The	Limit	Decision	Problem	and	Four-Dimensionalism* Abstract I	argue	that	medieval	solutions	to	the	limit	decision	problem	imply	fourdimensionalism,	i.e.	the	view	according	to	which	substances	that	persist through time	are	extended through time	as	well	as through	space,	and have	different	temporal	parts	at	different	times. Keywords Permanent and successive entities – threeand four-dimensionalism	– temporal	parts	–	limit	decision	problem	–	location Four-dimensionalism is the view according to which substances are extended	through	time	as	well	as	through	space;	just	like	a	substance	is thought	to	be	extended	through	space	by	having	different	spatial	parts	at different places, four-dimensionalists believe that substances are extended through time by having different temporal parts at different times. Boosted	by	a	growing	battery	of	arguments,	four-dimensionalism is	presumably	enjoying	its	Golden	Age,	and	has	forced	his	foes	on	the	back foot1. Somehow	unsurprisingly, things used to be different some eight hundred	years	ago,	when,	with	the	possible	exception	of	Bonaventure,	the overwhelming	majority	of scholastic	philosophers	maintained the	rival view – nowadays called three-dimensionalism – according to which substances persist through time without having temporal parts2. However, it is not unusual that philosophers commit themselves to doctrines	which	they	deny.	In	this	paper,	I	argue	that	standard	medieval accounts of change – standard solutions to the so-called limit decision problem	in	particular	–	trigger	an	argument	that	ultimately	leads	to	the four-dimensionalist	view	of	persistence. * I am grateful to Claudio Calosi, Ilaria Canavotto, Frédéric	Goubier, Can Loewe, Paolo Natali, Magali Roques, Cecilia Trifogli and to the members of the audience of the conference	Limit	Decision	Problems:	Medieval	and	Contemporary	Perspectives	in	Berlin	for useful	questions	and	fruitful	comments	on	earlier	drafts	of	this	paper. 1	For	a	sympathetic	review	of	the	arguents	in	favour	of	four-dimensionalism,	I	refer	the reader to T. Sider, Four-Dimensionalism: An Ontology of Persistence and Time (Oxford 2001). 2 R. Cross, "Four-Dimensionalism and Identity Across Time: Henry Ghent vs. Bonaventure,"	Journal	of	the	History	of	Philosophy	27	(1999),	393-414. The	Limit	Decision	Problem	and	Four-Dimensionalism 2 The	paper	is	in	two	main	parts.	The	first	one	sets	the	stage	by	presenting the limit decision problem and the standard solutions adopted by the scholastics during the period when controversies concerning this problem flourished the	most, i.e. during the thirteenth and fourteenth centuries.	Building	on	a	tension	in	these	standard	solutions	raised	by	Paul Vincent	Spade,	the	second	part	of	the	paper	argues	that	those	standard solutions make persistence without temporal parts impossible. The paper concludes then that if substances persist, they so by having temporal	parts. 1 Medieval	Solutions	to	the	Limit	Decision	Problem 1.1 The	Limit	Decision	Problem3 A light suddenly switches off. Consider the instant t at which the switching occurs. Before t, the light is on, after t, it is off, i.e. not on. However	at the instant t, is it still on	or already	off? In	principle, four options	offer	themselves: (1) At	t,	the	light	is	on. (2) At	t,	the	light	is	off. (3) At	t,	the	light	is	neither	on	nor	off. (4) At	t,	the	light	is	both	on	and	off. Many	philosophers	have	long	thought	that	options	(3)	and	(4)	must	be excluded, for they go against the law of non-contradiction, the law of excluded	middle	or	the	concept	of	negation	itself. Options	(3)	and	(4)	excluded,	one	may	think	that	there	is	no	real	problem here:	when	a	light	switches	from	on	to	off,	there	will	be	an	instant	t1	at which	it	is	on,	immediately	followed	by	another	instant	t2	at	which	is	off. In that	case,	a	choice	between	(1)	and	(2)	seems	not to	be	so	pressing after	all.	However,	this	easy	solution	is	excluded	if	one	thinks	that	time	is continuous. Indeed, if	we	assume	that time is	continuous,	between	any 3	The	following	account	of	the	Limit	Decision	Problem	and	its	treatment	is	mainly	based on G. Priest, In Contradiction. A Study of the Transconsistent (Dordrecht 1987): N. Kretzmann,	"Incipit/Desinit,"	in	Matter	and	Time,	Space	and	Motion,	eds.	J.	Machamer	and P. Turnbull (Columbus 1976), 101-136; S. Knuuttila and A. I. Lehtinen, "Change and Contradiction: A Fourteenth Century Controversy," Synthese 40 (1979), 189-207; P. Spade, "How to Start and Stop.	Walter	Burley on the Instant of Transition," Journal of Philosophical	Research	19	(1994),	193-221;	A.	de	Libera,	"La	problématique	de	l'	instant du	changement'	au	XIIIème	siècle:	contribution	à	l'histoire	des	Sophismata	Physicalia,"	in Studies	in	medieval	Natural	Philosophy,	ed.	S.	Caroti	(Firenze	1989)	and	N.	Strobach,	The Moment	of	Change.	A	Systematic	History in the	Philosphy	of	Space	and	Time (Dordrecht 1998). The	Limit	Decision	Problem	and	Four-Dimensionalism 3 two	instants	t1	and	t2	there	will	always	be	a	third	instant	t3:	no	instant	can immediately	follow	another. In light of the assumption that time is continuous, the choice	between option (1) and option (2) becomes indeed pressing. In	making such a choice,	we	are	choosing	whether	t	is	the	last	instant	at	which	the	light	is on	or	the	first instant	at	which it is	off.	And	given	that	there	cannot	be immediately adjacent instants, there cannot be both a last instant at which	the	light	is	on	and	a	first	instant	at	which	it is	off.	Hence,	we	are choosing	whether (1') there	is	a	last	instant	at	which	the	light	is	on	and	no	first	instant at	which	the	light	is	off or (2') there	is	no	last	instant	at	which	it	is	still	on	and	a	first	instant	at which	it	is	off. This	result	allows	us	to	understand	why	this	problem	is	sometimes	called the	limit	decision	problem.	The	problem	can	be	equivalently	described	in terms	of	the	kind	of	temporal	limits	that	a	state	–	such	as	the	light's	being on	–	has.	Let	us	introduce	a	bit	of	technical	terminology,	and	distinguish between extrinsic/open and intrinsic/closed temporal limits, and in particular	let	us	say	that	a	state	is	intrinsically	limited	at	its	beginning	– its	first	limit	is	closed	–	if	and	only	if	it	has	a	first	instant;	otherwise	it	is extrinsically	limited	at	its	beginning	–	its	first	limit	is	open.	Let	us	also	say that	a	state	is	intrinsically	limited	at	its	end	–	its	last	limit	is	closed	–	if	and only	if	it	has	a	last	instant;	otherwise	it	is	extrinsically	limited	at	its	end	– its	last	limit	is	open.	Choosing	between	option	(1)	and	(2)	means	choosing whether (1'') the	light's	being	on	is	intrinsically	limited	at	its	end	–	its	last	limit is	closed. or (2'') the	light's	being	on	is	extrinsically	limited	at	its	end	–	its	last	limit is	open. Square	brackets	have	been	conveniently	employed	to	represent	a	state's being extrinsically or intrinsically limited at one hand, where square The	Limit	Decision	Problem	and	Four-Dimensionalism 4 brackets	point	towards	a	state	that	is	intrinsically	limited.	For	example, we	will	represent	the	fact	that	the	light's	being	on	is	intrinsically	limited at	its	end,	i.e.	option	(1),	as	follows: Figure	1.	The	light's	being	on	is	intrinsically	limited	at	its	end. So	it	seems	that	we	should	indeed	make	a	choice	between	option	(1)	and option	(2).	However,	there	seems	to	be	no	reason	to	prefer	one	in	spite	of the	other.	If	there	are	no	clear	grounds	on	which	the	choice	can	be	made, the choice	will end	up	being	arbitrary, and	many find that this	kind	of arbitrariness is,	when	dealing	with	philosophical	problems,	completely out	of	place4.	Some	philosophers feel then	prompted	to find	reasons	to prefer	one	of	the	first	two	options	over	the	other,	while	others	think	that no	such	reason	can	be	found,	and	take	this	result	as	reason	to	take	the other	two	options	more	seriously5. 1.2 Medieval	Solutions	to	the	Limit	Decision	Problem:	the	Distinction between	Permanent	and	Successive	Entities The	limit	decision	problem	–	which	is,	for	obvious	reasons,	also	called	the problem	of	the	instant	of	change	–	is	known	and	discussed	at	least	since Aristotle's	Physics	and	Sofistical	Refutations6.	However	the	context	in	the history of philosophy in which it has been most hotly discussed is arguably	the	Latin	scholastics	of	the	thirteenth	and	fourteenth	century, when	the	relevant	Aristotelian	texts	made	their	reappearance	in	the	Latin world	after	their	diaspora	through	the	translatio	studiorum.	Scholastics processed	the	problem	and,	in	the	footsteps	of	Aristotle,	or	at	least	of	the Latin	Aristotle,	gave	it	a	distinctive	reading. 4	This	point	is	nicely	put	forward	in	Priest,	In	Contradiction. 5	For	a	summary	of	the	solutions	to	the	limit	decision	problem	adopted	during	the	last decades,	see	Strobach,	The	Moment	of	change,	part	II. 6	For	an	overview	of	the	ancient	debate	on	the	limit	decision	problem,	see	Strobach,	The Moment	of	Change,	part	I,	ch.	1	and	2. on off The	Limit	Decision	Problem	and	Four-Dimensionalism 5 Most	medieval	solutions	to	the	problem	of	the	instant	of	change	share a	common	default	setting.	First,	generally7	they	reject	options	(3)	and	(4). Second,	they	do	not	make	a	choice	between	option	(1)	and	option	(2)	that applies	to	all	possible	cases	of	change,	but	rather	think	that	both	options apply	in	different	cases. If one wants to understand what those different cases are, the most important distinction to be introduced is certainly the distinction between	permanent and successive entities,	which I	will explain in this section,	and	apply	in	the	next	section	to	the	limit	decision	problem. The	distinction	between	permanent	and	successive	entities	has	its	roots in Aristotle's Physics and, through Averroes commentaries, became a topos	of	medieval	metaphysics8.	The	distinction,	as	it	appears	through	the history	of	medieval	philosophy,	is	of	central	importance	for	the	solution of	the	limit	decision	problem	as	well	as	for	other	important	contexts,	such as	the	ones	dealing	with	the	relation	between	God	on	the	one	hand	and space	and	time	on	the	other9. The distinction is nicely introduced and applied to the limit decision problem	by	Walter	Burley	in	his	De	Primo	et	Ultimo	Instanti.	Given	that later	we	will	analyse	an	argument	originally	put	forward	against	Burley's view	on	the	limit	decision	problem,	it	will	be	a	good	starting	point	to	see how Burley himself introduces the distinction. The relevant passage reads: With respect to the first point, you must know that there is a difference	between	a	permanent	thing	and	a	successive	one.	For	a permanent thing, speaking about a	permanent thing in general, is one	for	which	it	is	not	inconsistent	from	the	nature	of	the	thing	to have	all	[its]	parts	simultaneously.	And	a	successive	thing	is	one	for which	it is inconsistent from	the	nature	of	the	thing	to	have	all its parts	simultaneously.	Indeed	it	belongs	to	its	nature	that	it	have	an earlier	and	another,	later	one.	And	when	the	earlier	part	exists,	the other	part	does	not	exist.	For	a	stone	is	a	permanent	thing,	because it	is	not	inconsistent	for	a	stone	to	have	all	its	parts	simultaneously in	the	same	measure.	But	a	day,	and	a	week,	and	so	on,	are	successive 7	With the notable exception of Quasi-Aristotelianism. See N. Kretzmann, "Continuity, Contrariety,	Contradiction,	and	Change,"	in	Infinity	and	Continuity	in	Ancient	and	Medieval Thought, ed. N. Kretzmann (London, 1982), 270-296 and P. V. Spade, "QuasiAristotelianism,"	in	Infinity	and	Continuity,	297-307. 8 A. de Libera, "La problématique de l'instant du changement", 51-52 and R. Pasnau, Metaphysical	Themes	1274-1671	(Oxford,	2011),	ch.	18. 9	Pasnau,	ibid.	and	R.	Pasnau,	"On	Existing	All	at	Once,"	in	God,	Eternity,	and	Time,	ed.	C. Tapp	(Ashgate	2011),	11-28. The	Limit	Decision	Problem	and	Four-Dimensionalism 6 things because it is inconsistent for them to have all their parts simultaneously.	For	it	is	inconsistent	with	a	day	that	when	it	is	the first	hour,	it	is	the	third	[hour]. 10 Burley says that successive things are by nature such that they have earlier	and	later	parts.	A	day,	for	example,	is	such	that	it	has	a	first	and	a third hour. Earlier and later parts are such that they cannot be had simultaneously (or that they	cannot	be	simultaneous	with	each	other). Indeed,	when	it	is	the	first	hour,	it	is	not	the	third	hour,	and	vice	versa.	On the	other	hand,	permanent	things	are	by	nature	such	that	they	can	have all their parts simultaneously (and therefore have no earlier and later parts, which must be had at different times). Burley mentions the example	of	a	stone,	the	nature	of	which	allows	it	to	have	all	its	parts	–	the physical	particles	that	make	up	the	stone	–	at	the	same	time. A	contemporary	philosopher	can	hardly	resist	the	temptation	to	see	here in play a distinction she is familiar with, i.e. the distinction between endurance and perdurance. Endurance and perdurance are notions invoked	nowadays	to	explain	how	things	persist	through	time,	or	in	other words,	how	is	it	possible	that	the	same	thing	exists	at	different	times.	On the one hand, something perdures if and only if it persists by having different	temporal	parts	at	different	times.	On	the	other	hand,	something endures if and only if it persists by being wholly present at different times11.	A	contemporary	philosopher	will	be	tempted	to	see	a	parallelism between Burley's notion of "earlier and later parts" and the contemporary notion of a temporal part, as well as between Burley's notion	of	"having	all	parts	simultaneously"	and	the	contemporary	notion of	being	wholly	present. 10	Here	I	quote	Paul	Vincent	Spade's	translation	of	Burley's	text	that	can	be	found	in	his "How	to	Start	and	Stop,"	199.	The	original	text	taken	from	Shapiro's	edition	reads:	"Circa primam est sciendum quod differentia est inter res permanentem et successivam, quoniam res permanens, communiter loquendo de re permanente, est illa cui non repugnat	ex	natura	rei	habere	omnes	partes	simuL	Et	res	successiva	est	illa	cui	repugnat ex	natura	rei	habere	omnes	suas	partes	simul;	ymo	est	de	natura	sui	quod	habeat	unam per temporem,	et aliam	posteriorem, et	quando	pars	prior est, pars	posterior	non	est. Lapis	enim	est	res	permanens,	quia lapidi	non	repugnat	habere	omnes	partes	simul in eadem mensura. Sed dies et septimana, et sie de aliis, sunt res successive, quia eis repugnat habere omnes partes suas simul. Repugnat enim diei quod quando est hora prima	quod	sit tertia. Intelligendum	ergo	est	per	rem	permanentem, illud	cuius	omnes partes	sunt	simul,	vel	cui	non	repugnat	habere	omnes	suas	partes	simul."	See	H.	and	C. Shapiro, "De Primo et Ultimo Istanti des Walter Burley," Archiv für Geschichte der Philosophie	47	(1965),	157-173. 11	See	T.	Sider,	Four-Dimensionalism	and	D.	K.	Lewis,	On	the	Plurality	of	Worlds	(Oxford, 1986),	199. The	Limit	Decision	Problem	and	Four-Dimensionalism 7 One of the central debates in contemporary metaphysics is the one concerning the	way in	which	substances	persist through time.	Do they persist	by	enduring	or	by	perduring?	Those	who	think that	substances endure – are wholly present at different times – are called threedimensionalists, or also endurantists, whereas those who think that substances	perdure	–	have	different	temporal	parts	at	different	times	– are called four-dimensionalists, or also perdurantists. Threedimensionalists	believe	that	substances	endure.	Yet,	they	may	–	and	often do	–	think	that	other	entities,	such	as	intervals	of	time	or	events,	persist by	perduring,	for	an	event	may	be	composed	by	different	phases	and	an interval	of	time	by	different	instants,	which	count	as	temporal	parts	of	the event and the interval, respectively. Given that Burley puts forward a stone	as	an	example	of	a	permanent	thing,	and	a	day	as	an	example	of	a successive thing, the contemporary philosopher will be tempted to identify	Burley	–	together	with	the	majority	of	its	contemporaries	which do,	or	would,	offer	similar	examples	–	as	a	three-dimensionalist. These	parallelisms	and identifications	have indeed	been	proposed	and endorsed by several contemporary scholars, and I add myself to that list12. 1.3 Medieval	Solutions	to	the	Limit	Decision	Problem:	the	Scholastics on	the	Temporal	Limits	of	Permanent	and	Successive	Entities Now	that	the	distinction	between	permanent	and	successive	entities	has been introduced, let us see how the scholastics applied it to the limit decision	problem. Recall	that	the	limit	decision	problem	has	to	do	with	the	temporal	limits of	entities	–	such	as	the	being	off	of	a light	–	and	in	particular	with	the question	of	whether	those	limits	are	open	or	closed.	In	that	context,	the distinction	between	permanent	and	successive	entities	comes	in	because most	medieval	scholars	though	that	whether	an	entity	has	open	or	closed limits is primarily dependent on whether the entity is permanent or successive. The	so-called	common	theory,	which	was	seen	by	medieval	scholars	as	the Aristotelian	theory,	claims	that	the	temporal	limits	of	permanent	entities 12 See in particular R. Cross, "Four-Dimensionalism and Identity Across Time" and A. Wood, "Mind the	Gap?	The	Principle	of	Non-Repeatability	and	Aquinas'	Account	of the Resurrection,"	Oxford	Studies	in	Medieval	Philosophy	3	(2015),	99-127.	On	the	other	hand, a more careful attitude would suggest	more investigation on the distinction between permanent and successive things before subscribing to these parallelisms and identifications.	However,	such	an	investigation	is	still lacking	in	the	literature	and	goes beyond	the	scope	of	this	paper. The	Limit	Decision	Problem	and	Four-Dimensionalism 8 are	both closed,	whereas the temporal limits	of successive entities are both open13. The scholastics proposed further distinctions in order to specify	whether	a	state	such	as	the	light's	being	on	counted	as	permanent and successive. However, they did not think that this kind of limit assignment	held	for	states	only.	They	though	that	every	permanent	entity has	both	limits	closed	and	every	successive	entity	has	both	limits	open. So,	for	example,	they	think	that	permanent	entities	such	as	human	beings and	stones	have	a	first	and	a	last	instant,	whereas	successive	things	such as	intervals	of	time	do	not.	Probably	we	can	safely	conceive	of	a	human being	as	having	a	first	instant,	or	as	having	her	or	his	first	limit	closed,	as equivalent	to	the	claim	that	a	state,	namely	the	existence	of	the	human being,	has	a	first	instant,	or	has	its	first	limit	closed. Figure 2. The so-called common theory about the limit decision problem. The	common	theory	dominated	almost	unrivalled	for	a	good	part	of	the thirteenth	and	fourteenth	centuries.	However,	an	impressive	number	of other	options	are	discussed	and,	with	the	passage	of time,	endorsed	as well14.	It	is	not	my	aim	here	to	enter	into	the	question	of	what	historic	and theoretical reasons have pushed	medieval authors to subscribe to the common theory	or variations thereof.	What is relevant for our aims is that, with the notable exceptions of Albert of Saxony and William of Ockham15,	one	can	see	a	clear	tendency	shared	by	most	medieval	scholars 13	A.	de	Libera,	"La	problématique	de	l'instant	du	changement,"	66.	See	also	N.	Kretmann, "Incipit/Desinit";	N.	Strobach,	The	Moment	of	Change,	ch.	3. 14 See again N. Kretzmann, "Incipit/Desinit;" S. Knuuttila and Lehtinen, "Change and Contradiction,"	and	A.	de	Libera,	"La	problématique	de	l'instant	du	changement." 15	See	A.	de	Libera,	"La	problématique	de	l'instant	du	changement,"	66. permanent entity successive entity The	Limit	Decision	Problem	and	Four-Dimensionalism 9 in	the	relevant	period,	to	ascribe	different	temporal	limits	to	permanent and	successive	entities,	respectively.	This	difference	between	permanent and successive entities as regards their temporal limits is the crucial element	that	will	allow	us	to	trigger	the	argument	that	ultimately	leads	to the	four-dimensional	view	of	persistence. 2 The	argument	for	four-dimensionalism The	first	part	of	this	paper	concluded	that	standard	medieval	solutions	to the	limit	decision	problem	claim	that	permanent	and	successive	entities have	different	temporal	limits.	For	example,	the	so-called	common	theory claims that permanent entities have both limits closed, whereas successive	entities	have	both	limits	open.	The	second	part	of	this	paper argues	that	if	permanent	and	successive	entities	have	different	temporal limits,	permanent	entities	are	impossible:	no	thing	can	be	permanent. I	will	begin	by introducing	an	objection,	moved	by	Paul	Vincent	Spade, against	theories	that	claim	that	permanent	and	successive	entities	have different temporal limits (§2.1). Then I will introduce several recent developments	in	contemporary	metaphysics	that	will	allow	us	to	realize the	full	scope	of	Spade's	objection	(§2.2	and	§2.3).	On	the	one	hand,	these developments will allow us to circumvent Spade's original objection. However,	at	the	same	time,	such	developments	will	put	us	in	a	position	to reinforce	Spade's	objection in	a	way that	will	ultimately lead	us to the impossibility	of	there	being	permanent	entities	(§2.4).	I	will	conclude	the section	by	discussing	a	possible	way	out	from	the	reinforced	objection, and	explain	why	it	is	ultimately	unsuccessful	(§2.5). 2.1 Spade's	Objection In a paper dedicated to Walter Burley's aforementioned De Primo et Ultimo	Istanti,	Paul	Vincent	Spade	discusses	at	length	Burley's	solution	to the limit decision problem. Burley shares the common medieval approach to the problem, and starts with distinguishing between permanent	and	successive	entities	in	the	passage	quoted	before.	He	then subscribes to the view according to which successive entities are extrinsically limited at both ends, whereas permanent entities are intrinsically limited	at their	beginning	but	extrinsically limited	at their end. In his paper, Spade not only investigates the reasons	why	Burley	may have	held	such	a	view,	but	also	engages	critically	with	it	by	raising	a	series The	Limit	Decision	Problem	and	Four-Dimensionalism 10 of	remarks	and	objections.	One	of	them	is	particularly	relevant	here.	In Spade's	words: If	every	interval	of	time	is	a	successive	entity,	as	Burley	says,	and	if enduring	permanent	entities	have	a	first	instant	of	their	duration,	as he	also	says, then	–	despite	what	Burley	explicitly	maintains	–	his theory is unavoidably committed to allowing that at least some successive	entities	have	a	first	instant	of	their	duration:	namely	the exact	intervals	of	enduring	permanent	entities'	durations.	Although	in such a case the thing that fills the time-interval is an enduring permanent	entity,	the	interval	of	its	duration	is	a	successive	one,	and it	is	that	to	which	we	are	assigning	limits.	No	theory	can	consistently assign	limits	to	temporal	intervals	in	a	way	different	than	it	assigns them	to	the	durations	of	things	that	exactly	fill	those	intervals.	But that	is	just	what	Burley's	theory	tries	to	do. 16 Spade's objection, in a nutshell, is that Burley's theory leads to a contradiction.	For	every	entity	that	is	in	time	there	should	be	an	instant or interval that counts as its duration, and the entity and its duration should	temporally	coincide.	A	permanent	entity	has	its	first	limit	closed. Hence the interval that is its duration should also have its first limit closed. However, given that a duration is an interval of time, it is successive,	and	hence	must	have	its	first	limit	open. It is	worth	noting	that	the	objection	as	such	does	not	apply	to	Burley's theory	alone.	It	generalizes	to	any	theory	that	prescribes	permanent	and successive	entities	to	have	non-coinciding	temporal	limits,	regardless	of whether	this	prescription	is	made	as	a	consequence	of	the	problem	of	the instant	of	change	or	not.	For	example,	the	same	objection	affects	the	socalled common theory according to which permanent entities are intrinsically limited at both ends, whereas successive entities are extrinsically	limited	at	both	ends. A	crucial	premise	of	Spade's	objection	is	the	following	one: (P) For	every	entity	that	is	in	time	there	is	an	instant	or	interval	that counts	as	its	duration,	and	the	entity	and	its	duration	should	temporally coincide, or,	in	other	words,	that	the	former	should	begin	and	end	exactly	when	the latter	does	and	vice	versa.	The	premise	seems	reasonable.	However	Spade 16	P.	Spade,	"How	to	Start	and	Stop,"	204. The	Limit	Decision	Problem	and	Four-Dimensionalism 11 does	not	spend	much	on	discussing	it.	Are	there	reasons	in	favour	of it apart	from	its	apparent	reasonableness? Let	us	have	a	look	at	the	same	problem	from	another	perspective.	Burley claims	that	intervals	are	open.	One	could	take	this	as	proof	that	there	is no	such	thing	as the interval	of the	duration	of	a	permanent	entity, for such an interval should be closed, but there are no closed intervals. Burley, of course, does	not talk about the interval of the	duration	of a permanent	entity;	he	only	talks	about	a	permanent	entity	having	a	first and	a	last	instant.	So	the	question	is:	why	think	that	for	every	permanent entity	there	should	be	an	interval	that	counts	as	its	duration? Interestingly enough, contemporary philosophers have investigated these	questions	under	a	slightly	different	terminology.	Spade's	premise concerns	the	relation	between	a	permanent	entity	and	the	interval	of	time at which it exists, and more generally an entity and the region of a dimension	where the entity can	be found. Contemporary	philosophers have	long	investigated	that	relation	between	an	entity	and	a	region	where it can be found under the name of location. In the next section, I will introduce some basic findings and principles of the contemporary metaphysics	of	location	that	will	help	us	answering	the	aforementioned questions	concerning	Spade's	premise. 2.2 The	contemporary	metaphysics	of	location Contemporary	philosophers	call	location	the	relation	between	an	entity and	a	region	of	a	dimension	where	the	entity	is	present.	The	term	location is	probably	best	suited	for	the	spatial	case	only,	but	it	is	not	hard	to	see that	the	relation	concerning	the	spatial	and	the	temporal	case,	if	not	the same, belong to one and the same family – or so contemporary philosophers think. Accordingly, we will say, for example, that I am located at the region of	my office, but also that	WWI is located at the twentieth century, and	more precisely at a four-year interval between 1914	and	191817. One of the points on which contemporary philosophers insist is that location	is	ambiguous.	Several	different	senses,	or	modes,	of	location	can be carefully distinguished and defined. At least six different	modes of location	have	been	distinguished	so	far,	but	here	we	will	focus	on	two	of them	only,	which	are	usually	considered	the	central	ones,	namely	weak and exact location. One the one hand, we say that an entity is	weakly 17	See	R.	Casati	and	A.	Varzi,	Parts	and	Places:	The	Structures	of	Spatial	Representations (Cambridge,	Mass.,	1999);	R.	Casati	and	A.	Varzi,	"The	Structure	of	Spatial	Localization," Philosophical Studies 82 (1996), 205-239; J. Parsons, "Theories of Location," in	Oxford Studies	in	Metaphysics,	Vol.	3,	eds.	K.	Bennett	and	D.	Zimmerman,	(Oxford,	2007),	201-232. The	Limit	Decision	Problem	and	Four-Dimensionalism 12 located	at	a	region	if	and	only	if	the	region	is	not	completely	free	of	it.	In this general sense an entity can be located at several regions of a dimension.	Consider	for	example	the	image	below,	depicting	a	grey	entity x and	a series	of red-dashed	regions	r1-r5.	The	grey	entity	x is	weakly located	at	regions	r1,	r2,	r3	and	r4,	for	all	those	regions	are	not	completely free	of	it,	but	it	is	not	weakly	located	at	region	r5,	which	is	completely	free of	it.	On	the	other	hand,	we	say	that	an	entity	is	exactly	located	at	a	region if	and	only if	x and	r	have the	same	shape, size,	and	stand in the	same distance	relations	with	other	entities18.	Looking	back	at	the	image	below, x is	exactly	located	at	r1	only,	because	r1	is	the	only	region	that	has	its same	shape,	size,	and	that	stands	in	its	same	distance	relations	with	other entities – for example the	other regions in the image	– in the relevant dimension. Figure 3. Weak and exact location. The squared grey entity x is weakly	located	at	regions	r1,	r2,	r3,	r4,	is	exactly	located	only	at	r1.	It is	neither	exactly	nor	weakly	located	at	r5. Having distinguished and informally characterized weak and exact location,	one	important	question	that	poses	itself	is:	what	is	the	relation between	these	two	modes	of	location?	Because	of	the	conceptual	vicinity between the two concepts of location, and	because	of other important 18 C. Gilmore, "Where in the	Relativistic	World	Are	We?"	Philosophical Perspectives 20 (2006), 199-236, C. Gilmore, "Persistence and Location in Relativistic Spacetime," Philosophy	Compass	3	(2008),	1224-1254. The	Limit	Decision	Problem	and	Four-Dimensionalism 13 reasons19,	most	philosophers	think	that	in	principle	one	mode	should	be definable	in	terms	of	the	other.	Probably,	the	most	intuitive	option	is	to take	exact	location	as	a	primitive	and	define	a	weak	location	as	any	region which	overlaps	an	exact	location20.	So	for	example	x	is	weakly	located	at region	r4	because	r4	overlaps	its	exact	location	r1. A	crucial	result	that	follows	from	this	definition	of	weak	location	in	terms of	exact location	is	that	anything	that is	weakly	located	at	a	region	will also	have an exact location, for a	weak location	of	x is any region that overlaps with x's exact location. This principle is usually called, for obvious	reasons,	exactness21: (E) anything	that	is	weakly	located	in	a	dimension	must	also	have	an exact	location in	that	dimension. Could exactness offer the much needed justification for the crucial premise	of	Spade's	objection?	At	first	glance,	it	seems	that	it	can.	Recall that the	premise	was	that	an	entity	and its	duration	should	temporally coincide.	The	premise	seems	to	rely	on	the	intuition	that	when	an	entity is located in time, there	should	be	a	region	of time	which	counts	as its exact	location,	and	this	is	exactly	what	is	required	by	the	aforementioned principle	of	exactness,	which	therefore,	I	suspect,	was	what	Spade	had	in mind. However	things	are	not	so	obvious	on	closer	look.	Spade's	premise	does not	only	require	that	a	permanent	entity	have	an	exact	temporal	location. It also requires that this location be an interval of time. This further requirement	cannot	properly	speaking	be	derived	from	exactness,	which only	requires	permanent	entities,	which	are	in	time,	to	have	at	least	an exact temporal location, without specifying that that exact temporal location should	be an interval.	This	difference is indeed crucial.	But in order	to	see	why	it	is,	we	should	now	introduce	some	basic	notions	in	the contemporary	metaphysics	of	persistence,	and	in	particular	explain	how three-dimensionalism	is	nowadays	defined. 2.3 What	is	three-dimensionalism? Recall that three-dimensionalism is the view according to which substances are not extended through time and persist through time without having temporal parts.	Prima facie the definition is somehow puzzling.	How	is	it	that	something	which	is	not	extended	through	time	still occupies,	in	a	certain	sense,	an	interval	of	time	which	is	extended,	namely 19	See	M.	Leonard,	"Locating	Gunky	Water	and	Wine,"	Ratio	27	(2014),	306-315. 20	J.	Parsons,	"Theories	of	Location,"	204. 21	J.	Parsons,	"Theories	of	location,"	205. The	Limit	Decision	Problem	and	Four-Dimensionalism 14 the	interval	of	its	persistence?	An	answer	to	this	question	can	be	and	has been	recently	given in light	of the	advancements in the	metaphysics	of location	presented	before. The	basic	idea	is	this.	According	to	three-dimensionalism,	substances	are not	extended	through	time	–	they	are	three-,	and	not	four-,	dimensional. If	a	substance	is	not	extended	through	time,	it	cannot	be	exactly	located at intervals of time. This is because intervals of time are temporally extended,	and	recall that	an	entity	shares	shape	and	size	with its	exact locations.	If	a	substance	is	not	exactly	located	at	intervals	of	time,	it	must be located	at instants	of time.	And it	can	cover	an	extended interval	of time	by	being	exactly	located	at	each	instant	that	makes	up	that	interval. In	other	words,	the	idea	is	that	a	three-dimensional	entity	that	persists through	time	is	an	entity	that	persists	by	being	exactly	located	at	several instants	of	time.	In	this	way	it	can	both	be	temporally	unextended	and	yet persist	through	an	extended	interval	of	time.	Defined	in	this	way,	threedimensional entities differ from four-dimensional ones in that fourdimensional entities are extended through time, and hence have a temporally	extended	interval	of	time	as	their	exact	temporal	location22. It	is	worth	noting	that	some	philosophers	prefer	to	talk	of	spatiotemporal instead of	merely temporal location. In that case, a three-dimensional entity	will cover	a four-dimensional spacetime region	by	being	exactly located	at	several, three-dimensional, instantaneous	spacetime	regions, whereas	a	four-dimensional	entity	will	cover	the	same	four-dimensional spacetime region by being exactly located at it. The spatiotemporal approach	delivers	immediately	why	three-dimensional	entities	and	fourdimensional	entities	are	called	in	this	way.	Recall	that	an	entity	shares	the shape of its exact location. Three-dimensional entities are located at three-dimensional	spacetime	regions,	and	hence	have	three-dimensional shapes, whereas four-dimensional entities are located at fourdimensional spacetime regions, and hence have four-dimensional shapes23. 22	T.	Bittner	and	M.	Donnelly,	"A	Classification	of	Spatio-temporal	Entities	Based	on	Their Location in Space-time," in International Workshop on Semantic-based Geographical Information Systems, ed. E. Zimanyi (Dordrecht, 2006), 1626-1635; T. Sattig, The Language	and	Reality	of	Time	(Oxford	2006);	C.	Gilmore,	"Where	in	the	Relativistic	World Are	We?." 23	C.	Gilmore,	"Where	in	the	Relativistic	World	Are	We?." The	Limit	Decision	Problem	and	Four-Dimensionalism 15 Figure 4. Threeand four-dimensionalism. On the left, a fourdimensional entity occupies a four-dimensional region by being exactly located at it. On the right, a three-dimensional entity occupies a four-dimensional region by being exactly located at several	instantaneous	regions	making	up	that	region. Now	we	are in	a	position to see	why	a three-dimensionalist can reject Spade's premise and yet keep exactness. First, recall we are working under	the	hypothesis	that	permanent	simply	means	three-dimensional.	A three-dimensionalist	will	then	reject	Spade's	premise,	because	it	requires that	a	permanent	–	hence	three-dimensional	–	entity	is	exactly	located	at an interval, whereas the three-dimensionalist thinks that threedimensional	entities	are	not	exactly	located	at	intervals.	Yet,	in	so	doing the	three-dimensionalist	will	keep	exactness,	because	three-dimensional entities	have	indeed	exact	temporal	locations,	namely	the	instants	of	time that	make	up	the	interval	of	their	persistence. 2.4 Spade's	objection	reinforced In the previous section, we have seen how in light of the recent developments in the	metaphysics of location and persistence, Spade's objection	–	or	at least the first reading that I	have	given to it – can	be ultimately	resisted.	Yet,	this	does	not	mean	that	Spade's	objection	cannot be reinforced in a	way to be successful. In this section, I shall indeed present	a	reinforced	version	of	Spade's	objection	which	will	conclude	that if	we	assign	temporal	limits	to	permanent	entities	in	a	different	way	from the one we use for successive ones, a contradiction follows and permanent	entities	are	thus	impossible.	I	shall	present	the	argument	as The	Limit	Decision	Problem	and	Four-Dimensionalism 16 an argument directed against the common theory but in principle the argument can be easily adapted to any theory according to which permanent	and	successive	entities	have	different	temporal	limits.	I	will leave	to	the	reader	the	task	of	adapting	the	argument	to	her	or	his	favorite version	of	such	a	theory. The argument runs as follows. Think again over the duration of a permanent	entity.	The	duration	of a	permanent	entity is an interval	of time.	It	is	not	the	exact	location	of	that	entity,	because	permanent	entities are	only	located	at	instants	and	not	at	intervals.	Rather,	the	duration	of	a permanent	entity	should	be	conceived	as the	sum	of	all the instants	at which	the	permanent	entity	is	exactly	located24. It	can	be	proven	that	this	sum	is	an	interval	of	time	which	is	intrinsically limited	at	both	ends.	First,	let	us	prove	that	it	is	an	interval	of	time.	It	is, because	an	interval	of	time	is	a	sum	of	instants	connected	by	the	order	of temporal succession that is limited by two instants and such that any instant	that	lies	between	the	two	limiting	instants	is	also	included	in	the sum25. The two limiting instants are the first and last instant of the existence	of	the	permanent	entity.	And	granted	that	the	existence	of	the permanent	entity	is	not	intermittent,	all	instants	that	lie	in	between	the limiting	instants	are	also	included	in	the	sum.	Second,	let	us	prove	that this	interval	of	time	has	both	limits	closed.	An	interval	of	time	has	both limits	closed if	and	only if the	two limiting instants	are included in	the sum	that	is	the	interval26.	And	the	limiting	instants	of	a	permanent	entity are included in that sum, for a permanent entity has a first and a last instant	of	its	existence	and	hence	at	which	it	is	exactly	located. Given	what	we	have	just	said,	we	can	conclude	that	the	interval	of	time which	is	the	duration	of	the	permanent	entity	is	an	interval	of	time	which is closed at both ends. However, since it is an interval of time, it is successive,	and	hence	it	is	open	at	both	ends.	Contradiction. 2.5 Transcendentist	endurantism	to	the	rescue? So far we have only considered the form of three-dimensionalism according to which substances persist through time without having temporal parts by being exactly located at several instants of time. Nowadays this is certainly considered the standard form of three- 24 This is indeed one of the six aforementioned modes of location distinguished by contemporary philosophers. Usually, it is called the path of an entity in the given dimension,	and	is	indeed	defined	as	the	mereological	sum	of	all	the	entity's	exact	locations in	that	dimension.	Cf.	C.	Gilmore,	"Where	in	the	Relativistic	World	Are	We?." 25	M.	C.	Gemignani,	Elementary	Topology	(New	York,	1990). 26	M.	C.	Gemignani,	Elementary	Topology	(New	York,	1990). The	Limit	Decision	Problem	and	Four-Dimensionalism 17 dimensionalism.	However,	it	is	not	the	only	form	of	three-dimensionalism on the	market.	There is another	way	of	defining three-dimensionalism that	does	not	require	substances	to	have	exact	temporal locations. It is the so called transcendentist theory of persistence, or transcendentist endurantism. The transcendentist theory of persistence denies that substances are weakly or exactly located at regions of time. In other words,	it	denies	that	the	relation	between	a	substance	and	the	times	at which	it	exist	is	a	relation	of	location.	Rather,	it	defines	such	a	relation	in terms	of	the	events	and	processes	in	which	such	a	substance	participates. Accordingly,	the	theory	claims	that	for	a	substance	to	exist	at	a	time	is	for it	to	participate	in	an	event	which	is	weakly	located	at	that	time27. The	transcendentist	theory	of	persistence	is	based	on	various	semantic and metaphysical grounds on which I will not focus here. What is important here is rather the question whether conceiving of threedimensionalism, and hence of permanence, in this way will help the three-dimensionalist	out	of	the	reinforced	objection.	The	answer	to	this question is in the negative. According to transcendentist endurantism, substances	exist	at	times	if	and	only	if	they	participate	in	events	that	are located	at	those	times.	For	example,	a	human	being	exists	at	the	interval of its persistence because it participates in her or his life, which is a temporally	extended	event	which	is	located	at	that	interval.	Now	either events	are	successive	or	permanent. If the former,	which	seems in	any case	to	be	the	most	plausible	option,	then	events	would	not	have	a	first and	a	last	instant,	and	so	also	the	life	of	the	human	being	would	not	have a	first	and	a	last	instant.	Hence,	also	the	existence	of	a	human	being	would not	have	a	first	and	a	last	instant,	given	that	for	a	substance	to	exist	at	a time	is	for	it	to	participate	in	an	event	located	at	that	time.	If	the	latter, then	events	would	be	three-dimensional	and	located	in	time,	so	the	view would	in	any	case	fall	prey	to	the	reinforced	objection	presented	before. 27	A.	Giordani	and	D.	Costa,	"From	Times	to	Worlds	and	Back	Again:	A	Transcendentist Theory	of	Persistence,"	Thought:	A	Journal	of	Philosophy	2	(2013),	210-220;	D.	Costa,	"The Transcendentist Theory of Persistence," The Journal of Philosophy (forthcoming); P. Simons,	Parts.	A	Study	in	Ontology	(Oxford,	1987);	P.	Simons,	"Where	is	it	At?	Modes	of Occupation and Kinds of Occupant," in Mereology and Location, ed. S. Kleinschmidt (Oxford	2014),	59-68	and	B.	van	Fraassen,	An	Introduction	to	the	Philosophy	of	Time	and Space (New York, 1970), when similar views, if not the same, are sympathetically discussed. The	Limit	Decision	Problem	and	Four-Dimensionalism 18 3 Conclusions To	sum	up,	recent	advancements in the	metaphysics	of location	and	of persistence led	us to see that even if Spade's	original objection can	be resisted, a reinforced version of it is successful, and shows that if the limits of permanent and successive entities are assigned differently, a contradiction	follows.	When	confronted	with	this	results,	several	options are	available.	Just	to	mention	one	obvious	option,	a	three-dimensionalist could take these results as a reductio ad absurdum of the idea that permanent and successive entities have different temporal limits. However, if one takes seriously the	aforementioned	medieval theories, and	in	particular	takes	seriously:	(i)	the	distinction	between	permanent and successive entities, (ii) that this distinction parallels the contemporary one between threeand four-dimensional entities, (iii) that	the	nature	of	permanent	and	successive	entities	require	them	to	have different temporal limits, (iv) that intervals of time are successive entities,	then	the	most	obvious	conclusion	seems	to	be	that	permanence, i.e. persistence	without temporal	parts, is impossible.	Moreover, if one takes seriously the idea that (v) there are substances, and that (vi) substances persist through time, the only possible conclusion to draw seems	to	be	that	substances	persist	by	being	successive	entities, i.e.	by having	different	temporal	parts	at	different	times.	In	other	words,	tenets (i)-(vi)	imply	the	four-dimensional	view	of	persistence. I	conclude	by	letting	the	reader	decide	whether,	if	confronted	with	this problem,	medieval	scholars	would	rather	embrace	four-dimensionalism, reject one out of tenets (i)-(vi), or attack	my argument in some other interesting	way.	Moreover, I	will	not	be	surprised if	some	historians	of philosophy	will	think	that	the	meddling	of	contemporary	notions	in	the medieval	lines	of	reasoning	constitutes	an	inadmissible	anachronism	that leads us to a fundamentally distorted interpretation of medieval philosophy	in	general.	On	the	contrary,	I	hope	that	the	close	interaction that	I	have	put	at	work	in	this	paper	will	show	the	impressive	possibilities that	are	open	once	contemporary	and	medieval	metaphysics	are	put	at work together, and that the case of the phenomenon of persistence through	time	is	an	emblematic	case	of	the	possibility	of	this	convergence –	one	that	remains,	as	yet,	largely	uncharted.