Existential	Instantiation, Arbitrary	Reference	and	Supposition Wylie	Breckenridge Existential instantiation is a rule of inference that allows us infer, from the proposition	that	there	are	some	p	things,	the	proposition	that	a	is	a	p	thing.	What role does 'a' play here? According to one account, recently defended by Breckenridge	and	Magidor,	we	use	'a'	to	refer	to	a	p	thing.	I	argue	that	this	cannot be	right. I	propose	an	alternative	account,	according	to	which	we	use	'a'	to	refer to	a	supposedly	p	thing. 1.	Existential	instantiation Suppose	that	p	and	q	are	properties.	Here	is	a	valid	argument: 1. There	are	some	p	things 2. All	p	things	are	q	things Therefore, C. There	are	some	q	things If its validity is not immediately apparent then we can make it more apparent by adding	some	intermediate	steps,	as	follows: 1. There	are	some	p	things	(Premise) 2. All	p	things	are	q	things	(Premise) 3. a	is	a	p	thing	(From	1,	by	existential	instantiation) 4. a	is	a	q	thing	(From	2	and	3,	by	universal	instantiation) C. There	are	some	q	things	(From	4,	by	existential	generalisation) Line	3	is	an	application	of	existential	instantiation.	When	presenting	this	argument	we might	more	naturally	say,	"Let	a	be	p	thing",	but	it	is	standardly	presented	in	writing	as "a	is	a	p	thing". 2.	The	role	of	'a'	and	the	p	thing	account What role does 'a' play in line 3, and then again in line 4? Although existential instantiation	is	discussed	in	standard	textbooks	on	first-order	logic,	typically	not	much is said about the role of 'a', other than that it 'stands for an arbitrary p thing' (or something	like	that).	Proving	that	the	rules	involving	the	introduction	and	elimination of	terms	such	as	'a'	are	sound	and	complete	tells	us	that	by	following	these	rules	we can	prove	exactly	what	we	should	be	able	to	prove,	but	it	does	not	tell	us	much	about what	'a'	means.	This	turns	out	to	be	complicated.	Various	accounts	have	been	offered. According	to	some	accounts	we	use	'a'	as	a	meaningless	symbol.	According	to	others we	use	it	as	a	variable.	According	to	Kit	Fine	(1985)	we	use	it	to	refer	to	the	arbitrary	p thing	–	a	special	kind	of	thing	that	is	not	an	ordinary	p	thing.	(For	details	and	further references	see	B&M	2012.) Breckenridge	and	Magidor	(2012)	have	recently	argued	against	each	of	these	accounts. They	defend	an	alternative	referential	account,	according	to	which	we	use	'a'	to	refer 2 to a p thing (an ordinary p thing, not Fine's arbitrary p thing). On this view line 3 actually	serves	a	dual	role	–	we	use	it	to	fix	the	reference	of	'a'	to	a	p	thing,	and	also	to express	the	proposition	that	a	is	a	p	thing.	By	writing	"a	is	a	p	thing"	we	emphasize	the second	of	these	two	roles;	we	could	instead	emphasize	the	first	role	by	writing	"Let	a be	a	p	thing". Defending	this	view	of 'a' is	not	the	main	aim	of	B&M's	paper	–	their	main	aim	is	to defend what they call 'arbitrary reference', the view that we can refer to things arbitrarily.	Nevertheless,	their	main	argument	for	arbitrary	reference	depends	on	this view	of	the	role	of	'a'.	They	argue:	In	line	3	we	get	'a'	to	refer	to	a	p	thing;	we	can	only get 'a' to refer to a p thing if we can arbitrarily refer; therefore, we can arbitrarily refer. 3.	Problems	for	the	p	thing	account I am persuaded by B&M's arguments against meaningless symbol accounts of 'a', against variable	accounts	of 'a', and	against Fine's referential account	of 'a', and for the	rest	of	this	paper	I	will	assume	that	these	accounts	are	wrong	and	that	some	other referential	account	is	right.	But	I'm	not	persuaded	by	the	referential	account	defended by	B&M.	In	fact,	I	think	it	must	be	wrong. B&M	do	not	say	such	much	about	what	kind	of	p	thing	we	get	'a'	to	refer	to	in	line	3, and	it	is	difficult	to	see	what	kind	of	p	thing	it	could	be.	It	cannot	be	an	actual	p	thing, because	for	some	properties	p	there	are	no	actual	p	things.	Nor	can	it	be	a	possible	p thing,	because	for	some	properties	p	there	are	no	possible	p	things (it	might	also	be impossible	to	refer	to	merely	possible	p	things,	at	least	for	some	properties	p).	So	they can't	claim	that	we	get	'a'	to	refer	to	an	actual	or	possible	p	thing. It	won't	do	for	them	to	say	that	these	properties	are	special	–	that	for	these	properties either	a	different	account	is	needed	or	that	existential	instantiation	does	not	work.	The original	argument	is	valid,	and	the	extended	argument	helps	to	show	that	it	is	valid,	no matter	which	properties	p	and	q	are; in	particular,	no	matter	which	property	p is.	So they	need	an	account	of	'a'	that	works	for	all	properties	p. There are also problems for the idea that 'a' refers to a p thing even for some properties	p	that	have	actual	instances. First, let p be the property of being male. This has actual instances. But if q is a property	that	is	inconsistent	with	p,	such	as	being	not	male,	then	there	are	no	possible circumstances in	which the	premises are	both true. So, if 'a' is supposed to refer to some	p	thing	in	some	circumstance	is	which	the	premises	are	both	true	then	'a'	cannot refer	to	anything	(because	there	are	no	such	circumstances). Second, let p be the property of being a never-referred-to thing (see Haze (Forthcoming)).	This	has	actual	instances.	But	'a'	cannot	refer	to	any	of	them,	because that	amounts	to	'a'	referring	to	a	never-referred-to	thing,	which	it	cannot	do,	because if	it	did	then	that	thing	would	not	be	a	never-referred-to	thing. 3 Again, it won't do for B&M to say that these properties are special. The original argument	is	valid,	and	the	extended	argument	helps	to	show	that	it	is	valid,	no	matter which	properties	p	and	q	are.	They	need	an	account	that	works	for	all	properties	p	and q. So	I	don't	see	how	a	referential	account	of	'a'	can	successfully	maintain	that	we	get	'a' to	refer	to	a	p	thing. 4.	A	proposal:	the	supposedly	p	thing	account I	have	a	suggestion,	and	presenting	this	suggestion	is	the	main	purpose	of	this	paper.	A referential	account	of	'a'	cannot	claim	that	we	use	'a'	to	refer	to	a	p	thing	(as	we	have just	seen).	But	it	might	be	able	to	claim	that	we	use	'a'	to	refer	to	a	supposedly	p	thing (that is, a thing that supposedly has p). Moreover, this might still give B&M an argument for arbitrary reference, as follows: In line 3 we get 'a' to refer to a supposedly p thing; we can only get 'a' to refer to a supposedly p thing if we can arbitrarily	refer; therefore,	we	can	arbitrarily	refer.	And it	might	allow	them	to	avoid the	problems	that	I	have	presented	above. My	task	now	is	to	explain	all	of	this. Suppose	that	we	are	developing	a	referential	account	of	the	role	of	'a'	in	the	extended argument	– that is,	an	account	according to	which	we	use 'a' to refer to something. Let's	think	through	how	the	account	should	go. One	thing	that	we'd	like	the	account	to	do	is	explain	how	the	extended	argument	helps us	to	see	that	the	original	argument	is	valid.	How	might	our	account	do	this? We might try saying this: by adding lines 3 and 4 to the argument we break the inference	down	into	some	smaller	steps,	each	of	which	is	more	clearly	valid	than	the step from	1	and	2 to	C. (i.e. lines	3	and	4	are intermediate	consequences.)	The idea would	be	this:	C	follows	from	1	and	2,	because	C	follows	from	4,	and	4	follows	from	2 and	3,	and	3	follows	from	1.	This	is	a	common	way	of	adding	lines	to	an	argument	to make	its	validity	more	apparent. But	we	can't say this. In	particular,	we can't say that	3 follows from	1 (although	we might	be	able	to	say	that	4	follows	from	2	and	3).	According	to	the	account	that	we	are developing,	'a'	refers	to	something.	For	3	to	follow	from	1	we	need	the	following	to	be the	case:	it	is	not	possible	for	1	to	be	true	while	3	is	false;	that	is,	it	is	not	possible	for there to be some p things without a being a p thing. So we need 'a' to refer to something x such that: necessarily, if there are any	p things then x is a p thing. For some	properties	p	there is	such	a	thing.	The	property	of	being	Bill	Gates is	one	such property	(necessarily,	if	there	are	any	things	that	have	this	property	then	Bill	Gates	has this	property).	So	is	the	property	of	being	a	number	(necessarily,	if	there	are	any	things that have this property then 2 has this property), and the property of being even (necessarily, if	there	are	any	things	that	have	this	property	then	2	has	this	property). 4 But,	and	this	is	the	problem,	for	some	properties	p	there	is	no	such	thing.	The	property of	being	human	is	one	such	property.	There	is	no	x	such	that:	necessarily,	if	there	are any humans then x is human. (i.e. every x is such that: possibly, there are some humans	but	x	is	not	human.)	We	want	an	account	that	works	for	all	properties	p,	so	it cannot	be	part	of	our	account	that	3	follows	from	1. Another common way of adding lines to an argument to make its validity more apparent	is	to	make	a	supposition.	Perhaps	we	can	say	that	in	line	3	we	are	making	a supposition? No, we cannot say this either. If 3 is a supposition then it has to be discharged before the argument is complete. There are two places that it	might be discharged:	either	after	line	4,	or	after	C. Suppose	that	it	is	discharged	after	line	4,	by	adding	line	5	as	follows: 1. There	are	some	p	things	(Premise) 2. All	p	things	are	q	things	(Premise) 3. a	is	a	p	thing	(Supposition) 4. a	is	a	q	thing	(From	2	and	3,	by	universal	instantiation) 5. If	a	is	a	p	thing	then	a	is	a	q	thing	(From	3	–	4	by	conditional	proof) C. There	are	some	q	things	(From	?) This cannot	be the	extended	argument. For from	which	of the	previous lines	does	C follow? We have lines 1, 2 and 5 available (3 and 4 are within the scope of the supposition).	It	is	no	help	to	say	that	it	follows	from	lines	1	and	2	because	that's	what we	are	trying	to	show.	It	does	not	follow	from	lines	1	and	5	(it	is	possible	for	1	and	5	to be true	without C being true)(remember that according to the account that	we are developing, 'a' is not a variable – it refers to something – and 5 is not a universal quantification). It	does	not follow	from lines	2	and	5 (it is	possible for	2	and	5 to	be true	without	C	being	true).	And	it	does	not	follow	from	line	5	(it	is	possible	for	5	to	be true	without	C	being	true). Suppose	that	it	is	discharged	it	after	C,	by	adding	C'	as	follows: 1. There	are	some	p	things	(Premise) 2. All	p	things	are	q	things	(Premise) 3. a	is	a	p	thing	(Supposition) 4. a	is	a	q	thing	(From	2	and	3,	by	universal	instantiation) C. There	are	some	q	things	(From	4,	by	existential	generalisation) C'.	If	a	is	a	p	thing	then	there	are	some	q	things	(From	3	–	C	by	conditional	proof) This cannot be the extended argument either, because it does not have the right conclusion. Perhaps there is an implicit further conclusion, C": There are some q things. (This is just a repetition of C, but this time it is outside the scope of the supposition.)	But	from	which	of	the	previous	lines	does	C"	follow?	We	have	lines	1,	2 and	C'	available	(3,	4	and	C	are	within	the	scope	of	the	supposition).	It	is	no	help	to	say that it follows	from	lines	1	and	2	because	that's	what	we	are	trying	to	show. It	does 5 not	follow	from	lines	1	and	C', it	does	not	follow	from	lines	2	and	C',	and	it	does	not follow	from	line	C'. So	we	have	this:	if	our	account	is	that	we	use	'a'	to	refer	to	something	then	to	explain why adding lines 3 and	4 is helpful	we cannot say that line 3 follows from	previous lines,	but	nor	can	we	say	that	it	is	a	supposition. So	what	can	we	say? I think	we	can	say	this: the	extended	argument	helps	us	to	see that	the	original	argument	is	valid	by	helping	us	to	see	that	C	is	true	on	the	supposition that	1	and	2	are	true.	And	this,	in	turn,	gives	our	account	a	possible	referent	for	'a'	–	a supposedly	p	thing. This	needs	some	explaining. We	start	by	supposing	that	the	premises	are	both	true. This	is	certainly	one	way	that	we	can	start	to	show	that	the	original	argument	is	valid. Showing	that	the	original	argument	is	valid	is	equivalent	to	showing	that	the	following proposition	is	necessarily	true:	if	there	are	some	p	things	and	all	p	things	are	q	things then there are some q things. One	way to do this is to start by supposing that the antecedent	is	true	and	then	use	conditional	proof.	(This	is	not	the	only	way	–	we	could start	by	supposing	that	the	proposition is false	and	then	use	proof	by	contradiction.) Now,	supposing	that	the	antecedent	is	true	amounts	to	supposing	that	the	premises	of the	original	argument	are	both	true.	So	we	can	see	that	the	original	argument	is	valid by	first	supposing	that	the	premises	are	both	true.	What	I'm	proposing	here	is	that	this is	what	we	do	when	we	use	the	extended	argument	to	see	that	the	original	argument is	valid. So	we start	by supposing that the	premises	are	both true. In	particular,	we suppose that	there	are	some	p	things. Next,	line	3.	According	to	the	referential	account	that	we	are	trying	to	develop,	in	this line	we	get	'a'	to	refer	to	something.	For	the	reasons	that	I	explained	earlier,	we	can't say	that	we	get	'a'	to	refer	to	a	p	thing,	either	actual	or	possible.	But	we	might	be	able to	say	this:	we	get 'a' to	refer	to	a	supposedly	p	thing.	The idea is	this:	by	supposing that there	are	some	p things	we	make it the	case that there	are	some	supposedly	p things (that is, some things that supposedly have p), no	matter	which property p is (and	no	matter	whether	there	are	any	actual	or	possible	p	things);	then	in	line	3	we	get 'a'	to	refer	to	one	of	these	supposedly	p	things. The crucial part of this proposal is the following claim: by supposing that there are some	p	things	we	make	it	the	case	that	there	are	some	supposedly	p	things.	We	need not claim that these supposedly	p things	are	always	created by the supposition.	We can	allow	that	we	can	suppose	that	there	are	some	p	things	by	thinking	about	some actual things	and supposing that they	have	p	– in this case	we	wouldn't	be creating these	things	(there	already	are	these	things),	we	would	just	be	making	it	the	case	that they	supposedly	have	p.	We	can	allow	this,	but	we	shouldn't require it	– it could	be 6 that	to	suppose	that	there	are	some	p	things	we	cannot	think	of	any	actual	things	and suppose	them	to	have	p.	This	is	what	we	should	say:	if	we	suppose	that	there	are	some p	things	then	there	are	some	things	that	supposedly	have	p	(which	we	may	or	may	not create by making the supposition). If we do create these things in making the supposition	then	their	existence	depends	on	the	supposition,	and	lasts	just	as	long	as the	supposition	lasts. This claim, that if we suppose that there are some p things then there are some supposedly p things, is obviously a controversial claim. But it is in good company – there is a precedent for this kind of claim in theories of fiction. According to some theories	of	fiction,	if	an	author	creates	a	work	of	fiction,	and	it	is	part	of	the	fiction	that there are some p things, then the author thereby	makes it the case that there are some	fictionally	p	things	(that	is,	some	things	that	fictionally	have	p)(see	for	example Kripke (2013), Searle (1975), van Inwagen (1977)). These things need	not be	merely fictional	things	–	they	could	be	real	things	which	feature	in	the	fiction.	Or	they	could	be merely	fictional	things,	things	that	are	created	when	the	fiction	is	created	and	last	as long	as	the	fiction	lasts.	Either	way,	in	creating	the	work	of	fiction	the	author	makes	it the case that there are some fictionally p things.	What I am suggesting is	much the same thing, just applied to the	making of suppositions rather than the authoring of fictions.	Making	a	supposition	is,	after	all,	very	much	like	telling	a	story.	In	fact,	we	can show	that	the	original	argument	above	is	valid	by	telling	a	story	instead:	"Once	upon	a time	there	were	some	p	things,	and	all	p	things	were	q	things.	Let's	call	one	of	these	p things	'a'.	So	a	was	a	p	thing.	And,	since	all	p	things	were	q	things,	a	was	also	a	q	thing. So	there	were	some	q	things." Note that the account of 'a' that I am suggesting here is different from the p thing account	defended	by	B&M.	Both	accounts	are	referential	accounts	(i.e.	both	claim	that we	use	'a'	to	refer	to	something),	but	whereas	the	B&M	account	says	that	we	use	it	to refer to a p thing, this account says that	we	use it to refer to a supposedly p thing. Although it is different from their account, it	might still give them an argument for arbitrary	reference.	For	we	might	still	need	to	appeal	to	arbitrary	reference	to	account for	how	we	can	refer	to	a	particular	one	of	these	supposedly	p	things in	the	cases in which	there	are	more	than	one.	If	so,	B&M	can	argue	as	follows:	In	line	3	we	get	'a'	to refer	to	a	supposedly	p	thing;	we	can	only	get	'a'	to	refer	to	a	supposedly	p	thing	if	we can	arbitrarily	refer;	therefore,	we	can	arbitrarily	refer. I	don't	know	whether	the	account	that	I	am	proposing	is	true	(I'm	worried	about	the crucial claim, that if	we suppose that there are some p things then there are some supposedly	p things).	But if	we	want to	maintain that in line	3	we	get 'a' to refer to something	then	I	don't	see	any	alternative. Actually,	there	is	an	alternative,	but	it	is	not	one	that	is	open	to	a	referential	account of	'a',	because	it	is	not	one	on	which	we	use	'a'	to	refer	to	something.	The	alternative is to say that in line 3 we	merely suppose that 'a' refers to a p thing (rather than getting	'a'	to	actually	refer	to	a	supposedly	p	thing).	On	this	account	we	do	not	get	'a' to	actually	refer	–	we	merely	suppose	that	it	does.	And	this	alternative	does	not	give B&M	an	argument	for	arbitrary	reference:	since	'a'	does	not	refer	it	does	not	have	its 7 reference	fixed,	so	there	is	no	need	to	appeal	to	arbitrary	reference	to	explain	how	it has	its	reference	fixed.	Note	that	on	this	account	we	don't	even	need	to	suppose	that 'a'	got	its	reference	fixed	arbitrarily	–	supposing	that	'a'	refers	to	one	of	many	p	things does	not	require	us	to	suppose	that	it	had	its	reference	fixed	arbitrarily	to	one	of	those things.	So	this	account	does	not	commit	us	in	any	way	to	an	ability	to	arbitrarily	refer to	things.	So	it	does	not	help	B&M's	case	for	arbitrary	reference. But	I	don't	think	that	this	account	is	right.	Here	is	one	argument	(I'll	give	another	one later,	a	better	one).	In	line	3,	rather	than	introduce	'a'	we	might	instead	just	direct	our thoughts	to	one	of	the	p	things,	and	argue	as	follows:	"Consider	one	of	these	p	things; it	is	a	q	thing;	so	there	are	some	q	things."	When	we	do	this	we	are	not	supposing	that we	are	considering	one	of these	p	things	–	we	actually	are considering	one	of them. But	by	introducing	'a'	in	line	3	all	that	we	are	doing	is	giving	ourselves	a	way	of	talking about the considering that we are doing. Since we actually are considering one of these	supposedly	p	things,	we	are	also	actually	are	referring	to	one	of	them	with	'a', not	merely	supposing	that	we	are. So	we	start	by	supposing	that	the	premises	are	both	true;	in	particular,	that	there	are some	p	things.	This	makes	it	the	case	that	there	are	some	things	that	supposedly	have p. Then, in line	3,	we	get 'a' to refer to	one	of these things that supposedly	have	p (fixing	the	reference	of	'a'	arbitrarily	to	one	of	them,	if	there	are	more	than	one). Having fixed the reference	of 'a' to	one	of the	supposedly	p things	we	can then talk about this thing,	and	say	what is true	of it	on the supposition that the	premises	are both true.	On this supposition line	3 ("a is	a	p thing") is true,	not	because it follows from	line	1	but	because	of the	way	that	we	have	fixed	the	reference	of 'a' (we	have fixed	its	reference	to	a	thing	that	supposedly	has	p).	And,	on	this	supposition,	line	4	is also	true,	since	2	and	3	are	true	and	4	follows	from	2	and	3.	And	C	is	true	too,	since	it follows	from	4.	This	establishes	that	C	is	true	on	the	supposition	that	1	and	2	are	both true.	And	this	is	how	lines	3	and	4	help	us	to	see	that	the	original	argument	is	valid	– they	help	us	to	see	that	C	is	true	on	the	supposition	that	1	and	2	are	both	true. 5.	The	problems	avoided We	are	now	in	a	position	to	see	how	the	account	that	I	am	proposing	would	work	for all	properties	p	and	q,	avoided	the	problems	for	the	p	thing	account	that I	described above. First,	what	if	p	is	such	that	there	are	no	p	things?	Answer:	if	we	suppose	that	there	are some	p	things	then	there	are	some	things	which	supposedly	have	p,	even	if	there	are no	actual	p	things,	and	these	are	available	for	'a'	to	refer	to. Second,	what if	p is	such	that	there	are	no	possible	p	things?	Answer: if	we	suppose that there are some p things then there are some things	which supposedly have p, even	if	there	are	no	possible	p	things,	and	these	are	available	for	'a'	to	refer	to. 8 Third, what if p and q are such that the premises are inconsistent? Answer: if we suppose that there	are some	p things then there	are some things	which supposedly have p, even if other assumptions are inconsistent	with this assumption, and these things	are	available for 'a' to	refer to.	Compare	the fictional	case: there is	a fictional character	Sherlock	Holmes,	even	if	the	stories	about	Sherlock	Holmes	are	inconsistent. Note	that	under inconsistent	suppositions	everything is true,	so	the	account	predicts that if p and q are such that the premises are inconsistent then we can use the extended	argument	to	derive	any	conclusion	from	these	premises.	But	that	is	the	right result. Fourth,	what if	p is the	property	of	being	a	never-referred-to	thing?	Answer:	we	can get 'a' to refer to something that supposedly is	never-referred-to,	because 'a' refers from	outside	the	supposition,	and	the	thing	it	refers	to	is	never-referred-to	only	inside the supposition. 'a' refers, from	outside the supposition, to something	which, inside the	supposition,	is	never-referred-to.	There	is	no	contradiction	in	that	(just	as	there	is no	contradiction	in	our	naming	and	referring	to	a	character	in	a	story	who,	in	the	story, has	no	name). You	might	wonder	whether	the	present	account	faces	something	like	this	last	problem but	with	a	different	property	– the	property	of	being	a	never-supposed-of thing.	For we	might	argue	as	follows: 1. There	are	some	never-supposed-of	things 2. All	never-supposed-of	things	are	q	things Therefore, C. There	are	some	q	things According	to	the	present	account,	we	use	'a'	to	refer	to	a	supposedly	never-supposedof thing. Isn't that impossible? No, and for the same reason. We can suppose, of something, that it is never-supposed-of, because the supposing occurs outside the supposition,	and	the	thing	of	which	we	are	supposing	is	never-supposed-of	only	inside the	supposition.	We	suppose, from	outside	the	supposition,	something	of	something which,	inside	the	supposition,	is	never-supposed-of.	There	is	no	contradiction	in	that. 6.	One	more	argument I can	now	give	another	argument	against the idea that in line	3	we suppose that 'a' refers	to	a	p	thing.	If	this	were	right	then	we	could	use	existential	instantiation	to	show that the proposition that there are some never-referred-to things entails the proposition that grass is green (or any other proposition). For we could argue as follows: 1. There	are	some	never-referred-to	things	(Premise) 2. a	is	a	never-referred-to	thing	(From	1,	by	existential	instantiation) C. Grass	is	green	(From	2) 9 In	line	2	we	suppose,	on	this	account,	that	'a'	refers	to	a	never-referred-to	thing.	But that	is	to	suppose	something	inconsistent.	So	on	this	supposition	everything	is	true.	In particular,	it	is	true	that	grass	is	green.	So	if	this	account	were	true	then	we	could	use existential instantiation to show that the proposition that there are some neverreferred-to	things	entails	the	proposition	that	grass	is	green.	But	we	cannot	do	that.	So this	account	is	not	true. 7.	Conclusion I	have	argued	that if, in	the	example	of	existential instantiation	that	we	started	with, we use 'a' to refer to something then it cannot be to a p thing, either actual or possible.	I	have	suggested	an	alternative	referential	account	–	that	that	we	use	'a'	to refer	to	a	supposedly	p	thing.	I	have	explained	how	using	'a'	in	this	way	helps	us	to	see that	the	original	argument	is	valid,	no	matter	which	properties	p	and	q	are.	And	I	have pointed	out	that	this	alternative	account	still	gives	B&M	their	hoped-for	argument	for arbitrary reference. But the alternative account relies on something controversial – that	if	we	suppose	that	there	are	some	p	things	there	are	some	things	that	supposedly have	p.	So	I'm	not	sure	whether	this	is	an	account	that	we	can	ultimately	maintain.	But if	we	want	a	referential	account	of	the	role	of	'a'	then	I	don't	see	any	alternative. References Breckenridge, W. & Magidor, O. (2012), 'Arbitrary Reference', Philosophical Studies 158,	pp.	377-400. Fine,	K.	(1985),	Reasoning	with	Arbitrary	Objects,	Oxford:	Basil	Blackwell. Haze,	T.	(2016),	'A	Counterexample	to	the	Breckenridge-Magidor	Account	of	Instantial Reasoning',	Journal	of	Philosophical	Research	41,	pp	539-541. Kripke,	S.	(2013),	Reference	and	Existence,	Oxford	University	Press. Searle,	J.	(1975),	'The	Logical	Status	of	Fictional	Discourse',	New	Literary	History	6,	pp. 319-332. Van	Inwagen,	P.	(1977),	'Creatures	of	Fiction',	American	Philosophical	Quarterly	14,	pp. 299-308.