forthcoming	in	Thought Speaks's	Reduction	of	Propositions	to	Properties:	a	Benacerraf Problem T.	Scott	Dixon Cody	Gilmore Ashoka	University University	of	California,	Davis ts.dixon@ashoka.edu.in gilmore@ucdavis.edu 1.	Introduction Jeff Speaks (2014) has recently defended the view that propositions are properties: the proposition that grass is green is the property being such that grass is green, the proposition	that	snow	is	white is	being	such	that	snow	is	white,	and	so	on.	We	argue	that there is no reason to prefer Speaks's theory to analogous but competing theories that identify propositions with, say, 2-adic or 17-adic relations of the relevant sort, and we conclude that none of these theories is true. This style of argument derives from Benacerraf's	(1965)	objection	to	the	view	that	numbers	are	sets,	and	it	has	been	deployed by,	e.g.,	Bealer	(1993),	Moore	(1999),	Moltmann	(2013),	and	King	(2007)	as	an	objection	to the	view	that	propositions	are	n-tuples, sets	of	worlds,	or functions from	worlds to truth values,	and	by	Caplan	and	Tillman	(2013)	as	an	objection	to	King's	view	that	propositions are	facts	of	a	special	sort.1	2	We	offer	our	argument	as	an	objection	to	any	Speaks-style	view according	to	which	propositions	are	unsaturated	(non-0-adic)	relations. 1 Moore (1999) claims that arbitrariness considerations count against every reductionist theory of propositions,	but	he	gives	little	support	for	the	full	generalization,	which	is	too	strong	on	its	face.	One	might, e.g., have independent reason to think that obtaining and non-obtaining states of affairs are abundant, hyperintensionally	individuated,	sui	generis	abstract	entities	(Plantinga	(1976)),	and	that	states	of	affairs	are the	best	candidates	to	which	propositions	might	be	reduced.	If	one	then	decided	to	identify	propositions	with states	of	affairs	(contra	Plantinga	(1976:	145)),	it	would	not	be	arbitrary	to	identify	the	proposition	that	grass is green with grass's being green, and so on (see Chisholm (1970) and Richard (2014)). So we hold that reductionist theories of propositions should typically be evaluated individually, and that it often takes detailed	investigation	to	find	out	whether	a	given	reduction	suffers	from	a	Benacerraf	problem. 2 Jubien (2001) argues that arbitrariness problems apply even to theories like those of King (2007), who replies (2007: 127-163), and Speaks (2014). As it applies to a Speaks-style theory, however, Jubien's argument depends on the assumption that propositions are representational, which Speaks (2014: 220) denies.	Our	argument	makes	no	such	assumption. 2 2.	Speaks's	Theory We	begin	by	comparing	Speaks's	theory	to	a	more	familiar	view,	which	we	call	PPR: (1) propositions, properties, and relations are (a) abundant and (b) hyperintensional:	there	are	necessarily	equivalent	propositions,	properties,	and relations	that	are	not	identical; (2) properties	are	1-adic	relations;	and (3) propositions	are	0-adic	relations.	3 It is	notoriously	hard	to	give	a	precise	characterization	of	the	abundance	thesis,	(1a),	and we	won't	attempt	it.4	For	our	purposes	it	will	be	enough	to	note	that	if	(1a)	is	true,	then (1a*)	almost5	every	open	sentence	expresses	a	relation.6 Speaks	is	favorably	disposed	towards	(1)	and	(2),7	but	he	rejects	(3)	in	favor	of (4) Propositions	are	properties;	they	are	1-adic,	not	0-adic,	relations. According	to	Speaks,	propositions	are	properties.	More	specifically:	(i)	not	every	property is	a	proposition.	The	property	being	an	x	such	that	x	is	red	is	not	a	proposition;	nor	is	being an	x	such	that	x	is	identical	with	x.	But	(ii)	every	proposition	is	a	property.	The	proposition 3	PPR	is	discussed	by	Quine	(1960:	165)	and	endorsed	by	Bealer	(1982:	1),	Zalta	(1983:	61),	Menzel	(1993), van Inwagen (2004: 131), and	Gilmore (2014). See	Dixon (Forthcoming: sect. 4) for a development of the analogies	between	properties,	relations,	and	instantiation,	on	the	one	hand,	and	propositions	and	truth	on	the other. 4	See	Orilia	and	Swoyer	(2016:	sections	5.2	and	8). 5	We	say	'almost	every'	to	avoid	saddling	PPRists	with	the	claim	that	'x	does	not	instantiate	itself'	expresses	a property. 6	We	assume	that	(i) if	an	open	sentence	contains	exactly	n free	variables	and	expresses	a	relation	of	some fixed	adicity,	it	expresses	an	n-adic	relation	and	that	(ii)	necessarily,	no	relation	is	both	mand	n-adic,	where m≠n This leaves open the hypothesis that some relations have no fixed adicity. On the incompatibility of different	adicities,	see	Gilmore	(2013). 7 Speaks needs abundant, hyperintensional properties, and it would be ad hoc not to endorse a parallel account	of	other	relations.	Speaks	expresses	no	reservations	about	(1)	or	(2)	but	does	not	explicitly	endorse them.	He	singles	out	van	Inwagen	(2004),	a	proponent	of	PPR,	as	providing	a	background	theory	of	properties that meets his needs –	modulo van Inwagen's commitment to (3). Regarding (3), Speaks writes, 'I don't understand	what	a	0-place	property	could	be'	(2014:	90,	note	36). 3 that	snow	is	white	=	being	an	x	such	that	snow	is	white;	the	proposition	that	Obama	is	male =	being	an	x	such	that	Obama	is	male;	and	so	on. Can	we	explicate	the	'and	so	on'?	It's	not	clear	that	we	must	in	order	to	evaluate	the theory.	Speaks	does	not	provide	a	fully	general,	precise	formulation	of	the	reductive	theory, but	he	argues	for	it	nonetheless,	and	we	will	not	object	to	this	policy.	One	can	take	a	step toward	increased	generality	by	noting	that if	Speaks's	theory,	call it	T1, is	true,	then	so	is the	following	meta-linguistic	thesis: M1 Every	instance	of	schema	S1	is	true. S1 the	proposition	that	φ	=	the	1-adic	relation	being	an	x	such	that	φ. Granted, M1 does not capture the full strength of T1, since it says nothing about propositions	that	are	not	expressed	by	any	sentence	of	English.	But, in	practice,	everyone who	endorses	M1	will	probably	also	endorse	T1.	So	M1	is	a	useful	proxy	for	T1. Here	is	a	final	exegetical	point.	The	properties	to	which	Speaks	reduces	propositions are	'necessarily	all-or-nothing'	properties.	In	the	idiom	of	possible	worlds:	for	any	possible world	w	and	any	proposition	p,	either	(i)	at	w,	everything	instantiates	p,	in	which	case	p	is true at w or (ii) at w, nothing instantiates p, in which case p is false at w. The true propositions at a world are the propositions that are instantiated by everything at that world,	while	the	false	propositions	at	a	world	are	those	that	aren't	instantiated	by	anything at	that	world.	But	some	necessarily	all-or-nothing	properties	–	e.g.,	being	self-identical	–	are not	propositions	(Speaks	(2014:	90)). 3.	The	Problem For	each	proposition	p,	Speaks	finds	some	1-adic	relation	r1	and	claims	that	p	=	r1.	But	why not	instead	find	some	2-adic	relation	r2	and	claim	that	p	=	r2?	Or	some	17-adic	relation	r17 and	claim	that	p	=	r17?	8 Let's look at the case of 2-adic relations. Let T2 be the theory that identifies propositions	with	certain	2-adic	relations	and	is	otherwise	parallel	to	T1.	According	to	T2, 8	A	different	Benacerraf	problem	for	Speaks:	why	not	identify	the	proposition	that	grass	is	green	with	the	1adic,	plural	property	being	some	xx	such	that	grass	is	green?	On	plural	properties,	see	Yi	(1999). 4 the	proposition	that	grass	is	green	=	the	2-adic	relation	being	an	x	and	a	y	such	that	grass	is green;	the	proposition	that	Obama	is	male	=	being	an	x	and	a	y	such	that	Obama	is	male;	and so	on.	Like	T1,	T2	is	hard	to	state	precisely,	but	we	can	take	a	step	in	that	direction	with M2 Every	instance	of	schema	S2	is	true. S2 the	proposition that	φ = the	2-adic relation	being	an x and	a y such that	φ. M2	fails	to	capture	the	full	strength	of	T2	for	the	same	reason	that	M1	fails	to	capture	the full strength of T1. But both	meta-linguistic theses can serve as useful proxies for their respective	theories. The	relations	to	which	T2	reduces	propositions	are	necessarily	all-or-nothing	2-adic relations:	each	such	relation	r2	is	such	that,	necessarily,	either	(i)	for	any	x	and	any	y,	r2	is instantiated	by	x	and	y,	in	that	order,	or	(ii)	for	any	x	and	any	y,	r2	is	not	instantiated	by	x and y, in that order. The true propositions at a world, according to this view, are the propositions	that	everything	bears	to	everything	at	that	world,	while	the	false	propositions are	those	that	nothing	bears	to	anything	at	that	world.	T2	says	that	all	propositions	are	2adic, necessarily all-or-nothing relations, but it does not say that all such relations are propositions. We	believe	that	the	case	to	be	made	for	T1	can	be	made	with	equal	plausibility	for T2. For example, each theory identifies propositions	with entities in	which	we allegedly have independent reason to believe. Speaks (2014: 77, note 12) assumes that we have independent reason to	believe in	abundant,	hyperintensional	properties,	on the	model	of van Inwagen (2004). If	he is right, then	presumably	we	also	have independent reason to believe	in	abundant,	hyperintensional	2-adic	relations,	on	the	same	model. Further, if one finds it antecedently plausible that the proposition that φ is a structured	complex	having	certain	entities,	ee,	as	constituents,	then	for	any	positive	integer n, one should also find it antecedently	plausible that the	n-adic relation	being	an x1	... xn such	that	φ is	a	similarly	structured	complex	having	ee	as	constituents.	So	neither	T1	nor 5 T2 forces us to depart	much from antecedently plausible views (whatever they	may be) about	the	constituents	and	internal	structure	of	propositions.9 Finally,	Speaks	says	that Propositions are properties which are true iff they are instantiated. Propositions are true with respect to	a	world	w iff,	were	w	actual, that	property	would	be instantiated. . . .	Given this	view	of truth	at	a	world,	thinking	of	propositions	as	properties	does	not	seem	to	require	any	serious	revision in the way that we think about entailment relations between propositions, or semantics more generally.	Propositions	are	necessary	iff	they	are	true	with	respect	to	every	possible	world; just	so, on	the	present	account,	propositions	are	necessary	iff	the	properties	which	they	are,	are	instantiated in	every	possible	world.	One	proposition	F	would	entail	another	proposition	G	iff	any	world	in	which F	is	instantiated	is	also	a	world	in	which	G	is	instantiated	(2014:	76-77). Likewise, T2 does not demand much revision in our views about entailment between propositions. It is also true according to	T2 that "propositions are necessary iff they are true	with	respect	to	every	possible	world".	According	to	T2,	propositions	are	necessary	iff the relations which they are, are "born by everything to everything in every possible world".	T2	also	guarantees	that	a	"proposition	F	would	entail	another	proposition	G	iff	any world	in	which	[something	bears	F	to	something]	is	also	a	world	in	which	[something	bears G	to	something]".	It	appears,	then,	that	T2	shares	these	crucial	virtues	with	T1. In	addition,	it	seems	that	for	any	obvious	objection	one	might	raise	to	T2,	there	will be	an	analogous	and	equally	forceful	objection	to	T1. Two	examples	should	suffice	to	make this	plausible. First, T2 faces a demarcation problem. Not all 2-adic, necessarily all-or-nothing relations	are	propositions.	For	example, the	relation	being	an	x	and	a	y	such	that	x=x	and y=y	is	not	a	proposition.	How,	then,	can	we	fill	in	the	blank	in	'necessarily,	x	is	a	proposition if	and	only	if	______'	so	as	to	make	the	resulting	sentence	both	true	and	informative?	Without an	answer,	T2	may	seem	incomplete.	But	as	Speaks	points	out	(2014:	89	–	90),	T1	faces	an analogous	demarcation	problem.	Any	solution	that	works	for	T1	should	have	an	analogue that	works	for	T2. Second, given that the proposition that 2>3 and the proposition that 1+1=5 both exist	and	are	not	identical,	T2	is	committed	to	the	existence	of	the	relations,	being	an	x	and a	y	such	that	2>3	and	being	an	x	and	a	y	such	that	1+1=5,	that	are	necessarily	uninstantiated 9	Speaks	does	not	make	this	point,	but	we	assume	that	he	would	not	object. 6 and hyperintensionally individuated. But analogous points apply to T1.10 Simply delete each	occurrence	of	'and	a	y'	in	the	italicized	expressions,	and	one	will	have	names	of	1-adic relations	to	which	T1	is	committed	and	which	share	the	potentially	problematic features. So	T2	is	no	worse	off	than	T1	in	these	respects. 4.	Objections	and	Replies Objection 1. What makes T1 preferable to T2 is that T1 attributes a lesser adicity to propositions	than	T2	does.	One	'adicity	minimization'	principle	that	supports	this	claim	is: AMp Theories according to	which propositions are	m-adic should, pro tanto, be preferred to theories according to	which propositions are n-adic, provided that	m	is	a	positive	natural	number	less	than	n. AMp	puts	T1	at	an	advantage	to	T2.	Adicity	aside,	T1	is	at	least	as	good	as	T2.	So,	given	AMp, it follows	that	T1 is	preferable	overall to	T2. In	support	of	AMp, (i) it is intuitive,	and	(ii) considerations	of	simplicity favor it: theories	according to	which	propositions	are	m-adic are pro tanto simpler than, hence pro tanto preferable to, theories according to which propositions	are	n-adic,	where	m	is	a	positive	natural	number	less	than	n. We	address	(i)	and	(ii)	separately. Reply	to	(i).	The	objector	takes	AMp	to	be	intuitive.	Perhaps	an	appeal	to	intuition	is out	of	place	in	a	case	like	this, in	which	the	subject	matter	(the	adicity	of	propositions)	is esoteric and theoretical. But suppose, with Speaks (2014: 72), that intuition does carry weight	even	on	such	topics.	Then	we	grant	that	some	principle	in	the	neighborhood	of	AMp is	intuitive.	But	we	deny	that	AMp	itself	is	intuitive. Consider	three	theories.	The	first,	T0,	says	that	propositions	are	0-adic	relations,	in accordance	with	component	(3)	of	PPR.	The	second,	T17,	says	that	propositions	are	17-adic relations of the appropriate sort. The third, T19, says that propositions are 19-adic relations of that sort. How should we rank these theories? AMp is silent as to how T0 10	On	necessarily	uninstantiated	properties,	see	(Speaks	2014:	77).	On	hyperintensional	properties,	see (2014:	89). 7 compares to T17 or T19, but it says that T17 is preferable to T19. Both facts deserve comment. First, it is plausible that, so far as considerations of adicity are concerned, T0 is preferable to	T17	or	T19. (More	on this below.) So if	AMp is intuitive, then so is a	more general	principle,	call	it	AM,	which	results	from	deleting	the	word	'positive'	in	AMp.	Second, it	is	not	intuitive	that	T17	is	preferable	to	T19.	As	we	see	it,	intuition	is	either	agnostic	on how to rank T17 and T19, or else it ranks them as equal, and equally bad. This	weighs against	the	intuitiveness	of	AMp	or	AM. If	any	principle	like	AMp	is	intuitive,	it	is	0-adicity	or	Bust: 0B Theories	according	to	which	propositions	are	0-adic,	fully	saturated	entities should,	pro tanto, be	preferred to theories	according to	which	propositions are	not	fully	saturated	entities,	i.e.,	are	n-adic,	for	some	n>0. 0B	harmonizes	with	the	view	that	T0	is	preferable	to	T17	and	T19,	and	that	T17	and	T19 are,	if	anything,	equally	implausible.	Further,	lest	0B	be	thought	to	beg	the	question	against Speaks,	we	can	argue	for	it	using	premises	acceptable	to	all	parties.11	Consider	these	four expressions: (a) ___	is	farther	from	---than	.	.	.	is	from	*** (b) ___	is	farther	from	---than	.	.	.	is	from	South	Bend (c) ___	is	farther	from	---than	Princeton	is	from	South	Bend (d) ___	is	farther	from	Delhi	than	Princeton	is	from	South	Bend They	express	4-adic,	3-adic,	2-adic,	and	1-adic	relations,	respectively.	We	assume	that	this is	granted	by	all	parties.	But	it	provides	inductive	evidence	that (e) Sacramento	is	farther	from	Delhi	than	Princeton	is	from	South	Bend 11	Moreover, 0B does not entail that T1 is false, only that it has a vice. Speaks concedes that T1 has vices (2014:	78;	83,	note	24;	90,	note	36).	So	he	seems	amenable	in	principle	to	0B. 8 expresses	a	0-adic	relation.	All	parties	also	grant	that	(e)	expresses	a	proposition.	So,	unless (e)	expresses	two	different	entities,12 it follows	that	at least	some	propositions	are	0-adic relations.	Considerations	of	uniformity	then	favor	the	idea	that	all	propositions	are	0-adic relations.	Call this the	plugging	argument.	Speaks	may find	some	step in the	argument to resist, but only,	we think, at some cost.	What the argument shows, then, is that there is theoretical	pressure	toward	0B. Might	one	hold,	with	AM,	that	while	it	would	be	ideal	to	treat	propositions	as	0-adic, it	would	still	be	better	to	treat	them	as	1-adic	than	as	2-adic	or	17-adic?	This	position	is	just as	unintuitive	as the	analogous	view that theories	according to	which	horses	have	n legs are	pro	tanto	preferable	to	theories	according	to	which	horses	have	m	legs,	provided	that	|4 –	n|	is	less	than	|4	–	m|. The	fact	that	17	is	closer	to	4	than	19	is	doesn't	make	the	view	that horses	have	17	legs	pro	tanto	preferable	to	the	view	that	horses	have	19	legs.	We	conclude that	it	is	0B,	not	AM	or	AMp,	that	is	intuitive	in	its	own	right	and	supported	by	nearby	data (via the plugging argument). And rather than breaking the tie between T2 and T1, 0B merely	puts	both	at	a	disadvantage	to	T0. Reply	to	(ii).	The	objector	claims	that	T1	is	simpler	than	T2.	There	are	several	things one	might	mean	by	this. (a)	One	might	mean	that	T1	is	more	ontologically	parsimonious	than	T2,	i.e.,	that	T1 posits fewer entities (or types of entity) than T2. But that is false. Strictly speaking, T2 posits	the	2-adic	relations	that	it identifies	with	propositions,	and	nothing	else, just	as	T1 posits the	1-adic	relations that it identifies	with	propositions,	and	nothing	else.13	So	both are	committed	to	an	abundance	of	entities	of	a	single	type.	Granted,	T2	is	plausible	only	to the	extent	that	there	is	independent	reason	to	believe	in	the	2-adic	relations	it	posits;	and any	such	reason	will also support the	1-adic relations	posited	by	T1.	But the	converse is equally true. T1 is plausible only to those who would have embraced an extreme abundance	of	hyperintensional	1-adic	relations	anyway,	even	if	these	relations	hadn't	been 12 If a 0-adic relation is expressed by (e), then, just as the 1-adic relation expressed by (d) is true of Sacramento,	the	0-adic	relation	expressed	by	(e)	is	true	simpliciter,	which	suggests	that	it	is	a	proposition. 13	Given that	T2	entails that the	2-adic relation	being	an	x	and	a	y such that	grass is	green exists,	doesn't it follow	that	T2	also	entails	that	the	property	being	an	x	such	that	grass	is	green	exists	and	is	1-adic?	We	doubt it. One	might reduce propositions to 2-adic relations via T2 and reduce properties to 2-adic relations, by saying	that	the	property	being	an	x	such	that	x	is	red	=	being	a	y	and	an	x	such	that	x	is	red,	and	so	on.	On	this (odd)	view,	propositions	and	properties	are	2-adic	relations	of	different	sorts. 9 used	in	a	reductive	theory	of	propositions.	But,	to	our	knowledge,	no	one	has	ever	held	that there	are	1-adic	relations	such	as	being	an	x	such	that	Obama	is	male	but	no	2-adic	relations such	as	being	an	x	and	a	y	such	that	Obama	is	male.	Such	a	view	would	be	highly	ad	hoc. So	T1	and	T2	are	committed	to	the	same	number	of	entities	and	to	the	same	number of types	of	entities.	And for	either theory to	be remotely	plausible, the	same	background ontology	of	properties	and	relations	is	required. (b)	One	might	instead	mean	that	T1	is	more	ideologically	parsimonious	than	T2,	i.e., that	T2 invokes	more	bits	of	primitive ideology than	T1.	This is	also	doubtful.	What	new primitive ideology	must	T2	use over and above that used	by	T1?	Both employ the same term-forming operator 'being an x1 ... xn such that	φ '. The mere fact that T2 uses two variables where T1 uses only one makes no difference with respect to the primitive ideological	machinery	of	the	two	theories.	And	even	if	it	did,	the	T1-ist	will	have	to	employ two	variables	when	forming	terms	that	refer	to	2-adic	relations. It's	just	that,	on	T1,	none of	these	relations	will	be	propositions. (c) Finally, one	might	mean that	T1 is shorter than	T2.	Consider some instance	of schema	S1	(e.g., 'the	proposition	that	grass	is	green	=	being	an	x	such	that	grass	is	green') and	the	corresponding	instance	of	schema	S2	('the	proposition	that	grass	is	green	=	being an x and a y such that grass is green'). The former sentence is shorter than the latter. Further,	consider	the	definitions	of	truth	at	a	world,	and	of	entailment,	associated	with	T1. These	are	shorter	than	the	corresponding	definitions	associated	with	T2. We	grant all this.	But this sort	of	brevity is	not	often cited	as a theoretical virtue. Moreover, if it	were	such	a	virtue,	Benacerraf's	original	problem	would	not	have	had the impact it	did.	Benacerraf argued that there is	no reason to identify the	ordinals	with the von	Neumann	ordinals	rather	than	the	Zermelo	ordinals.	Yet,	the	Zermelo	definitions	of	the numbers	(and	any	definitions	given	in	terms	of	them)	are	shorter	than	the	corresponding von	Neumann	definitions.	To	look	at	a	representative	example,	the	competing	definitions	of the	number	two	are: Zermelo: 2	=	{{∅}} von	Neumann: 2	=	{∅,	{∅}} 10 According to the Zermelo definition, 2 has only one member, which is {∅}; whereas, according the von Neumann definition, 2 has two members, {∅} and	∅. This makes it plausible that the Zermelo theory is shorter than the von	Neumann theory, in	whatever sense T1 is shorter than T2. But, to our knowledge, this has never been seriously considered	as	a	way	to	solve	the	Benacerraf	problem. This,	we	think,	is	because	brevity	of this	sort	should	not	play	a	significant	role	in	theory	choice.	Accordingly,	it	cannot	solve	our problem	for	Speaks's	theory.14 Objection	2.	According to	T1, the	proposition that	grass is	green	= being	an	x such that	grass	is	green.	According	to	T2,	that	proposition	=	being	an	x	and	a	y	such	that	grass	is green. Thus, T2 posits an additional argument place in the given proposition.	Moreover, that additional argument place does no work: it brings no advantages beyond those associated	with	the	single	argument	place	posited	by	T1.	Since	every	such	argument	place is	a	cost,	T2	is	costlier	all	things	considered	than	T1. Reply. We want to make two points. (i) Neither T1 nor T2 is stated in terms of argument	places.	As	far	as	T1	or	T2	go,	one	is	free	to	deny	that	there	are	argument	places. One is free, e.g., to say that loving is 2-adic but has no argument places. (Though see Gilmore:	2013	for	critical	discussion	of	various	'slot-free'	treatments	of	adicity.) In response, the objector might restate Objection 2 as follows: 'T2 increases the adicity	of	propositions, in comparison to	T1,	without	yielding	any corresponding	benefit. Since	theories	according	to	which	propositions	are	m-adic	should,	pro	tanto,	be	preferred to	theories	according	to	which	propositions	are	n-adic,	provided	that	m	is	a	positive	natural number less than	n,	T1 is	preferable	overall to	T2.'	But this is just	Objection	1,	which	we have	already	addressed. (ii)	Even	if	T1	and	T2	were	both	committed	to	argument	places,	Objection	2	would still fail, for familiar reasons. Each theory is plausible only against the backdrop of 14	A	fourth	version	of	the	simplicity	objection	is	that	T1	is	preferable	to	T2	because	T1	identifies	propositions with	entities	that	have	a lower	adicity,	and	hence	are	simpler,	than	those	associated	with	T2.	We	reply,	first, that	argument	is	needed	for	the	claim	that	lower	adicity	entities	are,	ceteris	paribus,	simpler	entities	in	some sense	relevant	to	theory	choice.	(If	n	has	fewer	prime	factors	than	m,	does	that	contribute	to	n's	being	simpler than	m?)	Second,	an	analogous	simplicity	objection	applies	just	as	forcefully	to	Benacerraf's	original	problem about	numbers	and	sets,	so	our	new	problem	for	Speaks	is,	at	worst,	on	par	with	the	original. 11 abundant	realism	about	n-adic	relations,	for	each	positive	natural	number	n.	Together	with this	backdrop,	T1	is	committed	to	the	same	number	of	argument	places	as	is	T2.	True,	the relations that T1 calls propositions have fewer argument places in them than do the relations	that	T2	calls	propositions,	but	both	overall	packages	have	the	same	ontology.	So the	'additional'	argument	places	posited	by	T2	are	not	'additional'	in	any	sense	that	harms T2 vis-à-vis T1. (Moreover, the finite Zermelo ordinals have fewer members than the corresponding finite von Neumann ordinals, but this doesn't solve Benacerraf's original problem.) 5.	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