Diamonds	are	Forever Cian	Dorr	and	Jeremy	Goodman Penultimate	draft	(December	2018) Forthcoming	in	Noûs Abstract We	defend	the	thesis that	every	necessarily true	proposition is	always	true. Since	not every	proposition	that	is	always	true	is	necessarily	true,	our	thesis	is	at	odds	with	theories of modality and time, such as those of Kit Fine and David Kaplan, which posit a fundamental	symmetry	between	modal	and	tense	operators. According	to	such	theories, just	as it is	a	contingent	matter	what is true	at	a	given	time, it is likewise	a temporary matter	what is true	at a	given	possible	world; so	a	proposition that is	now true	at all worlds,	and	thus	necessarily	true,	may	yet	at	some	past	or	future	time	be	false	in	the	actual world,	and	thus	not	always	true.	We	reconstruct	and	criticize	several	lines	of	argument	in favor of this picture, and then argue against the picture on the grounds that it is inconsistent	with	certain	sorts	of	contingency	in	the	structure	of	time. I. Nothing impossible has ever happened. If a proposition is	necessarily true, then it is always	true. Call	this	principle	Perpetuity. Obvious	as	it	sounds,	Perpetuity	has	fallen	on hard	times. We	aim	to	defend	it.1 Three	points	of	clarification. First:	Some	philosophers	are	propositional	eternalists- they	think	that	every	true	proposition	is	always	true. On	this	view,	Perpetuity	is	true	but uninteresting,	since	it	is	uncontroversial	that	every	necessarily	true	proposition	is	true. 1	Assuming	an	S5	logic	of	necessity,	Perpetuity	is	equivalent	to	the	principle	that	what	is possibly	true	is	always	possibly	true-hence	our	title. Proof:	Suppose	p	is	possibly	true; then	it	is	necessarily	possibly	true	(by	the	5	axiom),	and	is	therefore	always	possibly	true (by	Perpetuity). In	the	other	direction,	suppose	p	is	necessarily	true;	then	it	is	possibly necessarily	true	(by	the	T	axiom),	and	is	therefore	always	possibly	necessarily	true	(by the titular	principle). Since	possible	necessary truth	entails truth (by the	B	axiom), it follows	by	the	closure	of	'always'	under	entailment	that	p	is	always	true. 2 We think that even propositional temporalists-those who deny propositional eternalism-should	accept	Perpetuity. We	will	therefore	proceed	on	the	assumption	that there	are	some	temporarily	true	propositions. Second: Some philosophers are nominalists and think that there aren't any propositions	at	all. This	view	also	makes	Perpetuity	true	but	uninteresting. Yet	surely nominalists	should	want	to	find	some	way	of	reframing	the	debate	over	Perpetuity	that renders it	non-trivial	by their lights-one	need	not	believe in	propositions to	wonder whether	necessity	entails	eternity. One	attractive	reframing	strategy	involves	replacing quantification	over	propositions	with	quantification into	sentence	position.	The	debate about	Perpetuity	will	then	be	reframed	as	a	debate	about	whether,	for	all	p:	if	necessarily p, then always	p.2 We think this is also a perfectly intelligible debate. Since in	what follows	we	will	be	freely	intersubstituting	φ	and	⌜the	proposition	that	φ	is	true⌝,	it	would be a trivial exercise to rewrite our discussion of Perpetuity using quantification into sentence	position	in	place	of	quantification	over	propositions.3 Third: Perpetuity is a claim about metaphysical necessity. Given propositional temporalism,	there	are	certainly	other	notions	of	necessity	which	do	not	entail	eternity. For	example,	epistemic	necessity	does	not	entail	eternity,	since	the	proposition	that	there are	dogs is	known to	be true	but	has	not	always	been true. Similarly, the "historical" notion	of	necessity,	on	which	truths	about	the	past	are	automatically	necessary,	does	not entail	eternity,	since	there	are	non-eternal	truths	about	the	past	(e.g.,	that	it	has	rained	at least	once). But	how	could	something	that	was	true	yesterday	be	metaphysically	impossible	today, or	something	that	is	metaphysically	necessary	today	be	false	tomorrow? We	hope	you share	our	judgment	that	it	obviously	couldn't	be. If	you	do,	then	you	might	be	surprised 2	See	Prior	1971;	Williamson	2003,	§9;	Williamson	2013,	§5.9. 3	Even	those	who	harbor	no	doubts	about	the	existence	of	propositions	sometimes	reject the	intersubstitutability	of	φ	and	⌜the	proposition	that	φ	is	true⌝.	For	example,	some	think that	if	Socrates	had	not	existed,	then	the	proposition	that	Socrates	does	not	exist	would not	have	existed	either,	and	so	would	not	have	been	true. We	will	ignore	such	potential complications	in	what	follows,	since	they	do	not,	so	far	as	we	can	tell,	have	any	bearing	on Perpetuity. 3 (as	we	were) to learn	that this judgment	of	obviousness is far from	universal. In fact, Perpetuity is	inconsistent	with	most	well-developed	theories	of	the	interaction	of	tense and	modal	operators. Admittedly,	many	of	those	who	develop	combined	modal-temporal logics	do	not	specify	that	they	are	talking	about	metaphysical	necessity,	and	indeed	some (e.g.	Thomason	1984)	are	explicitly	concerned	with	historical	necessity,	which	seems	on its	face	to	be	something	altogether	different	from	metaphysical	necessity.4 But	two	of	the most prominent treatments of the interaction of tense and	modality-David Kaplan's logic of demonstratives (1989), which is the starting point for much subsequent theorizing	about	tense	and	modality	in	the	philosophy	of	language	and	formal	semantics, and	Kit	Fine's	modal tense logic	(1977),	which	has	been influential in the	metaphysics literature-are	explicitly	concerned	with	metaphysical	necessity.5 While	neither	of	these authors	explicitly	considers	Perpetuity,	as	we	shall	see	they	are	committed	to	its	falsity. Meanwhile, many treatments of tense and modality in the linguistics literature are intended	to	apply	to	all	available	readings	of	modal	adverbs	in	natural	language,	and	seem to entail that Perpetuity is false no matter how we resolve the context-sensitivity of 'necessarily'.6,	7 One	underlying	motivation	behind	these	views	is	the	desire	for	a	symmetric	treatment of	modality and tense. For consider the converse of	Perpetuity: the claim that every proposition	that	is	always	true	is	necessarily	true. This	claim	is	clearly	false,	since	there 4 This is not entirely obvious: some insist that "real" possibility requires historical possibility,	where the	only	alternatives	are "slippery subjective	or linguistic	or	merely mathematical	notions	of	possibility"	(Belnap	2012:	7).	Perhaps	given	these	remarks,	we should interpret these philosophers as holding that historical and metaphysical possibility	coincide. 5	Zalta	(1988)	also	clearly	embraces	principles	inconsistent	with	Perpetuity. 6	See,	e.g.,	Cresswell	1990;	Kaufmann,	Condoravdi,	and	Harizanov	2006;	Schlenker	2006; and	Rini	and	Cresswell	2012. More	carefully:	the	views	in	question	seem	to	entail	that	if propositional	temporalism	is	true	then	Perpetuity	is	false. It	is	a	vexed	question	whether or	not	such	views	take	a	stand	on	propositional	temporalism	(see	King	2003;	Ninan	2012; Schaffer	2012). 7	We	take	it	that	in	adopting	a	metaphysical	interpretation	of	'necessarily'	philosophers are	exploiting	the	normal	context-sensitivity	of	the	natural-language	'necessarily'	rather than coining something altogether alien. This does not require thinking that the metaphysical	interpretation	is	widespread	outside	of	philosophy. 4 are	metaphysically	contingent	propositions	that	are	always	true,	such	as	the	proposition that sometimes it rains.8 Thus	Perpetuity, if true, constitutes	an important	asymmetry between	necessity	and	eternity. Since all of the aforementioned authors articulate their views in model-theoretic terms,	getting	a	more	precise	understanding	of	their	view	will	require	a	model-theoretic detour. In	§II	we	define	a	class	of	"product	models"	for	a	simple	formal	language	with modal	and	tense	operators,	and	show	that	Perpetuity is	inconsistent	with	propositional temporalism in these models. In §III we present a more general class of "relational structures",	the	logic	of	which	we	will	treat	as	uncontroversial	in	what	follows.	This	will allow us to isolate a principle Symmetry, which explicitly captures the aspect of our opponents' view responsible for the incompatibility of Perpetuity and propositional temporalism. In	§IV–§VII	we	consider	and respond to four	arguments for	Symmetry: although	these	arguments	have	not	actually	been	given	in	the	literature,	they	are	valid	on the	class	of	relational	structures	and	their	premises	are	both	philosophically	interesting and	true	in	all	product	models. (We	will	not	appeal	to	model-theoretic	considerations	in assessing	these	arguments,	so	those	who	would	prefer	not	to	bother	with	model	theory can	skip	§§II–III,	referring	to	§III	only	for	the	statements	of	Symmetry	and	an	equivalent principle,	Supervenience.) In	§§VIII-IX	we	argue	against	Symmetry	for	reasons	having	to do	with contingency in	what times there are. In §X	we conclude by considering and rejecting	the	suggestion	that	the	whole	debate	is	merely	verbal. 8	It is	not	entirely	unprecedented: the	Hellenistic	philosopher	Diodorus	Cronus	argued that 'the possible is that which either is or will be', which entails (given a standard conception	of	necessity	as	dual	to	possibility)	that	the	necessarily	true	propositions	are exactly	those	that	are	true	and	always	will	be	true,	and	hence	include	all	the	eternal	truths; but he may not have been talking about metaphysical necessity. For an influential conjecture about	Diodorus's argument see Prior 1955. During the	medieval period it became customary to distinguish a notion of "possibility per accidens", conforming to Diodorus's	definition,	from	other,	less	demanding	notions	of	possibility-possibility	per se	and	God's	possibility-for	which	analogues	of	Perpetuity	seem	to	have	been	accepted: see	Knuutila	1982. 5 II. In	this	section	we	will	define	a	class	of	"product	models"	for	a	formal	language	L containing	the	Boolean	connectives	¬	and	∧,	the	modal	operator	□	('necessarily'),	the tense	operator	A	('always'),	and	infinitely	many	propositional	variables	pi	and corresponding	universal	quantifiers	∀pi. We	will	adopt	standard	abbreviations	for	∨,	→, ↔,	∃,	and	◇;	Sφ	('sometimes	φ')	abbreviates	¬A¬φ.9 We	can	formalize	Perpetuity	in	L as	∀p(□p	→	Ap),	assuming	a	scheme	for	translating	L	into	English	on	which	∀pi	is translated	as	'For	every	proposition	pi',	while	occurrences	of	pi	in	sentence	position	are translated	as	'pi	is	true'.10 A product frame is any ordered pair ⟨W×T,	⟨α,	η⟩⟩ such that ⟨α,	η⟩	∈	W×T. An assignment function on a product frame is a function from propositional variables to subsets	of	W×T. A	product	model is a	pair ⟨⟨W×T,	⟨α,	η⟩⟩,	⟦∙⟧⟩,	where ⟨W×T,	⟨α,η⟩⟩ is a product	frame	and	⟦∙⟧	is	the	unique	"interpretation	function"	that,	given	any	assignment function	g, maps each L-formula φ to a subset ⟦φ⟧g of W×T in accordance with the following	recursive	definition: ⟦pi⟧g	=	g(pi) ⟦¬φ⟧g	=	(W×T)	–	⟦φ⟧g ⟦φ	∧	ψ⟧g	=	⟦φ⟧g	∩	ψ⟧g ⟦□φ⟧g	=	{⟨w,	t⟩	:	⟨w′,	t⟩	∈	⟦φ⟧g	for	all	w′	∈	W} ⟦Aφ⟧g	=	{⟨w,	t⟩	:	⟨w,	t′⟩	∈	⟦φ⟧g	for	all	t′	∈	T} 9	L is in this	respect	simpler than	the language	of	standard	tense logic, in	which	Aφ	is usually	analyzed	in	terms	of	two	other	operators	H	('it	has	always	been	the	case	that')	and G	('it	will	always	be	the	case	that')	as	Hφ	∧	φ	∧	Gφ. 10 It is important to distinguish accepting Perpetuity-a quantified sentence-from accepting	all	instances	of	the	schema	□φ	→	Aφ,	since	some	parties	to	the	debate	accept Perpetuity	without	accepting	the	schema. For	example,	Salmon	(1983,	Appendix	C)	rejects the schema □φ	→	Aφ but is a propositional eternalist and so certainly accepts Perpetuity. One	way	of	making	sense	of	such	a	position	(although	not	Salmon's	way)	is	to think	of	tense	operators	as	binding	unpronounced	time	variables,	which	deictically	pick out	particular	times	when	they	occur	free. □φ	∧	¬Aφ	will	then	fail	to	entail	∃p(□p	∧	¬Ap) for	the	same	reason	that	the	consistent	reading	of	'He	is	happy	but	not	every	man	is	such that	he	is	happy'	fails	to	entail	'∃p(p	but	not	every	man	is	such	that	p)'. 6 ⟦∀piφ⟧g	=	{⟨w,	t⟩	:	⟨w,	t⟩	∈	⟦φ⟧h	for	all	assignment	functions	h	that	agree	with	g	on all	variables	distinct	from	pi}. φ is true in the product model ⟨⟨W×T,	⟨α,	η⟩⟩,	⟦∙⟧⟩ on an assignment g just in case ⟨α,	η⟩	∈	⟦φ⟧g. A	closed	sentence	is	true	in	a	product	model	just	in	case	it	is	true	on	some (or	equivalently,	on	every)	assignment.	The	product	logic	is	the	set	of	sentences	true	in	all product	models. Perpetuity	is	true	in	a	product	model	⟨⟨W×T,	⟨α,	η⟩⟩,	⟦∙⟧⟩	just	in	case	T	=	{η},	since	when W×{η}	is	assigned	to	p,	□p	is	true,	while	Ap	is	false	if	T	has	more	than	one	member. This is	also	the	condition	for	∀p(p	→	Ap)	('Every	true	proposition	is	always	true')	to	be	true	in a	product	model. So,	as	advertised,	Perpetuity	turns	out	to	be	equivalent	to	propositional eternalism	in	the	product	logic.11, There are some additions to L that look natural from the perspective of product models,	although	(as	we	shall	see)	their	interpretation	in	other	kinds	of	models	we	discuss below	raises	difficult	philosophical	issues. Most	straightforwardly,	we	can	add	operators @	('actually')	and	N	('now'),	with	the	following	semantic	clauses: ⟦@φ⟧g	=	{⟨w,	t⟩	:	⟨α,	t⟩	∈	⟦φ⟧g} ⟦Nφ⟧g	=	{⟨w,	t⟩	:	⟨w,	η⟩	∈	⟦φ⟧g} In §IV and §V we consider arguments against Perpetuity formulated using these operators.	We	could	also	enrich	L	with	devices for	explicitly talking	about	worlds	and times,	as follows.	We	add	countably	many	"world	variables"	wi,	and	"time	variables"	ti have	assignment	functions	assign	a	member	of	W	to	each	world	variable	and	a	member	of T	to	each	time	variable	and	have	∀wi	and	∀ti	vary	these	assignments	in	the	obvious	way. We	can	then	add	new	predicates	'Actualized'	and	'Present'	that	take,	respectively,	a	world variable	or	a	time	variable	as	arguments,	interpreted	as	follows: ⟦Actualized(wi)⟧g	=	{⟨g(wi),	t⟩:	t	∈	T} 11	Perpetuity and propositional eternalism	would still be equivalent if product	models were	enriched	with	a	reflexive	accessibility	relation	R	on	W	and	the	clause	for	□	modified accordingly. More	generally	still,	we	can	relativize	R	to	members	of	T,	defining	⟦□φ⟧	as {⟨w,	t⟩:	⟨w′,	t⟩	∈	⟦φ⟧	for	all	w′	such	that	wRtw′} (see	Thomasson	1984). 7 ⟦Present(ti)⟧g	=	{⟨w,	g(ti)⟩:	w	∈	W} Instead,	or	in	addition,	we	could	add	an	'At'	operator	that	takes	a	world	or	time	variable and	a	formula	as	arguments,	interpreted	as	follows: ⟦At	wi	φ⟧g	=	{⟨w,	t⟩:	⟨g(wi),	t⟩	∈	⟦φ⟧g} ⟦At	ti	φ⟧g	=	{⟨w,	t⟩:	⟨w,	g(ti)⟩	∈	⟦φ⟧g} If	we	have	both 'Present'	and 'At' in the language, the	above	clauses	make 'Present(ti)' logically	equivalent	to	'∀p(p	↔	At	ti	p)',	and	also	make	'At	ti	φ'	logically	equivalent	to	both 'A(Present(ti)	→	φ)'	and	'S(Present(ti)	∧	φ)';	similarly	for	'Actualized'. But	be	warned	that such equivalences are not uncontroversial, since taken together they impose severe constraints	on	the	extent	to	which	there	can	be	contingency	in	the	composition	of	the	time series,	as	we	discuss	in	§IX.12 Adding	worldand time-quantifiers	makes	@ and	N redundant in a certain sense: every	sentence	φ	containing	N	is	logically	equivalent	to	∃t(Present(t)	∧	φ*),	where	t	is	a 12 An alternative to explicit quantification over times is to enrich the language with devices	of	"temporal	anaphora"	(Kamp	1971,	Vlach	1973,	Cresswell	1990). We	add	to	L countably many "time-storage" operators ↑0, ↑1,... and "then" operators ↓0, ↓1.... To interpret	them,	let	assignment	functions	assign	each	"then"	operator	a	member	of	T,	and extend	the	interpretation	function	as	follows: ⟦↓iφ⟧g	=	{⟨w,	t⟩:	⟨w,	g(↓i)⟩	∈	⟦φ⟧g} ⟦↑iφ⟧g={⟨w,	t⟩:	⟨w,	t⟩	∈	⟦φ⟧g[↓i↦t]} where	g[↓i↦t]	is	an	assignment	like	g	except	that	it	maps	↓i	to	t. As	Cresswell	(1990) points	out,	this	apparatus	is	expressively	equivalent	to	explicit	quantification	over	times, assuming	time	variables	can	only	occur	as	arguments	of	'At'	and	'Present'. Consider	the translation	functions	*	and	†	defined	as	follows: Present(ti)†	=	∀p(p	↔	↓ip) (↓iφ)*	=	At	ti	φ* (At	ti	φ)†	=	↓iφ† (↑i	φ)*	=	∃ti(Present(ti)	∧	φ*) (∀ti	φ)†	=	↑jA↑i↓j	φ†	(where	j	is	a	new	index) (*	and	†	commute	with	all	other	operators	and	quantifiers.) For	any	formula	φ,	φ†	contains no	time	variables	and	φ*	contains	no	time-storage	or	'then'	operators. It	is	easily	shown that,	whenever	φ	is	a	closed	sentence	in	which	each	'then'	operator	is	in	the	scope	of	the corresponding	storage	operator,	φ	is	logically	equivalent	to	any	formula	obtained	from	φ by	first	relabeling	its	time	variables	so	that	they	don't	share	any	indices	with	any	timestorage	or	'then'	operators	and	then	applying	either	*	or	†. All	of	the	above	applies	mutatis mutandis	to	quantification	over	worlds	(see	Correia	2007),	for	which	one	would	want	a separate	set	of	"world-storage"	operators	⇑i	and	"world-retrieval"	operators	⇓i. 8 time-variable	that	doesn't	occur	in	φ	and	φ*	is	the	result	of	substituting	'At	t'	for	every occurrence	of	N	in	φ. Similarly	for	@	and	'At	w'. Moreover,	as	Fine	(1977)	shows,	the	addition	of	quantification	over	times	and	worlds is	also	expressively	redundant	in	the	product	logic. In	place	of	such	quantification,	we	can quantify over 'time-propositions'-intuitively, propositions equivalent to a particular time's	being	present-and	'world-propositions'-intuitively,	propositions	equivalent	to particular	world's	being	actualized-where	these	notions	are	defined	as	follows: WorldProp(φ)	=df	◇A(φ	∧	∀p(p	→	□(φ	→	p))) TimeProp(φ)	=df	S□(φ	∧	∀p(p	→	A(φ	→	p))) The	upshot	of	these	definitions	is	that	⟦WorldProp(φ)⟧g	=	W×T	if,	for	some	w	∈	W,	⟦φ⟧g	= {⟨w,	t⟩	:	t	∈	T},	and	=	∅	otherwise;	likewise,	⟦TimeProp(φ)⟧g	=	W×T	if,	for	some	t	∈	T,	⟦φ⟧g =	{⟨w,t⟩	:	w	∈	W},	and	=	∅	otherwise. As	Fine	(1977:	167)	puts	it,	'the	instant-propositions and	the	world-propositions	are	the	temporal	and	modal	cross-sections,	respectively,	of the	two-dimensional instant-world	manifold'. To	eliminate	world	and	time	quantifiers from	a	formula	without	free	world	or	time	variables,	we	proceed	as	follows: First,	relabel the	variables so that	no index is shared	between	variables	of	different types. Second, eliminate	'At'	in	favor	of	'Present'	and	'Actualized'	as	described	above. Third,	replace	each subformula of the form ∀wi(...)	with ∀pi(WorldProp(pi)	→	...), each subformula of the form	∀ti(...)	with	∀pi(TimeProp(pi)	→	...),	and	each	subformula	of	the	form	Present(ti)	or Actualized(wi)	with	pi. It	is	straightforward	to	verify	that	the	resulting	formula	has	the same	semantic	value	as	the	original	formula	on	every	assignment	in	every	model. But	be forewarned	that	these	equivalences	do	not	hold	in	any	of	the	other	classes	of	models	we will	be	considering	below. The fact the negation of	Perpetuity follows from propositional temporalism in the product	logic	does	not	by	itself	constitute	anything	like	an	argument	against	Perpetuity that	could	be	set	against	its	seemingly	obvious	truth. Perhaps	model-theoretic	elegance is	some	sort	of	guide	to	truth;	but	there	are	comparably	elegant	classes	of	models	which validate	Perpetuity	without	validating	propositional	eternalism. For	example,	Montague 9 (1973) uses what we will call 'Montagovian models'-models also based on product frames,	but	with	a	clause	for	□	equivalent	in	our	notation	to: ⟦□φ⟧g	=	{⟨w,	t⟩	:	⟨w′,	t′⟩	∈	⟦φ⟧g	for	all	w′	∈	W	and	t′	∈	T}. That is, ⟦□φ⟧g =	W×T if ⟦φ⟧g =	W×T and = ∅ otherwise; so clearly, Ap is true on an assignment	whenever	□p is. In a footnote to	Montague's clause, his editor Richmond Thomason	writes	'Here,	□	is	interpreted	in	the	sense	of	"necessarily	always"'	(Montague 1974:	259).	We	disagree:	Montague	explicitly	states	that	'necessarily'	is	to	be	translated as	'□',	and	the	natural	explanation	for	his	interpreting	□	in	the	above	way	is	a	desire	to respect	Perpetuity. The	additions	to	L	discussed	above	can	be	interpreted	in	Montagovian	models	in	the same	way	as in	product	models. These interpretations	are completely	natural for the temporal additions (N, ∀t, 'Present', and 'At	t'). They are less natural for the modal additions	(@,	∀w,	'Actualized'	and	'At	w'),	since	for	example	we	will	lose	the	equivalence of	□p	and	∀w	At	w	p. (In	§V	and	§VI	we	will	discuss	some	other	ways	of interpreting possible-worlds talk.) Note too that Fine's reduction of the expanded language to L behaves	pathologically	in	Montagovian	models:	∃p(TimeProp(p))	and	∃p(WorldProp(p)) are	both false in	any	Montagovian	model	where	T	has	more than	one	member. More generally, when N, or ∀t and 'Present', or ∀t and 'At	t', or @, or ∀w and 'At	w' are interpreted	in	Montagovian	models	in	the	ways	described	above,	some	formulae	without free time or world variables are not equivalent to any L-formula. 13 This suggests a 13 When f	:	W	→	TT maps each member of W to a permutation of T, define f*	:	P(W×T)	→	P(W×T)	by	f*(X)	=	{⟨w,	f(w)(t)⟩	:	⟨w,	t⟩	∈	X}.	A	straightforward	induction shows	that	for	any	L-formula	φ	and	assignment	g,	⟦φ⟧f*∘g	=	f*(⟦φ⟧g). But	this	is	not	true for	any	of	the	listed	expansions	of	L. For	example,	consider	the	Montagovian	model	where W = T = {0,1} and α=η=0. For each x	∈	{0,1}, let f(0)(x)=x and f(1)(x)=1–x. Let g(p)	=	{⟨0,0⟩,	⟨1,0⟩} so	f*∘g(p)	=	{⟨0,0⟩,	⟨1,1⟩}. Then ⟦Np⟧g	=	W×T, so ⟦□Np⟧g	=	W×T = f*(W×T) = f*(⟦□Np⟧), but	⟦Np⟧f*∘g	=	{⟨0,0⟩,	⟨0,1⟩}, so ⟦□Np⟧f*∘g	=	∅	≠	f*(⟦□Np⟧). Similarly for ∃t(Present(t)	∧	□A(Present(t)	→	p), ∃t(∀q(q	↔	At t q)	∧	□At t p), □(p	↔	@p),	and	∃w(∀q(q	↔	At	w	q)	∧	□(p	↔	At	w	q)). By	contrast,	if	we	just	add	worldquantifiers	and	'Actualized',	with	the	obvious	interpretations,	every	world-variable	free formula	is	equivalent	to	an	L-formula,	namely	one	derived	from	it	by	eliminating	indexsharing, replacing Actualized(wi) with pi, and replacing ∀wi(...) with ∀pi(WorldProp*(pi)	→	...),	where	WorldProp*(φ)	=df	◇(Aφ	∧	∀q(Aq	→	□(φ	→	q))). 10 broader	lesson:	for	proponents	of	Perpetuity,	quantification	over	times	provides	a	kind	of co-ordination	between	different	possible	world-histories	that	cannot	be	expressed	using only	standard	modal	and	temporal	operators. It	is	open	to	them	to	think	that	sentences that ineliminably involve these additional resources are problematic in a	way that	Lsentences	are	not; for	example,	perhaps time-quantification	gives	rise to	vagueness	or indeterminacy,	in	a	way	that	temporal	operators	do	not,	when	quantifying	into	the	scope of	modal	operators. III. In	this	section	we	will	define	a	more	general	class	of	models	whose	logic	we	will	treat	as uncontroversial in	what follows. Exploring the connection between these	models and product	models	leads	us	to	a	principle	-	Symmetry	-	which	is	valid	in	the	product	logic and whose conjunction with propositional temporalism uncontroversially implies the falsity of	Perpetuity. Isolating this principle thereby allows us to prescind from	model theoretic considerations in the rest of the paper, considering instead the comparative merits	of	Symmetry	and	Perpetuity. A	relational	structure	is	a	quadruple	S	=	⟨I,	≈□,	≈A,	ι⟩,	where	≈□	(the	'modal	accessibility relation')	and	≈A	(the	'temporal	accessibility	relation')	are	each	equivalence	relations	on I,	and	ι	(the	'home	point')	is	a	member	of	I. Given	a	relational	structure	⟨I,	≈□,	≈A,	ι⟩	and	an assignment function	g mapping propositional variables to subsets of I, we define the interpretation	function	⟦∙⟧g	from	formulae	of	L	to	subsets	of	I	as	follows: ⟦pi⟧g	=	g(pi) ⟦¬φ⟧g	=	I	–	⟦φ⟧g ⟦φ	∧	ψ⟧g	=	⟦φ⟧g	∩	⟦ψ⟧g ⟦□φ⟧g	=	{x	∈	I	:	y	∈	⟦φ⟧g	for	all	y	such	that	x	≈□	y} ⟦Aφ⟧g	=	{x	∈	I	:	y	∈	⟦φ⟧g	for	all	y	such	that	x	≈A	y} ⟦∀piφ⟧g	=	{j	∈	I	:	j	∈	⟦φ⟧h	for	all	h	that	agree	with	g	on	variables	other	than	pi} 11 φ	is	true	on	an	assignment	g	in	⟨I,	≈□,	≈A,	ι⟩	if	and	only	if	ι	∈	⟦φ⟧g;	a	closed	sentence	is	true in	a	relational	structure	if	and	only	if	it	is	true	on	some	assignment;	the	background	logic is	the	set	of	all	sentences	true	in	every	relational	structure. We	will	now	consider	some	important	subclasses	of	relational	structures. (i) Let	a	connected	relational	structure	be	one	in	which	every	point	in	I	can	be	reached from	ι	by	some	finite	sequence	of	points,	each	bearing ≈□	or ≈A	to	its	predecessor. Clearly the same sentences are true in a relational structure as are true in the connected	relational	structure	derived	from	it	by	throwing	away	all	the	points	not reachable	from	ι	by	such	a	finite	sequence;	thus	the	background	logic	is	also	the	logic of	the	class	of	connected	relational	structures. (ii) Let a	product structure be a relational structure of the form ⟨W×T,	=2, =1, ⟨α,	η⟩⟩, where	α	∈	W,	η	∈	T,	⟨w,	t⟩	=1	⟨w′,	t′⟩ if	and	only if	w=w′,	and	⟨w,	t⟩	=2	⟨w′,	t′⟩ if	and only if t=t′. 14 A product structure has the same interpretation function as the product	model	based	on	⟨W×T,	⟨α,	η⟩⟩. So	the	background	logic	is	contained	in	the product	logic. (iii) Let a Montagovian structure be a relational structure of the form ⟨W×T, (W×T)×(W×T),	=1,	⟨α,	η⟩⟩-i.e.	the	same	as	a	product	structure	but	with	a	universal modal	accessibility	relation. A	Montagovian	structure	has	the	same	interpretation function	as	the	Montagovian	model	based	on	the	product	frame	⟨W×T,	⟨α	,η⟩⟩. So	the background	logic	is	contained	in	the	logic	of	Montagovian	models. (iv) Let a generalized product structure be a relational structure of the form ⟨D,	=2,	=1,	⟨α,	η⟩⟩,	where	D	is	some	set	of	ordered	pairs	and	⟨α,	η⟩	∈	D.15 (v) Let a generalized Montagovian structure be one of the form ⟨D,	D×D,	=1,	⟨α,	η⟩⟩, where	D	is	some	set	of	ordered	pairs	and	⟨α,	η⟩	∈	D. 14	A	product	structure	⟨W×T,	=2,	=1,	⟨α,	η⟩⟩	is	in	a	natural	sense	the	product	of	the	pointed Kripke	structures ⟨W,	UW,	α⟩	and	⟨T,	UT,	η⟩,	where	UW	and	UT	are	the	universal	accessibility relations	on	W	and	T:	see	Kurucz	2007 15 The interpretation function on generalized product structures can be naturally extended to interpret sentences containing 'Present', 'Actualized', '∀t', and '∀w' in the same way as in the case of product	models. By contrast, there are several different possible	ways	of	interpreting	N,	@,	At	t,	and	At	w	in	such	structures,	for	reasons	that	will emerge	in	§IX. 12 Every	relational	structure	with	a	universal	modal	accessibility	relation	is	isomorphic to a generalized Montagovian structure. (The function mapping each point x to the ordered pair ⟨[x]≈A,	x⟩, where [x]≈A is the ≈A-equivalence class containing x, is an isomorphism	between	the	original	structure	and	the	generalized	Montagovian	structure whose	set	of	points	is	the	image	of	that	function.) And	every	relational	structure	with	a universal	modal	accessibility	relation	in	which	all	the	equivalence	classes	under	≈A	are equinumerous is isomorphic to a Montagovian structure. (Choose an arbitrary equivalence	relation	≈?	such	that,	for	all	x	and	y	there	is	a	unique	y′	in	[y]≈A	such	that	x	≈? y′,	and	then	consider	the	function	mapping	each	point	x	to	the	ordered	pair	⟨[x]≈A,	[x]≈?⟩.) We	can	similarly	give	intrinsic	characterizations,	up	to	isomorphism,	of	generalized product	structures	and	product	structures. Say	that	a	point	x	in	a	relational	structure	is unaccompanied if x is the only point both	modally and temporally accessible from x. Clearly, every point in a generalized product structure is unaccompanied. Moreover, every relational structure in which every point is unaccompanied is isomorphic to a generalized	product	structure. Given	any	relational	structure	S	=	⟨I,	≈□,	≈A,	ι⟩,	let	WS	=	I∕≈□ (the set of equivalence classes of I under ≈□), TS = I∕≈A, αS = [ι]≈□ (the	member of I∕≈□ containing ι), and ηS = [ι]≈A. Define fS	: I	→	WS×TS by fS(j)	= ⟨[j]≈□,	[j]≈A⟩, and let	DS = {fS(j)	:	j	∈	I}. If	every	point	in	S	is	unaccompanied,	fS	is	one-to-one. Moreover	x	≈□	y	if	and only if fS(x) =2 fS(y), x	≈A	y if and only if fS(x) =1 fS(y), and fS(ι) = ⟨αS,	ηS⟩, so fS is an isomorphism	from	S	to	the	generalized	product	structure	⟨DS,	=2,	=1,	⟨αS,	ηS⟩⟩. Turning	to	product	structures:	say	that	a	point	x in	a	relational	structure	is	squarecompleting	just	in	case	whenever	x	≈□	y1	and	x	≈A	y2,	there	is	some	z	such	that	y1	≈A	z	and y2	≈□	z. Every point in a product structure is square-completing (as well as unaccompanied). Moreover,	every	connected	relational	structure	in	which	every	point	is both	unaccompanied	and	square-completing	is	isomorphic	to	a	product	structure. This follows from the fact that	whenever	x and	y are two	points in a connected relational structure	S	where	every	point	is	square-completing,	there	is	a	point	modally	accessible 13 from	y	and	temporally	accessible	from	x.16 Hence	every	member	of	WS	overlaps	every member	of	TS,	so	that	DS	(the	image	of	the	function	fS	defined	above)	is	the	full	Cartesian product	WS×TS. The	two	properties	of	points	we	have	just	isolated	correspond	to	the	following	pair	of sentences	of	L: Symmetry:	Every	falsehood	necessitates	something	that	is	never	true	when	it	is. ∀p(¬p	→	∃q(□(p	→	q)	∧	A(p	→	¬q))) Church-Rosser:	Whatever	could	always	be	true	always	could	be	true. ∀p(◇Ap	→	A◇p) In any relational structure, the unaccompanied points are all and only those where Symmetry	is	true,	and	the	square-completing	points	are	all	and	only	those	where	ChurchRosser	are	true.17 (Proofs: (i) Suppose	Symmetry is true at	x,	y	≈□	x, and	y	≈A	x. Let	g(p)={y}. Then x	∉	⟦∃q(□(p	→	q)	∧	A(p	→	¬q))⟧g, since for any q-variant g′ of g, x	∉	⟦□(p	→	q)⟧g′ if y	∉	g′(q)	while	x	∉	⟦A(p	→	¬q)⟧g′ if	y	∈	g′(q). Since	Symmetry is	true	at	x,	x	∉	⟦¬p⟧g,	so x	∈	g(p), so x	=	y. Hence x is unaccompanied. (ii) Suppose x is unaccompanied and x	∉	g(p). Let	g′	be	the	q-variant	of	g	that	assigns	to	q	the	set	of	all	points	that	are	either modally	accessible	from	x	and	in	g(p),	or	temporally	accessible	from	x	and	not	in	g(p). Clearly	x	∈	⟦□(p	→	q)⟧g′. Also	x	∈	⟦A(p	→	¬q)⟧g′,	since	the	only	points	in	both	g′(p)	and 16 Proof: we first show that in a relational structure where every point is squarecompleting,	whenever	y	is	reachable	from	x	in	three	steps	it	is	also	reachable	from	x	in two	steps. Suppose	x	≈□	z1,	z1	≈A	z2,	and	z2	≈□	y. Since	z2	is	square-completing,	there	is some	point	z3	such	that	z1	≈□	z3	and	z3	≈A	y;	then	by	the	transitivity	of	≈□,	x	≈□	z3,	so	y	is just	two	steps	from	x. (Similarly	when	the	initial	sequence	contains	two	temporal	and	one modal	step.)	We	can	then	show	by	induction	that	whenever	y	is	reachable	from	x	in	any finite	number	of	steps	it	is	reachable	in	two	steps. If	the	structure	is	connected,	this	means that	any	two	points	are just two	steps	apart-and	given	that the intermediate	point is square-completing,	the	steps	can	be	taken	in	either	order. 17	Notice	that	both	Symmetry	and	Church-Rosser	are	symmetric	with	respect	to	modality and	tense	in	the	sense	that	they	are	equivalent	(in	the	background	logic)	to	their	"mirror images",	i.e.	the	results	of	interchanging	□	and	A	in	them. The	mirror-image	of	Symmetry, ∀p(¬p	→	∃q(A(p	→	q)	∧	□(p	→	¬q))), is	witnessed for a given	p by the negation of any proposition	q that	witnesses	Symmetry for that	p. The	mirror-image	of	Church-Rosser, ∀p(S□p	→	□Sp),	is	equivalent	to	∀p(¬A◇¬p	→	¬◇A¬p)	and	hence	to	∀p(◇A¬p	→	A◇¬p), which	is	equivalent	to	Church-Rosser	by	quantificational	logic. 14 g′(q)	are	modally	accessible,	but	also	distinct,	from	x,	and	hence	not	temporally	accessible from	x. So	x	∈	⟦∃q(□(p	→	q)	∧	A(p	→	¬q)⟧g,	and	thus	Symmetry	is	true	at	x. (iii)	Suppose Church-Rosser is true at x, y1	≈□	x, and y2	≈A	x. Let g(p)	=	[y1]≈A. Since y1	∈	⟦Ap⟧g, x	∈	⟦◇Ap⟧g,	so	by	Church-Rosser,	x	∈	⟦A◇p⟧g,	and	thus	y2	∈	⟦◇p⟧g. Hence	there	is	some z	modally	accessible	from	y2	that	belongs	to	g(p),	and	so	is	temporally	accessible	from	y1: thus	x	is	square-completing. (iv)	Suppose	x	is	square-completing	and	x	∈	⟦◇Ap⟧g. Then there	is	a	point	y1	modally	accessible	from	x	such	that	every	point	temporally	accessible from y1 is in	g(p). Consider any y2 temporally accessible from x. Since x is squarecompleting,	there	is	a	point	z	that	is	both	temporally	accessible	from	y1	(so	z	∈	g(p))	and modally	accessible	from	y2	(so	y2	∈	⟦◇p⟧g).	Since	this	holds	for	all	such	y2,	x	∈	⟦A◇p⟧g; hence	Church-Rosser	is	true	at	x.) It follows immediately from	this	result that	every	connected	relational	structure in which	all	the	sentences	that	result	from	prefixing	any	sequence	of	□s	and	As	to	Symmetry or	Church-Rosser are true is one in	which every point is unaccompanied and squarecompleting,	and	hence	isomorphic	to	a	product	structure. As	it	turns	out,	three	of	these sentences,	namely	□A(Symmetry),	□(Church-Rosser),	and	A(Church-Rosser),	suffice	in	the background	logic	to	imply	all	the	rest-and	thus,	to	imply	the	entire	product	logic.18,	19 Perpetuity is true in a relational structure just in case every point temporally accessible	from	ι	is	also	modally	accessible	from	ι. Propositional	eternalism,	meanwhile, 18 Proof: From □(Church-Rosser) we get ∀p□(◇Ap → A◇p) (by the Converse Barcan Formula), hence ∀p□(◇A□p	→	A◇□p) by quantificational reasoning, hence ∀p(□◇A□p	→	□A◇□p)	by	the	modal	K	schema,	hence	∀p(A□p	→	□Ap)	by	the	modal	B schema. Similarly, A(Church-Rosser) implies ∀p(□Ap	→	A□p). The two together thus yield the commutativity principle ∀p(□Ap ↔ A□p), which by an induction using the temporal	and	modal	4	schemas	implies	∀p(□Ap	→	O1...Onp)	for	any	string	O1...On	of	As	and □s. In particular we have □A(Symmetry) → O1...On(Symmetry). The combination of commutativity with A(Church-Rosser) also implies □A(Church-Rosser), and hence O1...On(Church-Rosser)	for	any	O1...On. For	consider	any	proposition	p. If	A◇p	is	false	then A¬A◇p,	so	A¬◇Ap	by	A(Church-Rosser),	so	A□¬◇Ap,	so	□A¬◇Ap	by	commutativity,	so □A(◇Ap	→	A◇p). If on the other hand A◇p is true, we have A□◇p, so □A◇p by commutativity,	so	□AA◇p,	so	again	□A(◇Ap	→	A◇p). 19 Fritz (forthcoming) proves that the product logic (his ΛPU) is not recursively axiomatizable;	given	its	axiomatizability	relative	to	the	background	logic,	it	follows	that the background logic (Fritz's ΛPF) is also not recursively axiomatizable, a fact already proved	by	Antonelli	and	Thomasson	(2002). 15 is	true	just	in	case	no	point	other	than	ι	is	temporally	accessible	from	ι. These	conditions will	coincide	in	any	relational	structure	whose	home	point	is	unaccompanied,	i.e.	in	any relational structure in which Symmetry is true. This fact makes Symmetry a natural principle	to	focus	on,	since	it	seems	to	at	least	partially	articulate	the	idea	that	time	and modality	interact	in	a	symmetric	way,	it	is	valid	in	the	product	logic,	and	in	the	background logic	it	implies	the	collapse	of	Perpetuity	to	propositional	eternalism. In	fact	very	little	of	the	background	logic	is	needed	to	establish	this	implication,	as	we can	see	from	the	following	direct	argument.	Suppose	Symmetry	and	Perpetuity	are	both true. Let	p	be	any	falsehood. By	Perpetuity,	∀q(□(p	→	q)	→	A(p	→	q)),	while	by	Symmetry, ∃q(□(p	→	q)	∧	A(p	→	¬q)). Combining these two formulae using standard quantifier reasoning, we have ∃q(A(p	→	q)	∧	A(p	→	¬q))). Given the agglomeration of A over conjunction, this implies ∃q(A((p	→	q)	∧	(p	→	¬q))), which is equivalent to ∃qA¬p by substitution	of	tautological	equivalents. But	∃qA¬p	obviously	entails	A¬p. Generalizing, we	have	∀p(¬p	→	A¬p),	which	is	equivalent	to	propositional	eternalism. Symmetry is	not	the	easiest	principle	to	think	about. Fortunately,	it	is	equivalent	in the	background	logic	to	the	following	structurally	more	familiar	principle: Supervenience:	Every	truth	is	necessitated	by	a	permanent	truth. ∀p(p	→	∃q(□(q	→	p)	∧	Aq)) (Proof:	∃q(□(q	→	p)	∧	Aq)	is	logically	equivalent	to	∃q(□(q	→	p)	∧	A(¬q	→	p)),	since	any	q that witnesses the former witnesses the latter, while if q witnesses the latter, p∨q witnesses the former. Supervenience is therefore equivalent to ∀p(p	→	∃q(□(q	→	p)	∧	A(¬q	→	p)));	substituting	¬p for	p	and	¬q for	q	and	contraposing the	conditionals	yields	Symmetry.) Supervenience	doesn't	wear	its	symmetry	on	its	sleeve	in	the	way	that	Symmetry	does. Nevertheless, since Supervenience is equivalent to something equivalent to its mirror image	(namely	Symmetry),	and	the	mirror	image	of	anything	valid	in	the	background	logic is	also	valid	in	the	background	logic,	Supervenience	is	equivalent	in	the	background	logic to	its	mirror	image: 16 Supervenience*:	For	every	truth,	there	is	a	necessary	truth	that	is	never	true	without it	being	true. ∀p(p	→	∃q(A(q	→	p)	∧	□q)) This	equivalence	helps	to	bring	out	how	Symmetry	goes	beyond	the	mere	denial	of	the combination of Perpetuity with propositional temporalism: not only are there are temporary necessary truths, there are necessary truths of arbitrarily short temporal extent. But	why	would	anyone	believe	these	principles? In the	next four	sections,	we	will consider	some	possible	arguments. IV. Perhaps the most straightforward argument for Symmetry turns on the following principle: NOW Every	proposition	is,	necessarily,	true	just	in	case	it	is	now	true. ∀p□(p	↔	Np) Supervenience, and hence Symmetry, follows from NOW together with the following relatively	uncontroversial	additional	premise: RIGN Everything	true	now	is	always	true	now. ∀p(Np	→	ANp) For	suppose	that	p	is	true.	Then	by	NOW,	the	proposition	that	p	is	now	true	is	true;	by RIGN it is a permanent truth, and by NOW it necessitates p. Thus every truth is necessitated	by	a	permanent	truth. Given NOW, RIGN, and propositional temporalism, we can generate specific counterexamples	to	Perpetuity. Suppose	p	is	temporarily	true. Then	the	proposition	that p	is	true	if	and	only	if	p	is	now	true	is	also	temporarily	true,	since	it	is	true	exactly	when	p 17 is true. But by NOW, this biconditional is necessarily true, and is therefore a counterexample	to	Perpetuity.20 Those	who	accept	NOW	will	not	think	that	the	proposition	it	expresses	is	always	true. Rather,	the	interesting,	non-arbitrary	view	in	the	vicinity	is	that	the	modal	status	which NOW	attributes	to	the	present	time	is	one	which,	necessarily,	every	time	has	at	itself. We can	express	this	more	general	idea	using	quantification	over	times,	as	follows: NOW+	Necessarily,	at	each	time	t,	every	proposition	is,	necessarily,	true	just	in	case it	is	true	at	t. □∀t	At	t	∀p□(p	↔	At	t	p)21 As	Fine	(1977:	169)	puts	it:	'At	each	time,	the	same	present	runs	through	each	possible world'.	The	necessary	permanent	truth	of	Supervenience	is	an	immediate	consequence	of NOW+:	necessarily,	at	any	time	t,	for	any	true	proposition	p,	the	proposition	that	p	is	true at	t	is	a	permanent	truth	that	necessitates	p.22 As	defenders	of	Perpetuity,	we	of course think there	are counterexamples to	NOW. Consider	the	proposition	that	dinosaurs	roam	the	Earth. It	is	not	true,	so	it	has	never	been now	true. But	it	has	been	true,	so	it	has	been	true	without	being	now	true. So	by	Perpetuity it	must be possible for it to be true	without being now true. With such propositions squarely	in	view,	we	find	that	Perpetuity	remains	compelling,	and	the	plausibility	of	NOW correspondingly diminishes. Without some further argument, then, rejecting NOW doesn't strike us as	particularly costly. But there are some interesting arguments for NOW. We	will	consider	two	of	them. 20	NOW	is	valid	on	the	logic	of	product	models	extended	to	interpret	the	N	operator	in	the manner	described	in	§II,	but	not	on	the	logic	of	Montagovian	models	extended	in	the	same way. RIGN	is	valid	on	both	classes	of	models. 21	NOW+	is	also	valid	on	the	class	of	product	models	enriched	to	interpret	quantification over	times.	If	you	are	a	fan	of	the	approach	to	temporal	anaphora	discussed	in	footnote	12, then,	for	your	eyes	only,	we	can	reformulate	NOW+	as	follows:	□A↑1∀p□(p	↔	↓1p). 22	By	contrast,	there	is	no	straightforward	derivation	of	NOW+	from	Supervenience. We can see this fact model-theoretically, by considering a generalization of product structures	in	which	the	modal	accessibility	relation	need	not	be	=2	(sharing	the	same	time co-ordinate), but can be any equivalence relation each of whose equivalence classes contains exactly one point with each world co-ordinate. Symmetry (and hence Supervenience)	is	still	valid	on	this	class	of	structures,	but	NOW	and	NOW+	no	longer	are. 18 First:	NOW	can	be	derived	from	the	following	claims:	(a)	'necessarily'	commutes	with 'now' (i.e. the propositions that are now necessarily true are exactly those that are necessarily	now	true);	(b)	every	proposition	is	necessarily	now:	true	if	and	only	if	now true;	and	(c)	every	proposition	that	is	now	true	is	true. Given	(a),	(b)	implies	that	every proposition	is	now	necessarily:	true	if	and	only	if	now	true. And	given	(c),	we	can	delete the	initial	'now'	from	this	claim	to	derive	NOW. We	see	no	convincing	grounds	to	accept	(a),	on	its	intended	interpretation	as	a	claim about	metaphysical	necessity.23 While	'necessarily	now	φ'	and	'now	necessarily	φ'	may be	interchangeable	in	typical	contexts,	it	is	also	clear	that	in	typical	contexts,	prefixing	a context-sensitive sentence with 'now' will lead us to favor resolutions of its contextsensitivity	on	which	it	expresses	a	non-eternal	proposition-otherwise,	the	'now'	would be	pointless. But	given	Perpetuity	and	modal	S5,	all	attributions	of	metaphysical	necessity are eternal: if true they are necessarily true, and hence always true; if false they are necessarily	false,	and	hence	always	false. So	in	typical	contexts	it	will	not	be	natural	to interpret the 'necessarily' in 'now	necessarily	φ' as	expressing	metaphysical	necessity. For this reason, one should be wary of drawing any conclusions about metaphysical necessity	from	our	intuitive	reactions	to	the	schema	'Necessarily	now	φ	just	in	case	now necessarily	φ'.24 Second:	NOW	can	be	derived	from	certain	premises	about	counterfactuals:	(i)	Every proposition	would	have	been	true	now	if	it	had	been	true;	(ii)	No	possibly	true	proposition is	such	that	a	contradiction	would	have	been	true	if	it	had	been	true;	(iii)	'Now'	commutes with truth-functional connectives (even under counterfactual suppositions); (iv) 'Now now'	is	intersubstitutable	with	'now'	(even	under	counterfactual	suppositions). (Proof: 23	We	also	have	doubts	about	(b)	which	will	emerge	in	§IX. 24 It would be a bit uncomfortable for us if, outside of special contexts like that of philosophy,	modals	in	ordinary	language	always	commuted	with	'now':	one	might	then worry	that	metaphysical	necessity	as	we	conceive it	does	not	really	count	as	a	kind	of necessity	at	all. But	there	are	cases	where	ordinary	modals	seem	not	to	commute	with 'now'. Consider	for	example	the	use	of	'might'	on	which	it	concerns	the	epistemic	state	of some	salient	person	other	than	the	speaker. If	the	salient	person	is	not	sure	of	the	time, we	can	say	things	like	'Although	she	knows	that	it	is	raining,	it	might	not	(as	far	as	she knows)	be	raining	now,	since	she	is	still	trying	to	find	out	what	time	it	is'. 19 Let	>	abbreviate	the	counterfactual	conditional. By	(i),	∀p(¬(p	↔	Np)	>	N¬(p	↔	Np)). By (iii) this is equivalent to ∀p(¬(p	↔	Np) > ¬(Np	↔	NNp)), and hence, given (iv), to ∀p(¬(p	↔	Np) > ¬(Np	↔	Np)). But ¬(Np	↔	Np) is a contradiction, so by (ii), ∀p□(p	↔	Np).)25 To	resist	this	argument,	proponents	of	Perpetuity	must	deny	one	of	(i)–(iv). We	will focus	on	(i),	although	we	also	have	some	doubts	about	(ii)	and	(iii). Granted,	it	generally seems	fine	to	insert	and	delete	'now's	in	the	consequents	of	ordinary	counterfactuals. But since	most	of	the	counterfactuals	we	ordinarily	consider	have	antecedents	which	could have	been	true	now,	it	is	tendentious	to	abstract	from	this	pattern	a	rule	which	licenses such insertions and deletions even in counterfactuals whose antecedents could not possibly	have	been	true	now. For	example,	'now'	seems	not	to	be	redundant	in	'If	we	were at	a	philosophy	conference	in	the	year	2500,	we	would	probably	not	be	talking	about	any issue	now	regarded	as	important.'	Analogously,	in	evaluating	ordinary	counterfactuals	we freely	help	ourselves	to	the	actual	laws	of	nature	(see	Goodman	2015	and	Dorr	2016),	but it	would	be	very	tendentious	to	extract	from	this	practice	a	general	principle	to	the	effect that	the	actual	laws	would	have	been	true	no	matter	what.26 V. A	second	argument	for	Symmetry	appeals	to	the	following	two	premises: ACT Every	proposition	is,	always,	true	if	and	only	if	actually	true. ∀p	A(p	↔	@p) RIG@ Every	actually	true	proposition	is	necessarily	actually	true. ∀p(@p	→	□@p) For	the	same	reason	that	NOW	and	RIGN	jointly	imply	Supervenience,	ACT	and	RIG@	jointly imply Supervenience* (the result of interchanging temporal and modal operators in 25	Thanks	to	John	Hawthorne	for	suggesting	this	argument. 26	Note	too	that	there	is	no	need	for	defenders	of	Perpetuity	to	make	a	once-and-for-all choice as regards which premise to give up: given the context-sensitivity of counterfactuals,	one	could	reasonably	maintain	that	(i)	holds	in	some	contexts	while	(ii) holds	in	others. 20 Supervenience),	and	hence	also	imply	Supervenience	and	Symmetry	(which	we	showed	to be	equivalent	to	Supervenience*	in	§III).27 Assuming	propositional	temporalism,	they	also can	be	used	to	generate	explicit	counterexamples	to	Perpetuity:	whenever	p	is	temporarily true,	the	proposition	that	p	is	actually	true	is	necessarily	true,	by	ACT	and	RIG@,	and	yet sometimes	false,	by	ACT. We	do	not	deny	that	both	(the	English	versions	of)	ACT	and	RIG@	have	readings	on which	they	express	obvious	truths.28 But	English	sentences	involving	'actually'	often	have several	readings,	and	we	see	little	reason	to	think	there	is	any	non-equivocating	reading of	ACT	and	RIG@	on	which	both	are	true.29 The	acceptability	of	sentences	like	'I	could	have actually run you over!' shows that adding 'actually' sometimes makes no discernible difference	to	truth	conditions. Read	in	the	corresponding	way,	the	following	principle	is true: ACT□ Every	proposition	is,	necessarily,	true	if	and	only	actually	true. But	on the interpretation corresponding to this	one,	RIG@ is false, since together	with ACT□	it	would	entail	that	every	truth	is	a	necessary	truth. Our opponents might object to our assimilation of ACT to ACT□ by pointing to a contrast between the ways 'actually' embeds in temporal and modal environments. Consider: (1) a. The	climate	could	be	warmer	than	it	actually	is. 27	ACT	and	RIG@	are	also	both	valid	on	the	class	of	product	models	enriched	to	interpret @	in	the	manner	described	in	§II. 28	However, see Yalcin 2015 for reasons to doubt that RIG@ admits a true reading in ordinary	English. 29	There	are	a	number	of	ways	in	which	this	multiplicity	might	be	generated. On	one	view (Crossley	and	Humberstone	1977),	the	word	'actually'	is	ambiguous,	having	a	'rhetorical' use	on	which	it	is	semantically	inert,	and	a	separate	'logical'	use	characterized	by	RIG@. On	a	second	view	(Correia	2007;	inspired	by	Vlach	1973),	'actually'	is	more	flexible,	and has	the	effect	that	the	sentence	it	embeds	is	evaluated	as	if	it	occurred	at	some	wider	scope in the sentence; on its most natural implementation, this view treats 'actually' as a bindable	operator,	introducing	structural	ambiguities	analogous	to	those	introduced	by variables. On	a	third	view	(Yalcin	2015),	'actually'	is	strictly	speaking	semantically	inert, although	it	may	provide	clues	to	the	resolution	of	certain	structural	ambiguities	which	are present	in	the	'actually'-free	sentence. 21 b. The	climate	will	be	warmer	than	it	actually	is. (1a) is fine,	whereas (1b)	sounds	odd, at least	out	of the	blue.30 (Our informants	said things	like	'That	is	not	the	right	way	to	say	it	in	English-you	should	say	"now"	rather	than "actually"'.) So	our	opponents	might	argue	as	follows:	(1b)	is	infelicitous	because	it	is	false on	all	readings;	the	falsity	of	(1b)	is	best	explained	by	a	more	general	principle	from	which it	follows	that	ACT	is	true	on	all	readings;	hence,	since	it	is	agreed	that	RIG@	is	true	on	at least	one	reading,	it	follows	that	there	is	at	least	one	reading	on	which	both	are	true,	so that	the	argument	against	Perpetuity	goes	through. We	think	that	this	argument	fails	at	the	first	step	(which	is	not	to	concede	that	the	rest of	the	argument	is	unproblematic). Although	there	is	a	contrast	between	(1a)	and	(1b)	as regards	the	ease	of	accessing	true	readings,	with	the	right	setup-for	example,	when	a certain	fiction,	misapprehension,	or	only	recently	ruled	out	hypothesis	about	the	current climate	is	salient-(1b)	can	sound	fine.	There	are	also	theoretical	reasons	to	think	that (1b)	has	a	true	reading.	Consider	the	results	of	deleting	'actually'	from	(1a)	and	(1b): (2) a. The	climate	could	be	warmer	than	it	is. b. The	climate	will	be	warmer	than	it	is. (2b)	clearly	has	a	true	reading,	and	in	view	of	the	equivalence	of	(2a)	and	(1a),	it	is	hard to see how the addition of 'actually' in (1b) could prevent it from having a parallel reading.31 The	infelicity	of	(1b)	discourse	initially	is	an	instance	of	a	general	feature	of	'actually' that	has	nothing	to	do	with	temporal	environments. In	response	to	the	question	'What	do you	do?',	it	would	be	odd	to	reply	'Actually	I	am	a	doctor'. No	doubt	this	is	because	the usual	role	of	'actually'	is	to	signal	some	kind	of	surprise	or	contrast,	or	that	something	is 30	If	you	are	having	trouble	accessing	the	required	non-epistemic	reading	of	'could'	in	(1a), consider	instead	'He	could	work	harder	than	he	actually	does'. 31	Of	course,	(2b)	has	a	trivially	false	reading	too.	But	when	the	present	tense	occurs	in	a past-tense environment (with or without 'actually') a reading analogous to the true reading	of	(2b)	is	forced-e.g. 'It	used	to	be	colder	than	it	[actually]	is'. See	Wehmeier 2004	and	Mackay	2013. 22 being	corrected,	or	the	like	–	when	it	is	not	clear	from	context	how	it	could	play	any	of these	roles,	it	is	typically	infelicitous.	In	view	of	this	generalization,	the	striking	fact	is	that (1a)	is	felicitous	out	of	the	blue,	or,	more	generally,	that	'actually'	embeds	more	happily under	certain	modals	than	elsewhere.	But	we	will	not	speculate	about	the	explanation	of this	fact,	since	we	see	no	reason	to	think	that	it	bears	on	the	question	of	this	paper.32 One	might	reply	that	even	if	the	use	of	@	that	makes	RIG@	unambiguously	true	is	a philosopher's invention, it	is intelligible	and	useful. We	agree. But	clearly	we	have	no business	having	pre-theoretic	judgments	about	the	truth	of	ACT	when	@	is	introduced	in part	by	the	stipulation	that	RIG@	is	true. Another	way	to	introduce	@	as	a	piece	of	philosopher's	jargon	is	to	appeal	explicitly to	the	metaphysics	of	possible	worlds:	we	could	introduce	a	name	'α'	for	the	actual	world, and	let	'@p'	abbreviate	'p	is	true	at	α'. But	on	this	interpretation,	the	right	way	to	assess ACT	and	RIG@	is	again	not	on	the	basis	of	their	pre-theoretical	plausibility,	but	as	part	of some	broader	theory	about	possible	worlds. In	the	next	section	we	consider	how	such theories	bear	on	Perpetuity. VI. Philosophers writing about possible worlds often take the following principles for granted: Leibnizian	Possibility:	A	proposition	is	possibly	true	if	and	only	if	it	is	true	at	some possible	world. ◇p	↔	∃w(At	w	p) Conjunction:	The	conjunction	of	two	propositions	is	true	at	a	possible	world	if	and only	if	both	of	those	propositions	are	true	at	that	world. At	w	(p	∧	q)	↔	(At	w	p	∧	At	w	q) Negation:	The	negation	of	a	proposition	is	true	at	a	possible	world	if	and	only	if	that proposition	is	not	true	at	that	world. 32	See	Mackay	2017	for	a	presupposition-theoretic	account	of	the	felicity-conditions	for 'actually'. 23 At	w	¬p	↔	¬At	w	p Historicity: Possible	worlds that agree about how things always are agree about everything. ∀p(At	w	Ap	↔	At	w′	Ap)	→	∀p(At	w	p	↔	At	w′	p) These principles jointly imply Symmetry, given a further principle that is valid in the background	logic: Encylopedia:	Some	permanent	truth	necessitates	every	permanent	truth. ∃q(Aq	∧	∀p(Ap	→	□(q	→	p))) (Proof:	We	will	derive	Supervenience,	which	is	equivalent	to	Symmetry. Let	h	witness	the truth	of	Encylopedia. We	first	show	that	any	two	worlds	w	and	w′	at	which	h	is	true	agree about	everything. Let	p	be	any	proposition. If	p	is	always	true,	then	the	proposition	that p	is	always	true	is	itself	always	true,	and	thus	necessitated	by	h. So	h	∧	¬Ap	is	not	possible; by	Leibnizian	Possibility	it	is	true	at	no	world;	so	by	Conjunction	and	Negation,	Ap	is	true at	every	world	where	h	is	true,	and	in	particular	at	w	and	at	w′. If	on	the	other	hand	p	is not	always	true,	then	the	proposition	that	p	is	not	always	true	is	always	true,	and	thus necessitated	by	h. So	h	∧	Ap	is	not	possible;	by	Leibnizian	Possibility	it	is	true	at	no	world; so	by	Conjunction,	Ap	is	neither	true	at	w	nor	true	at	w′. So	for	any	p,	Ap	is	true	at	both	or neither	of	w	and	w′,	which	by	Historicity implies	that	w	and	w′	agree	about	everything. Next, to establish Supervenience, suppose	p is true. Then	p	∧	h is true, and therefore possible,	and	therefore	true	at	some	world	w	by	Leibnizian	Possibility. By	Conjunction,	h and	p	are	both	true	at	w,	so	by	the	result	just	established,	p	is	true	at	every	possible	world at	which	h	is	true. So	by	Negation	and	Conjunction,	there	is	no	possible	world	at	which h	∧	¬p	is	true,	and	hence	by	Leibnizian	Possibility	h	∧	¬p	is	not	possible:	p	is	necessitated by	the	permanent	truth	h.) Since propositional eternalism is a common presupposition of theorizing about possible	worlds,	it	is	worth	checking	that	the	four	principles	above	are	compatible	with propositional temporalism. This can be seen by observing that, unlike propositional eternalism, all four principles are valid on the class of product models from §II. In 24 combination with propositional temporalism, these principles suggest a "changing pluriverse"	way	of thinking	about time	and	modality. A	proposition is	metaphysically possible	just	in	case	it	is	true	at	some	world. But	the	facts	about	what	is	true	at	which worlds are temporary, so some of the propositions that are possible today will be impossible	tomorrow.33 Like	many	other	metaphysicians,	we	think	it	is	dangerous	to	let	one's	opinions	about modal	questions	be	driven	by	one's	theory	of	possible	worlds,	rather	than	the	other	way around.	So	we	do	not	think	it	is	much	of	a	count	against	the	conjunction	of	Perpetuity	and propositional	temporalism	that	it	requires	giving	up	at	least	one	of	Leibnizian	Possibility, Conjunction,	Negation,	and	Historicity. How	should	friends	of	Perpetuity	and	propositional	temporalism	think	about	worlds and	truth	at	a	world? Perhaps	the	simplest	option	is	to	give	up	Historicity,	keep	the	other three	principles,	and	think	of	possible	worlds	as	"points	in	logical	space",	corresponding to propositions p which are possibly modally maximal truths, in the sense that ◇(p	∧	∀q(q	→	□(p	→	q))).34 Alternatively,	we	could	preserve	Historicity,	thinking	of	possible	worlds	as	"possible histories", corresponding to propositions p which are possibly modally maximal permanent	truths:	◇(Ap	∧	∀q(Aq	→	□(p	→	q))).	When,	in	the	course	of	a	possible	history, a	proposition	is	sometimes	true	and	sometimes	false,	the	world	is	not	enough	to	settle	its truth	value.35 To	figure	out	which	of	the	other	three	principles	fails,	we	need	to	decide what	it	means	for	a	proposition	to	be	true	"at"	a	world. There	are	two	obvious	options. The first holds that	p is true at	w just in case	p is necessitated by the	world-history 33 The changing pluriverse picture is not the only way of combining propositional temporalism	with the four	principles. One could instead	adopt	a	view	on	which facts about	what	is	true	at	a	given	possible	world	are	both	permanent	and	necessary,	but	which things	are	possible	worlds	is	a	temporary	matter,	so	that	any	given	possible	world	is	only possible	for	an	instant. 34	In	the	generalized	Montagovian	structures	of	§III,	this	condition	is	satisfied	by	all	and only	the	singleton	subsets	of	the	domain. 35	In	generalized	Montagovian structures this condition is satisfied	by	all and	only the equivalence	classes	of	temporally	accessible	points. 25 corresponding to	w. This preserves	Conjunction	but requires giving up	Negation and Leibnizian	Possibility.36 This	view	also	rejects	the	following	principle: Disjunction:	The	disjunction	of two	propositions is true	at	a	possible	world if	and only	if	one	of	those	propositions	is	true	at	that	world. At	w	(p	∨	q)	↔	(At	w	p	∨	At	w	q) The	second	option	holds	that	p	is	true	at	w	just	in	case	p	is	compossible	with	the	worldhistory	corresponding	to	w. On	this	approach,	Leibnizian	Possibility	and	Disjunction	hold, but	Conjunction	and	Negation	fail.37 Moreover,	for	any	p,	p	∨	A¬p	will	be	compossible	with every	world-history	despite	being	contingent,	and	will	thus	be	a	counterexample	to: Leibnizian	Necessity:	A	proposition	is	necessarily	true	if	and	only	if	it	is	true	at	every possible	world. □p	↔	∀w(At	w	p) On both the "pointy" and "historic" conceptions, worlds correspond to certain propositions,	only	one	of	which	is	true. Call	this	world	'α'	('the	actual	world')	and	interpret '@'	as	synonymous	with	'At	α'. Assuming	Perpetuity	and	propositional	temporalism,	we know from the previous section that at least one of ACT and RIG@	must fail on this interpretation. In	fact,	since	the	all	the	views	we	have	considered	make	facts	about	what is true	at a	world	non-contingent, they	all	preserve	RIG@	and thus reject	ACT. On the pointy	conception,	and	on	the	historic	conception	with	the	compossibility	interpretation of	'At',	any	temporarily	true	p	is	a	counterexample	to	ACT:	@p	is	necessarily	true,	hence always true, so	p	↔	@p is only temporarily true. On the historic conception	with the necessitating	interpretation	of	'At',	any	temporarily	false	p	is	a	counterexample	to	ACT: @p	is	necessarily	false,	hence	always	false,	so	again	p	↔	@p	is	only	temporarily	true. 36	Let	p be some temporary truth. Then	p	∧	¬Ap is true, and so	possibly true,	but	not possibly always true, and hence not necessitated by any possible world-history, in violation	of	Leibnizian	Possibility;	and	at	least	one	possible	history,	viz.	the	true	one,	only necessitates	truths	and	hence	fails	to	necessitate	the	negation	of	p	∧	¬Ap,	in	violation	of Negation. 37	Where	p	is	a	temporary	truth,	the	true	world-history	is	compossible	both	with	p	and with ¬p, in violation of Negation, and not compossible with p	∧	¬p, in violation of Conjunction. 26 Our	attitude	towards	the	argument	for	Symmetry	from	ACT	and	RIG@	on	the	worldtheoretic	interpretation	is	the	same	as	our	attitude	to	the	argument	considered	earlier	in this	section: 'possible	world'	is	enough	of	a	term	of	art	that	we	are	unperturbed	by	the prospect	of	giving	up	one	of	these	two	principles. Indeed, in view of the potential for equivocation between pointy and historic conceptions	of	worlds,	we	think	it	best	to	avoid	'world'-talk	altogether	in	theorizing	about temporary	matters. But	if	forced	to	choose,	we	would	favor	the	pointy	conception,	since Leibnizian Possibility and Leibnizian Necessity seem particularly central for the usual applications	of	world-talk. Moreover	there	are	three	further	reasons	to	dislike	the	historic conception,	at	least	when	combined	with	a	necessitating	or	compossibility	interpretation of	truth	at	a	world:	first,	it	seems	arbitrary	to	choose	between	Leibnizian	Possibility	and Leibnizian Necessity; second, it seems arbitrary to choose between Conjunction and Disjunction;	and	third,	the	failure	of	truth	to	be	coextensive	with	truth	at	the	actual	world runs	completely	counter	to	the	way	philosophers	are	used	to	talking	about	worlds.38 38 These three further problematic features are not inevitable consequences of the historic	conception	of	worlds. Instead	of	a	necessitating	or	compossibility	interpretation of	'At	w',	one	could	adopt	a	counterfactual	interpretation,	on	which	a	proposition	is	true at	a	world	if	and	only	if	it	would	have	been	true	had	the	corresponding	world-history	been true. Given	the	Strong	Centering	principle,	according	to	which	counterfactuals	with	true antecedents	have	the	same	truth	value	as	their	consequents,	this	analysis	preserves	the coextensiveness of truth with truth at the actual world. Conjunction holds; given Perpetuity and propositional temporalism, Leibnizian Possibility fails. In order to get Disjunction to hold we need Conditional Excluded Middle, in which case Leibnizian Necessity will also fail. Negation holds assuming that only counterfactuals with metaphysically	impossible	antecedents	are	vacuously	true. Another	interesting	feature of	the	view	is	that,	unlike	the	views	considered	in	the	main	text,	it	preserves	ACT. On	the other hand RIG@ and its converse both fail (assuming Perpetuity and propositional temporalism),	since	the	property	of	being	true	at	the	actual	world	is	one	that	temporary truths	have	only	contingently. Another	analysis	yielding	similar	results	holds	that	p	is	true	at	w	just	in	case	whichever time	is	present	is	such	that	possibly,	w's	world-history	and	p	are	both	true	and	that	time is	present. But	if	the	present	time	could	fail	to	be	ever	present,	as	we	will	argue	in	§IX, this	view	will	require	giving	up	Negation. 27 VII. We turn finally to a	more abstract and theoretical strategy for arguing for Symmetry. Consider	the	following	schematic	premises: F-Supervenience:	Every	truth	is	necessitated	by	some	type-F	truth. ∀p(p	→	∃q(q	∧	Fq	∧	□(q	→	p)) F-Eternalism:	Every	type-F	truth	is	always	true. ∀p((p	∧	Fp)	→	Ap) Together these premises immediately imply Supervenience, and hence Symmetry. Moreover, there are various interpretations of 'type-F propositions' on which both premises	of	this	argument	have	some	plausibility,	such	as	propositions	about	how	things are	in	"metaphysically	fundamental"	respects,	propositions	about	microphysics,	and	(of particular	interest)	spacetime-theoretic	propositions. Since	Einstein	and	Minkowski,	we have	learnt	to	characterize	the	physical	world	in	terms	of	the	distribution	of	field-values in four-dimensional spacetime, rather than the evolution of field-values in threedimensional	space. It	is	natural	to	think	that	there	is	already	something	temporal	about points	of	spacetime,	so	that	it	makes	no	sense	to	suppose,	for	example,	that	a	particular spacetime point will change from having a low mass-density to having a high mass density,	just	as	it	would	make	no	sense	to	suppose	that	a	particular	instant	of	time	will change	from	being	one	at	which	it	is	raining	to	being	one	at	which	it	is	not	raining. And given	a	broadly	physicalistic	outlook,	the	success	of	this	way	of	doing	physics	supports the	thesis	that	all	truths	supervene	on	truths	about	the	distribution	of	physical	fields	in spacetime,	perhaps	by	way	of	the	thought	that	only	such	truths	are	fundamental	and	all truths	supervene	on	the	fundamental	truths.39 This strikes us as an important line of argument. But it also looks to us like an argument for propositional eternalism. It is hard to see how it could persuade a propositional	temporalist	to	accept	Symmetry,	since	it	is	unclear	what	could	motivate	F- 39	Bacon	(2018)	defends	a	view	on	which	all	fundamental	truths	are	eternal,	and	on	which they	jointly	necessitate,	but	do	not	eternally	imply,	the	non-fundamental	truths. 28 Supervenience	without	also	motivating	the	stronger	thesis	that	for	every	truth	there	is	a type-F	truth	that	always	necessitates it. One	obviously	cannot	appeal to the	claim	that every	truth	is	a	type-F	truth. One	might	try	appealing	to	the	thesis	that	every	truth	is	in some	sense "determined"	or "grounded"	by	a type-F truth. But if	grounding is	simply identified	with	necessitation,	this	is	question-begging,	while	if	it	is	understood	in	some other	way,	it	is	hard	to	see	why	grounded	truths	should	be	necessitated	by	the	truths	that ground them,	given that they	need	not	be eternally implied	by the truths that	ground them. F-Supervenience and F-Eternalism have further puzzling consequences on the assumption that there are temporarily true qualitative propositions. 40 Let an Fpossibility	be	a	possibly-true	proposition	that	conjoins	F-Supervenience	with	a	maximally specific type-F proposition. Assuming that being type-F is a necessary property of propositions, every F-possibility necessitates every proposition with which it is compossible.41 For example, each F-possibility	either necessitates that the number of stars	is	odd,	or	necessitates	that	it	is	not	odd. But	if	there	is	qualitative	change,	there	will presumably	be	many	F-possibilities	that	necessitate	that	the	number	of	stars	is	sometimes but	not	always	odd. The	division	of	these	F-possibilities	into	those	that	necessitate	that the	number	of	stars	is	odd	and	those	that	necessitate	that	it	isn't	odd	seems	like	it	must exhibit a certain arbitrariness that we should hope to avoid when such purported necessary	connections	are	at	issue. One	might try to	mitigate such arbitrariness by postulating that there is a certain three-dimensional	slice	s	through	spacetime,	such	that	what	propositions	an	F-possibility necessitates	is	a	function	of	what	it	says	about	s. For	example,	perhaps	an	F-possibility necessitates	that	the	number	of	stars	is	odd	when	it	entails	that	s	contains	an	odd	number 40	This	is	not	an	inevitable	commitment	of	propositional	temporalism:	Dorr	(MS)	develops a form of propositional temporalism on which only non-qualitative (haecceitistic) propositions	can	be	temporarily	true. The	arbitrariness	worries	we	are	about	to	raise	do not	arise	in	any	obvious	way	for	proponents	of	F-Supervenience	who	accept	this	view. 41 If being type-F is a contingent property of propositions, we could redefine an Fpossibility	as	a	possibly-true	conjunction	which,	for	each	proposition,	specifies	whether it	is	type-F	or	not,	and	if	it	is	specified	to	be	type-F,	specifies	whether	it	is	true	or	not,	and also	has	F-Supervenience	itself	as	a	conjunct. 29 of	appropriately	shaped,	high-temperature	subregions. (Presumably	this	proposal	will	be combined	with	the	claim	that	which	region	of	spacetime	is	modally	distinguished	in	this way is constantly changing.) But this suggestion introduces new problems. First, in distinguishing	one	slice	through	spacetime	as	special	in	this	way,	the	proposal	conflicts with	what	is	widely	taken	to	be	a	basic	moral	of	relativity	physics.42 Second,	it	is	in	tension with	the	following	claims:	(i)	if	F-Supervenience	is	true,	then	so	too	is	the	analogous	thesis restricted to purely qualitative propositions, namely that every qualitative truth is necessitated	by	a	qualitative spacetime-theoretic	proposition; (ii) there	are	pairs	of	Fpossibilities	that	agree	on	all	qualitative	spacetime-theoretic	propositions	but	disagree	as regards	the	qualitative	spacetime-theoretic	role	played	by	s.43 For	example,	if	two	such F-possibilities disagree as regards whether s contains an odd number of hightemperature	star-shaped	regions,	the	proposed	special	status	for	s	will	require	them	to necessitate	incompatible	qualitative	propositions,	in	violation	of	(i). Thirdly,	the	proposal breaks	down	when	we	consider	metaphysical	possibilities	in	which	s is	not	part	of	the spacetime	manifold	at	all:	for	example,	possibilities	where	spacetime	undergoes	an	early gravitational	collapse. (We	will	have	more	to	say	about	such	possibilities	in	§IX.) 42	Bacon	(2018)	suggests	a	response	to	this	sort	of	worry:	although	s	is	specially	related to	a	large	family	of	properties	and	relations	which	includes	being	a	star,	being	spherical, being	more	massive than, etc., this is just one of	many structurally similar families of properties	and	relations,	and	every	slice	is	specially	related	to	its	own	such	family.	Bacon further	claims	that	these	families	of	properties	and	relations	are	all	'on	a	par',	so	that	s	is not	special	in	any	objectionable	sense. But	we	think	that	in	the	relevant	sense	of	'on	a	par', namely	being	equally	natural	in	the	sense	of	Lewis	(1983),	the	different	families	are	not on	a	par.	For	the	members	of	the	familiar	family	that	contains	being	a	star	are	easier	to refer	to	than	their	counterparts	in	other	families	(which	typically	require	special	devices like	metric	tense	operators	to	express),	which	is	good	reason	to	think	them	more	natural: see	Dorr	and	Hawthorne	2012. (The fact that	reference	also	has	counterparts in	other families, each	of	which is similarly	easy for	people to	stand in	to the	members	of that family,	does	not	undermine	this	point.) 43 (ii) will be denied by "modal anti-haecceitists", according to whom all truths are necessitated	by	qualitative	truths,	and	perhaps	also,	depending	on	the	shape	of	spacetime, by	"metric	essentialists"	like	Maudlin	(1990). 30 VIII. Having	addressed	a	range	of	arguments	for	Symmetry,	in	this	section	and	the	next	we	will go	on the	offensive	and	develop two	arguments	against	Symmetry. The	basic thought behind	both	arguments	is that	it is	contingent	what	times	there	are. This	is	a	plausible idea. Moreover,	its	plausibility	is	independent	of	the	plausibility	of	Perpetuity,	and	so	it	is dialectically appropriate in the context of a defence of Perpetuity. Indeed Fine, immediately after presenting the product model theory, suggests that the account is 'perhaps over-simple', and that a 'more sophisticated' account would allow for contingency	as	regards	what	times	there	are	(Fine	1977:	167). Our	first	argument	turns	on	there	being	contingency	as	regards	the	cardinality	of	the time	series. A	nice	feature	of	this	argument	is	that	it	can	be	formalized	in	a	way	that	does not	explicitly	talk	about	times	at	all,	using	instead	only	tense	operators	and	propositional quantifiers. For	any	natural	number	n,	the	claim	that	the	time-series	has	a	cardinality	of at	least	n	can	be	formalized	as	the	claim	that	there	are	n	propositions,	each	of	which	is sometimes	true,	and	no	two	of	which	are	ever	both	true. And	although	our	basic	language L does not contain the resources to distinguish different infinite cardinalities that the time-series	might	have, there	are several	natural extensions	of	L	which	do	allow	such distinctions	to	be	drawn. For	example,	we	could	add	higher-order	quantifiers,	e.g.	into	the position of n-ary sentential operators, which would allow us to adapt standard settheoretic	ways	of	characterizing	different	infinite	cardinalities. Or	we	could	move	to	an infinitary	language	in	which	the	analogues	of	the	above	finite	quantified	claim	could	be expressed	directly. Or	we	could	simply	add	the	standard	future	and	past	tense	operators G and H, which make it possible to formulate various claims about the before-after structure	of	the	time-series	which	(intuitively	and	model-theoretically)	have	cardinalitytheoretic	implications,	although	they	are	strictly	stronger	than	any	mere	cardinality	claim. For	example,	using	these	tense	operators	together	with	propositional	quantifiers,	we	can say	that	the	time-series	is	a	discrete,	dense,	or	continuous	linear	ordering.44 44	For the	expression	of such	claims see	Burgess (2002). Relational structures can	be straightforwardly	extended	to	interpret	G	and	H	by	supplementing	them	with	a	transitive, 31 We	maintain	that	such	cardinality-theoretic	claims	are	metaphysically	contingent. For example,	it	is	metaphysically	possible	for	time	to	be	structured	either	like	the	integers, like	the	rationals,	or	like	the	real	numbers.	This	judgment	is	bolstered	by	the	judgment that	the	structure	of	spacetime	is	contingent	in	parallel	ways:	spacetime	could	have	been continuous,	discrete,	dense	but	countable,	etc. Admittedly,	such	possibility	claims	are controversial: they will be denied, for example, by those who identify metaphysical possibility	with	physical	possibility. Moreover, it is	a	vexed	question	exactly	what the connection	is	between	the	metaphysics	of	spacetime	and	the	metaphysics	of	time. But	it would be extremely surprising if there were no such connection: it is not a mere coincidence	that	spacetime	and	time	are	both	continuous	(or	both	discrete,	or	both	densebut-countable,	as	the	case	may	be). In addition to contingency in the cardinality of the time-series, our argument	will require	two	other	premises.	The	first-which	we	don't	expect	to	be	controversial-is	that if	Symmetry is true, then it is	necessarily	always	true. The	second is that	metaphysical modality is not a source of temporariness in its own right. The thought is that the attribution	of	metaphysical	possibility	or	necessity to	an	eternal	proposition	results in another eternal proposition. Intuitively, an eternal proposition is one that is nontemporary	not	just	de	facto	but	of	its	nature. However	this	status	is	understood,	clearly Ap	should	count	as	eternal	for	any	p,	and	clearly	□A(p	→	Ap)	should	be	true	for	any	eternal p. Thus, if the underlying thought about eternalness is true under any reasonable interpretation,	so	is asymmetric	relation	<	on	the	domain,	required	to	be	such	that	x	≈A	y	just	in	case	x	=	y	or x	<	y	or	y	<	x: ⟦Gφ⟧g	=	{x	∈	I	:	y	∈	⟦φ⟧g	for	all	y	such	that	x	<	y} ⟦Hφ⟧g	=	{x	∈	I	:	y	∈	⟦φ⟧g	for	all	y	such	that	y	<	x} This	language	contains	sentences	which	are	true	at	a	point	if	and	only	if	the	<-ordering restricted	to	points	temporally	accessible	from	it	is	discrete;	similarly	for	being	discrete and	bounded,	being	unbounded,	being	dense,	being	dense	and	Dedekind-complete,	and other	properties	which	settle	the	cardinality	of	temporally	accessible	points. 32 Eternity:	Necessarily	always:	if	a	proposition	is	necessarily	always	true,	it	is	always necessarily	always	true. ∀p(□A(□Ap	→	A□Ap)) We think that Eternity is attractive in its own right, and not merely because it is a consequence	of	(the	necessary	permanent	truth	of)	Perpetuity. For	example,	Eternity	will appeal	to	those	attracted	to	the	idea	that	eternal	truths	are	"accessible	from	a	God's	eye point	of	view"	in	a	way	that	temporary	truths	are	not,	since	one	would	expect	that	if	the truth	of	a	proposition	is	open	to	God's	view,	then	so	too	is	its	modal	status.45 To	see	why	Eternity	and	the	necessary	eternal	truth	of	Symmetry	are	inconsistent	with contingency in the cardinality of the time series, first note that that	Eternity logically implies	that	the	Church-Rosser	principle	from	§III-∀p(◇Ap	→	A◇p)-is	true	necessarily and	always. (Proof:	suppose	that	◇Ap. Then	by	temporal	S5,	AS◇SAp,	i.e.	A¬A□A¬Ap. So by the contrapositive form of Eternity, A¬□A¬Ap, i.e. A◇SAp, which implies A◇p by temporal	S5.46) Given	this	result,	the	inconsistency	we	are	interested	in	follows	from	a result proved in §III, namely that any relational structure in which Symmetry is necessarily always true and Church-Rosser is both necessarily and always true is isomorphic to a product structure. In any such structure, all	histories	have the same cardinality, so any sentence entailing contingency in the cardinality of the time series must	be	false.47 45	The	tension	between	this	way	of	thinking	and	Symmetry	is	perhaps	unsurprising,	since the	idea	that	contingent	questions	can	be	"objective"	in	a	way	that	temporary	questions cannot	be	involves	a	breaking	of	the	symmetry	between	time	and	modality. 46 The reverse is true as well: Eternity follows in the background logic from the combination of □(Church-Rosser) and A(Church-Rosser), since as shown in §III these jointly imply the commutativity principle ∀p□A(□Ap	↔	A□p) and hence ∀p□A(□AAp	↔	A□Ap),	which	is	trivially	equivalent	to	Eternity. 47	In fact, just from the truth	of	Church-Rosser and	A(Symmetry) in	a	model, it already follows	that	there	is	no	point	x	modally	accessible	from	ι	such	that	x's	history	is	smaller than	ι's	(where	the	history	of	a	point	is	the	set	of	all	points	temporally	accessible	from	it). For	by	Church-Rosser,	ι	is	square-completing,	so	for	every	point	in	ι's	history,	there	is	some point	in	x's	history	is	modally	accessible. If	ι's	history	had	a	larger	cardinality	than	x's,	it would	have	to	contain	two	distinct	points	from	which	the	same	point	in	x's	history	was modally accessible. But since	modal accessibility and temporal accessibility are both equivalence	relations,	these	two	points	would	be	both	modally	and	temporally	accessible to one another, and would therefore not be unaccompanied, which cannot happen if 33 As	noted	above,	our	simple	language	L	lacks	the	resources	to	say	that	the	cardinality of the time series is infinite, or to distinguish between different possible infinite cardinalities	of	the	time	series.	But	it	does	allow	us	to	talk	about	finite	cardinalities,	and since some	might	want to escape the present objection to Symmetry by rejecting our background logic, it may be instructive to consider object-language arguments from Eternity	and	(the	necessary	permanent	truth	of)	Symmetry	to	the	falsehood	of	particular claims	to	the	effect	that	there	are	n	times	and	could	have	been	more	or	fewer	than	n	times. An	especially	easy	case	is	that	of	cardinality	1. Insofar	as	the	general	idea	of	contingency in	the	cardinality	of	the	time	series	is	well-motivated,	it	is	hard	to	deny	that,	as	a	limiting case,	there	could	have	been	exactly	one	time:	i.e.	that	it	could	have	been	that	everything true was always true (◇∀p(p	→	Ap)). But given Eternity, this claim directly implies Perpetuity,	and	is	thus	inconsistent	with	the	combination	of	Symmetry	and	propositional temporalism. For	suppose	□q. Then	□□q;	so	◇(□q	∧	∀p(p	→	Ap)),	so	◇A□q. By	Eternity this	yields	A◇A□q;	but this implies	A◇□q (by	the factivity	of	A)	and	hence	Aq	(by	the modal	B	schema). Similar	arguments can	be	given for finite cardinalities	greater than one.48 A(Symmetry)	is	true	in	the	model. By	similar	reasoning,	the	truth	of	□(Church-Rosser)	and □A(Symmetry) is enough to guarantee that no point modally accessible from ι has a history	greater	in	cardinality	than	that	of	ι. 48	Suppose,	for	example,	that	there	are	in	fact	at	least	three	times,	but	there	could	have been	at	most	two	times:	it	is	possible	that	although	you	only	live	twice,	you	live	forever. That	is: (i) ∃p1∃p2∃p3(Sp1	∧	Sp2	∧	Sp3	∧	¬S(p1	∧	p2)	∧	¬S(p1	∧	p3)	∧	¬S(p2	∧	p3)) (ii) ◇∀q1∀q2∀q3((Sq1	∧	Sq2	∧	Sq3)	→	(S(q1	∧	q2)	∨	S(q1	∧	q3)	∨	S(q2	∧	q3)). We will derive a contradiction from (i), (ii), Eternity, and the permanent truth of Symmetry. Since	Symmetry	implies	Supervenience*	(as	explained	in	§III),	Supervenience* is	always	true:	A∀p(p	→	∃q(A(q	→	p)	∧	□q)). Hence ∀pA(p	→	∃q(A(q	→	p)	∧	□q))	(by	the	temporal	Converse	Barcan	Formula);	so ∀p(Sp	→	S∃q(A(q	→	p)	∧	□q))	(by	the	normal	modal	logic	of	A);	so ∀p(Sp	→	∃qS(A(q	→	p)	∧	□q))	(by	the	temporal	Barcan	Formula);	so ∀p(Sp	→	∃q(A(q	→	p)	∧	S□q))	(by	temporal	S5). Instantiating	this	generalization	with	the	three	propositions	p1,	p2,	p3	that	exist	according to	(i)	guarantees	the	existence	of	corresponding	sometimes-necessary	propositions	q1,	q2, q3	which	respectively	always	materially	imply	p1,	p2,	and	p3. By	the	modal	4	axiom,	they are sometimes necessarily necessary: S□□q1	∧	S□□q2	∧	S□□q3. By the mirror-image 34 In	summary,	proponents	of	Symmetry	face	a	choice. They	can	accept	Eternity,	and	deny that	it	is	contingent	how	many	times	there	are. Or	they	can	reject	Eternity,	and	hold	that metaphysical	possibility	and	necessity	are	sources	of	temporariness	in	their	own	right. The implausibility of both options constitutes an argument against Symmetry that is independent	of	judgments	about	Perpetuity. IX. Let	us	now	turn	our	attention	to	contingency	in	the	composition	of	the	time	series-i.e. which	particular	times	are	ever	present. There	are	three	possible	views: Tomorrow	Never	Dies: Every	time	is	necessarily	sometimes	present. ∀t(□S	Present(t)) Die	Another	Day: Some	times,	but	not	the	present	time,	are	possibly	never	present. ∃t(◇¬S	Present(t))	∧	∀t(Present(t)	→	□S	Present(t)) Live	And	Let	Die:	The	present	time	is	possibly	never	present. ∃t(Present(t)	∧	◇¬S	Present(t)) In	this	section	we	will	first	argue	against	Tomorrow	Never	Dies	and	Die	Another	Day. We will	then	argue	that	if	Live	and	Let	Die	is	true,	NOW	(discussed	in	§V)	must	be	rejected, thereby	undermining the	most	intuitive	argument	for	Symmetry. The previous section discussed one good reason to reject Tomorrow Never Dies, namely	that	it	is	possible	that	the	time-series	has	a	smaller	cardinality	than	it	in	fact	has. Church-Rosser principle ∀p(S□p	→	□Sp) (which follows from Eternity as explained above),	□S□q1	∧	□S□q2	∧	□S□q3,	hence	□(S□q1	∧	S□q2	∧	S□q3).	So	by	(ii)	(using	the	modal Converse Barcan Formula), ◇(S(□q1∧□q2) ∨ S(□q1∧□q3) ∨ S(□q2∧□q3)), and hence ◇S□(q1∧q2)	∨	◇S□(q1∧q3)	∨	◇S□(q2∧q3). By	□(Church-Rosser)	(another	consequence	of Eternity), this implies ◇□S(q1	∧	q2)	∨	◇□S(q1	∧	q3)	∨	◇□S(q2	∧	q3), and hence S(q1	∧	q2)	∨	S(q1	∧	q3)	∨	S(q2	∧	q3). But since q1, q2, q3 respectively always materially imply	p1,	p2	and	p3,	this	implies	that	S(p1	∧	p2)	∨	S(p1	∧	p3)	∨	S(p2	∧	p3),	contradicting	(i). Unlike the argument about cardinality 1 in the main text, this reasoning generalizes straightforwardly	to	other	finite	cardinalities,	but	could	be	resisted	by	rejecting	appeals to the temporal or	modal Barcan or Converse Barcan formulae. However it unclear whether	this	way	of	resisting	the	argument	leads	to	a	stable	view. Even	if	we	say	that none	of	q1,	q2,	and	q3	would	have	existed	had	there	been	only	two	times,	it	is	hard	to	see how	they	could	nevertheless	all	have	been	sometimes	necessary	without	any	two	of	them having	been	necessary	together. 35 But there are other arguments against Tomorrow Never Dies that don't turn on that possibility,	and	which	can	also	easily	be	turned	into	arguments	against	Die	Another	Day. First,	one	might	argue	from	the	premise	that	there	could	have	been	a	cosmic	catastrophe in	which	history	itself	came	to	an	end,	so	that	some	times	which	are	in	fact	a	proper	initial segment	of	the	time	series	would	instead	have	been	the	totality	of	the	time	series. For example:	should	the	sky	fall	tomorrow,	times	that	will	in	fact	be	present	next	week	would never	get	to	be	present. This	argument	generalizes	to	an	argument	against	Die	Another Day,	assuming	that	if	such	a	cosmic	catastrophe	is	possible	at	all,	then	there	could	have been	such	a	catastrophe	some	time	in	the	past,	so	that	the	present	time	would	have	been absent from the time series. Second, one	might appeal to the idea that the temporal relations	between	times	are	essential	to	them,	in	the	sense	that	necessarily,	if	t	is	present before	t′,	then	it	is	not	possible	that	t	or	t′	ever	be	present	without	t	being	present	before t′. Combined	with	the	claim	that	the	before-after	structure	of	the	time	series	could	have been	different-e.g.	because	it	is	contingent	whether	time	is	circular,	or	whether	there	is a	first	or	last	moment-this	essentialist	thesis	implies	that	every	time	is	possibly	never present,	thus	ruling	out	both	Tomorrow	Never	Dies	and	Die	Another	Day.	(Note	that	the premises	of	these	two	arguments	are	incompatible,	since	the	essentialist	premise	rules out the	possibility	of the	time-series	being	a	proper initial	segment	of the	actual timeseries.) A	third	kind	of	argument	turns	on	theses	about	the	relation	of	time	to	spacetime. For	example,	one	might	think	that	for	every	time	t	there	is	a	region	of	spacetime	st	such that	necessarily,	t	is	part	of	the	time	series	just	in	case	st	is	an	appropriate	kind	of	part	(a "simultaneity	slice")	of	the	spacetime	manifold. (One	version	of	this	view	identifies	each time t	with the corresponding region	st.) Contingency in the composition	of the time series then follows from contingency in the composition of the spacetime manifold: Tomorrow	Never	Dies	is	ruled	out	if	some	time	corresponds	to	a	region	which	could	have failed to be a simultaneity slice, and	Die Another Day is ruled out if the present time corresponds	to	such	a	region.49 Note	that all three	of these	arguments	are	compatible 49 Even someone who thought that the composition and topological structure of spacetime was metaphysically non-contingent might think that which regions of spacetime count as "simultaneity slices" is a contingent matter-e.g. if being a 36 with	the	necessity	of	the	cardinality	of	the	time-series,	as	well	as	with	the	necessity	of	the laws	of	physics.50 Those	who	want to	maintain	Tomorrow	Never	Dies in the face of these arguments might	offer	a	rejoinder	inspired	by	Timothy	Williamson's	defense	of	the	claim	that	it	is necessary	what	things	there	are	(Williamson	2013). According	to	Williamson,	there	are	a great	many very boring things in addition to the interesting ones	with	which	we are familiar. Since	Wittgenstein	could	have	had	a	child,	there	are	things	that	could	have	been children	of	Wittgenstein;	but	each	is	a	mere	spectre-it	is	not	a	person,	and	has	no	mass, has	no	spatial	location,	etc. Williamson	thinks	that	his	opponents	are,	by	and	large,	right about	the	ways	in	which	it	is	contingent	what	interesting	things	there	are,	but	fall	into error	by failing	to take	boring	things into	account. Defenders	of	Tomorrow	Never	Dies might	analogously	suggest	that	we	have	fallen	into	error	by	confusing	the	true	claim	that it is a contingent matter which times are interesting with the false claim that it is a contingent	matter	which times are sometimes present. "Boring" times could then be characterized	as	times	at	which	all	objects	are	boring,	in	the	sense	in	which	Wittgenstein's possible	children	are	boring	on	Williamson's	view. Even	those	who	are	happy	with	Williamson's	defense	of	necessitism	have	reasons	to be	cautious	about this	strategy. Just	as	Williamson	counters the	claim	that there	could have	been	things	that	are	not	actually	anything	by	saying	that	there	actually	are	boring simultaneity	slice is	being	a	Cauchy	surface	of	constant	mean	curvature	(see	Belot	and Earman 2001: 239–40 and references therein) and there is a reasonable amount of contingency	as	regards	the	distribution	of	matter	in	spacetime. 50	One	deviant	way	of	reconciling	the	above	considerations	with	the	letter	of	Tomorrow Never	Dies	is	to	adopt	an	eliminativist	view	on	which	there	aren't	any	times	at	all. The	less radical	version	of	this	view	rejects	the	ontology	of	times	but	not	the	devices	of	temporal anaphora	discussed	in	note	12. Assuming	such	devices	are	legitimate,	we	can	use	them	to formulate	eliminativism-friendly	analogues	of	Tomorrow	Never	Dies,	Die	Another	Day,	and Live	And	Let	Die,	to	which	the	arguments	of	the	present	section	are	equally	applicable. A more radical version of eliminativism holds that the entire notion of trans-history simultaneity	is	unintelligible,	so	that	devices	of	temporal	anaphora	cannot	meaningfully be embedded under modal operators. This picture renders the present section's arguments	redundant,	since	its	proponents	already	reject	NOW	(as	unintelligible).	Similar points	apply	to	proposals	to	save	the	letter	of	Tomorrow	Never	Dies	by	holding	that	there is	only	one	time,	namely	the	present. 37 things,	so	too	proponents	of	the	"boring	times"	strategy	will	counter	our	claim	that	there could	have	been	times	that	are	actually	never	present	by	saying	that	there	actually	are sometimes-present	boring	times	(and	indeed,	infinitely	many	of	them). In	other	words: sometimes,	nothing	is	interesting. But	isn't	the	question	whether	there	have	always	been, and	always	will	be,	interesting	things	(such	as	electrons)	a	question	for	physics,	not	to	be answered	by	philosophers	from	the	armchair? Furthermore,	whereas	Williamson	can	and does	deny	that	boring	objects	bear	nontrivial	spatial	relations	to	interesting	ones	or	to one	another,	proponents	of	the	"boring	times"	strategy	cannot	likewise	deny	that	boring times	bear	nontrivial	temporal	relations	to	the	interesting	times	in	which	we	live,	or	to one another-at least holding fixed the truisms that anything that is sometimes true either	is,	was,	or	will	be	true,	and	that	no	two	times	are	ever	both	present	at	once. So: have	you	been	interesting	ever	since	you	were	born?	Are	the	boring	times	clustered	at one or other end of the time series? And how are boring times ordered-densely, discretely,	continuously,	or	in	some	other	way? These	embarrassing	questions	suggest that	the	present	strategy	is	unlikely	to	provide	a	solid	dialectical	basis	for	a	defense	of Tomorrow	Never	Dies. Let	us	now	return	to	Die	Another	Day. Even	setting	aside	the	arguments	above,	there is	something	bizarre	about	the	proposal	that	some	times	are	modally	robust	(necessarily sometimes	present)	in	a	way	that	other	times	are	not. Moreover,	on	pain	of	arbitrariness, anyone	who	thinks	that	the	present	time	is	modally	robust	should	accept	the	more	general principle	that	every	time	is	modally	robust	when	it	is	present.51 In	combination	with	Die Another	Day,	this	general	principle	entails	that	modal	robustness	is	a	temporary	feature of	times. And	this	consequence	is	inconsistent	with	Eternity,	which	we	defended	in	§VIII. That	principle	says	that	metaphysical	necessity	is	not	itself	a	source	of	temporariness	- 51	Note	that	∀tA(Present(t)	→	□Present(t)),	and	hence	the	disjunction	of	Tomorrow	Never Dies	and	Die	Another	Day,	is	valid	in	the	generalized	product	structures	described	in	§III (interpreting	'Present'	and	'∀t'	as	explained	in	§II),	since	in	such	models	⟨w,	t⟩	is	modally accessible from ⟨w′,	t′⟩ only when t	=	t′. These models provide the natural way of generalizing	product	models	to	allow	for	contingency	in	the	composition	or	cardinality	of the	time	series. Whereas	Tomorrow	Never	Dies is	valid	in	product	models,	it	can	fail	in generalized	product	structures. For	example,	in	a	generalized	product	structure	where I	=	{⟨0,1⟩,	⟨0,2⟩,	⟨1,0⟩,	⟨1,2⟩,	⟨2,0⟩,	⟨2,1⟩},	Tomorrow	Never	Dies	is	false	at	every	point. 38 necessitation	never	turns	an	eternal	proposition	into	a	temporary	one.	But	according	to the	view	under	consideration,	each	time	t	that	is	only	contingently	ever	present	generates a	counterexample	to	Eternity,	since	although	the	proposition	that	t	is	sometimes	present is	eternal,	its	necessitation	is	temporary	(false	now,	but	true	when	t	is	present). Having	just	argued	for	Live	And	Let	Die	by	arguing	against	its	two	alternatives, we	will now	argue	that	it	implies	the	falsity	of	NOW,	as	follows: (1) The	present	time	is	possibly	never	present. ∃t(Present(t)	∧	◇¬S	Present(t)) (2) So	the	present	time	is	not	necessarily	present. ∃t(Present(t)	∧	¬□Present(t)) (3) So it is not necessary that all and only true propositions are true at the present	time. ∃t(Present(t)	∧	¬□∀p(p	↔	At	t	p)) (4) So	not	every	proposition	is,	necessarily,	true	just	in	case	true	at	the	present time. ∃t(Present(t)	∧	¬∀p□(p	↔	At	t	p)) (5) So	not	every	proposition	is,	necessarily,	true	just	in	case	now	true. ¬∀p□(p	↔	Np) (1)	is	Live	and	Let	Die;	(5)	is	the	negation	of	NOW.	We	will	discuss	each	step	in	turn. From	(1)	to	(2):	The	validity	of	this	inference	follows	from	the	plausible	claim	that	for something	to	be	never	true	just	is	for	it	to	neither	be	true,	have	been	true,	nor	be	going	to be	true. Thus	nothing	could	possibly	be	both	present	and	never	present. If	you	disagree with	us	about	this,	we	can	without	any	loss	of	plausibility	replace	Live	and	Let	Die	with	the claim	that	the	present	time	could	have	been	neither	present,	formerly	present,	nor	ever going	to	be	present,	and	never	say	'never'	again. From	(2)	to	(3):	Say	that	t	is	accurate	just	in	case	all	and	only	the	true	propositions	are true	at	t. (2)	implies	(3)	so	long	as	being	accurate	necessitates	being	present. This	is	true on	all	of	the	most	natural	accounts	of	the	connection	between	'present'	and	'At	t': (a) To	be	present	is	to	be	accurate. 39 (b) For	p	to	be	true	at	t	is	for	it	always	to	be	the	case	that	if	t	is	present,	p	is	true. (c) For	p	to	be	true	at	t	is	for	it	sometimes	to	be	the	case	that	t	is	present	and	p is	true. (d) For	p to	be	true	at	t is for it to	be	the	case	that	p	would	be	true if	t	were present.52 For	(a),	the	implication	from	accuracy	to	presence	is	immediate. For	(b)	and	(d),	it	follows from	the fact that the	proposition	that	t is	present	cannot fail to	be	true	at	t, so if	t is accurate	this	proposition	must	be	true.	For	(c),	it	follows	from	the	fact	that	the	proposition that	t	is	not	present	cannot	be	true	at	t,	so	if	t	is	accurate	this	proposition	must	be	false. Are	there	any	principled	views	about	the	connection	between	'present'	and	'at	t'	that could	allow	for	the	possibility	of	accuracy	without	presence? One	might	suggest	a	view on	which	things	that	are	never	present	get	to	count	as	accurate	'by	courtesy',	as	in	the following	variant	of	(c): (e) For	p	to	be	true	at	t	is	for	it	to	be	the	case	that	either	it	is	sometimes	the	case that	(p	is	true	and	t	is	present),	or	p	is	true	and	t	is	never	present. This	view	blocks	the	implication	from	accuracy	to	presence	and	thus	from	(2)	to	(3). The problem	with it is that it	is incompatible	with	the	principle that	what is true	at	t is	an eternal	matter:	∀t□∀p(At	t	p	→	A	At	t	p). For	presumably,	if	the	present	time	could	have been	never	present,	then	propositional	temporalism	could	still	have	been	true	in	such	a possibility: that is, there could	have	been some temporarily true	proposition. (e) then entails	that	the	present	time	is	such	there	could	have	been	a	proposition	that	was	true	at 52 (b), (c), and (d) are temporal analogues to the three accounts of truth at a world compatible	with	the	combination	of	Perpetuity	and	Historicity	considered	in	§VI. If	it	is contingent	what	times	are	sometimes	present,	(b)	and	(c)	entail	that	'At	t'	fails	to	commute with	negation	in	modal	contexts,	whereas	(d)	preserves	commutativity	given	Conditional Excluded	Middle. 40 it	but	not	always	true	at	it. This	is	bizarre:	how	could	it	have	been	sometimes	raining	now and	sometimes	not	raining	now?53,	54 From	(3)	to	(4):	The	validity	of	this	inference	follows	from	the	validity	of	the	modal Barcan Formula, ∀p□φ	→	□∀pφ. Those who think that there could have been propositions	in	addition	to	those	there	actually	are	might	thus	consider	resisting	at	this step.	They	might	think	that	the	present	time	is	necessarily	accurate	with	regard	to	all	the propositions	there	actually	are	(as	required for	(4) to	be false)	while	maintaining	that there could have been new	propositions	whose truth values differed from their truth values	at	the	present	time	(which	suffices	for	(3)	to	be	true). However,	there	is	a	different way	of	arguing	from	(3)	to	(4),	relying	not	on	the	Barcan	Formula	but	instead	on	two	hardto-deny	principles	about	the	logic	of	'At	t':	(i)	'At	t'	is	closed	under	classical	consequence; (ii) 'At	t' is	redundant	when	it	occurs	within	the	scope	of 'At	t'	without	any	intervening modal	or	temporal	operators. (Both	principles	are	consequences	of	the	analyses	(b)–(e) of	'At	t'	in	terms	of	'Present'	considered	above.) Let	t	be	the	present	time,	and	let	q	be	the proposition	that	t	is	accurate.	Necessarily,	if	anything	is	true	at	t,	∀p(p	↔	p)	is	true	at	t	by 53	Those	who	reject	the	factivity	of	'always'	have	another	way	of	rejecting	the	implication from	accuracy	to	presence:	they	can	say	that	for	a	time	to	be	present	is	for	it	to	be	both accurate	and	sometimes	accurate. We	have	already	said	what	we	have	to	say	against	this maneuver	in	connection	with	the	step	from	(1)	to	(2). 54	The	views	we	have	just	considered	all	also	entail	that	being	present	necessitates	being accurate. For	(a)	this	is	obvious;	for	(b),	(c),	and	(e)	it	follows	from	the	principle	that	no time	is	present	more	than	once	(i.e.	that	if	sometimes	t	is	present	and	p	is	true,	then	always if t is	present	p is true); for (d) it follows from	"strong	centering" for counterfactuals, according	to	which	a	counterfactual	with	a	true	antecedent	has	the	same	truth	value	as	its consequent. Relying	only	on the implication from	presence to	accuracy,	we	can run	a different	argument	against	NOW,	replacing	Live	And	Let	Die	with	the	stronger	premise that	the	present	time	could	have	been	absent	from	the	time	series	without	any	new	times being	included	in	it	(e.g.	because	time	came	to	an	end	before	now). We	can	articulate	this possibility	in	a	language	with	plural	quantification	over	times,	as	follows: The	Living	Daylights:	There	are	some	tt	such	that	none	of	tt	is	present;	each	of	tt	is sometimes	present;	and	possibly	always	one	of	tt	is	present. If	such	tt	exist,	then	at	least	one	of	them	t	is	possibly	present,	and	so	possibly	accurate: ◇∀p(p	↔	At	t	p). Instantiating NOW+ (□∀t	At	t	∀p□(p	↔	At	t	p)) with t, we have ◇∀p□(p	↔	At	t	p), hence	◇□(¬Present(t)	↔	At	t ¬Present(t)), hence (by the modal B schema),	¬Present(t)	↔	At	t	¬Present(t). Since	t	is	not	present	this	gives	At	t	¬Present(t). But, uncontroversially, every time is present at itself. So proponents of The Living Daylights	must	reject	NOW+,	and	hence,	as	we	argued	in	§V,	should	also	reject	NOW. 41 (i), in which case ∀p(p	↔	At	t	p), i.e. q, is also true at t by (ii). Now suppose for contradiction that (3) is true	and (4) is false. By (3),	¬q is	possibly true. And	by the negation	of	(4),	every	proposition	(including	¬q)	is,	necessarily,	true	just	in	case	true	at	t. Hence	it	is	possible	that	¬q	is	true	at	t. But	by	the	earlier	result,	it	is	necessary	that	q	is true	at	t	if	anything	is. So	by	(i),	it	is	possible	that	q	∧	¬q	is	true	at	t;	hence	by	the	falsehood of	(4),	it	is	possible	that	q	∧	¬q:	contradiction. We	thus	see	no	plausible	way	to	resist	the step	from	(3)	to	(4). From (4) to (5): This step should be uncontroversial. (Some might reject the equivalence	of	(4)	and	(5)	on	the	grounds	that	that	there	are	no	times,	but	even	they	will accept	the	relevant	material	implication.) This	concludes	our	argument	against	NOW. It is	addressed,	in the first instance, to those	who,	when	confronted	with	the	tension	between	NOW	and	Perpetuity,	were	initially inclined	to	find	NOW	the	more	plausible	principle.55 But	the	falsity	of	NOW	also	causes 55	Several	people	have	suggested	to	us	that	proponents	of	Symmetry	should	retreat	from NOW	to	the	following	weaker	claim: WEAK	NOW	Necessarily,	if	things	are	ever	as	they	now	are,	things	are	as	they	now are. □(S∀p(p	↔	Np)	→	∀p(p	↔	Np)) Advocates	of	Symmetry	who	are	worried	about	NOW	might	take	a	quantum	of	solace	in the	fact	that	Symmetry	can	also	be	derived	from	WEAK	NOW	together	with	RIGN. (Proof: uncontroversially,	∀q(q	↔	Nq). So	AS∀q(q	↔	Nq). Moreover,	for	any	true	p,	we	have	Np, hence	by	RIGN,	ANp. Thus	Np	∧	S∀q(q	↔	Nq)	is	an	eternal	truth,	and	by	WEAK	NOW,	one that	necessitates	p. So	Supervenience is true:	every	truth is	necessitated	by	an	eternal truth. Symmetry follows.) But	WEAK	NOW	seems	completely	unmotivated	apart from being a consequence of	NOW. Moreover, even	WEAK	NOW is in tension	with certain natural claims about the extent of contingency in the composition of the time series, specifically	The	Living	Daylights (see	note	52). For just as	proponents	of	NOW	should accept the more general NOW+, proponents of WEAK NOW should accept the more general	WEAK	NOW+: WEAK	NOW+:	Necessarily,	at	each	time,	it	is	necessary	that	if	things	are	ever	as	they then	are,	things	are	as	they	then	are. □∀t	At	t	□(S∀p(p	↔	At	t	p)	→	∀p(p	↔	At	t	p)) But	WEAK	NOW+	is	inconsistent	with	The	Living	Daylights	for	the	same	reason	that	NOW+ is	(see	note	54):	The	Living	Daylights	entails	that	some	temporarily	non-present	time	t	is possibly	present,	hence	possibly	accurate,	hence	by	WEAK	NOW+	possibly	necessarily accurate-if-sometimes-accurate, hence in fact accurate, which no temporarily nonpresent	time	is. 42 trouble	for	those	who	accept	Symmetry	on	some	other	grounds,	and	thus	reject	Perpetuity. Consider the	question:	which	possibly sometimes-true	propositions	are	possibly true? According to proponents	of	Perpetuity: all of them.	According to proponents	of	NOW: those	that	are	possibly	true	now.	But	what	about	people	who	reject	both	Perpetuity	and NOW? They must think that some but not all propositions that could have been sometimes	true	but	could	not	have	been	true	now	are	possibly	true.	If	they	accept	(the necessary	eternal	truth	of)	Symmetry,	they	must	also	think	that,	for	each	possible	worldhistory, there is	a	unique	time in	that	history	which	could	have	been	present	had	that history	obtained.	Given	the	falsity	of	NOW,	this	function	from	possible	world-histories	to members	of	their	respective	time-series	cannot	be	the	constant	function	that	maps	every history	to	the	present	time. But	it	is	hard	to	see	how	this	function	could	then	fail	to	draw arbitrary	distinctions	of	a	sort	that	ought	to	disqualify	it	from	marking	the	boundaries	of metaphysical	possibility. X. Someone might accept everything we have said up to this point while nevertheless regarding the dispute we have been engaged in as "merely verbal". Perhaps our opponents mean something different from us by the term of art 'metaphysically necessary',	such	that	(a)	'Every	metaphysically	necessary	truth	is	always	true'	is	false	as used	by them,	and (b)	both	our	way	of talking	and	our	opponents'	way	of talking	are "equally	good". In	particular,	one	might	suggest	that	our	opponents	use	'metaphysically necessary' to express the notion of immediate necessity,	where this can be defined in terms	of	our	notion	of	metaphysical	necessity	as	follows:	it	is	immediately	necessary	that φ	just	in	case	the	truth	of	φ	is	a	metaphysically	necessary	consequence	of	the	truth	about which	time	is	present. (In	symbols:	□Iφ	=df	∃t(Present(t)	∧	□(Present(t)	→	φ)),	where	t is	some	time	variable	not	free	in	φ). For	any	formula	φ,	let	φI	be	the	result	of	substituting immediate	for	metaphysical	necessity	in	φ	(i.e.	the	result	of	replacing	each	subformula	of the	form	□ψ	with	□Iψ).	We	can	now	express	the	interpretative	hypothesis	as	follows:	For any	φ,	φ	as	used	by	our	opponents	is	equivalent	to	φI	as	used	by	us. 43 In	favor	of	this	hypothesis,	one	might	appeal	to	the	following	facts: (i) Any formula	φ is logically equivalent to	φI	on the class of generalized product structures	from	§III,	interpreting	quantification	over	times	as	discussed	in	§II. (ii) PerpetuityI-∀p(∃t(Present(t)	∧	□(Present(t)	→	p))	→	Ap)-is uncontroversially false	assuming	propositional	temporalism,	since	it	entails	that	the	present	time	is always	present. (iii) SymmetryI-∀p(¬p	→	∃q(∃t(Present(t)	∧	□((Present(t)	∧	p)	→	q)	∧ A(p	→	¬q)))-is uncontroversially true, since the existential generalization is witnessed	by	the	proposition	attributing	presentness	to	the	present	time. (iv) NOWI	is	uncontroversially	true,	since	it	is	uncontroversial	that	the	present	time	is such	that	necessarily,	if	it	is	present,	then	the	true	propositions	are	exactly	those propositions	that	are	true	now. Since	our	opponents	think	that	Symmetry	is	not	only	true	but	necessarily	true,	necessarily always	true,	and	so	on,	(i)	makes	it	natural	to	think	of	them	as	recognizing	only	one	notion of necessity	where	we recognize two. It	might therefore be thought that	we face an interpretative	dilemma	that	should	be	resolved	by	considerations	of	charity. And	given (ii)–(iv),	such	considerations	seem	to	support	the	present	interpretative	hypothesis. We	are	unmoved	by	this	mode	of	argument. Philosophers	do	regularly	make	mistakes about	general	metaphysical	principles,	and	equate	statuses	that	are	in	fact	distinct,	so	the fact	that	a	certain	interpretation	avoids	attributing	such	errors	is	very	weak	evidence	that the interpretation is	correct. But	even if	we	were	convinced	that there	was	a	practice afoot	of	using	'metaphysically	necessary'	to	express	immediate	necessity,	we	would	still emphatically	reject	the	claim	that	this	way	of	speaking	was	"just	as	good"	as	ours. For there are hypotheses about the possible structures of time that simply cannot be expressed in	the	language	of	tense	operators,	propositional	quantifiers,	and	an	operator expressing	immediate	necessity. For	example,	consider	the	following	pair	of	hypotheses: H1 It	is	metaphysically	necessary	that	time	is	dense. 44 H2 Although	it	is	only	contingently	true	that	time	is	dense,	it	is	necessary	that	for each	time	t,	either	it	is	necessary	that	if	t	is	ever	present,	time	is	dense,	or	else it	is	necessary	that	if	t	is	ever	present,	time	is	not	dense. (Intuitively:	whether	a	time	belongs	to	a	dense	time-series	is	an	essential	property	of	it.) H1 and H2 strike us as perfectly respectable competing hypotheses about the modal metaphysics	of	time,	about	which	there	could	be	a	substantive	debate. But	there	seems to	be	no	way	of	conducting	such	a	debate	using	immediate	necessity	as	one's	basic	modal notion. For	one	thing,	H2I is flatly inconsistent	and	so	clearly fails	as	a	way	of	making sense	of	H2. Nor	would	it	help	to	instead	replace	'metaphysically	necessary'	in	H2	with 'always immediately	necessary',	or 'immediately	necessarily	always',	or	with	any finite string of 'always' and 'immediately necessarily' operators, or even with the infinite conjunction	of	all	such	strings. We	will	focus	on	this	last	proposal. Model-theoretically, the	effect	of	such	a	replacement	is	an	operator	corresponding	to	an	accessibility	relation which	is	the	transitive	closure	of	the	union	of	the	temporal	accessibility	relation	and	the "immediate	possibility"	accessibility	relation	(the	relation	one	point	bears	to	another	if and only if is modally accessible and agrees about which time is present). Such an operator is not semantically equivalent to metaphysical necessity; the accessibility relation	so-defined	can	fail	to	be	universal	even	in	a	structure	where	modal	accessibility is	universal. In	particular,	H2	is	only	true	in	structures	where	this	relation	fails	to	be	a universal	relation:	if	H2	is	true,	there	are	possibilities	where	time	is	not	dense,	but	they cannot	be	reached	by	any	sequence	of	steps	each	of	which	either	takes	you	to	a	different point	in	the	same	possible	history	or	to	a	different	possible	history	with	the	same	present time. By	our	lights,	these	expressive	limitations	constitute	an	important	respect	in	which	a language	whose	only	modal	operator	is	immediate	necessity	is	inferior	to	ours. Moreover, this inferiority strikes us as a strong consideration against the hypothesis that our opponents	are	in	fact	speaking	such	a	language. To be clear, we are not advancing this "expressive power" consideration as an argument	for	Perpetuity. Our	current	target	is	the	view	that	the	debate	over	Perpetuity	is 45 merely	verbal:	we	have	argued	that	is	not	the	case	that	both	ways	of	speaking	are	equally good,	on	the	grounds	that	if	our	way	of	speaking	is	good,	our	opponents'	way	of	speaking is	worse,	since	it	is	expressively	impoverished	relative	to	ours. We	are	not	claiming	that it	is	an	important	desideratum	that	a	view	be	neutral	as	regards	hypotheses	like	H1	and H2;	on	the	contrary,	we	see	settling	such	difficult	questions	as	an	advantage	of	a	theory about	time	and	modality. In	metaphysics	as	in	other	areas	of	theory-building,	strength	is a	virtue. Our impulse	to	defend	Perpetuity	was initially triggered	by incredulity that	anyone would deny something so obvious. But our considered view is that this is an area of metaphysics	where	surprises	may	well	be	in	store,	and	that	competing	theories	should	be adjudicated	on	broader theoretical grounds.	This is	why	our	defense	of	Perpetuity has taken the form of a critical investigation of opposing views-in particular, those that combine	propositional	temporalism	with	Symmetry.	We	have	argued	that	such	views	are both	under-motivated	and	incompatible	with	plausible	claims	about	contingency	in	the time-series.	The	competing	picture	that	accepts	Perpetuity	does	not	face	these	problems. Moreover,	Perpetuity	offers	a	simple	and	compelling	explanation	of	its	enormously	many obviously	and	uncontroversially	true	instances. The	balance	of	considerations	thus	tells firmly	in	its	favor. 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