sound	and	persuasive	reasoning	will	be	unsuccessful, and	we	will	sense	a	lack	of	knowledge.	Whether	belief	in	p	is	in	these	contexts	intuitively	rational	is a	subtler	question:	the	apparent	rationality	of	a	belief	depends	not	only	on	the	content	of	what	is believed,	but	also	on	the	manner	in	which	the	subject	is	arriving	at	that	content.	Our	initial	statement	of	the	problem	left	it	somewhat	open	how	the	subject	is	thinking.	After	worries	have	been raised	about	the	possibility	of	a	very	recent	death,	and	the	mortality	statistics	have	been	taken	on board,	one	might	imagine	a	subject	reasoning	as	follows: (R1)	I	have	not	checked	the	news	in	the	last	hour.	Given	the	pope's	age	and	regional mortality	statistics,	he	has	a	.0000081616	chance	of	having	died	in	this	past	hour. I	have	no	further	information	to	settle	the	question	of	whether	Francis	is	still	the	current pope,	beyond	his	age	and	regional	mortality	statistics. 1.0	–	.0000081616	=	.9999918384. Assuming	he	was	alive	an	hour	ago,	the	pope	has	a	.9999918384	chance	of	being	alive	now. If	he	is	alive,	Francis	is	the	current	pope. Therefore,	Francis	is	the	current	pope. 18 This	way	of	reasoning	looks	problematic:	in	particular,	the	conclusion	that	seems	best	warranted	by this	effort	at	precise	reasoning	is	not	that	Francis	is	the	current	pope,	but	that	there	is	at	most a	.9999918384	chance	that	Francis	is	the	current	pope. However,	this	is	not	the	only	way	we	could	imagine	our	subject	as	reasoning,	on	hearing	the mortality	statistics.	We	do	not	have	to	see	the	subject	as	attempting	to	demonstrate	that	Pope Francis	is	alive	on	the	basis	of	precise	reasoning	from	the	statistical	information	provided. We could	instead	imagine	the	subject	recognizing	that	there	is	a	small	chance	that	the	pope	has	very recently	died,	but	then	stepping	back	to	a	larger	perspective: (R2)	Last	time	I	checked,	Francis	was	the	pope.	If	he	had	died,	other	than	in	the	last	hour,	I would	have	heard. Given	Francis's	age	and	regional	mortality	statistics,	he	has	a	.0000081616	chance	of	having died	in	this	past	hour. It	is	very	unlikely	that	the	pope	has	died	in	this	past	hour. It	is	generally	reasonable	to	assume	that	nothing	very	unlikely	has	happened. It	is	reasonable	to	assume	that	the	pope	is	alive,	and,	assuming	he	is	alive,	Francis	is	the current	pope. Therefore,	Francis	is	the	current	pope. This	argument	is	incomplete	in	a	number	of	ways.	For	example,	there	is	no	mention	of	the	possibility	that	the	regional	mortality	statistics	are	inaccurate,	or	the	possibility	of	Francis's	having	relinquished	the	Papacy,	and	so	forth.	However,	(R2)	is	not	obviously	problematic	in	the	way	that	the precise	argument	(R1)	was.	We	could	even	see	this	argument	as	a	reflection	of	the	thinking	behind a	deliberate	adoption	of	a	gist	representation:	note	the	shift	from	numerical	chances	to	the qualitative	"very	unlikely".	In	situations	in	which	the	pope	has	in	fact	just	died,	reasoning	of	this type	will	obviously	not	secure	the	subject	knowledge	of	the	conclusion.	But	even	if	we	consider	the subject	who	reaches	a	false	conclusion	by	reasoning	in	this	manner	just	after	the	pope's	demise,	we can	allow	that	the	subject	was	relatively	blameless,	relying	at	the	crucial	juncture	on	an	assumption she	could	not	have	known	to	be	false.	The	argument	works	as	a	partial	defense	of	a	style	of	thought that	will	generally	produce	accurate	answers	to	questions	of	the	type	under	consideration,	and	that overall	epistemic	rationality	of	thinking	in	this	manner	is	one	way	of	defending	the	subject's 19 epistemic	rationality	in	holding	this	conclusion.	The	argument	does	not	serve	to	defend	the legitimacy	of	relying	on	gist	in	all	circumstances:	for	example,	in	the	context	of	a	much	larger argument	stringing	together	judgments	about	many	world	leaders,	the	thought	that	it	is	generally reasonable	to	assume	nothing	unlikely	has	happened	would	not	so	easily	warrant	the	legitimacy	of a	series	of	assumptions	about	each	of	many	hundreds	of	particular	world	leaders,	where	we	are actively	working	to	conjoin	these	assumptions	in	a	pattern	of	precise	reasoning. By	contrast	with	the	case	about	the	pope,	one	interesting	feature	of	the	original	lottery	case is	that	the	very	idea	of	a	ticket	in	a	fair	lottery	carries	with	it	a	suggestion	of	chances	and	precise reasoning.	One	might	easily	have	thoughts	of	Pope	Frances	without	reflecting	on	mortality statistics,	but	thoughts	of	lottery	tickets	with	specified	odds	naturally	demand	verbatim representation	and	reflective	cognition.	This	feature	of	lottery	tickets	plays	a	key	role	in	Igor Douven's	(2012)	treatment	of	the	paradox.	Douven	holds	that	high	probability	generally	suffices	for rational	belief,	but	he	argues	that	it	is	irrational	to	believe	outright	that	one's	ticket	will	lose, because	such	a	belief	would	violate	principles	of	what	he	calls	'epistemic	hygienics'.	Just	as	we	can say	things	that	are	strictly	speaking	true	but	misleading	to	others,	Douven	argues,	we	can	also	form beliefs	that	are	strictly	speaking	warranted	but	potentially	misleading	to	our	future	selves: rationality	requires	us	to	avoid	forming	beliefs	that	are	risky	in	this	way	(Douven	2010). In	a standard	context	where	the	high	odds	of	loss	are	publically	known,	one	cannot	assert	"your	lottery ticket	will	lose"	just	on	the	strength	of	those	odds,	Douven	argues,	because	to	do	so	implicates	at least	weakly	that	you	have	inside	information	about	the	lottery	outcome.	A	cooperative	speaker	will not	chose	to	make	the	categorical	statement	about	loss	unless	he	wishes	to	suggest	that	he	has information	going	beyond	the	known	odds;	if	the	cooperative	speaker	simply	wanted	to	remind	the hearer	of	the	low	chance	of	winning,	Douven	observes	that	he	could	easily	say	something	weaker, such	as	"your	ticket	is	even	less	likely	to	win	than	you	may	think"	(2012,	359-60).	If	we	think	of belief	formation	as	a	type	of	communication	with	one's	future	self,	inscribing	a	particular	sentence in	one's	belief	box	for	later	retrieval,	then	given	that	one's	awareness	of	the	past	grounds	of	a particular	belief	may	fade	over	time,	it	is	also	risky	to	encode	a	plain	categorical	belief	to	the	effect that	one's	ticket	will	lose.	At	the	point	of	recalling	that	categorical	belief,	Douven	argues,	we	risk gaining	the	impression	that	we	must	have	had	inside	information	about	the	lottery	warranting	a higher	degree	of	certainty	about	the	loss	than	we	would	have	had	from	our	standing	awareness	of the	long	odds	of	lotteries. 20 Several	objections	come	to	mind.	First,	if	one	openly	maintains	that	high	probability	is sufficient	for	outright	belief,	then	this	does	something	to	diminish	the	sense	that	believing	one's ticket	will	lose	is	really	stronger	than	believing	one's	ticket	is	very	likely	to	lose:	given	the	perceived weakness	of	the	categorical	belief,	it	is	not	entirely	clear	why	exactly	it	implicates	the	possession	of inside	information.	True,	there	is	some	redundancy	in	having	not	only	beliefs	about	the probabilistic	setup	of	the	lottery	but	also	the	further	belief	that	one's	ticket	will	lose,	but	in	a context	where	forgetting	is	expected,	some	redundancy	seems	fine,	and	clarity	about	the	weakness of	belief	should	to	some	extent	weaken	the	impression	that	one	must	have	had	inside	information, in	a	context	where	one	happens	to	recall	just	the	belief	about	loss.	Second,	one	might	wonder	about the	picture	of	belief	as	inscription	of	a	particular	sentence,	to	be	recalled	verbatim	at	a	later	time:	it seems	that	our	capacity	for	verbatim	recall	is	generally	limited	to	short	durations	(Goldsmith, Koriat	et	al.	2002,	Britt,	Kurby	et	al.	2007,	Reyna,	Corbin	et	al.	2016).	Over	time,	when	one	has forgotten	the	grounding	of	one's	original	judgment,	it	is	likely	that	one	will	also	have	forgotten	its original	verbatim	formulation,	and	recall	only	the	gist,	which	on	Douven's	theory	should	not distinguish	between	loss	and	very	likely	loss. However,	there	is	something	in	Douven's	approach	which	is	very	plausible:	he	notes	that the	evaluation	of	what	we	say	or	believe	seems	to	depend	in	part	on	how	it	compares	to	the	field	of rival	propositions	we	might	have	chosen.	The	most	natural	way	in	which	these	fields	of	rivals	can differ	is	as	a	function	of	the	various	ways	in	which	one	might	come	to	a	belief	that	the	ticket	will lose. For	example,	a	person	might	be	represented	as	arriving	at	the	judgment	that	the	ticket	will lose	on	the	basis	of	questionable	precise	reasoning: (R3)	999,999	out	of	the	1,000,000	tickets	in	this	lottery	will	lose. Therefore,	this	particular	ticket	will	lose. As	I	am	evaluating	this	way	of	thinking,	if	the	arithmetic	of	the	set-up	provokes	me	to	engage	in precise	reasoning	that	the	ticket	has	a	.999999	chance	of	losing,	then	it	looks	to	me	like	a	mistake for	the	subject	to	endorse	the	absolute	conclusion	that	the	ticket	will	lose,	over	that	more	precise rival.	Framing	things	like	this	will	move	us	towards	Staffel's	intuitive	sense	of	the	problem:	the rational	person	does	not	outright	believe	that	his	ticket	will	lose.	To	speak	of	outright	belief	here,	is perhaps	to	underscore	the	way	the	subject	is	represented	as	decisively	asserting	the	conclusion	of this	effort	at	precise	reasoning.	If	Timothy	Williamson	(2000,	ch.	11)	is	right	that	knowledge	is	the norm	of	assertion,	then	our	intuitive	sense	that	the	subject	fails	to	know	that	his	ticket	will	lose 21 arguably	plays	a	role	in	our	discomfort	with	the	subject's	assertion	of	this	conclusion.	If	we	take	the possibility	of	direct	assertion	of	this	kind	to	be	the	thing	we	should	check	for	in	belief	attribution, and	precise	reasoning	as	our	model	for	ideal	belief	formation,	then	we	are	well	on	our	way	to	the theory	according	to	which	knowledge	is	the	norm	of	belief. However,	there	are	other	ways	of	reasoning	about	the	lottery. One	might,	for	example,	just highlight	the	contrast	between	winning	and	losing: (R4)	999,999	out	of	the	1,000,000	tickets	in	this	lottery	will	lose,	and	only	one	will	win. It	is	vastly	more	likely	that	this	ticket	will	lose	than	that	it	will	win. It	is	reasonable	to	expect	that	this	ticket	will	lose;	so,	I	believe	it	will	lose. If	the	choice	is	between	winning	and	losing,	rather	than	winning	and	being	.999999	likely	to	win, the	rational	person	picks	loss.	This	argument	shifts	into	qualitative	talk	of	probabilities	in	the second	line,	nudging	us	towards	a	gist	representation	of	the	outcome. The	conclusion	does	not state	baldly	that	the	ticket	will	lose,	and	if	it	did,	we	might	feel	more	resistance	to	it,	especially	if Williamson	is	right	about	the	norm	of	assertion.	However,	there	are	multiple	ways	of	attributing	a belief	that	p	to	a	subject,	and	hearing	the	subject	declare	that	p	is	the	case	is	only	one	of	these	ways. In	this	case,	the	subject	moves	from	a	judgment	about	reasonable	expectation	to	a	self-attribution	of belief,	rather	than	a	frank	embrace	of	the	key	content. Still,	we	can	read	this	self-attribution	as	good evidence	for	what	the	subject	actually	believes,	in	the	sense	that	it	is	the	kind	of	declaration	that sets	up	systematic	expectations	for	how	the	subject	will	plan	and	act	in	the	future.	This	subject	is not	figuring	out	what	to	believe	on	the	basis	of	precise	numerical	reasoning-indeed	the	subject might	appear	less	rational	if	she	were	represented	as	embarking	on	a	mission	of	precise	numerical reasoning	here-but	human	courses	of	action	are	not	always	driven	by	precise	numerical	reasoning.	For	many	of	the	purposes	behind	the	formation	of	beliefs,	the	subject's	way	of	thinking	here	is entirely	sensible. If	we	represent	subjects	as	thinking	this	way	about	the	lottery,	then	we	can	shift towards	Hawthorne's	sense	of	what	is	intuitive. The	person	who	believes	that	her	lottery	ticket	will	lose	may	be	reasoning	precisely	but poorly,	or	in	a	qualitative,	gist-based	manner,	and	well.	If	one	has	an	uneasy	sense	that	there	is some	shift	between	what	is	meant	by	'belief'	in	these	two	possible	ways	of	forming	a	belief,	the	shift can	be	explained	by	the	complexity	of	our	natural	resources	for	believing,	and	the	uneasy	feeling can	perhaps	be	traced	to	our	instinctively	simplified	self-models,	according	to	which	there	should 22 always	be	a	clean	and	transparent	answer	to	the	question	of	what	we	believe.	In	fact,	the	psychological	reality	of	belief	may	be	more	complex	still:	I	have	alluded	to	a	distinction	between	verbatim and	gist	representation,	but	in	fact	we	seem	to	encode	several	levels	of	gist	on	encountering	a situation	or	narrative,	alongside	any	verbatim	representations,	and	there	can	be	subtle	dissociations	between	these	various	levels	(for	a	review,	see	Reyna,	Corbin	et	al.	2016). Which	is	the optimal	way	to	think? Here	again	there	is	no	simple	answer:	in	practice,	given	the	limits	on	our capacities	to	reason	accurately,	each	of	our	ways	of	believing	has	a	complex	array	of	strengths	and weaknesses.	It	seems	advantageous	for	time-pressured	and	resource-limited	creatures	like	us	to have	a	capacity	to	switch	between	these	ways	of	thinking,	but	we	are	a	long	way	from understanding	the	principles	which	would	best	guide	these	switches. 7.	Kyburg's	sunrise,	and	the	problem	of	avoiding	skepticism When	Kyburg	encouraged	us	to	see	high	probability	as	sufficient	for	acceptance,	he	argued	that	the alternative	was	a	skepticism	in	which	we	would	be	unable	to	accept	that	the	sun	would	rise	tomorrow.	He	placed	the	proposition	that	the	sun	will	rise	(or	that	one's	lottery	ticket	will	lose)	in	a basket	alongside	"any	statistical	hypothesis	of	the	sort	that	everyone	simply	accepts"	(1961,	197); in	his	view,	such	propositions	are	accepted	rationally,	despite	having	"nothing	but	their	probability to	recommend	them"	(1961,	87).	According	to	Kyburg,	because	"anyone	has	enough	evidence	to render	it	highly	probable	that	the	sun	will	rise	tomorrow",	this	is	a	statement	that	everyone	should accept	(1961,	82).	Indeed,	he	sees	the	intuition	that	this	belief	is	rational	as	a	key	source	of	motivation	for	his	system:	"if	someone	maintains	that	the	sun	will	not	rise	tomorrow,	we	want	to	be	able to	say	that	his	belief	is	irrational"	(1961,	82). The	reliance	on	intuitive	motivation	here	makes	it more	awkward	for	Kyburg	to	advance	his	proposed	solution	to	the	lottery	paradox	later:	in	his	own treatment	of	the	lottery	paradox,	Kyburg	concluded	that	we	need	to	place	restrictions	on	the practice	of	conjoining	rational	beliefs,	noting	in	passing	that	this	is	"one	of	the	most	curious	things about	the	system"	(1961,	196).	It	seems	worse	than	curious:	Kyburg	may	well	be	right	that	we	will want	to	accuse	someone	of	being	irrational	if	they	will	not	agree	that	the	sun	will	rise	tomorrow,5 but	we	arguably	have	an	even	stronger	desire	to	accuse	someone	of	irrationality	if	they	accept	that p,	accept	that	q,	and	refuse	to	accept	that	p	and	q. 5	Or	that	an	apparent	sunrise	will	happen	somewhere	on	earth	tomorrow;	we	set	aside	complications involving	the	heliocentric	solar	system,	circumpolar	latitudes,	etc. 23 Kyburg	is	interestingly	neutral	on	questions	of	cognitive	procedure.	In	arguing	that "everyone	simply	accepts"	what	he	calls	"statistical	hypotheses",	he	was	not	suggesting	that	we	had to	be	following	a	special	style	of	precise	statistical	reasoning	in	doing	so.	He	does	not	mandate	a rule	of	explicit	reasoning	that	obliges	us	to	"round	upwards"	and	accept	any	proposition	one	sees	as highly	probable.	As	Gregory	Wheeler	has	emphasized	in	his	(2007)	review,	what	Kyburg	sees	as making	it	rational	for	a	subject	to	accept	that	the	sun	will	rise	tomorrow	is	just	the	proposition's high	objective	probability	given	the	subject's	evidence;	it	is	not	that	the	subject	is	rational	in	virtue of	first	assigning	this	proposition	a	high	subjective	probability,	and	then	reasoning	that	the proposition	is	acceptable	on	this	basis.	Indeed,	Kyburg	focuses	consistently	on	what	we	would	now call	propositional	as	opposed	to	doxastic	justification.	His	account	is	neutral	on	the	manner	in which	a	person	must	be	thinking	in	order	to	count	as	rational:	it	is	a	substantive,	rather	than procedural,	account	of	rationality.	If	we	now	think	that	a	full	account	of	rational	acceptance	will need	to	assess	the	way	in	which	a	proposition	is	accepted-so	that	it	is	possible	to	embrace	a tautology	irrationally,	for	example-then	we	will	need	to	enrich	Kyburg's	account.	At	the	same time,	a	closer	look	at	the	foundation	of	Kyburg's	epistemology	may	give	us	fresh	ideas	about	the avoidance	of	skepticism. Kyburg	introduces	the	expression	'rational	corpus'	for	the	set	of	beliefs	an	individual	has	at any	given	time,	where	it	is	assumed	to	be	transparent	to	an	agent	what	he	believes.	Membership	in this	corpus	could	in	principle	be	settled	for	any	individual,	Kyburg	says,	"by	presenting	him	with	a list	of	statements	...	and	asking	him	to	check	off	the	ones	that	he	believes."	(Kyburg	1961,	83) Whether	any	such	belief	is	rational	depends	on	its	relationship	to	the	'basis'	of	the	corpus,	which Kyburg	describes	as	a	set	of	"beliefs	which	are	based	on	the	'immediate	experience'	of	the individual	or	group	whose	rational	corpus	we	are	discussing",	and	then	glosses	immediately afterwards	as	"those	statements	of	matters	of	fact	(whatever	kinds	of	statements	these	may	be)	that are	known	without	inference	to	the	bearer	of	the	rational	corpus	under	discussion."	(Kyburg	1961, 82)	Any	member	of	the	basis	set	of	noninferential	knowledge	is	rationally	believed,	and	any	further belief	is	rational	or	irrational	as	a	function	of	its	probability	given	this	basis,	with	high	probability sufficient	for	rationality. The	shift	from	beliefs	based	on	immediate	experience	to	matters	of	fact	that	are	known without	inference	is	more	controversial	now	than	it	was	in	1961.	It	is	not	universally	granted	that beliefs	based	on	immediate	experience	necessarily	constitute	knowledge,	nor	that	they	are	even guaranteed	to	be	true.	The	other	direction	of	the	claimed	equivalence	has	also	been	challenged:	it	is 24 not	obvious	that	what	is	known	without	inference	must	always	be	based	on	immediate	experience. Certainly,	if	we	mean	person-level	explicit	inference,	the	type	of	inference	that	distinguishes reflective	from	intuitive	cognition,	then	most	epistemologists	take	many	facts	that	go	beyond immediate	experience	to	be	known	without	inference,	including,	for	example,	the	identity	of	the current	pope	and	the	fact	that	the	sun	will	rise	tomorrow.	Do	our	judgments	on	these	matters	have anything	more	than	their	probability	to	recommend	them,	or	to	confer	a	positive	epistemic	status upon	them?	Kyburg	saw	judgments	of	immediate	experience	as	the	necessary	foundation	of	all empirical	justification,	but	here	again	many	contemporary	epistemologists	will	disagree,	not	least out	of	fears	that	such	a	way	of	thinking	could	lead	us	to	a	skeptical	phenomenalism	of	the	present moment.	In	particular,	externalists	could	point	to	features	of	the	environmental	history	of	the judgment	as	contributing	to	its	positive	epistemic	status:	for	example,	that	one's	judgment	about the	identity	of	the	pope	originally	came	from	a	knowledgeable	source,	and	that	it	has	been	well preserved	in	memory	in	a	relevantly	stable	world,	and	so	on.	If	externalists	are	right	that	subjects can	have	noninferential	knowledge	of	such	matters	as	the	identity	of	world	leaders,	the	factors	in virtue	of	which	they	have	this	knowledge	are	not	guaranteed	to	be	first-person	accessible	or retrievable	by	a	subject	who	is	challenged	to	give	an	argumentative	defense	of	her	judgment. The belief	that	the	sun	will	rise	tomorrow	can	have	more	to	recommend	it	than	the	subject	can	state,	not to	mention	more	in	its	favor,	epistemically,	than	its	probability	on	the	subject's	immediate experience. With	a	more	expansive	notion	of	what	is	known	without	inference	(not	restricted	to immediate	experience),	we	could	accept	that	rationality	was	a	function	of	the	relationship	to	this basis	without	courting	skepticism. At	the	extreme,	one	could	hold	a	position	in	which	only	what	is known	is	fully	rational,	allowing	just	the	basis	of	what	is	known	without	inference	and	further additions	of	inferential	knowledge	gained	by	sound	precise	reasoning	from	that	basis.	On	this	way of	thinking,	it	would	make	no	sense	to	characterize	"the	sun	will	rise	tomorrow"	as	a	"statistical hypothesis."	We	could	certainly	accept	propositions	on	the	basis	of	sound	statistical	reasoning,	but these	would	be	propositions	like	"this	ticket	has	a	.999999	chance	of	winning",	with	the	precise character	of	the	statistical	reasoning	appropriately	leaving	its	mark	on	the	conclusion,	and	this conclusion	believed	outright.	Less	radical	views	are	also	possible,	however.	One	might	grant	the	full rationality	of	what	is	known	without	inference,	and	also	allow	that	the	stock	of	what	it	is	rational	to believe	could	then	be	enlarged	not	only	precise	reasoning,	but	also	by	various	manipulations	of	the gist	of	what	is	known.	To	the	extent	that	gist	representations	approximate	verbatim	originals,	this proposal	could	maintain	the	idea	that	rational	belief	is	best	understood	in	terms	of	a	relationship	to 25 knowledge.	There	are	limits	to	the	rational	employment	of	gist	in	governing	action	and	reasoning:	it can	be	irrational	to	rely	on	gist	in	making	insurance	decisions	or	stringing	together	multiple conjuncts,	but	it	does	not	follow	that	epistemic	rationality	generally	forbids	the	employment	of	gist representations.	Rougher	gist	representations	can	stray	further	from	knowledge	than	the	sharp outright	beliefs	we	gain	from	reasoning	precisely	from	what	is	known,	but	in	virtue	of	their	role	in guiding	thought	and	action,	they	can	still	count	as	beliefs,	and	in	virtue	of	their	role	in	keeping limited	creatures	like	us	in	touch	with	the	truth,	they	can	still	count	as	rational. As	epistemologists,	we	are	chiefly	concerned	with	normative	questions,	but	in	my	view	a deeper	descriptive	understanding	of	the	nature	of	belief	can	serve	as	a	valuable	guide	in	decoding the	significance	of	our	epistemic	intuitions,	and	the	appropriateness	of	various	possible	norms. 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