COMPOSITION	AS	IDENTITY	AND	PLURAL	CANTOR'S	THEOREM Einar	Duenger	Bohn University	of	Agder [Forthcoming	in	a	special	issue	of	Logic	and	Logical	Philosophy] Let	Composition	as	Identity	(CAI)	be	the	thesis	that	a	whole	is	identical	with	all	its parts collectively, not individually.1	For a toy example, let	my	body	be a	whole composed of some	parts, say	my arms, legs, head and torso. Then, by CAI,	my body	is	identical	with	my	arms,	legs,	head	and	torso	collectively,	but	not	with	any one	of	them	individually. Plural Cantor's Theorem (PCT) is the proposition that for any plurality containing two or more members, there are more sub-pluralities of it than members.2 For a toy example, consider you and me. That plurality has 2 members: you and	me, but 22-1 sub-pluralities: you,	me, and you-and-me. The point	generalizes: for	any	plurality	with	n	members, it	has	2n-1	sub-pluralities, which	is	strictly	greater	than	n,	provided	n>1. It seems to be a well-known fact among philosophers working on the topic	that	CAI	blocks	PCT,	but,	unfortunately,	it	has	so	far	been	neither	formally shown	nor fully appreciated in	print.3	So, in	what follows, I first show in some detail	how	my	favorite	version	of	CAI	blocks	PCT	(section	1).	Second,	to	see	some 1	See	especially	Lewis	(1991:3.6);	Sider	(2007);	Wallace	(2009,	2011);	Bohn (2009,	2011,	2014);	Cotnoir	(2013);	and	Bricker	(forthcoming);	as	well	as	the various	essays	in	Baxter	&	Cotnoir	(2014).	For	criticisms	of	CAI,	see	especially van	Inwagen	(1994),	Yi	(1999),	Merricks	(2003:20-28),	and	Sider	(2007,	2014). 2	See	especially	Florio	(forthcoming). 3	The	closest	we	get	are	Sider	(2007,	2014),	Cotnoir	(2013)	and	Hovda	(2014); but	see	also	Saucedo	(ms).	Sider	seems	to	think	it's	a	problem	for	CAI,	rather than	a	virtue,	but,	arguably,	Sider's	problems	for	CAI	are	avoided	by	the relational	units	employed	below;	cf.	Bohn	(2011,	2014)	and	Cotnoir	(2013). 2 of	its	philosophical	importance,	I	show	how	this	in	turn	blocks	a	recent	argument against both modal realism and necessitism, and how this latter fact can be turned into	an	abductive	argument for	CAI,	given	modal	realism	or	necessitism (section	2).	All	in	all,	I	thus	hope	to	show	that	and	how	my	favorite	version	of	CAI blocks	PCT,	and	that	it's	a	philosophically	important	fact	we	need	to	recognize,	a fact	that	can	be	put	to	some	interesting	philosophical	work. To	avoid	some	potential	confusion	from	the	outset,	three	things	are	worth noting at this point already. First, I am not defending CAI. I'm here simply assuming	that	CAI	is	a	coherent	view,	in	order	to	show	that	(i)	if	CAI	is	true,	then PCT is	not a	universal truth, and that (ii) this fact	has important	philosophical consequences,	which	I	illustrate	by	the	examples	with	respect	to	modal	realism and	necessitism (presumably there are other examples too).	Hence, it's a nonstarter	to	object to	the	thesis	of this	paper	by	objecting	to	CAI	as	such.	Second, since I am not here defending CAI, neither will I here attempt to develop a version	of	CAI	in	full	details.	I	only	develop	a	version	to	the	extent	needed	to	see that it will block PCT, and how. I intentionally leave the various directions of further developments of it open. Third, to the extent I do develop CAI, I only develop	one	version	of	it	(the	version	I	find	the	most	plausible).	Now,	there	are other versions of it as	well, some of them	with the same consequences as the ones	I	show	below,	but,	of	these	other	versions,	I	say	nothing.	These	assumptions and	omissions	are	justified	by	the	fact	that	CAI	is	an	ongoing	research	program,4 arguably	at	a	stage	of	maturity	that	allows	taking	this	program,	or	at	least	some aspects	of	it	for	granted	in	order	to	explore	its	consequences. 4	Witnessed	by	the	growing	amount	of	work	on	CAI	over	the	last	5-8	years.	For some	references,	see	fn.1. 3 1.	CAI	BLOCKS	PCT Let	CAI first	and foremost	amount to the following	stipulative	definition	of the mereological	term	'compose':5 (CAI):	xx	compose	y	=df	xx	are	(collectively)	identical	with	y where	the	semantics	of	the	right-hand	side	is	as	expected,	namely	'α=β'	is	true	iff v(α)	is	the	same	as	v(β),	where	α	and	β	are	schematic	meta-variables	for	either singular	or	plural	object-variables, and	v is the	assignment	of a referent to the object-variables.6	The corresponding laws of identity are as expected, namely Reflexivity: ∀α(α=α), and Leibniz's Law: ∀α∀β(α=β→(Φ(α)↔Φ(β))), from which	we	can	easily	derive	Symmetry	and	Transitivity,	where	again	α	and	β	are schematic meta-variables for singular or plural object-variables. Mereological composition is thus intended to be just one among four possible forms of (informative) identity: one-one, one-many, many-one and many-many ('x=y', 'x=yy', 'xx=y' and 'xx=yy'). CAI is thus committed to a generalized concept of identity, of	which the	ordinary	one-one ('x=y') is just one among four	possible 5	As	such	we	no	longer	need	any	mereological	term	as	a	primitive,	since	all classical	mereological	predicates	can	be	defined	in	terms	of	'compose',	which	we here	define	in	terms	of	a	primitive	generalized	notion	of	identity.	For	the	kind	of irreducibly	plural	logic	used	throughout	this	paper,	see	Yi	(2005,	2006)	and Oliver	&	Smiley	(2013). 6	See	Cotnoir	(2013),	Bohn	(2014)	and	Bricker	(forthcoming).	This	stipulative definition	of	composition	will	of	course	not	convince	anyone	of	the	general coherency	of	CAI	unless	already	convinced	of	the	coherency	of	the	underlying primitive	generalized	notion	of	identity	and	its	corresponding	semantics,	but, recall,	CAI	is	an	ongoing	research	program,	and	convincing	people	of	that program	is	not	our	present	aim.	We	here	only	explore	some	things	that	follow from	its	supposed	success. 4 cases,	another	which	is	composition	('xx=y').	The	general	idea	is	just	that	a	whole and all its parts collectively is one and the same ontological constituent, or "portion	of	reality",	just	conceptualized	in	two	different	ways.7	We	thus	see	a	first sense in	which	CAI is committed to a revisionary language, namely a language allowing	each	side	of	its	identity	sign	to	be	flanked	by	either	a	singular	or	a	plural term,	independently	of	each	other. But	consider	my	body.	Let	a	be	my	body	and	bb	be	my	arms, legs,	head and	torso,	and	assume	bb	compose	a.	Then,	according	to	CAI,	bb=a.	But	a	has	the cardinal	property	one,	which	bb	does	not;	and	bb	has the	cardinal	property	six, which a does not. So, by Leibniz's Law (and the assumption that the cardinal properties	one	and six exclude each	other),	we get a contradiction. Likewise,	a forms a singleton set, but bb does not, so, by Leibniz's Law, we get another contradiction,	assuming	bb=a.	Also,	my	left	arm	is	one	of	bb,	but	not	one	of	a,	so, by	Leibniz's	Law,	we	again	get	a	contradiction,	assuming	bb=a.	What	such	cases have	in	common	is	that	the	properties	in	question	(e.g.	cardinality,	set-formation, and	being	one	of)	only	hold	relative	to	a	unique	kind	of	"division"	of	their	subject. For	obvious	reasons,	Sider	(2007)	calls	such	properties	set-like. To solve for the kind of contradictions we get from such set-like properties,	we	let	CAI	be	committed	to	all	such	properties	being,	contra	what	we might	have initially thought,	relational	properties, i.e.	properties that	hold	only relative	to	a	unit,	which	I	henceforth	(non-essentially)	assume	is	a	concept.8	So, 7	For	a	discussion	of	the	notion	of	"portion	of	reality",	see	especially	Hawley (2013)	and	Bricker	(forthcoming). 8	Though	I	will	henceforth	take	concepts	to	be	my	relational	units,	note	that	for logical	purposes,	any	kind	of	relational	unit	will	do;	e.g.	modes	of	presentation,	or perhaps	just	contexts.	The	general	idea	is	of	course	a	modification	of	Frege's 5 for	example,	the	property	of	being	one	in	number	is	relative	to	a	concept	C,	and the	property	of	being	some	number	larger	than	one	in	number,	say	six,	is	relative to	some	other	concept	C*.	We	then	get	that	a	(and	bb)	is	one	relative	to	C	(being	a body),	but	bb	(and	a)	is	not	one	relative	to	C*	(being	arms,	legs,	head	and	torso), which resolves the contradiction. The solution generalizes to all other such contradictions	that	are	due	to	set-like	properties,	e.g.	those	due	to	forming	a	set and	being	one	of	some	things	mentioned	above.9	We	thus	see	a	second	sense	in which	CAI	is	committed	to	a	revisionary	language,	namely	a	language	in	which	a predicate	we	might	have	initially	thought	was	n-place,	is	really	m-place,	for	some m>n, with concepts filling the "new" places (assuming the predicates are to match	the	structure	of	the	properties	they	express).10 The cardinality of something (as	well as any other set-like property) is thus	always	relative	to	a	concept	used	to	present	it	with,	a	concept	that	provides us	with	a	certain	"division"	of	the	referent	of	the	subject	term.	For	example,	using the	concept	of	being	a	deck	of	cards,	what's	in	my	hand	has	the	cardinality	one, famous	idea	in	his	Grundlagen	(1884)	that	cardinality	is	a	property	attaching	to	a concept.	Cf.	Bohn	(2009,	2011);	Wallace	(2009,	2011);	Cotnoir	(2013). 9	In	short,	and	in	general,	while	F(x)	&	∼F(x)	is	a	contradiction,	F(x,c)	&	∼F(x,c*) is	not,	provided	c≠c*.	For	the	general	strategy,	see	Bohn	(2009,	2011,	2014)	and Cotnoir	(2013);	see	also	Appendix.	Wallace	(2009,	2011)	suggests	a	similar strategy,	but	it	is	unclear	to	me	how	much	her	suggestion	generalizes	beyond purely	numerical	predication.	McDaniel	(2013)	suggests	that	a	proponent	of	CAI should	not	relativize	numerical	predication	as	above,	but	just	accept	that	one	and the	same	thing	can	have	two	different	cardinal	numbers.	But	McDaniel's	solution is	insufficiently	general;	it	becomes	simply	incoherent	in	other	cases,	e.g.	in	the cases	of	forming	a	set	and	being	one	of	some	things. 10	Note	that	by	thus	relativizing	the	set-like	properties,	there	is	no	need	to	put	a restriction	on	Leibniz's	Law,	and,	as	pointed	out	in	Bohn	(2009,	2011,	2014)	and Cotnoir	(2013),	we	also	block	the	devastating	results	of	CAI	for	plural	logic shown	in	Yi	(1999)	and	Sider	(2007,	2014).	Yi	and	Sider's	results	rest	on	the derivation	of	the	principle	Sider	(2007)	calls	Collapse	(x	is	part	of	the	fusion	of	yy iff	x	is	one	of	yy),	but	the	derivation	of	that	principle	equivocates	on	the relational	aspects	of	set-like	predications.	In	my	mind,	any	satisfactory	version	of CAI	must	block	Collapse.	See	also	Bricker	(forthcoming). 6 but	using	the	concept	of	being	cards,	it,	the	very	same	thing,	has	the	cardinality fifty-two.	None	of	them	is	privileged	in	the	sense	of	being	the	cardinality	of	it.	It has both cardinalities, but relative to different concepts providing different "divisions"	of it.11	Examples	of	properties that	are	not thus	relative (i.e.	are	not set-like)	are	mass,	spatial	location,	and	identity.12 A legitimate worry at this point is how to individuate the set-like properties, as opposed to the non-set-like properties. For example, why is cardinality (or forming a set, or being one of) a relational property, but selfidentity (or mass, or spatial location) not? The obvious, but perhaps not too informative answer is that the former holds of a subject only relative to a particular	kind	of	"division"	of	the	subject,	but	the	latter	holds	independently	of any	such	particular	kind	of	"division".	For	example,	the	deck	of	cards	in	my	hand counts fifty-two only relative to a particular kind of "division" (divide it differently	and	you	get	a	different count),	but it is self-identical relative to	any kind of "division" (divide it however you want, self-identity holds no matter what13). 11	Cf.	Frege	(1884).	One	might	of	course	also	appeal	to	the	idea	of	some properties	being	more	natural	than	others	(Lewis,	1986:59-69),	and	hence	argue that	it	has	one	of	the	cardinalities	more	fundamentally	(in	some	sense	or	other) than	the	other.	Though	I	am	sympathetic	to	this	idea,	I	ignore	it	for	present purposes. 12	Note	that	CAI	is	not	committed	to	the	thesis	of	relative	identity.	Cf.	Geach (1967). 13	More	precisely,	"divide"	it	into	one	thing,	a,	and	a	is	self-identical;	"divide"	it into	three	things,	bcd,	and	bcd	are	self-identical,	both	individually	and collectively.	The	point	generalizes	to	any	kind	of	"division",	so	self-identity	is	not set-like	in	the	relevant	sense.	Note	also	that	if	a=bcd,	and	a	is	one	self-identical object	and	bcd	are	three	self-identical	objects,	then,	assuming	cardinalities exclude	each	other,	one	might	be	tempted	to	conclude	that	a	both	is	and	is	not one	self-identical	object,	which	is	a	contradiction.	But	this	contradiction	is	solved for	by	the	fact	that	the	cardinalities	are	relative,	not	the	self-identity. 7 Now, it	would take	us too far	afield to fully	explore the individuation	of set-like properties here, but note that, plausibly, there	might not be any fully satisfactory	such	criterion	of individuation	of	set-like	properties.	They	might in the end have to be individuated simply by our linguistic intuitions concerning the	relevant truth-conditions:	does the truth	of this	or that	predication	depend on	a	particular	kind	of	"division"	of	the	subject	of	predication?	If	yes,	it's	set-like; if no, it's not set-like. But note also that, assuming CAI, the fact that	we get a contradiction with respect to some properties, but not with respect to others should	be	taken	to	be	a	good	indicator	of	the	former	being	set-like,	but	the	latter being	non-set-like.14 Note	finally,	and	importantly	for	what's	to	come,	that	if	I	counted	what's in my hand as being fifty-two cards and one deck of cards and from that concluded	that	I	have	fifty-three	things	in	my	hand,	then,	given	CAI,	I	would	have double	counted	the	content in	my	hand.	That is, I	would	have	counted	the	same thing	under two	different	concepts, summed	up	both	counts, ignored that	each count is	of	one	and	the	same	thing,	and	as	a	result falsely	concluded	that there are fifty-three things in my hand. It is as if I count the morning star and the evening star and conclude that there are two different things there. Given the identity	between the	morning	star	and the	evening	star, that conclusion is just 14	Bricker	(forthcoming)	denies	that	CAI	needs	such	relational	properties,	so	he avoids	the	above	individuation	problem	altogether.	But,	in	return,	he	gets	a	much weaker	thesis,	which	is,	in	my	mind,	harder	to	see	as	a	coherent	picture.	In	any case,	I	take	the	individuation	problem	to	be	one	of	many	interesting	problems	to be	further	explored	in	the	ongoing	research	program	at	hand. 8 false;	likewise	in	the	case	of	a	deck	of	cards	and	its	cards,	as	well	as	in	the	case	of my	body	and	its	arms,	legs,	head	and	torso,	given	CAI.15 Of	course,	CAI	has	many	problems	yet	to	be	resolved,	but	I	take	it	we	now	have	a sufficient characterization of it to see how it blocks PCT. As just shown, CAI comes	with a revisionary language in the sense that (i) it contains a primitive generalized	identity-predicate,	and	(ii)	many	predicates	only	hold	of	something, or	some	things,	relative	to	a	concept	that	"divides"	up	its	subject	in	a	certain	way. (See	Appendix.)	I	now	present	a	simple	counterexample	to	PCT,	as	re-interpreted in this revisionary language	of	CAI. That should suffice to show that given CAI, PCT	fails	to	be	a	universal	truth.16 The basic idea behind the counterexample is simply that given CAI,	we need	to	be	careful	when	we	count	our	ontology.	By	CAI,	two	overlapping	things are	not	wholly	distinct	things,	so	counting	them	both	amounts	to	at	least	partly double counting ones ontology. Of course, for many purposes, such double counting	is	harmless,	but	not	so	for	the	purposes	of	what's	in	one's	ontology,	in which case it is harmful to the truth. In counting one's ontology, one must therefore	count	by	concepts	with	disjoint	extensions	on	pain	of	harmful	double counting. It's not that, according to CAI, we cannot count by concepts with 15	I	here	ignore	in	my	mind	exotic	metaphysical	positions	according	to	which fusions	constitute	(but	not	compose)	further	objects,	or	substances.	Such structures	of	constitution	can	be	added	on	top	of	CAI's	mereological	structures,	if wanted. 16	More	specifically,	I	do	this	by	providing	a	domain	over	which	we	can	derive	a contradiction	from	the	conjunction	of	CAI	and	PCT	(as	understood	in	the	slightly revisionary	language	of	CAI),	which	suffices	to	show	that	if	CAI	is	true,	then	PCT fails	to	hold	for	all	domain.	I	here	intentionally	stay	neutral	on	the	more constructive	side	of	things,	e.g.	the	exact	plural	logic	that	should	accompany	CAI (though	see	fn.10). 9 overlapping extensions (we often truly do); it's just that when it comes to purposes	of	what's in	ones	ontology, it	would	give	us	a	false	answer:	we	would count	as	distinct	what's	not	distinct.	As	we'll	now	see,	given	CAI,	PCT	is	guilty	of such	harmful	double	counting;	so	PCT	fails	as	a	universal	truth. We	write	'f(xx)'	for	the	fusion	of	xx	(i.e.	the	unique	thing	xx	composes17); and	'<x,y>'	for	the	ordered	pair	of	x	and	y.18	Letting	bb	be	a	plurality	of	ordered pairs,	we	define	the	domain	of	bb	–	dom(bb)	–	as	the	plurality	of	all	and	only	the first	members	of	the	pairs	in	bb.	We	say	that	some	x	in	the	domain	of	bb	codes	the plurality	of	all	and	only	the	second	members	of the	ordered	pairs	of	which	x is the	first	member,	and	bb	codes	a	plurality	xx	iff	some	x	in	the	domain	of	bb	codes xx.	We	define	the	predicate	'among':	xx	are	among	yy	iff	for	any	z,	if	z	is	one	of	xx, then	z	is	one	of	yy;	where	'is	one	of'	is	understood	as	expected:	x	is	one	of	y1y2... iff	x=y1	or	x=y2	or....	Call	this	definition	of	'among',	D1.	Note	that	all	the	pluralities among	yy	are	all	and	only	the	sub-pluralities	of	yy. PCT	can	then	be	more	precisely	formulated	as	follows:	there	is	no	plurality of	pairs that	codes	every sub-plurality	of its	domain, if the	domain is larger than one: (PCT):	∼∃xx(|dom(xx)|>1	&	∀yy(yy	are	among	dom(xx)	→	∃x∀y(<x,y>	is	one	of xx	↔	y	is	one	of	yy))) 17	I	here	assume	uniqueness	of	composition,	though	it	follows	from	CAI;	cf.	Sider (2007). 18	The	following	terminology,	as	well	as	the	more	precise	formulation	of	PCT below,	is	adopted	from	Florio	(forthcoming),	which	is,	as	far	as	I	know,	the	only place	that	gives	the	proof	of	PCT	in	full	details.	The	sketch	in	Hawthorne	& Uzquiano	(2011)	is	too	rough	to	use	to	show	exactly	how	CAI	blocks	PCT. 10 Let	cc	be the three-membered	plurality	of	ordered	pairs	<f,f>,	<a,a>	and	<b,b>, where	f	is	short	for	f(ab),	i.e.	the	fusion	of	ab.	We	officially	state	CAI	as	before:	xx compose y =df xx=y.	We know from our earlier definitions that	dom(cc) is the three	(and	only	three)	membered	plurality	fab.	So,	we	also	know	that	there	are seven	(and	only	seven)	sub-pluralities	of	dom(cc):	fab,	fa,	fb,	ab,	a,	b,	and	f.	Since we	know	that	dom(cc)	contains	more	than	one	thing,	we	universally	instantiate PCT	by	our	case	cc,	drop	the	first	conjunct,	and	perform	the	following	derivation: 1. ∼∀yy(yy	are	among	dom(cc)→∃x∀y(<x,y>	is	one	of	cc	↔	y	is	one	of	yy)) 2. ∃yy∼(yy	are	among	dom(cc)→∃x∀y(<x,y>	is	one	of	cc	↔	y	is	one	of	yy)) 3. ∼(aa	are	among	dom(cc)→∃x∀y(<x,y>	is	one	of	cc	↔	y	is	one	of	aa)) 4. aa	are	among	dom(cc)	&	∼∃x∀y(<x,y>	is	one	of	cc	↔	y	is	one	of	aa) 5. aa	are	among	dom(cc) 6. aa=fab	∨	aa=fa	∨	aa=fb	∨	aa=ab	∨	aa=a	∨	aa=b	∨	aa=f Lines	1-5	are	obtained	by	standard	plural	and	singular	quantificational	logic,	and line	6 follows from	5	by	D1	plus	our	knowledge	of the seven (and	only seven) sub-pluralities of dom(cc). Tediously running through each one of the seven cases	of line	6, and instantiate the second conjunct of line	4 appropriately,	we find	a	direct contradiction in	each	of the	cases	aa=a,	aa=b	and	aa=f,	but	not in any	of the	cases	aa=fab,	aa=fa,	aa=fb,	aa=ab. I	here	only	show	the	two	cases	of aa=f	and	aa=ab. Assume aa=f. We universally instantiate the second conjunct of line 4: ∃y∼(<f,y>	is	one	of	cc	↔	y	is	one	of	aa).	There	are	three	and	only	three	cases	to consider:	a,	b	and	f.	Both	a	and	b	make	both	sides	of	the	latter	bi-conditional	false, 11 and	hence	the	entire	bi-conditional	true,	and	hence	its	negation	false,	and	hence contradicts	the	second	conjunct	of	line	4.	But	f	makes	both	sides	of	the	latter	biconditional true, and hence the entire bi-conditional true too, and hence its negation	false,	and	hence	contradicts	the	second	conjunct	of	line	4.	But	there	are no other possible instantiations. So,	∼∃y∼(<f,y> is one of cc	↔ y is one of	aa), which	contradicts	the	second	conjunct	of	line	4.	So,	if	aa=f,	we	get	a	contradiction. We	get	the	same	kind	of	contradiction	if	aa=a	or	aa=b. Assume	aa=ab,	and	universally instantiate	the	second	conjunct	of line	4: ∃y∼(<f,y> is one	of	cc	↔ y is one	of	aa).	Again, there are three	and	only three cases	to	consider	as	possible	instantiations	(we	just	need	one	of	course,	but	let's go	for	all	three):	a,	b	and	f.	Both	a	and	b	make	the	latter	bi-conditional	false	(by making its left-hand side false, but its right-hand side true), and hence its negation true; hence no contradiction. But f makes the bi-conditional false as well	(by	making	its left-hand	side	true,	but	its	right-hand	side	false),	and	hence its negation true; hence no contradiction. Neither is a contradiction found if aa=fb,	aa=fa	or	aa=fab. So,	all	in	all,	a	contradiction	is	found	in	the	cases	aa=a,	aa=b	and	aa=f,	but not	in	any	of	the	cases	aa=fab,	aa=fa,	aa=fb	or	aa=ab.	But	then,	so	far	our	case	cc satisfies	PCT	by	at least	one	of	the	disjuncts in line	6	being	true,	and	hence	the whole	disjunction	being	true.	But	by	CAI,	together	with	the	laws	of	(generalized) identity	and	collapse	of	redundant	plural	listing,19	line	6	collapses20	into: 19	By	'the	laws	of	(generalized)	identity'	I	mean	as	before	the	appropriately generalized	versions	of	both	Leibniz's	Law:	∀α∀β(α=β→(Φ(α)↔Φ(β))),	and Reflexivity:	∀α(α=α),	where	each	one	of	α	and	β	is	a	singular	or	plural	term, independently	of	each	other.	From	those	two	laws,	we	can	derive	Symmetry	and Transitivity	as	expected.	Note	again	that	it's	not	Leibniz's	Law	as	such	that	needs to	be	relativized	or	restricted	because	the	relational	units	are	built	into	the 12 7. aa=a	∨	aa=b	∨	aa=f But,	as	we	saw	above,	the	cases	aa=a,	aa=b	and	aa=f	are	all	and	only	the	cases	in which	we	get	a	contradiction	with	the	second	conjunct	of	line	4,	so,	since	we	get	a contradiction	from	each	one	of	the	disjuncts	of	line	7,	we	also	get	a	contradiction from the entire disjunction, i.e. line 7. We have thus established our counterexample	to	PCT,	within	the	revisionary	language	of	CAI. The acute reader will have noticed that in providing the above counterexample we never appealed to relational predicates. However, by CAI together	with the standard laws of (plural) identity and collapse of redundant plural	listing	again,	line	6	also	collapses	into: 7.* aa=a	∨	aa=b	∨	aa=ab And by CAI alone there is no reason to accept 7 over 7* because ab=f. Interestingly,	there	is	no	contradiction	arising	from	7*,	because,	as	we	have	seen, aa=ab verifies the entire disjunction, i.e. line 7*. So, by collapsing 6 into 7* instead	of	into	7,	we	don't	get	our	counterexample	to	PCT	on	the	basis	of	CAI.	But, of	course,	according	to	CAI,	the	difference	between	7	and	7*	is	a	mere	change	of conceptualization	of one and the same thing.	That is,	we	have	merely changed substitutions	for	Φ.	So,	since	everything	in	Φ	except	α	and	β	is	to	be	constant across	both	sides	of	the	biconditional,	LL	holds	as	expected.	By	collapse	of redundant	plural	listing,	I	mean	that	any	plural	list	containing	the	same	term more	than	once,	collapses	into	a	list	that	contains	that	term	only	once,	e.g.	abcb collapses	into	abc. 20	Because:	fab=fa=fb=f=ab. 13 the	relational	units	hidden	in	the	underlying	predicates	involved	in	the	proof	(e.g. in	the	predicates	'|	|>1',	'is	one	of',	'are	among'	and	'<	,	>').	By	tediously	unpacking definitions based on the official language (cf. Appendix), and filling in the relational	units,	we	can	see	that	CAI	blocks	PCT	relative	to	one	set	of	concepts, namely one according to which we double count the whole in addition to its parts (cf. the third disjunct of line 7), but not relative to another, namely one according	to	which	we	don't	double	count	the	whole	in	addition	to	its	parts	(cf. the	third	disjunct	of	line	7*).	In	fact,	we	can	see	this	almost	directly	from	lines	7 and	7*.	More	generally,	it	can	be	shown	that	by	accepting	CAI,	we	can	accept	PCT just	in	case	we	count	our	ontology	on	the	basis	of	a	partitioning	of	it	into	disjoint members;	that's	the	only	way	to	avoid	double-counting.	Given	CAI,	as	soon	as	we let overlapping members into our ontology, PCT no longer holds due to illegitimate	double	counting,	i.e.	counting	the	same	twice	over.21 In sum, assuming	CAI, there	will be	pluralities such that there is a	map from	its	members	onto	all	its	sub-pluralities,	and	this	is	so	because	by	CAI	those pluralities	are	such	that	some	of	their	members	are	identical	with	some	of	their sub-pluralities	such	that	we	get	that	map.	PCT	thus	holds	only	if	we	either	ignore those	identities	and	double	count	(which	is	ontologically	misleading)	or	we	don't ignore those identities but only consider pluralities with no overlapping members. 21	Arguably,	our	denial	of	PCT	also	amounts	to	a	denial	of	Plural	Comprehension (PC)	as	a	universal	truth,	the	proposition	that	for	any	non-empty	predicate,	there are	some	things	that	are	all	and	only	the	things	that	satisfy	that	predicate.	PC	is formulated	in	terms	of	the	predicate	'is	one	of',	which	is	one	of	the	predicates that,	according	to	CAI,	need	to	be	appropriately	relativized.	Unfortunately,	a	full discussion	of	this	must	wait	for	another	time;	though	see	fn.23.	On	CAI	and	PC, see	Sider	(2014);	though	note	that	Sider's	conclusions	look	very	different,	much less	worrisome,	when	we	invoke	relativized	predicates	as	above. 14 I	take	it	this	much	suffices	to	show	that	if	CAI	is	true,	then	PCT	is	blocked as	a	universal truth.	We	now	turn	to two	examples	of interesting	philosophical upshots	of	this	fact	(presumably	there	are	other	such	examples	as	well). 2.	BLOCKING	ONE	THING	BLOCKS	ANOTHER Hawthorne	and	Uzquiano	(2011)	present	us	with	the following	puzzle.	Assume there	can	be	at	least	two	co-located	point-sized	concrete	objects	in	a	point-sized region of space.22	How	many such co-located points can there be? Given that there	can	be	at	least	two,	any	particular	number	above	two	seems	objectionably ad hoc. For any such particular number, finite or transfinite, the question immediately arises: why not	more? But then, since	we grant at least two, but accept	no	particular	number	above two,	the following two	answers	seem	to	be the	only	viable	options: (P):	at	least	as	many	as	the	alephs (IE):	not	as	many	as	the	alephs,	but	for	each	aleph	there	can	be	at	least	as	many as	that where	the	alephs	is	assumed	to	be	the	entire	series	of	all	the	cardinal	numbers, having the absolute size	Ω, into which all things can be 1-1 mapped, i.e. the 22	By	'can'	I	here	mean	metaphysically	possible;	by	'co-located'	I	mean	exactly	colocated;	by	'point-sized'	I	mean	zero-dimensional	(though	this	is	inessential;	we only	need	that	it	is	mereologically	atomic);	'concrete	object'	I	take	to	be	a primitive,	but	opposed	to	'abstract	object';	I	have	no	idea	what	a	region	of	space is,	nor	what	more	exactly	it	is	to	be	located	in	one.	But	let's	not	quarrel	about	any of	this	here.	I	henceforth	use	'point'	to	mean	zero-dimensional	concrete	object. The	less	heretic	among	us	could	think	of	these	zero-dimensional	objects	as concrete	angels	dancing	on	the	point	of	a	needle,	instead	of	as	concrete	points. 15 alephs are assumed to be that than which nothing larger is or can be.23	P (plenitude) then says that there can	be	as	many	co-located	points as there	are alephs,	while	IE	(indefinite	extensibility)	says	that	is	not	the	case,	but	that	there can	nonetheless	be	indefinitely	many. Hawthorne	and	Uzquiano	(H&U)	presents	two	different	arguments	to	the effect	that	P	is	false,	so,	given	that	P	and	IE	are	the	only	viable	options,	IE	is	true; but	IE	in	turn	contradicts	modal	realism	and	necessitism,	so	modal	realism	and necessitism	must	be	false.24	In	what	follows	I	show	how	to	block	one	(but	not	the other)	of	their	two	arguments	against	modal	realism	and	necessitism	by	virtue	of the	results	from	section	1	above.25,26 We assume both that composition is unrestricted: any plurality xx compose something, and that composition is unique: if xx compose y, then xx compose	nothing	but	y.27	If	xx	compose	y,	we	also	say	that	y is	the fusion	of	xx. 23	I	assume	that	Ω	is	an	amodal	matter	of	metaphysical	necessity	in	the	sense	of the	alephs	not	being	tied	to	any	possible	world,	but	rather	being	"outside"	all possible	worlds	(hence	'amodal'),	but	nonetheless	holding	true	in	all	possible worlds	(hence	'metaphysical	necessity').	Perhaps	contra	H&U,	I	intentionally avoid	thinking	of	them	in	terms	of	set	theory.	I	am	also	not	comfortable	with talking	as	if	Ω	is	a	definite	size,	or	a	mathematical	object	in	its	own	right (because	then	it	seems	something	could	be	larger),	but	H&U	talk	this	way (though	presumably	without	any	particular	commitments),	so	for	present purposes	we	can	and	do	too.	In	any	case,	Ω	is	an	absolute	limit	on	size,	what	so ever. 24	For	modal	realism,	see	Lewis	(1986);	for	necessitism,	see	Williamson	(2013). 25	Their	other	argument	rests	on	wholly	different	(set-theoretical	rather	than mereological)	premises,	and	so	demands	separate	treatment,	and	so	must	be	left for	another	time.	But	just	to	put	my	cards	on	the	table,	I	reject	their	second argument	too	on	the	basis	of	accepting	unrestricted	set-formation,	but	denying unrestricted	plural	comprehension. 26	Note	that	the	argument	is	basically	a	generalization	of	the	Russell-Myhill Paradox.	See	Klement	(1995).	The	general	problem	is	also	lurking	in	Lewis (1991),	Rosen	(1995)	and	Nolan	(1996). 27	Arguably,	both	these	assumptions	follow	from	CAI.	See	Sider	(2007)	and	Bohn (2014).	McDaniel	(2010)	and	Cameron	(2012)	argue	that	the	first	assumption does	not	thus	follow;	Bohn	(2014)	replies. 16 We	say that	a fusion is	based	on a	plurality iff it is the fusion	of	one	of its	subpluralities;	and	we	say	that	a	plurality	is	disperse	iff	no	two	of	its	sub-pluralities have	one	and	the	same	fusion.	For	example,	a	plurality	of	mereological	atoms	is disperse;	so	is	the	plurality	of	all	and	only	the	cats.	A	plurality	of	a	single	cat	and its left	and	right	halves	is	not	disperse,	since	the	plurality	of	the	cat	and	its left half	has	one	and	the	same	fusion	as	that	of	the	cat	and	its	right	half. From these assumptions and definitions, we get what H&U calls the mereological	result: (MR):	there	are	more	fusions	based	on	a	disperse	plurality	of	two	or	more	things than	there	are	members	of	that	disperse	plurality Proof:	consider	an	arbitrary	disperse	plurality	xx	of	two	or	more	things.	By PCT,	there	are	more	sub-pluralities	of	xx	than	there	are	members	of	xx;	but by	UC, each of these sub-pluralities has a fusion; so, since xx is disperse, there	must	be	more fusions	based	on	xx	than	there	are	members	of	xx	as well.	Q.E.D. Based	on	MR,	H&U	presents	the	following	argument	against	P: (A):	assume	P, i.e. that there	can	be	at least	as	many	co-located	points	as there are alephs. By MR, there are strictly more fusions based on those co-located points than there	are	co-located	points.	But then there	are	strictly	more things altogether	than	co-located	points,	contradicting	P	or	the	assumption	that	all	the alephs	is	that	than	which	nothing	larger	is	or	can	be. 17 By	virtue	of	A	(and	the	assumption	that	all	the	alephs	is	that	than	which	nothing larger is or can	be), P is false, leaving	us	with IE as the correct answer to our initial question (given that those two were the only viable options). But, according	to	H&U,	IE	contradicts	modal	realism	and	necessitism. It suffices for present purposes to say that	modal realism is the view according to which all possible worlds and objects exist on a par with (or as concretely as) the actual world and objects, though the possible worlds are presumably	spatiotemporally	and	causally isolated from	each	other.28	But if	we assume such modal realism together with unrestricted quantification over all possible	worlds	and	objects,	i.e.	over	the	entire	pluriverse,	and	the	result	that	IE is	the	correct	answer	to	our	initial	question,	then	a	version	of	argument	A	arises all	over	again.	The	argument rests	on the following	observation:	by IE, for	any aleph,	there	is	a	possible	world	having	that	many	co-located	points,	so	by	modal realism,	there	will	be	as	many	such	points	across	the	entire	pluriverse	as	there	are alephs	(after	all,	since	there is	a	world	for	any	arbitrary	aleph	many	co-located points,	there	are	as	many	worlds	as	alephs!). (A*):	assume	IE	and	modal	realism.	Then,	by	the	above	observation,	there	are	at least as many points as there are alephs across the entire pluriverse. By MR, there	are	strictly	more	fusions	based	on	those	points	than	the	points	themselves, which means there are strictly more things altogether across the entire pluriverse than alephs, which is impossible on pain of contradicting IE or the assumption	that	all	the	alephs	is	that	than	which	nothing	larger	is	or	can	be. 28	For	more	details,	see	Lewis	(1986). 18 It suffices for present purposes to say that necessitism is the view that necessarily,	everything	exists	necessarily.	In	terms	of	possible	worlds,	that	is	to say that for any possible world, everything in it exists in any other possible world as	well (though it	might switch	between	being abstract and concrete).29 But	if	we	assume	such	necessitism	together	with	unrestricted	quantification	over all	existents,	and	the	result	that	IE	is	the	correct	answer	to	our	initial	question, then	a	version	of argument	A	arises	all over	again.	The	argument, rests	on the following	observation:	by	IE,	for	any	aleph,	there	is	a	possible	world	having	that many	co-located	points,	so	by	necessitism,	there	will	be	as	many	such	(concrete or	abstract)	points	in	actuality	as	there	are	alephs	(after	all,	since	there	is	a	world for any arbitrary aleph	many co-located points, and all those points in all the worlds	must	also	be	something	actual,	there	are	actually	as	many	such	(abstract or	concrete)	points	as	alephs!). (A**):	assume	IE	and	necessitism.	Then,	by	the	above	observation, there	are	as many points in actuality as there are alephs. By MR, there are strictly more fusions	based	on	those	points	than	the	points	themselves,	which	means	there	are strictly	more things	altogether in	actuality than	alephs,	which is impossible	on pain	of	contradicting	IE	or	the	assumption	that	all	the	alephs	is	that	than	which nothing	larger	is	or	can	be. So,	if	everything	is	correct	so	far,	both	modal	realism	and	necessitism	are	false. But, of course, given	CAI, not everything is correct so far.	As	we saw in section	1, if CAI is true, then	PCT is false; but as	we	have seen in this section, 29	For	more	details,	see	Williamson	(2013). 19 H&U's	arguments	against	modal	realism	and	necessitism	both	rest	on	MR,	which in turn rests on PCT; so, if CAI is true, then H&U's arguments are blocked by virtue	of	CAI	blocking	PCT,	which	in	turn	blocks	MR.	(Note	that	CAI	thus	not	only blocks	A*	and	A**,	but	A	as	well;	so	P	might	very	well	be	true	after	all.)	CAI	thus provides a way of blocking H&U's arguments against modal realism and necessitism. But then	we in effect have an abductive argument for CAI, given either modal realism	or	necessitism.	Either composition is identity	or it is	not. If it is not, then H&U's argument presumably goes through as it is intended, since counting overlapping things don't then amount to double counting; after all, overlapping	things	are	then	not	in	any	way	identical	things,	and	hence	they	are distinct,	and	hence	ought	to	be	counted	as	distinct.	If	so,	both	modal	realism	and necessitism	are	false	on	pain	of	paradox,	as	argued	by	H&U.	But	if	composition	is (an	instance	of	generalized)	identity,	and	we	understand	that	along	the lines	of CAI	as	articulated	in	section	1	above	(though	presumably	there	are	other	ways	of understanding	it	too,	which	gives	the	same	result),	then,	as	we	have	seen,	PCT	is blocked,	and	hence	MR	is	blocked,	and	hence	H&U's	arguments	are	blocked.	So, given	modal	realism	or	necessitism	(and	the	two	assumptions	of	the	alephs	being that	than	which	nothing	larger	is	or	can	be,	and	unrestricted	composition),	one should	accept	CAI	on	pain	of	paradox! Of	course,	there	are	many	possible	replies	to	such	an	abductive	argument, but	my	point	here	is	only	that	we	now	at	least	have	a	debate	up	and	running	due 20 to	the	fact	shown	in	section	1,	namely	that	given	CAI,	PCT	fails	to	be	a	universal truth.	In	general,	PCT	can	no	longer	be	uncritically	appealed	to.30 Appendix:	a	sketch	of	the	language	of	CAI Below	is	a	sketch	of	the	language	of	CAI.	I	also	provide	an	intuitive	translation-function	from	an ordinary	plural	first-order	language	into	the	language	of	CAI,	to	facilitate	understanding. Alphabet: Constants:	∼,	∧,	∃ Non-Logical	Predicates:	Fni Logical	Predicates:	= Object-variables:	xi,	xxi Object-constants:	ai,	aai Concept-variables:	Yi for any 0<i, where n indicates the number of places of F. We call the object-variables and constants (i.e. excluding concept-variables), terms.	We also have complex terms: if	α and	β are terms,	then	so	is:	αβ.	Complex	terms	are	associative	and	commutative. Well-formed	formulas: Atomic:	(i)	Π2nα1,...,αn,Y1,...,Yn	is	an	atomic	wff,	where	Π2n	is	a	2n-place	non-logical	predicate,	αi	is a	term,	and	Yi	is	a	concept-variable	indexed	to	αi;	and	(ii)	α=β	is	an	atomic	wff,	where	each	one	of α	and	β	can	be	any	term	(simple	or	complex),	independently	of	each	other. Non-Atomic:	(iii)	∼Φ; (iv)	Φ∧Ψ; and (v)	∃αΦ; if	Φ and	Ψ are	wffs,	and	α is	a singular	or	plural object-variable.	(The	other	logical	connectives	and	quantifiers	are	defined	in	the	usual	way.) Truth-conditions: Let d be our denotation-function on predicates and terms, d* be our denotation-function on concept-variables,	and	let	v	be	our	evaluation-function	on	wffs.	Then: Atomic:	(i)	Π2nα1,...,αn,Y1,...,Yn	is	true	iff	dΠ2n	is	instantiated	by	<dα1,...,dαn,d*Y1,...,d*Yn>;	(ii)	α=β is	true	iff	dα	is	identical	with	dβ. Non-Atomic:	(iii)	∼Φ	is	true	iff	vΦ	is	not	true;	(iv)	Φ∧Ψ	is	true	iff	vΦ	is	true	and	vΨ	is	true;	and	(v) ∃αΦ	is	true	iff	Φ	is	true	of	some	dα. Translation-function	from	a	more	standard	plural	first-order	language: Tr(Πnα1,...,αn)	=	Π2nα1,...,αn,Y1,...,Yn Tr(α=β)	=	α=β,	where	=	is	hybrid	n-m	identity	instead	of	the	usual	n-n	identity Tr(∼Φ)	=	∼Tr(Φ) Tr(Φ∧Ψ)	=	Tr(Φ)∧Tr(Ψ) Tr(∃αΦ)	=	∃αTr(Φ) where, intuitively,	Π2n holds of <α1,...,αn> relative to Yi holding of αi. (Note that I have been sloppy by, among other things, saying that open	wffs are true, but if you are still reading you probably get the point.) In particular, there are, according to CAI, cases where	Π2n holds of <α1,...,αn> just	in	case	dαi	has	a	unique type	of "decomposition",	or "division". 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