History and Philosophy of Logic

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Sellars, Second-order Quantification, and

Ontological Commitment

Andrew Parisi

To cite this article: Andrew Parisi (2018): Sellars, Second-order Quantification, and Ontological
Commitment, History and Philosophy of Logic, DOI: 10.1080/01445340.2018.1474427

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Published online: 24 May 2018.

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HISTORY AND PHILOSOPHY OF LOGIC, 2018
https://doi.org/10.1080/01445340.2018.1474427

Sellars, Second-order Quantification, and

Ontological Commitment
ANDREW PARISI
University of Connecticut, USA
andrew.parisi@uconn.edu

Received 7 March 2017 Accepted 6 May 2018

Sellars [1960. ‘Grammar and existence: A preface to ontology’, Mind, 69 (276), 499–533; 1979. Naturalism
and Ontology, Reseda, CA: Ridgeview Publishing Company] argues that the truth of a second-order sentence,
e.g. ∃ffa, does not incur commitment to there being any sort of abstract entity. This paper begins by exploring
the arguments that Sellars offers for the above claim. It then develops those arguments by pointing out places
where Sellars has been unclear or ought to have said more. In particular, Sellars’s arguments rely on there being
a means by which language users could come to understand sentences of a second-order language wherein the
truth of sentences of the form ∃ffa do not require there to be abstract entities. In addition to this, as Sellars [1979.
Naturalism and Ontology, Reseda, CA: Ridgeview Publishing Company] notes, a formal account of quantification
is required that does not make use of the apparatus of sequences. Both a translation of ∃ffa and a formal account
of quantification are provided by this paper.

1. Introduction
In ‘On what there is’, Quine (1948) argues that the ontological commitments of a theory
are laid bare by first formalizing that theory and then examining what entities must be
in the range of that theory’s bound variables. This account of ontological commitment
places special emphasis on quantifiers and the variables bound by them. Quine famously
stated his view as ‘To be assumed as an entity is, purely and simply, to be reckoned as
the value of a variable’. This account of ontological commitment entails that any language
with second-order quantifiers will require there to be entities which can be the values of
the second-order variables of quantification. Twelve years after the publication of ‘On
What There Is’, Sellars published ‘Grammar and Existence: A Preface to Ontology’. In
that paper he argues that a theory making use of second-order quantifiers need not be
committed to any entities which are the values of its second-order variables. In ‘Naturalism
and Ontology’, he elaborates on this position by offering further arguments in support
of it. This paper explores Sellars’s arguments from those two works and develops novel
arguments for the same conclusions.
The first half of this paper explores Sellars’s arguments that the introduction of second-
order quantifiers into a language does not, of itself, incur ontological commitments to there
being abstract entities. Two arguments for this conclusion are considered: the first attempts
to cast doubt on the claim that the only natural language translation of the formal sentence
‘∃ffa’, is one that clearly incurs ontological to some abstract entity or other. The second
argument examines the standard model-theoretic definition of the consequence relation—
which Sellars clearly ascribes to Quine—and concludes that adopting that definition of
the consequence relation entails that the introduction of second-order quantifiers into a
language incurs commitment to there being abstract entities. This explains why Sellars
takes Quine to be one of his main disputants on the topic and why he thinks he is required
to offer an alternative to the model-theoretic definition of the consequence relation. The
© 2018 Informa UK Limited, trading as Taylor & Francis Group
2 A. Parisi

second half of the article addresses weakneses in Sellars’s arguments. The problems are
dissolved by offering an alternative translation of the formal sentence, ‘∃ffa’, into English
and providing proof-theoretic alternative to the standard model-theoretic account of the
consequence relation. The paper concludes by showing that the proof-theoretic alternative
introduced in the previous section does not require entities to be assigned as the values of
second-order variables.

2. A Quinean argument
This paper is concerned with arguments offered by Sellars. It is clear in ‘Naturalism
and Ontology’ that he takes his disputant to be Quine. This does not entail that Sellars
has correctly interpreted Quine’s position. In what follows, Sellars arguments are said to
be leveled against a ‘Quinean’ view. This Quinean view may not have been Quine’s own
in detail, but it is the view that Sellars argues against in ‘Grammar and Existence’ and
‘Naturalism and Ontology’ and it is faithful to Quine’s views in broad strokes.
Sellars (1979) makes use of the term ‘reference’. This is unfortunate because that term
has been used variously by philosophers and it is unclear how Sellars’s use of it there
fits with his other work. ‘Reference’ is also unfortunately used by Quine (1948) in the
expression ‘range of reference’, while Sellars (1979) talks of ‘determinate reference’ and
‘indeterminate reference’. Whether there is a unified account of the meaning of that term
is set aside. To make the argument under consideration clearer, new terminology is intro-
duced. The term ‘supposition’ describes the relation that an expression must have to the
world when that expression occurs in a true sentence.1 For instance, if the sentence ‘Helen
is happy’ is true, then ‘Helen’ must supposit for Helen. Similarly, if the sentence ‘Some
donkey is running’ is true, then the expression ‘Some’ must supposit for some objects.2
Call the following Quinean3 argument the Argument from Abstracta:
(1) Commitment to the truth of a sentence containing an expressions that supposits
for an entity entails an ontological commitment to that entity.—supposition entails
ontological commitment.
(2) For any quantifier, if it is the main operator in a true sentence, it supposits for
something.
(3) Second-order quantifiers could only supposit for properties, attributes, classes, etc.
(4) Properties, attributes, classes, etc. do not exist.
(C) Therefore, introducing second-order quantification into a language incurs commit-
ment to entities that do not exist.
Suppose that second-order quantifiers were introduced into a language. In such a lan-
guage ‘∃ffa’ is a logical truth, for any term ‘a’. By premise (2), the quantifier ∃f supposits
in that sentence. By premise (3) that entity is a property, attribute, class, or other sort of
abstract entity. Since any theory is committed to the logical truths, by premise (1), every
theory is committed to there being some property, attribute, class or other abstract entity.
By premise (4), those do not exist. So any theory in a second-order language is committed
to there being entities that do not exist.
Premise 1 is designed to be true. Its truth is built into the notion of supposition that was
introduced above. Since the sort of entity that is assigned to first-order variables is the sort

1
The medieval term is used both because it is long enough out of use that it can be re-purposed for this paper, and it may offer
clarity to anyone familiar with the term to grasp the concept aimed at by its use.
2
Medieval philosophers also made a distinction between determinate and indeterminate supposition. It is crucial to note that
these notions of ‘determinate’ and ‘indeterminate’ have nothing to do with the notions discussed below. ‘Supposition’ is a term
used only to indicate a relation between expressions in true sentences and entities in the world.
3
It is emphasized that this argument may not be the one actually given by Quine, but it is one to which Sellars responds.
 Sellars, Second-order Quantification, and Ontological Commitment 3

of entity that names stand for, the sort of entity that is assigned to a second-order variable
is the sort of entity that predicates stand for. Sellars is committed to premise (4) by his
nominalism.4
Premise (3) is worth an aside. Boolos (1984) argued that second-order quantifiers do
supposit but do so multiply. They supposit for the same objects as first-order quantifiers,
but do so plurally. As he says in ‘To Be is to Be the Value of a Variable (or some Values of
Some Variables)’, ‘There are, rather, two (at least) different ways of referring to the same
things’ (Boolos 1984, p. 449). The purpose of this paper is not to refute Boolos (1984), but
to offer an alternative Sellarsian justification of second-order logic. Premise (3) is taken
for granted in what follows.
This leaves only premise (2) to be denied. Premise (2) follows from the Quinean claim
introduced above that variables of quantification are assigned entities when assessing the
ontological commitments of a theory. Call the set of entities so assigned to a variable
the range of that variable. Second-order quantifiers bind second-order variables and so if
a second-order quantifier is the main operator of a sentence, there must be some entity
assigned to the variable it binds. If premise (2) is false, then the Quinean idea that the
ontological commitments of a theory are laid bare by examining the entities in the ranges
of variables of that theory must also be false.
It is easier to make explicit what it would take for a sentence of a formal language to
be true than to make explicit what it would take for a sentence of English to be true. This
paper examines two ways of doing this, a model-theoretic method and a proof-theoretic
method. If formal methods are to offer clarification in metaphysical or other discussion,
there must be a way of translating discourse in those areas into the formal language and
vice versa. Since the truth conditions of a set of sentences of a formal language are explicit,
the translation from the formal language into natural language provides truth conditions
for the translated sentences of natural language. Similarly, if S is a sentence of natural
language that does not obviously carry ontological commitment when accepted, and S is
an adequate translation of the sentence S  of a formal language, then this provides reason to
doubt that S  carries ontological commitment when accepted. The process of coordinating
the two languages will ultimately lay bare the ontological commitments of both languages.
One argument that the sentence ‘∃ffa’ incurs commitment to some abstract entity rests on
the premise that there is no English translation of that sentence that does not obviously
incur ontological commitment to some sot of abstract entity. The goal of Section 3.1 is to
suggest that there is a natural language translation of the formal sentence ‘∃ffa’ that does
not incur ontological commitment to there being some abstract entity, except possibly what
is named by ‘a’. Sellars suggests that there is a fallacy in this argument, but does not offer
an adequate translation of ‘∃ffa’ into English. This is provided in Section 5 which also
offers an explanation of how other sentences with second-order quantifiers may come to
be understood. Before this section another concern for Sellars’s position and his response
to this concern are examined. Even if Sellars’s efforts described in Section 3.1 and mine of
Section 5 are successful, if there is no available account of the commitments of the formal
sentence ‘∃ffa’ that rules out that the second-order quantifier supposits then the proposed

4
There are nuances of Sellars’s view of abstract objects that are set to the side for the purposes of this paper. Sellars (1963a,
1979) tries to preserve the grammar of sentences involving apparent reference to abstract entities so that it turns out that it is
true that ‘There are properties’. But he also acknowledges that there really are no abstract entities, as he says in ‘Naturalism and
Ontology’ ‘I shall, however, as you might expect, go on to argue that although there are attributes, there really are no attributes.’
[emphasis in original] (Sellars 1979, p. 41). There are interpretations of Quine (1986) according to which second-order quanti-
fiers are illegitimate for use in theorizing because they mask the true set-theoretic nature of second-order quantification. Quine
(1986) calls second-order logic ‘set theory in sheep’s clothing’. Whether this argument can be sustained is a contentious issue.
For arguments that it cannot, see Boolos 1975, Shapiro 1991. This argument is set aside for the purposes of this paper
4 A. Parisi

translation cannot be correct. Section 3.2 shows that if the model-theoretic account of the
consequence relation is accepted, commitment to the truth of ‘∃ffa’ results in commitment
to an entity to be the value of the varable ‘f ’. Section 4 provides an alternative account of
the consequence relation and shows that that account does not require that second-order
quantifiers supposit. This clears the way for the translation that is offered in Section 5. The
paper concludes that there is a coherent account of second-order quantification on which
philosophers need not be hesitant to make use of them.
Quine’s (1948, 1986) official position is that variables are the expressions of a first-order
quantified sentence that supposit. Sellars (1979), on the other hand, holds that quantifiers
are the expressions of a first-order quantified sentence that supposit. When there is no
loss of clarity this distinction between the two philosophers is ignored. Sellars (1979) dis-
cusses the difference between himself and Quine as being over how the truth conditions of
quantified sentences of a formal language are to be determined.
Sellars (1960) attempts to cast doubt on Premise (2) of the Argument from Abstracta
by fixing what sentential forms of natural language carry ontological commitment when
accepted. He then suggests that the proper translation of ‘∃ffa’ is neither one of those forms
nor commits one to a sentence of one of those forms. If a translation of ‘∃ffa’ can be found
which is neither of an ontologically committing form nor entails commitment to a sentence
with such a form, then the sentence ‘∃ffa’ does not carry ontological commitment in virtue
of the quantifier ‘∃f ’. Under these circumstances, by Premise (1) of the Argument from
Abstracta, the quantifier ‘∃f ’ does not supposit in ‘∃ffa’. From this it follows that premise
(2) of the Argument from Abstracta is false.
Sellars points out that the paradigm case of an ontologically committing sentence is
‘There is a K’ and the equivalent sentence ‘There are Ks’. This is because they are the most
straightforward answer to the question ‘What is there?’, the central question of ontology.
Commitment to the truth of such sentences results in commitment to there being Ks. Any
sentence of this form, or that entails a sentence of this form, is ontologically committing.
These forms are taken from (Sellars 1960, p. 235)
[ . . . ] among the forms by the use of which one most clearly and explicitly asserts
the existence of objects of a certain sort [ . . . ] are the forms ‘There is an N’,
‘Something is an N’, and ‘There are N’s.’

3. Not all quantifiers supposit

3.1. The first argument
For convenience call the sentences that have one of the above forms ontologically com-
mitting. The question under discussion then is whether the formal sentence ‘∃ffa’, when
translated into natural language, is or entails an ontologically committing sentence. In
order to answer, in the first case, that the sentence ‘∃ffa’ is not itself of an ontologically
committing form, Sellars reasons in analogy with the accepted translation of first-order
sentences.
Consider the pair of sentences

(A) ∃xx is warm.

(B) ∃f S is f .

A standard translation of sentence (A) is ‘There is an x such that x is warm’. This,

however, cannot be an adequate translation. The first occurrence of ‘x’ is a common noun,
as ‘donkey’ in ‘Fred is a donkey’ is. It must be a common noun because it is preceded
by the indefinite article ‘a’. But the second occurrence of ‘x’ is a singular term, the sort
of grammatical item that names an entity, as ‘Fred’ is in ‘Fred is a frog’. No expression
 Sellars, Second-order Quantification, and Ontological Commitment 5

used univocally can serve both purposes. Grammatically, that translation of (A) has the
same form as either ‘There is an apple such that apple is warm’ or ‘There is a Fred such
that Fred is warm’. Neither sentence is grammatically correct. A similar issue arises if the
same technique is used to translate (B). The analogous translation is ‘There is an f such
that S is f ’. The first occurrence of ‘f ’ is again a common noun such as ‘apple’, but the
second is a predicate adjective such as ‘red’, ‘cold’, or ‘loud’. That ‘f ’ is not being treated
univocally is shown by the sentences substituting a common noun and predicate adjective
for ‘f ’: ‘There is an apple such that S is apple’, and ‘There is a red such that S is red’. Since
neither of these sentences is grammatical, the original translation of (B) is not grammatical.
The most immediate remedy is to introduce a Carnapian (1934) universal word, or what
Sellars calls a category word, such as ‘thing’. (A) could then be translated as ‘There is
a thing, x, such that x is warm’. ‘x’ is being treated univocally in both positions of that
sentence. It is a singular term in both. The analogous translation of (B) would then be
‘There is a property, f, such that S is f ’. Importantly, the first occurrence of ‘f ’ renames
the property that is supposited for by the quantifier —Sellars has granted that ‘There is a
property’ is of an ontologically committing form —whereas the second occurrence of ‘f ’
is an adjective. If this sentence were grammatical the sentence, ‘There is a property, loud,
such that S is loud’ would have to be. The proper names of properties are generally words
with ‘-ness’, ‘-ity’, and ‘-hood’ suffixes. The properly grammatical sentence is ‘There is a
property, loudness, such that S is loud’.
If Quine (1986) thought that the correct translation of (A) was ‘There is a thing, x, such
that x is warm’, and by analogy concluded that the correct translation of any existentially
quantified sentence required the introduction of a Carnapian universal word, followed by
a name for an entity of that category, then it is easy to make sense of arguments he gives
in ‘Philosophy of Logic’ [Quine (p. 66, 1986)]

Consider first some ordinary quantifications: ‘∃x (x walks)’, ‘∀x (x walks)’,

‘∃x (x is prime)’ . The open sentence after the quantifiers shows ‘x’ in a position
where a name could stand; [ . . . ] To put the predicate letter ‘F’ in a quantifier,
then, is to treat predicate positions suddenly as name positions, and hence to treat
predicates as names of entities of some sort.

The second sentence of this passage is an indication that Quine takes the correct trans-
lation of ‘x is warm’ to require ‘x’ to be a name. When ‘x’ occurs ‘in a quantifier’, it must
also be treated as a name to preserve grammaticality. Since ∃x and ∃F are both quantifiers,
‘F’ must have the same grammatical category as ‘x’. The only grammatical treatment of
‘∃F’ is to treat ‘F’ as a name, and so the only translation of ‘∃ffa’ is ‘There is a property,
f -ness, such that S is f ’.
Boolos (p. 511, 1975) criticizes Quine’s argument here on the grounds that he does not
consider the analogous argument:

Consider some extraordinary quantification: ‘(∃F)(Aristotle F)’ [ . . . ] the open

sentence after the quantifier shows ‘F’ in a position where a predicate could stand
[ . . . ] to put the variable ‘x’ in a quantifier, then is to treat name positions suddenly
as predicate positions, and hence to treat names as predicates with expansions of
some sort.

If Quine’s reasons for making the above argument were that he could only see how to
translate the sentence ‘∃f S is f ’ as ‘There is a property, f -ness, such that S is f ’, then
Boolos’s argument is off the mark. The open sentence ‘Aristotle is F’ does have F in a
place where a predicate could stand, but the quantifier ‘∃F’ , and for similar reasons ‘∀F’,
6 A. Parisi

cannot. It must have F in a place where a name could go. If the above translation is the
only one available, that phrase must be translated as ‘There is a property F-ness . . . ’
If, contrary to the above assumption, another translation of ‘∃f S is f ’ is available then
it may be that second-order quantifiers do not suddenly treat predicate positions as name
positions. Sellars attempts to provide such a translation of (A), one that is not itself of
an ontologically committing form, though in the first-order case it entails a sentence of
such a form. Sellars offers ‘Something is warm’ as a translation of (A). This sentence is
not among the forms listed above that were ontologically committing. However, it entails
a sentence which is of an ontologically committing form, ‘There are warm things’. This
explains the potentially perplexing remark he makes in ‘Grammar and Existence’
[ . . . ] not even quantification over singular term variables of type 0 makes, as such,
an existence commitment involving an ontological category, i.e. says ‘There are
particulars’5
‘Something is warm’ does not differ ‘only graphologically’ from ‘There is a warm
thing’. They are two importantly distinct sentences of English. The two sentences, though
of different form, are logically equivalent. Commitment to the difference being merely
the way the sentences are written or said may tempt one to think that all uses of the word
‘something’ can be exchanged for the ontologically committing ‘There is a . . . thing’. This
deprives formal language from the resources to merely generalize without a corresponding
ontological commitment. It sets the role of quantifiers as expressions which can supposit
in the forefront. This paper suggests that the generalizing role of quantification should be
emphasized. The role of a quantifier as an expression for suppositing is a secondary fea-
ture of those quantifiers that generalize over grammatical categories that are themselves
used for suppositing. This is contrary to the received opinion that the role of quantifiers
is primarily for suppositing and only incidentally that they are used to generalize. There
are other philosophers who deny Quine’s Dictum. For instance, Azzouni (p. 4, 2004) also
denies that quantification is the main vehicle of ontological commitment. The view in this
paper is importantly different from that account. Sellars, and the view espoused here, take
it that true sentences of the form ‘There are K’s’ are ontologically committing. There is
nothing on the view under consideration that wishes to ‘free ontology from its linguis-
tic straitjacket.’ Wright (2007) has also argued against Quine’s dictum. As is suggested at
the end of this paper, the view of quantification offered here complements Wright’s views
nicely. Sellars (1979) offers an outline of how an alternative to Quine’s Dictum might
be articulated. As mentioned in the introduction, this elaborates and defends a Sellarsian
account of quantification and ontological commitment.
The analogous way of translating (B) that Sellars first recommends is ‘S is something’.
If the translation of (A) as ‘Something is warm’ is not itself of an ontologically committing
form, but only entails a sentence of that form, then there is, at first glance, reason to deny
that commitment to the truth of a second-order sentence incurs commitment to there being
properties, attributes, sets, etc.
A possible concern is that the sentence ‘S is something’ is not grammatical. The above
argument relies heavily enough on finding a grammatical translation of ‘∃ffa’ that if the
translation Sellars recommends is not grammatical, one might accuse Sellars of not being
even-handed in his treatment of the issue. This criticism as it is currently stated is too
strong. One reply is that the ungrammaticality of ‘S is something’ is of a different sort from
the ungrammaticality of ‘There is an f such that S is f ’. If ‘S is something’ is ungrammat-
ical this is not because it can only be understood to be treating expressions of different

5
All emphasis is in the original text.
 Sellars, Second-order Quantification, and Ontological Commitment 7

grammatical categories as if they were the same expression. Sellars may argue that since
natural language has some flexibility to it, all that is required is a way of using natural
language to teach others the meaning of the newly introduced sentence ‘S is something’
and so to understand the meaning of ‘∃ffa’. This would explain why Sellars says, ‘Now it
is easy enough, if I may be permitted a paradox, to invent an “ordinary language” equiva-
lent of [(B)]’ (Sellars 1960, p. 502). It is, however, preferable to avoid such paradoxes. A
grammatical translation of ‘∃ffa’ is presented below.6
A second concern is that just as ‘Something is warm’ entails a sentence of an ontolog-
ically committing form, so too might ‘S is something’. Sellars addresses this concern at
length in both ‘Grammar and Existence’ and ‘Naturalism and Ontology’.
The sentence ‘Something is warm’, as noted above, entails the sentence ‘There is a warm
thing’. So commitment to the truth of (A) translated as ‘Something is warm’ would commit
one to there being warm things. Note, however, commitment to the sentence ‘Something is
warm’ would not commit one to there being warms in the way that commitment to the sen-
tence ‘Something is a cat’ commits one to there being cats. If commitment to ‘Something
is warm’ did not entail a sentence of ontologically committing form, then by premise one
of the Argument from Abstracta ‘Something’ in ‘Something is warm’ would not supposit
for anything.
An argument that second-order quantifiers supposit can be generated by analogy with the
case of first-order quantifiers. Suppose that ‘∃f S is f ’ is true and consider the following
series of inferences:

(1) ‘∃f S is f ’ is translated as ‘S is something’.

(2) Thus, there is something that S is.
(3) Thus, there is a property which S is.
(C) Therefore, there are properties.

Sellars denies that the second sentence entails the third sentence in this case. While
‘There is something . . . ’ appears to be an ontologically committing form, it is not. ‘Some-
thing’ does not appear in that sentence as a kind term, or common noun. In that sentence,
‘something’ takes the place of a predicate adjective. Sentences of an ontologically commit-
ting form require a kind term to follow the ‘There is’, as in ‘There is a K’. The inference
from the second sentence to the third substitutes ‘a property’ for ‘something’ but ‘property’
is a common noun. This substitution makes a grammatical mistake akin to the grammati-
cal mistakes discussed above. In order for the substitution to be valid, it must be valid to
transform ‘something’ into a common noun. The above argument requires this inference
in the second-order case:
S is something
S is some thing
‘There is a property which S is’ does follow from ‘S is some thing’. But in order to go
from ‘S is something’ to ‘There is a property which S is’ relies on the above inference.
The inference might be made more explicit with the use of formal notation. It could then
be represented as

∃x S is f
∃f (f is a thing ∧ S is f )

6
See Section 5.
8 A. Parisi

Since in the corresponding first-order inference the instances of the quantified sentence
will replace the variable with names of things the inference is innocent. But the above
inference relies on holding that predicates stand for things. If the reasoning above with
respect to grammaticality is correct, then this conclusion of the inference may not even
be grammatical. The second instance of ‘f ’ is a name, while the third is an adjective. The
sentence should be ‘∃f (f −ness is a thing ∧ S is f )’.
If translating ‘∃f S is f ’ as ‘S is something’ avoids grammaticality worries and does not
entail ‘There is a property which S is’, then Sellars has taken the first steps in divorcing
quantification from supposition. If correct, this shows that to put a predicate-variable after
a quantifier is not to treat predicates as names for objects.
This section offered an understanding of formal sentences in terms of natural language
in such a way that premise two of the Argument from Abstracta is false. It relied on the
claim that the only sentences of natural language that are of an ontologically committing
form are ‘Something is a K’, ‘There is a K’, and ‘There are Ks’. Sellars offers no real
argument for this claim. As was suggested above these forms take center stage because
they are the most perspicuous answers to the central question of ontology, ‘What is there?’.
There is, however, an argument that the best understanding of the formal language requires
a particular interpretation of our natural language sentences such that premise two of the
Argument from Abstracta is true.
3.2. The second argument
Another argument for premise two of the Argument from Abstracta is as follows. The
best account of quantifiers in a formal language is the familiar Tarskian model-theoretic
account. This theory requires first-order quantifiers that are the main operators of true
sentences to supposit. There is no reason to suppose that the account should be any dif-
ferent from that of any higher-order quantifier. Therefore, the best theory of quantification
requires that all quantifiers supposit.
Sellars (Chapter 1, 1979) considers some form of this argument. There he considers
two different accounts of the existential quantifier, call them (a) and (b). The first account,
(a), again bifurcates, into (a1) and (a2). The second account, (b), is the standard Tarskian
model-theoretic account. On (a1) a sentence of the form ‘∃xFx’ is true iff either ‘Fa1 ’ is true
or ‘Fa2 ’ is true and so on for each expression ‘ai ’. This is standardly called substitutional
quantification. On (a2) a sentence of the from ‘∃xFx’ is true iff some statement of the form
‘Fc’ is true. It is crucial that the right hand side of the biconditional is not to be interpreted
as referring to a list as in the strategy (a1). The crucial feature of strategy (a2) is that
[ . . . ] a language not only consists of more than the grammatical strings which
are actually deployed at any one time [ . . . ] It also includes, in a sense difficult
to define, the resources by which the language could be enriched through being
extended in specific ways.7
Sellars ultimately opts for strategy (a2), but does not give an explicit formal account of
this theory of quantification. A formal account is proposed in Section 4 below.
The Tarskian model-theoretic account of quantification, (b) above, gives the truth condi-
tions for existential sentences by defining their truth in a model conditions. Fix a language.
A model for that language is a pair of a domain of objects and an interpretation function.
The interpretation function assigns each n-ary predicate in the language a subset of the nth -
Cartesian product of the domain of the model and each term an element of the domain of
the model. Along with each model comes a set of sequences. A sequence is a function that

7
For more information see Sellars (1979, p. 8).
 Sellars, Second-order Quantification, and Ontological Commitment 9

assigns each variable of the language an object of the domain. The set of sequences for a
model is the set of all such functions. For convenience consider only the sentence ‘∃xFx’.
This sentence is said to be satisfied by a sequence, s, in a model, M, if some sequence, s ,
exactly like s except possibly for what it assigns to x, satisfies Fx. A sequence satisfies Fx
iff the object assigned to x by that sequence is in the set that the interpretation function
assigns to F. A sentence is true iff it is satisfied by every sequence.
Sequences, therefore, do the main work in making quantified sentences true. It is the
variables of quantified formulas that are assigned objects in the domain of a model. There is
no need for a quantified sentence to get its truth conditions via a sentence involving a name.
Sellars (p. 12, 1979) says that this is not a helpful model of natural language. But the details
of his concerns would take this paper off topic. The goal of this paper is merely to show
that there is a viable alternative to the Tarskian model-theoretic account of quantification.
It is not here argued that the Tarskian model-theoretic account of quantification will not
serve the purposes of a philosophical logician in trying to model natural language.
The Tarskian model-theoretic account of quantification, as stated above, entails that if
a person is committed to the truth of ‘∃xFx’, then there is an entity assigned to ‘x’ by
some sequence, and that entity is F. Similarly, if the account of quantification is to be
generalized, it will hold that a quantified sentence of the form ‘∃ffa’ is true if and only if
there is an entity assigned to ‘f ’ by some sequence such that the entity denoted by ‘a’ is or
has whatever entity is denoted by ‘f ’. It follows that the Tarskian model-theoretic account
of quantification entails that all quantifiers, if they are the main operators of a sentence,
supposit.
This means that if the response to Argument from Abstracta given in Section 3 is
to succeed, an alternative to the Tarskian model-theoretic account of quantification is
required.

4. An alternative account of quantification

The account of quantification presented here is embedded in a framework first developed
by Restall (2009).8 Restall offers an explication of the notion of a position used in the
question ‘what is Y ’s position on topic T?’ or the statement ‘My position on T is X ’. A
position is an ordered pair of sets of sentences. Positions can be coherent or incoherent
depending on the sentences that they contain. The rules governing the assertion and denial
conditions of sentences featuring logical vocabulary determine the coherent and incoherent
positions. Those positions in which a sentence can be (in)coherently asserted or denied are
the primary semantic machinery of the framework. The rules governing the structure of
positions and the conditions of assertion and denial for a sentence determine whether or
not it is coherent to assert or deny that sentence in a position. The semantic contribution
that a logical expression, such as ¬ or ∃, makes to a sentence is given by the rules governing
coherent assertion and denial of that sentence. This inferentialist theory of the meaning of
sentences follows the spirit of Sellars (1974, 1979).
It is important to note that the truth of this framework is not argued for here. It is pre-
sented only as an alternative to the Tarskian model-theoretic picture presented above. The
goal of this paper is only to show that there is a prima facie viable picture of quantifica-
tion and ontological commitment that can germinate from the kernel planted by Sellars
(1960, 1979). In what follows a formal account both of positions and of ontological com-
mitment are developed. It is then shown that on this view not all quantifiers which are the
main operators in a true sentence must supposit. More precisely, if an expression in a true

8
This is similar to work done by Koslow (1992).
10 A. Parisi

atomic sentence does not supposit, then the quantified sentence that generalizes over that
expression does not supposit in any way that the original atomic sentence does not.
Fix a formal language. Let there be an infinite set of names c1 , . . . , cn , . . ., an infi-
nite set of variables, x1 , . . . , xn , . . ., and for each arity, m, an infinite set of predicates,
Rm1 , . . . , Rn , . . . Call the union of the set of names and the set of variables the set of terms.
m

The set of logical vocabulary is {¬, ∧, ∃}. If ϕ is a formula, then ϕ[ti /tj ] is the result of
replacing every occurrence of tj in ϕ by ti . The set of formulas of the language is defined
recursively by

• n is an m-ary predicate, and t1 , . . . , tm are m-many terms, then Rn t1 , . . . , tn is a

If Rm m

formula.
• If ϕ and ψ are formulas then so are ¬(ϕ), (ϕ ∧ ψ), and ∃xi (ϕ)9
• Nothing else is a formula.

A sentence is a formula with no unbound variables.

Definition 1 (Position) Let  and  be sets of sentences.  ⇒  in the formal language

represents the position a person would be in if they were to assert all of  and deny all
of .

A deduction is also defined inductively

• All instances of Id are deductions.

• If δ1 , . . . , δn are deductions, R an n-premise rule that has the last positions of
δ1 , . . . , δn as premises and  ⇒  as a conclusion, then
δ1 δn
.. ..
. ... .
R
⇒
is a deduction.

A position,  ⇒  is incoherent iff there is a deduction of  ⇒ , written   ⇒ .

This generates the reading a rule of Figure 1, e.g. R
⇒
R
⇒
as indicating that if it is incoherent to assert all of  and deny all of , then it is incoherent
to assert all of  and deny all of . On this understanding, Id corresponds to the fact that
assertion and denial are exclusive speech acts. It is incoherent to assert and deny the same
sentence. WL and WR indicate that if a position is incoherent, then asserting or denying
more sentences will not change that.
R read contrapositively indicates that if it is coherent to assert all of  and deny all of
, then it is coherent to assert all of  and deny all of . Cut, on this reading, indicates
that if it is coherent to assert all of  and deny all of , then for any sentence ϕ, either it is
coherent to assert all of  and assert ϕ and deny all of , or it is coherent to assert all of 
and deny ϕ and all of . Cut thus entails that any coherent position can be expanded to a
coherent position that either asserts or denies each sentence of the language.

9
Parenthesis are dropped when convenient.
 Sellars, Second-order Quantification, and Ontological Commitment 11

Figure 1. First-order logic. ∗ t does not occur in the conclusion of L∃.

Similar to the Cut rule, the L∃ rule is a rule for the expansion of a position. The L∃ rule
read from bottom to top indicates that if , ∃xϕ ⇒  is coherent, then it is coherent to add
a new term, t, to one’s language and assert ϕ[t/x]. On this account of quantification, the
contribution that a quantifier makes to the sense of a sentence is to mark what is coherent
or incoherent in expansions of the language. The general role of quantifiers in a language,
on the view under consideration, is to account for the ways in which that language can be
expanded. L∃ indicates that if it is coherent to assert ∃xϕ, then it is coherent to expand the
resources of the language by a new name, c, and assert ϕ[c/x]. This is embodied in the
familiar pattern of reasoning that names what is existentially quantified over, e.g. moving
from ‘There is a set containing such and such numbers’ to ‘Call it ‘A’.’ and asserting ‘A
contains such and such numbers’. R∃ indicates that if it is coherent to deny ∃xϕ, then for
any term, t—even terms in an expansion of the language—it is coherent to deny ϕ[t/x].
This account of quantifiers thus offers a way of understanding one sense in which a lan-
guage that includes quantification has the resources to put restrictions on expansions of the
language. In the first-order case, quantifiers mark what is coherent and incoherent when the
language is expanded by new names. Second-order quantifiers mark what is coherent and
incoherent given expansions of the language by predicates. Sentential quantifiers cover the
case of expansions of the language by sentences.
It is more appropriate, on this account of quantification, to assign quantifiers the role of
generalizing rather than of marking the number of entities that fall under a kind, property,
attribute, etc. This particular role of first-order quantifiers is due to the fact that they are
generalizing over names. As is discussed below, that quantifiers have such ontological
significance is not essential to quantifiers as such but is a feature of those quantifiers that
generalize over grammatical expressions that have ontological significance.
Ontological commitment is a species of commitment. This paper agrees with the stan-
dard picture that the most straightforward way to be committed to Fs is to be committed
to the sentence ∃xFx. A position,  ⇒  is said to be committed to a sentence, ϕ iff
  ⇒ ϕ, , i.e. a position is committed to ϕ when it is incoherent to take up that position
and deny ϕ.
To come to an explication of the notion of an ontological commitment it is necessary to
consider all the ways that a coherent position can be expanded.  ⇒  is an expansion
12 A. Parisi

Figure 2. A sample tree with labeled lists.

of  ⇒  iff  ⊆  and  ⊆ . A position is maximal iff for each sentence, ϕ, it either

asserts ϕ or it denies ϕ. A maximal coherent expansion of a position is a way of com-
pleting that position’s account of everything. A maximal position takes a stand on every
sentence of the language. An expansion of a position asserts and denies all the sentences
that the original position asserts and denies, and possibly others. A coherent expansion of
a position is thus a coherent way of continuing the description that the original position
makes. The maximal coherent expansions of some position thus correspond to complete
characterizations of ways things could be given that the original position is adopted.
Let  ⇒  be a coherent position. The set of maximal coherent expansions of  ⇒ ,
ME( ⇒ ) is defined by a construction that makes use of both the Cut rule and the L∃
rule. Both rules guarantee ways of coherently expanding a position. The L∃ rule offers
a way of expanding a position by a new term, and the Cut rule guarantees that given a
coherent position, there is a maximal coherent expansion of that position. Using these
two rules a tree is constructed. The set of maximal coherent expansions of a position are
constructed using this tree.
In order to construct the tree, let ϕ1 , . . . , ϕn , . . . be an ordering on the sentences of L .
Let W be a denumerably infinite set of witnesses, and let w1 , . . . , wn , . . . be an ordering
on W. W is the set of expressions that account for the dynamic nature of language. They
are the terms that are not in the language of a coherent position, but are added to witness
the truth of existential sentences. Let LW and L∃ be two lists that at the beginning of the
procedure are empty. Each branch will have its own L∃ list. This list will be identified by a
superscript of a sequence of ‘l’s and ‘r’s. This index corresponds to the branch of the tree
to which that list belongs. For instance, the list Lrr ∃ is thus associated with the right branch
of the right branch of the tree. An example is given in Figure 2.
Sequences of ‘l’s and ‘r’s are abbreviated by ‘d’s. To determine the set ME( ⇒ ),
begin a tree whose root is  ⇒ . The leaves of the tree are those positions that do not
have any positions above them. A leaf of the tree,  ⇒ , is open iff   ⇒ , otherwise
it is closed. At stage n consider sentence ϕn and do the following for each open leaf  ⇒ 
of the tree:

• If for any sentence of the form ∃xϕ, ∃xϕ ∈  and ∃xϕ ∈ Ld∃ , then take the least
element of W ∈ LW , wn , expand the branch by

, ϕ[wn /x] ⇒ 
⇒
 Sellars, Second-order Quantification, and Ontological Commitment 13

Add wn to LW , ϕ to Ld∃ , and add the set of sentences formed using wn to the end of
the list of sentences.
• If  , ϕn ⇒  and   ⇒ ϕn , , the tree branches and becomes
, ϕn ⇒   ⇒ ϕn , 
⇒
If Ld∃ was associated with that branch, then Ldl
∃ is associated with the left branch, and
Ldr
∃ is associated with the right branch.
• If exactly one of  , ϕn ⇒  or   ⇒ ϕn ,  holds extend the branch by that
sequent.

Cycle through the sentences of the language in this way until no new sentences are added
to any open leaves of the tree. 
 Let  ⇒ , 1 ⇒ 1 , . . . , n ⇒ n , . . . be an open branch on this tree. Call i i ⇒
i i the maximal leaf of this branch.

Definition 2 (Maximal Coherent expansions) The set of maximal coherent expansions of

a position  ⇒ , ME( ⇒ ) is the set of maximal leaves after the process described
above is completed.

As discussed above, the most natural answer to the question ‘What is there?’ is ‘There
are K’s’, or some variant. The central question of ontology is thus best answered by indi-
cating what non-empty kinds there are. An account of ontological commitment requires an
account of what non-empty kinds a position is committed to. If ϕ is a sentence in which
a term t occurs, let ϕ[ξ/t] be a kind term. Intuitively, the kind-term ‘Fξ ’ corresponds to
being an F.
The maximal coherent expansions of a position are all the ways that things could
be given the truth of that position, i.e. that was it asserts is true and what it denies is
false. A position  ⇒  is said to be committed to a sentence ϕ iff for each  ⇒
 ∈ ME( ⇒ ), ϕ ∈ . A position is thus committed to a sentence if on any way of
coherently filling it in, ϕ is asserted. It is worth remarking that   ⇒ ϕ,  iff for any
 ⇒  ∈ ME( ⇒ ), ϕ ∈ . From this it follows that a position is committed to ϕ iff it
is incoherent, given that position, to deny ϕ. A position  ⇒  is committed to there being
entities of kind K iff for each  ⇒  ∈ ME( ⇒ ), K[t/ξ ] ∈  for some term or wit-
ness t. Given the construction of ME( ⇒ ) it follows from this definition that  ⇒ 
is committed to there being K’s iff it is committed to ∃xK[x/ξ ] iff   ⇒ ∃xK[x/ξ ], .10
Thus a position  ⇒  is committed to there being K’s iff   ⇒ ∃xK[x/ξ ], . The first-
order existential sentences to which a position is committed are those closely tied to the
ontological commitments of a position.
The ontological commitments of a position  ⇒  on this account, depend on what
sentences are in the set of maximal coherent expansions of  ⇒ , and what parts of
unquantified sentences are taken to be ontologically significant. On the above definition
kinds were taken to be of primary ontological significance. But it is consistent with this
view that whatever is named by a predicate has ontological significance as well. Whether
or not commitment to the sentence ‘∃ffa’ carries commitment to there being something
which the entity named by ‘a’ is or has, depends on whether commitment to sentences
of the form ‘Ha’ carry such a commitment. That will depend on what the right account
of the supposition of atomic sentences is. What determines the ontological significance of

10
A proof of these is offered in the appendix.
14 A. Parisi

a quantified sentence depends on what the ontological significance of its instances in an

expanded language are.
Sellars (1963b, 1983) goes to great lengths to provide a theory of predication whereby
predicates occurring in true sentences do not supposit. If both Sellars’s account of pred-
ication and the present treatment of quantification are correct, then there is no reason to
think that second-order quantifiers, when occurring as the main operator of a true sen-
tence, supposit. Thus, premise two of the Argument from Abstracta on this account is false.
The sentence ‘∃ffa’ does not mask commitment to properties, attributes, sets, classes, etc.
Coherently asserting ‘∃ffa’, does not require that there be some entity which a is or has. A
coherent assertion of ‘∃ffa’ only requires that there is a coherent expansion of the language
by a newly defined predicate H, such that it is coherent to assert ‘Ha’. This account of
quantification and ontological commitment serve Sellars’s purposes.
It is worth noting that it can also be seen as an account that can serve the purposes of
Wright (2007), who claims that the ontological commitments of a quantified sentence are
reducible to the ontological commitments of its instance. He calls this position ‘Neutral-
ism’. Wright’s larger abstractionist project requires that some sentences of second-order
logic such as Hume’s Principle,11 be analytically true. As noted above, on the model-
theoretic interpretation of quantification introducing second-order quantification into a
language requires the introduction of new entities. The argument against Wright’s position
can be summed up as follows:
(1) If the introduction of some expression into a langauge requires an expansion of the
ontology of that language then sentences involving that expression are not analytic.
(2) Introducing second-order quantification into a language requires the introduction
of abstract entities of some sort into the ontology of that langauge.
C. Therefore, sentences involving second-order quantification are not analytic.
It follows from the conclusion of this argument that sentences like Hume’s Principle are
not analytic. Since that argument is valid, Wright is forced to reject one of its premises. If
what I have argued in this section is correct, premise 2 is false. The introduction of second-
order quantification into a language does not require the introduction of abstract entities of
some sort into the intology of that language.

5. A grammatical interpretation of ‘∃ffa’

Sellars argues against several translations of ‘∃ffa’ into English on the grounds that they
are not grammatical. He offers various alternatives as translations, such as ‘a is some-
thing’, ‘a is somequale’, and ‘a is somehow’.12 It may be that none of these sentences are

11
Let # be a function from predicates to individuals, and H ≈ G be shorthand for

∃F(∀xy(Fxy → Hx ∧ Gy)∧

(∀xy(Fxy → ¬∃z(Fxz ∧ z = y)))∧

(∀yx(Fxy → ¬∃z(Fzy ∧ z = x))))

H ≈ G expresses in English that there is a one-one correspondence between the Hs and the Gs. Hume’s principle is stated as
∀HG(#H = #G ↔ H ≈ G)
12
In Grammar and Existence, Sellars says

To the first of each of these pairs there would correspond a general statement which would bear the mark of its origin,
thus,
(34) There is something which is quale Tom is (i.e. tall).
(35) There is something which is quid Tom is (i.e. a man)
or, more concisely,
 Sellars, Second-order Quantification, and Ontological Commitment 15

grammatical either. Their ungrammaticality, however, does not consist in translating ‘f ’

equivocally, sometimes as an adjective, sometimes as a common noun, and sometimes as a
name. Sellars’s suggested translations do not commit that grammatical error. Instead, they
are not obviously sentences of English. This may be acceptable given that natural language
is flexible, and they could be introduced as sentences of English with little trouble.
New sentences such as ‘a is somehow’ can be introduced into English and given a coher-
ent meaning. The introduction of such sentences would provide correlates for the sentence
‘∃ffa’. These sentences could then be used to assess whether the second-order quantifier,
when the main operator of a true sentence, supposits. If ‘a is somehow’ could be intro-
duced so that it is clearly a translation of ‘∃ffa’ and does not clearly entail a sentence like
‘There is some how or way that a is’, this could defuse worries that the sentence ‘∃ffa’ had
problematic ontological commitments.
Even with the introduction of new sentences, it is helpful to point out that such resources
are already available in English. It may be difficult to make clear what one might express by
asserting the sentence ‘Homer is somequale’ or ‘Fred is somehow’. The latter is worrying
because it looks like an incomplete sentence of English to which an interlocutor might
respond ‘Fred is somehow what?’13 These sentences are indicated in question and answer
dialogs of the sort that Sellars considers at the end of ‘Grammar and Existence’. Sellars
did not leave any clues that he considered the following translation. This should therefore
be seen as an amendment to Sellars’s views. Consider the following:
A : Bart is doing something that you won’t like.
B : What is Bart doing?
A : He’s being mean to Lisa.
The first sentence of the dialogue is the one containing the translation of the second-order
quantifier. For a small set of cases it is appropriate to translate ‘∃ffa’ as ‘a is doing some-
thing’. Appropriate specifications of the first sentence are phrases such as ‘being mean’,
‘being kind’, etc. The expression ‘is doing something’ is as much a part of English as the
expressions, ‘Lisa is doing something about the issue’, or ‘Gil is doing something new’.
The above dialogue suggests that it is not inappropriate to use the expression ‘doing some-
thing’ to generalize over ways that persons might be. ‘a is doing something’ is not itself
of an ontologically committing form as defined above. It also does not entail the sentence
‘There is some thing that a is doing’ which is of an ontologically committing form. The
latter is equivalent to the sentence ‘a is doing some thing’ the former is not.
If ‘a is doing something’ is a legitimate translation of ‘∃ffa’, this would dispel the worry
introduced at the end of Section 3.1 that there is no translation of ‘∃ffa’ into English that is
not of an ontologically committing form.

(341) There is somequale which Tom is (i.e. tall).

(351) There is somequid which Tom is (i.e. a man).

In Naturalism And Ontology he writes

Thus, instead of
Jones is something and Smith is also it
I will write
Jones is somehow and Smith is also it.

The latter of these was first suggested by Prior (1971), and later used by Rayo and Yablo (2001) as a translation of second-order
sentences.
13
One such interlocutor posed exactly this question to Sellars in the discussion period following Sellars’s first Dewey lecture see
https://www.youtube.com/watch?v = 6UiV-vMOueY. Sellars’s response is that ‘somehow’ is being used in a non-standard
way.
16 A. Parisi

6. Discussion
In Grammar and Existence and Naturalism and Ontology Sellars attempts to free up
second-order quantification from ontological commitments to abstract entities. His broader
nominalist views require that no abstract entities really exist. But, he makes use of higher-
order quantification in at least two parts of his philosophy: he gives the beginning of an
account of causation in Counterfactuals, Dispositions, and the Causal Modalities (see Sel-
lars 1957, p. 305), and an account of denotation in Science and Metaphysics14 These two
points require Sellars to offer some argument that higher-order quantification is ontologi-
cally innocent, or at least that it does not incur commitment to abstract entities. The first
and second sections of this article lay out Quine’s objection to the claim that second-order
quantification is ontologically innocent. There are two interconnected arguments for this
claim. The first rests on the claim that there is no translation of ∃ffa into a sentence of
English that avoids commitment to there being abstract entities of some sort. The second
argues that on the model theoretic understanding of ontological commitment all quantifiers
supposit. Since the most natural candidate for what second-order quantifiers supposit for is
abstract entities, a language that makes use of second-order quantification is thereby com-
mitted to there being abstract entities. Sellars argues against the first objection by showing
that it rests on a false premise. He argues against the second by sketching a non-model-
theoretic conception of quantification, according to which quantifiers have an important
connection to ways that our language is constrained to expand or contract.
In the second half of the paper I offer a formal proof-theoretic account of the meaning of
quantifiers that underwrites remarks made by Sellars in Naturalism and Ontology regard-
ing the meaning of quantifiers. I prove that on this account of quantification it is possible to
introduce second-order quantifiers into a language without thereby incurring commitment
to abstract entities, or any new entities at all. Sellars’s objection to the argument resting on
the premise that there is no translation of second-order quantified expressions into English
rests on the claim that the word ‘something’ does not always supposit. Sellars gives vari-
ous translations that require the introduction of new vocabulary into English. In the final
section of the paper I offer my own translation of second-order quantifiers using resources
available in English to dispel the concern that there is no grammatical translation of ∃ffa
into English.

References
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Philosophy, 81 (8), 430–49.
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Language, Amethe Smeaton. (1937). Routledge.
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14
For more information see Sellars (1968, §63, p. 84).
 Sellars, Second-order Quantification, and Ontological Commitment 17

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Appendix A. Proof of claims

Lemma A.1 If  ⇒  ∈ ME( ⇒ ), then  ⊆  and  ⊆ 

Proof There is no step in the construction of ME( ⇒ ) that removes sentences from  ⇒ . So every
element of ME( ⇒ ) is an expansion of it. 

Theorem A.1   ⇒ ϕ,  iff for every  ⇒  ∈ ME( ⇒ ), ϕ ∈ , for a sentence ϕ containing no
witnesses.

Proof For the left to right direction, suppose that   ⇒ ϕ, . Suppose that there is a  ⇒  ∈ ME( ⇒ )
such that ϕ ∈ . By Lemma A.1,  ⊆  and  ∪ {ϕ} ⊆ . Since   ⇒ ϕ,  by weakening   ⇒ . But
then  ⇒  ∈ ME( ⇒ ). So there is no  ⇒  ∈ ME( ⇒ ) such that ϕ ∈ . Let  ⇒  ∈ ME( ⇒
). Since ϕ ∈  and  ⇒  is maximal, ϕ ∈ .
For the right to left direction suppose that for every  ⇒  ∈ ME( ⇒ ), ϕ ∈ . But also suppose that
  ⇒ ϕ, . The only way for this to be the case is that at the stage, call it n, where ϕ was considered, there was
no open leaf, ⇒ such that  ⇒ ϕ, . Since   ⇒ ϕ, , there must be a stage earlier than n, m, such
that it was still the case that   ⇒ ϕ,  , but that at stage m + 1   ⇒ ϕ,  . Let each time a witness is
added for an existential sentence count as a sub-stage. The sub-stages of a stage i are i.1, i.2, . . .. Let the sentence
in consideration at stage m + 1 be ψ. There are four cases to consider.
Case 1. Suppose that this happens at stage m.i. Since witnesses are added there is a sentence, ∃xψ ∈  , such
that   , ∃xψ ⇒ ϕ, but   , ψ[wi /x] ⇒ ϕ, . However, since, wi does not occur in ϕ, it is guaranteed by
L∃ that   , ψ[w/x] ⇒ ϕ, . So this cannot be where the leaf became committed to ϕ.
Case 2. Suppose that   ⇒ ϕ,  , that both   , ψ ⇒  and   ⇒ ψ,  . From the assumption it
follows that both   , ψ ⇒ ϕ,  and   ⇒ ψ, ϕ,  . But then by Cut   ⇒ ϕ, , contradicting the
assumption.
Case 3. Suppose that   ⇒ ϕ,  , and that both   , ψ ⇒  and   ⇒ ψ,  . By assumption 
 , ψ ⇒ ϕ,  . But then by Cut   ⇒ ϕ, , a contradiction.
Case 4. Suppose that   ⇒ ϕ,  , and that both   , ψ ⇒  and   ⇒ ψ,  . By assumption   ⇒
ψ, ϕ,  . But then by Cut   ⇒ ϕ, , a contradiction. 

Theorem A.2 If for each  ⇒  ∈ ME( ⇒ ), K[t/ξ ] ∈  for some t, then   ⇒ ∃xK[x/ξ ], .

Proof Suppose that for each  ⇒  ∈ ME( ⇒ ), K[t/ξ ] ∈  for some t. Suppose that there is a  ⇒  ∈
ME( ⇒ ) such that ∃xK[x/ξ ] ∈ . Since  ⇒  is maximal, ∃xK[x/ξ ] ∈ . Thus, there is some finite stage
on that branch with position, ⇒ , where K[t/ξ ] ∈ for some t and ∃xK[x/ξ ] ∈ . The following deduction
establishes that that branch is closed

Id
K[t/ξ ] ⇒ K[t/ξ ]
R∃
K[t/ξ ] ⇒ ∃xK[x/ξ ]
WL/WR
, K[t/ξ ] ⇒ ∃xK[x/ξ ],

This contradiction the assumption that  ⇒  ∈ ME( ⇒ ). Thus, there is no such  ⇒ . So for each
 ⇒  ∈ ME( ⇒ ), ∃xK[x/ξ ] ∈ . By Theorem A.1,   ⇒ ∃xK[x/ξ ], .