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Published in:

Existence as a property of individuals, 2014, in: Erkenntnis, 79, 503-523 [peer reviewed]. Online:http://dx.doi.org/10.1007/s10670-013-9505-9.

Existence as a property of individualsDolf Rami

Georg-August-Universität Göttingen

[penultimate draft; please only cite the published version]

ABSTRACT – In this paper I aim to defend a version of the view that ‘exist’ expresses primarily a property of

individual objects, a property that each of them has. In the first section, I will distinguish the three main types

of rival conceptions concerning the semantic status of ‘exist’ that will define the subsequent discussion. In the

second section it will be shown that the best explanation of our overall use of ‘exist’ in natural language

requires the treatment of ‘exist’ as a predicate that can be applied to individual objects. The third section will

be concerned with the problem of the semantic analysis of negative singular existential sentences that contain

proper names. I will briefly defend a specific solution to this problem that makes use of a single-domain free

logic. In the following sections, I will try to defend this view of existence in more detail. That is, I will deal with

three different specific challenges that prima facie seem to speak against the proposed view and for some rival

alternatives; and I will show how these challenges can be met.

(1) Different general views on existence

There are three main rival conceptions concerning the semantic status of the verb ‘exist’.

Two distinctions are required to draw the main differences between these three

conceptions. Firstly, a first-order property is a property of objects, but not of properties. A

second-order property is a property of first-order properties. Secondly, a property is a

discriminating property if it is possible that (there is something that exemplifies this property

and something that does not exemplify this property). A non-discriminating property is a

property that can only be exemplified by everything or by nothing.

Against this background, we can now distinguish three general views on existence: (i) the

first-order non-discriminating property view of existence, (ii) the first-order discriminating

property view of existence and (iii) the second-order discriminating property view of

existence.

There are several different sub-varieties especially of the first two views. Let me now

introduce those versions of these views that will be relevant for our subsequent discussions.

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A much-discussed version of the first-order non-discriminating property view of existence

holds that ‘exist’ is semantically equivalent to ‘identical with something’ interpreted as

‘x(x=y)’. It is important to distinguish at least two different variants of this view: A first

variant that is based on classical first-order predicate logic with identity; a second that is

based on some variant of a universally free single-domain predicate logic.

A single-domain free logic shares with classical predicate logic the view that there is only one

kind of domain of discourse, but a universally free variant of free logic allows in opposition

to classical predicate logic that this domain is empty and it makes use of a partial instead of a

total interpretation function. In opposition to classical logic, every free logic assumes that

the existence predicate, like the identity predicate, has a distinctive logical role to make

certain inferences valid that contain this predicate.1 Against this background, it is possible to

introduce a primitive logical existence predicate ‘E!’ with a fixed contribution to truth-

conditions. For our purposes we must specify the contribution of ‘E!’ to the truth-conditions

of atomic sentences in such a way that it is compatible with the desired interpretation of the

existence predicate. That is, such a specification must contribute to the vindication of the

logical truth of the equivalence:

(1) x(E!x $y(y=x)).

There are different ways to achieve this goal. The two most notable are either based on a

bivalent negative free logic2 or a specific three-valued non-standard neutral free logic. Both

of these views make use of the following interpretation of the logical predicate ‘E!’:

V(E!t)=T iff I(t) is designated.

V(E!t)=F iff I(t) is not designated.

They complement this interpretation with the following moderate modification of the

classical truth- and falsity-conditions of atomic sentences that contain the identity predicate:

1 In classical predicate logic, we can directly derive instances from universal generalizations. So for example‘xFx’ implies ‘Fa’. This rule of universal instantiation is not valid in single-domain free logic. It requires theadditional existential assumption ‘E!a’ to derive ‘Fa’ from ‘xFx’. In classical predicate logic, we can also deriveexistential generalizations from atomic sentences – for example ‘Fa’ implies ‘xFx’. This rule of existentialgeneralisation is not valid in single-domain free logic. It requires the additional existential assumption ‘E!a’ toderive ‘xFx’ from ‘Fa’. These examples show why the existence predicate ‘E!’ is required for logical reasonsand therefore has to be conceived as a logical predicate in single-domain free logic.2 C.f.: Burge (1974); Sainsbury (2005, 64-75); Nolt (2010, §3.1).

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V(s=t)=T iff I(s)=I(t).

V(s=t)=F iff I(s) is not designated or I(t) is not designated or I(s)I(t).

There are significant differences between these two variants of single-domain free logic

when it comes to the interpretation of the semantics of (i) quantifications, (ii) logical

connectives and (iii) non-logical atomic sentences.3 Negative free logic interprets

quantifications and logical connectives in the very same way as classical first order predicate

logic.4 Non-standard neutral free logic interprets quantifications like Bochvar's three-valued

predicate logic5 and the logical connectives in the same way as Bochvar's external three-

valued logic.6 Negative free logic conceives of every atomic sentences that contains at least

one individual constant without a denotation as false. And it therefore slightly alters the

falsity-conditions of atomic sentences according to classical predicate logic. Non-standard

neutral free logic conceives of every atomic sentence that contains a non-logical predicate

and at least one individual constant without a denotation as neither true nor false. And it

therefore adds a neutrality-condition to the truth- and falsity-condition of classical logic for

non-logical atomic sentences.7

According to the first-order non-discriminating property view of existence based on classical

first-order predicate logic, we can identify the extension of the existence predicate ‘E!’ with

the domain of discourse or alternatively we can define it by means of the equivalence (1).

Against this classical background, the existence predicate is not a logical predicate in the

strict sense, because it does not fulfil any significant inferential role. But it can be called a

logical predicate in a looser sense, because it can be defined by means of purely logical

notions.

There are also different versions of the first-order discriminating property view of existence

based on different logical frameworks. The most notable two variants are either based on

classical first-order predicate logic or a dual-domain free predicate logic. This kind of free

logic assumes an outer and an inner domain of discourse. The inner domain is possibly

empty, the outer domain cannot be empty. The range of bound variables is identical with the

3 A non-logical atomic sentence is an atomic sentence that contains a non-logical predicate.4 C.f.: Nolt (2010, §3.1).5 C.f.: Malinowksi (1993, 80).6 C.f.: Malinowksi (1993, 55).7 The main differences between our two distinguished versions of single-domain free logic concern the truth-values of non-logical atomic sentences that contain constants without denotations like ‘Vulcan is a planet’ andspecific quantifications that also contain constants without denotations like ‘No planet is larger than Vulcan’. Inany other case, they provide equivalent truth- and falsity-conditions.

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inner domain of discourse. The interpretation function is a total function that assigns objects

of the total domain, which is the union of the inner and the outer domain, to individual

constants.8 We can specify the contribution of a logical existence predicate ‘E!’ to the truth-

and falsity-conditions of atomic sentences that contain this predicate in the following way:

V(E!t)=T iff I(t) of the inner domain of discourse.

V(E!t)=F iff I(t) of the inner domain of discourse.

This dual-domain free logic interprets the identity predicate in the classical way. The truth-

and falsity-conditions of quantifications of classical predicate logic are only slightly altered,

insofar as their range is restricted to the inner domain. Against this background, the

equivalence (1) holds again, but it again has a different meaning. That is, we can distinguish

at least three different views that claim that ‘exist’ is semantically equivalent with ‘is

identical with something’, because there are at least three rival interpretations of the

semantics of the expression ‘is identical with something’.

The classical variant of the first-order discriminating property view of existence assumes that

the particular and the universal quantifier have their classical semantics, nevertheless they

both do not have existential import, because they range over a non-empty domain that may

contain existent and non-existent objects. The first-order existence-predicate ‘E!’ is

identified with a possibly empty proper sub-set of the domain of discourse. On this basis, the

existence predicate is neither conceived of as a logical predicate in the strict nor in the loose

sense. It is possible to introduce relative to such a framework a second set of standard

quantifiers with existential import, whose range is restricted to the extension of the

predicate ‘E!’.9

There is only one variant of our third view that can be found in the relevant literature, but

this view can be introduced in different but equivalent ways. According to the first

formulation, ‘exist’ is identified with a second-order predicate ‘E!!(P)’ which is defined in the

following way: If applied to a first-order predicate ‘P’, ‘E!!(P)’ is true if ‘P’ has a non-empty

extension. ‘E!!(P)’ is false, if ‘P’ has an empty extension. An alternative, but equivalent

formulation holds that ‘exist’ is represented on the level of logical form by the classical first-

order existential quantifier interpreted in the standard model-theoretic way. On this basis, it

8 Cf.: Priest (2008, 190-191); Nolt (2010, §3.2).9 C.f.: Priest (2008, 295-297).

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is possible to define the second order predicate ‘E!!()’ in the following way: E!!(P) =df xPx.

This shows that there are at least two equivalent ways to characterize this third view.

Our distinction between three different general views on existence is not exhaustive. The

distinguished views have one thing in common: they assume that ‘exist’ is a non-ambiguous,

context-insensitive expression that expresses relative to each sentential context and each

context of use the same property. Therefore, we can at least distinguish a notable fourth

view on existence that holds that ‘exist’ is either ambiguous or context-sensitive and may

therefore express different properties relative to different uses of ‘exist’. One might for

example distinguish a logical from an ontological reading of ‘exist’ and identify the first

reading with a first-order non-discriminating property and the second with a first-order

discriminating property.10 Alternatively one might hold that ‘exist’, if it is combined with

definite noun phrases, expresses a first-order property and if it is combined with any, or only

certain, indefinite noun phrases, it expresses a second-order discriminating property.

Methodologically, such a view seems to be only acceptable if there are good reasons for

such disunification and no defensible alternative can be provided by the three uniform

views. In the following, I will try to show that there are good reasons that favour the first-

order non-discriminating view of existence based on single-domain free logic over any of its

possible rivals.

(2) Existence as a first-order property

From a purely syntactic point of view there is no significant difference between the

expression ‘exist’ and any other verb that clearly functions semantically as first-order

predicate. Normally, if there are two expressions of a certain syntactic category that have a

significantly different kind of contribution to the truth-conditions of a sentence this

difference is marked by diverging syntactic properties. So it would be very surprising if ‘exist’

provides an exception in this respect.

Let us now focus on this matter from a semantic point of view: Are there any good reasons

that in principle favour a first-order analysis of ‘exist’ over a second-order analysis? As Evans

has pointed out there seems to be strong evidence in favour of the first-order analysis. He

10 C.f.: Williamson (2002, 244-245).

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even makes the stronger claim that there is evidence that favours a first-order non-

discriminating view. 11

The following examples seem to provide a challenge for a second-order analysis, but a first-

level analysis can easily account for the intuitive truth-conditions of these sentences:

(2) Something exists.

(3) Everything exists.

(4) Something both exists and is concrete.

(5) Every man exists.

(6) Pedro is an existent donkey.

According to our proposed first-order analysis these sentences receive the following logical

forms:

(2’) xE!x

(3’)xE!x

(4’) x(E!x Cx)

(5’)x(Mx E!x)

(6’) (Dp E!p)

There are, nevertheless, significant differences concerning the evaluation of the truth-values

of these logical forms relative to classical, single-domain or dual-domain free logic.

According to our classical variant of the first-order non-discriminating view, both (2’) and (3’)

come out as logical truths. This is a counterintuitive result. Independently of the fact

whether we favour a first-order discriminating or non-discriminating view it should be

avoided.

If we apply our classical variant of the first-order discriminating view to (2’) and (3’), it is

possible to find an interpretation where both are true or false. Similar results can be

achieved by a specific interpretation of particular and universal generalisations based on a

single-domain or dual-domain free logic.12 There are no significant differences between our

11 C.f.: Evans (1982, 345).12 We have to alter the truth-conditions for universal generalisations as follows: VI(xA) = T iff (D/DI= and forevery interpretation I* differing from I at most in what it assigns to t, VI*(A[t/x]) = T) or (D/DI ≠ and for everydefined interpretation I* differing from I at most in what it assigns to t, VI*(A[t/x]) = T); otherwise VI(xA) = F.

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different logical frameworks when it comes to the evaluation of the truth-values of (4’) and

(6’). (5’) comes out as true according to both versions of the first-order non-discriminating

view and the sketched dual-domain free logic. There are variants of the classical version of

the first-order discriminating view that assign truth to (5’), others that assign falsity to (5’)

because they assume that there are also non-existent men.

In any case, a defender of a first-order analysis of ‘exist’ has the resources to account for the

meaningfulness of sentences like (2)-(6) and at least in most cases also to account for their

intuitive truth-values. Does a second-order view concerning existence have the same

explanatory resources? A second-order view obviously cannot make use of the logical forms

(2’)-(6’) to account for the truth-conditions of (2)-(6). Are there alternative ways to achieve

the same result?

Intuitively, there is no significant structural semantic difference between (2) and ‘Something

is extended’ and (3) and ‘Everything is extended’. But a defender of the second-order view

cannot accept the correctness of this intuition. He has two options: Firstly, he might question

that claims like (2) and (3) are well-formed and meaningful. Secondly, he might claim that

‘something’ and ‘everything’ are ambiguous. The first option does not seem to be viable. It is

undeniable that (2) and (3) are meaningful and well-formed. But also the second option is

problematic. Surely, there are the formal resources to substantiate this claim. A defender of

the second-order approach might claim that ‘something’ and ‘everything’ may either

represent first-order or second-order quantifiers. In case of (2) and (3) they represent

second-order quantifiers. Against this background we can assign to (2) either the logical form

‘FxFx’ or ‘F(E!!(F))’; and to (3) either ‘FxFx’ or ‘F(E!!(F))’. But there are two problems

with such an account. Firstly, such a move seems to be ad hoc. A defender of this view must

provide further evidence for this kind of ambiguity. But it is not clear whether such evidence

really exists. Secondly, this view cannot account for a reading according to which (2) is false

and (3) is true. On the basis of the given interpretation (3) is false, because there are

predicates with empty extension, and (2) is a logical truth. This is a significant disadvantage

of the given account, because intuitively (2) seems to have a false and (3) seems to have a

true reading and every alternative view can account for these readings.

The problems of the second-order view become more severe, when it comes to the analysis

of sentences like (4)-(6). The mentioned ambiguity view cannot be used to account for the

truth-conditions of (4), because ‘is concrete’ is without doubt a first-order predicate and so if

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(4) is meaningful, then ‘something’ has to be interpreted as a first-order quantifier. And the

same holds in case of (5). ‘is a man’ is unquestionably a first-order predicate. (5) is without

doubt a meaningful sentence. But the defender of the second-order view does not have the

resources to account for this fact and the correct truth-conditions of (4) and (5).

Furthermore, the second-order view cannot explain why the adjective ‘existent’ that can be

derived from the verb ‘exist’ has an intersective reading in (6) that can be represented by

(6’).

So far things seem to be pretty bad for a defender of the second-order view of existence. But

maybe things are not as bad as they look at first sight if one takes into account an interesting

interdependence between first-order and second-order properties of existence. It is possible

to define a first-order predicate ‘E!’ in terms of our second-order predicate ‘E!!’ in the

following way:13

(7) x(E!x E!!(y(y=x)))

On this basis we can now reformulate the logical forms (2’) - (6’) in such a way that these

forms explicitly only contain our second-order predicate ‘E!!’. Does this show that the

second-order view is in a better position than it seemed at first sight? Not at all. It only

shows that the second level view in fact seems to be based on a misconception.

Firstly, (7) is only a stylistic variant of (1). That is, even on the basis of (1) there is some sort

of conceptual connection between the notion that the second-order view identifies with the

notion of existence and our proposed first-order view of existence. Secondly, the expressions

‘E!!(y(y=x))’ and ‘$y(y=x)’ are themselves first-order predicates. The first expression

contains a second order predicate, the second an existential quantifier. But if we represent

the natural language predicate ‘exist’ by means of one of these two first-order predicates on

the level of logical firm we thereby confirm the view that ‘exist’ expresses a first-order

property. So the mentioned transformations of (2’) - (6’) cannot be used in favour of a

second-order view of existence. But this possible transformations show that the second-

order view made a mistake by identifying our ordinary expression ‘exist’ with the second-

order predicate ‘E!!’ or the existential quantifier. They cannot be used alone to represent

this expression on the level of logical form. They only can be used as part of a more complex

expression to do this job. So, in some sense it is also misleading to call ‘E!!’ an existence

13 Thanks to an anonymous referee for pointing this out.

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predicate, because if we conceive our ordinary notion of existence as the defining notion of

what existence is, then ‘E!!’ should not be regarded as an existence predicate. We get further

confirmation for this view if we compare (7) with a natural language paraphrase of (7):

(8) x(x exists the property of being identical with x has a non-empty extension).

This paraphrase contains the expression ‘has a non-empty extension’ as an informal

counterpart of the second-order predicate ‘E!!’. This expression is related in an important

way to our ordinary notion of existence, as shown by (8), but it is not itself a representation

of our ordinary notion of existence. It is our theoretical goal to analyse the semantics of our

ordinary notion of existence as it is represented by the verb ‘exist’ and in this respect the

second-order view fails.

Interestingly, one can also clearly identify the source of this misconception that led

defenders of the second-order analysis to their erroneous view. Firstly, defenders of this

view mainly focus on a specific set of examples to justify their view explicitly. Secondly, they

provide an analysis of these examples on the basis of a false analogy with other predicates

with a peculiar semantics. Thirdly, they use the problem of singular existential sentences as

an implicit justification for a second-order view. But we have already shown that the

problem of singular existentials does not provide any good reasons for a second-order view

of existence and that there are better ways for a first-order view to deal with this problem.

In order to justify their semantic views defenders of a second-order view mainly cite bare

plural sentences containing ‘exist’ such as:

(9) Horses exist.

They justify their analysis of sentences like (9) by comparing them to other bare plural

sentences. Namely, sentences like the following:

(10) Horses are numerous.

But it is an error to claim that (9) and (10) have a similar semantics. ‘is numerous’ clearly is

not a predicate that one can meaningfully ascribe to individuals. The sentence ‘Peter is

numerous’, for example, is odd. It is not only singular sentences with ‘is numerous’ that are

problematic; generalisations like ‘Something is numerous’ or ‘Everything is numerous’ are

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not meaningful either. This shows that ‘is numerous’ has a very specific semantics and that

the semantic peculiarity of this expression is also reflected in its syntactic properties. That

not only distinguishes ‘is numerous’ from other normal first-order predicates, but it also

distinguishes this expression from ‘exist’. We can conceive of ‘is numerous’ as second-order

predicate and analyse (10) roughly in the following terms:

(10’) The property of being a horse has a relatively large number of instances.

But there is no good evidence for transferring such a second-order analysis to sentences like

(9). On the contrary, the significantly different syntactic profile of ‘exist’ and ‘is numerous’

suggests that there is an important semantic difference between these expressions.

Therefore, the second-order view seems to be based on a one-sided diet of examples and

false analogy. This fact also explains why a second-order view has problems providing an

analysis of sentences like (2)-(6).

(3) The problem of singular existential sentences

The semantic analysis of singular existential claims that contain proper names provides a

challenge for each of the mentioned views on existence. Such an analysis should be able to

account for the truth-values that we intuitively assign to sentences like:

(11) Vulcan does not exist.

(12) Barack Obama does not exist.

Intuitively, (11) is true, while (12) is false. But it is not easy to account for this data if we take

further plausible assumptions into consideration. Firstly, ‘exist’ seems to have the same

syntactic properties as any typical first-order predicate – like ‘sleep’ for example. Secondly,

according to the classical semantics of simple atomic sentences that contain a first-order

monadic predicate and a proper name, it is a precondition of the truth and the falsity of such

a sentence that this proper name refers to something. Thirdly, if a simple atomic sentence is

true then its negation is false and vice versa. Fourthly, existence seems to be a necessary

precondition of reference.

These four prima facie plausible assumptions are not compatible with the mentioned data

concerning singular existential sentences like (11) and (12). If we apply, for example, our

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classical variant of the first-order non-discriminating view on existence to these examples,

both claims come out as logically false. The best a defender of this view may do to account

for our data is to go for some hybrid view concerning singular existential sentences. But such

a view needs a good independent justification to avoid the accusation of being ad hoc. And it

seems to be difficult to provide such justification, because there seems to be no good

evidence that sentences like (11) or (12) have, in general, two different readings.

A defender of a first-order discriminating view on existence may try to solve our problem by

sacrificing the assumption that existence is a precondition of reference. Intuitively, this

seems by itself difficult to swallow, and it also has some problematic consequences. Firstly, it

requires postulating certain specific entities that can be used to fill the anti-extension of

‘exist’. And it is difficult to find an independent motivation for such an ontologically

extravagant move. Secondly, such a view excludes the existence of proper names without a

semantic referent for conceptual reasons. This seems again to require a good independent

justification. Furthermore, there seems to be a specific class of empty names that an account

based on a first-order discriminating view on existence cannot plausibly explain away: Empty

names that are introduced into use by an unrecognized unsuccessful descriptive act of

naming.14 Such names can have an established use and it seems to be implausible and ad hoc

to claim that these names have a semantic referent. Thirdly, to account for several

consequences of this view it might be necessary to postulate certain ambiguities or a

problematic semantics concerning expressions like ‘some’, ‘something’ and ‘there are’. That

is, the rejection of our fourth assumption seems to lead to a bunch of further problems.

Therefore, neither of our two distinguished variants of the first-order discriminating view on

existence seem to be good candidates when it comes to an analysis of (11) and (12).

A more promising strategy to solve the problem concerning singular existential sentences

can be provided by the rejection of the second of our four intuitive assumptions. From a

syntactic point of view ‘exist’ might be indistinguishable from other first-order predicates

like ‘sleep’ or ‘run’, but this does not mean that these predicates that form a homogenous

syntactic class of expressions share the same kind of semantics. The contribution of ‘exist’ to

the truth-conditions of simple atomic sentences might be in significant ways different from

the contribution of ‘sleep’, for example. Such differences would not be a surprising

14 A descriptive act of naming is an act of naming where an attributively used definite description is used to pickout the intended bearer of a name. Such an act is unsuccessful if there is no object that satisfies thisdescription.

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exception. From a syntactic point of view the relational expressions ‘hit’ and ‘admire’ seem

to belong to the same syntactic category, nevertheless there are important semantic

differences between an extensional verb like ‘hit’ and a hyper-intensional verb like ‘admire’.

Any of the two mentioned versions of a single-domain free logic provides the tools to

account for the desired semantic differences between ‘exist’ and other monadic predicates

in a plausible way. The two most notable features of such an account are: (a) it can be used

to provide a formal semantic analysis of sentences like (11) and (12) that can account for

their intuitive truth-values and (b) it can be used to put forward a first-order non-

discriminating view of existence that provides a plausible and economical solution to the

problem of singular existential sentences.

The main question that I now want to address in the remainder of the paper is whether the

proposed analysis of the existence predicate can be used not only to provide an analysis of

singular existential sentences that contain proper names, but whether it can also be used to

provide an analysis of the overall use of ‘exist’ in natural language. That is, we separately

have to assess whether the proposed analysis of ‘exist’ can be used to provide an analysis of

non-singular existential sentence that make use of quantifier phrases like ‘something’ or

bare plurals like ‘horses’.

(4) The first challenge: Existential sentences with bare plurals

The first challenge for our view that existence is a first-order non-discriminating property

based on a single-domain free logic concerns the use of ‘exist’ in connection with bare

plurals noun phrases like ‘horses’ or ‘dogs’. Bare plurals, as I use this expression here, are

unmodified nouns that function as grammatical subjects of sentences in their plural form.

Defenders of a second-order view on existence have claimed that (i) a sentence like ‘Horses

exist’ is semantically equivalent to ‘There are horses’ and that (ii) the latter claim has the

logical form ‘xHx’.15 The acceptance of these two claims seems to vindicate an analysis of

‘exist’ as a second-order predicate.

In principle, there are two possible reactions to the proposed second-order view: Firstly, one

may reject the equivalence thesis (i) on the basis of a semantic analysis that draws a

significant difference between the truth-conditional contributions of ‘exist’ and ‘there are’.

15 Or: alternatively the logical form ‘E!!(H)’.

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Secondly, one may accept the equivalence thesis (i), but doubt that the acceptance (i)

commits one to the proposed semantic analysis of sentences like ‘Horses exist’ and explain

the equivalence on the basis of an alternative analysis. I will now try to show why the second

choice is the best rejection to our challenge.

‘Horses exist’ is an example of a sentence that contains a bare plurals as its grammatical

subject. The analysis of the truth-conditions of sentences of this kind is in general a difficult

issue, because sentences of this grammatical variety can have very different kinds of truth-

conditions. Firstly, we can distinguish sentences of this kind with a second-order reading like

‘Horses are rare’. Such a sentence does not attribute a certain property to individual horses,

but it says of the property of horses that it presently has very few instances. Secondly, we

can distinguish different first-order readings. Philosophers, typically, distinguish three kinds

of readings of this variety: universal, generic and existential (or indefinite) readings.16

Linguists, in general, distinguish only two main kinds of reading: generic and existential (or

indefinite), because they conceive the universal reading a specific sub-variety of the generic

reading.17 ‘Horses are animals’ is a paradigm example of a sentence with a universal reading,

because it is necessarily equivalent with the sentence ‘Every horse is an animal’. ‘Horses are

nice pets’ is an example of a sentence with a generic reading, because it is necessarily

equivalent with the sentence ‘Usually, if something is a horse, it is a nice pet’. A sentence like

‘Horses run around on Peter’s farm’ is a paradigm example of a sentence with an existential

reading, because it is necessarily equivalent with the sentence ‘Some horses run around on

Peter’s farm’.

So which kind of reading should we attribute on this basis to bare-plural sentences that

contain ‘exist’? A defender of the second-order view holds that sentences of this kind always

have a specific second-order reading. I will now present two arguments in favour of the

thesis that these kinds of sentences always have a first-order existential reading and

vindicate on this basis the proposed reaction to our challenge.

The first argument runs as follows: Firstly, only bare plural sentences with an existential

reading are upward entailing.18 A sentences of the form ‘Fs are G’ is upward entailing iff any

argument of the following form that contains such a sentence as first premise has only valid

instances:

16 C.f.: Priest (2009, 238).17 C.f.: Leslie (2012, 355-357).18 C.f.: Leslie (2012, 356).

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Fs are G.

Every F is an H.

Hs are G.

I leave it to the reader to check why arguments of this kind are only in the case of bare plural

sentences with an existential reading valid and have in the case of the other three

mentioned readings invalid instances. Secondly, sentences of the form ‘Fs exist’ are upward-

entailing. Therefore, sentences of this form always have an existential reading.

My second argument is based on the following assumptions: Firstly, only in the case of a

sentence of the form ‘Fs are G’ that has an existential reading the predicative negation of

this sentence ‘Fs are not G’ is semantically equivalent with a corresponding sentence of the

form ‘No F is a G’. I leave it to the reader to check why this criterion can only be applied to

bare plural sentences with an existential reading and not to those with a second-order,

universal, or generic reading. Secondly, a sentence of the form ‘Fs do not exist’ is the

predicative negation of a sentence of the form ‘Fs exist’. Thirdly, sentences of the form ‘Fs

do not exist’ are semantically equivalent with corresponding sentences of the form ‘No F

exists’. Therefore, bare plural sentence with ‘exist’ have an existential reading.

This analysis of bare plural sentences with ‘exist’ perfectly fits with a first-order view of

existence based on a single-domain free logic. Against the background of this framework,

sentences of the form ‘Fs exist’ are not only semantically equivalent with sentences of the

form ‘Some Fs exist’, they also have the following logical form: x(Hx E!x). Such an analysis

also fits more properly than the proposed second-order analysis with the general semantic

and syntactic features of ‘exist’.

On this basis, we can now also explain why the second-order analysis at least seemed to

provide a plausible analysis of the truth-conditions of a sentence like (12); namely because

the logical form of our proposed solution of such a sentence is necessarily and logically

equivalent to the logical form that is proposed by the second-order view. In this sense both

analyses make correct predictions concerning the truth-values of sentences like (12), but

only our analysis is compatible with the overall use of ‘exist’ in natural language.

15

(5) The second challenge: particular generalisations with the existence predicate

In this section I will focus on a challenge that can be put forward by two possible variants of

the classical version of the first-order discriminating view of existence against our favoured

first-order non-discriminating view of existence.

The first variant holds the following additional theses: Firstly, sentences of the grammatical

form ‘There are Fs’ or ‘Something is an F’ have the logical form ‘xFx’. Secondly, sentences of

the grammatical form ‘There are Fs that are G’ or ‘Some Fs are G’ have the logical form

‘x(Fx Gx)’.19

The second variant makes different assumptions concerning the formal interpretation of

claims that contain ‘there are’, ‘some’ and ‘something’: Firstly, sentences of the grammatical

form ‘There are Fs’ have the logical form ‘x(Fx E!x)’. Secondly, sentences of the

grammatical form ‘Something is an F’ have the logical form ‘xFx’. Thirdly, sentences of the

grammatical form ‘There are Fs that are G’ have the logical form ‘x(E!x (Fx Gx))’.

Fourthly, sentences of the grammatical form ‘Some Fs are G’ have the logical form ‘x(Fx

Gx)’.20

Defenders of the first variant of the classical version of the first-order discriminating view of

existence draw our attention to specific example sentences that we intuitively accept as true

and that seem to vindicate their proposed analysis of the existence predicate. These are

sentences like the following:

(13) There are things that do not exist; namely, witches and unicorns.

They claim that a sentences like (13) has a true reading and the best way to explain the truth

of this reading is provided by the semantic framework they advocate. That is, we can assign

to a claim of the form ‘There are Fs that are not G’ like (13) the following logical form:

(13’) x(Tx E!x)

Such a claim has a true interpretation on the basis of the outlined classical version of the

first-order discriminating view of existence.

19 C.f.: Parsons (1982, 365-366).20 C.f.: Priest (2005, 13-14); Priest (2008, 295-296).

16

Defenders of the second variant of the classical version of the first-order discriminating view

of existence deny that a claim of the grammatical form (13) has a true reading, but they offer

a similar example to vindicate their specific view. Namely, sentences like the following:

(14) Some things do not exist; namely, witches and unicorns.

According to them, such a sentence has a true reading and the best way to explain the truth

of this reading is provided by assigning to this sentence the logical form (13’) and

interpreting this form on the basis of their proposed semantic framework. Defenders of the

first variant of the classical version of the first-order discriminating view of existence agree

with them concerning the semantic interpretation of (14). For them there is no significant

logical difference between (13) and (14).

How should a defender of a first-order non-discriminating view of existence react to the

challenge provided by examples like (13) and (14)? It has to be noticed that intuitions about

truth-values are problematic. For one thing, they can be wrong. For another thing, they can

be right, but misleading. They need not necessarily concern the semantic content of a

sentence. They rather can be intuitions about the truth-value of a content that is only

pragmatically conveyed by a sentence relative to certain contexts of its use. I will argue that

sentences like (13) and (14) cannot be conceived as true according to their semantic content,

but they can be used relative to specific occasions to convey true contents. Therefore, the

proposed analyses of (13) and (14) should be rejected as incorrect.

Let me now show why the proposed analysis (13’) of (13) is incorrect. People who claim that

‘There are horses’ can be represented by the logical form ‘xHx’ tend to justify this claim by

a certain interpretation of the original sentence. According to them, ‘There are horses’ is a

condensed version of either ‘There is at least one thing such that it is a horse’ or ‘There is

something such that it is a horse’. On this basis it is then claimed that ‘There is at least one

thing such that’ is an informal counterpart of ‘x’ and ‘it is a horse’ the informal counterpart

of ‘Hx’.21 In this sense ‘There are horses’ is an elliptical sentence. But in fact, none of the

mentioned claims is elliptical and the function of ‘there’ in all these claims is the very same.

In all these cases ‘there’ is not the bearer of any semantic content, it is a grammatical

placeholder for the real and logical subject of a sentence.22 There is no difference between

21 Quine is a famous proponent of this view; c.f.: Quine (1990, 26-28).22 C.f.: Moro (2006, 212; 216-222).

17

our cases and an example like the following: ‘There are two men in the room’ is only a

syntactic variant of ‘Two men are in the room’ and ‘there’ in the first sentence is a

placeholder for the logical subject ‘two men’. But there is a pragmatic reason why these two

forms exist and they have a different communicative function. Such ‘there’-transformations

allow us to change the informative focus of a sentence. In ‘Two men are in the room’ the

informative focus is on the expression ‘two men’, but in ‘There are two men in the room’ the

informative focus is on the verb ‘are’.

In the same sense ‘There are horses’ is a syntactic variant of ‘Horses are’. ‘Horses are’ might

seem to be elliptical, but it is not, because ‘are’ can mean different things and in the context

of our sentence the best way to interpret it is to regard it as synonymous with ‘exist’. In this

sense ‘There are horses’ and ‘There exist horses’ are expressions of the very same content

and ‘There exist horses’ is again a syntactic variant of ‘Horses exist’. The very same role of

‘there’ can also be identified in case of the proposed paraphrases of ‘There are horses’:

‘There is at least one thing such that it is a horse’ is a syntactic variant of ‘At least one thing is

such that it is a horse’ and ‘There is something such that it is a horse’ is a syntactic variant of

‘Something is such that it is a horse’. In each of these sentences the function of ‘there’ is

exactly the same; it is a grammatical device without a specific semantic function, but with a

pragmatic purpose.

There are different reasons why these sentences can be conceived of as existential

generalisations on the level of logical form: in case of ‘Horses exist’ it is the indefinite reading

of ‘horses’ that requires an existential quantifier on the level of logical form. In the case of

‘At least one thing is such that it is a horse’ it is the expression ‘at least one’ and in the case

of ‘Something is such that it is a horse’ it is ‘something’ that represents the existential

quantifier in natural language. On this basis we can assign the following different logical

forms to these different sentences:

(15) Horses are/exist. x(Hx E!x)

(16) At least one thing is such that it is a horse. x(Tx Hx)

(17) Something is such that it is a horse. xHx

These are different sentences with different semantic properties, but the three mentioned

logical forms are necessarily equivalent if we make use of any of our two proposed variants

of a single-domain free logic. This explains why there might be a temptation of assign to (12)

18

and (13) the same logical form as to (14). And this equivalence also explains why there might

be some temptation to declare ‘There is a horse’ to be an elliptical version of some other

necessarily equivalent claim.

If we apply these observations and our explanation of the function of ‘there’ and the status

of ‘are’ in a sentence like ‘There are horses’ to a sentence like (13), then we have to accept

that this claim is semantically equivalent to the following claim with the following logical

form:

(18) There exist things that do not exist. x((Tx E!x) E!x)

Whatever interpretation of the existential quantifier and the predicate ‘E!’ we chose, there is

no possible true interpretation of the proposed alternative logical form of (18). This shows

why we should not accept the claim that (13) or (18) have a true semantic content.

How can we now show that (13) and (18) can nevertheless be used to convey true contents?

One explanation for this phenomenon could be that (13) and (18) can be seen as examples

of loose talk. That is, by using (13) and (18) relative a certain communicative setting we only

speak loosely, and speaker and addressee in this situation know that and they know how to

substitute (13) and (18) with a more accurate and precise expression to communicate the

desired content. Possible examples of substitutes for (13) and (18) are the following

sentences:

(19) There are concepts/kinds of things that do not exist; namely of witches and of unicorns.

(20) There are examples of things that do not exist; namely witches and unicorns. 23

Both of these claims have a true reading on the basis of our favoured variant of the first-

order non-discriminating view of existence and so it seems to be plausible to conceive (13)

and (18) as examples of loose talk that convey contents that can be more accurately

expressed by sentences like (19) and (20).

Alternatively, we could claim that one can make use of Grice’s conversational maxim of

quality to convey a true content.24 The truth-conditions of a sentence like (13) and (18)

cannot be satisfied, but someone might convey, on the basis of an exploitation of this maxim23 C.f.: Sainsbury (2010, 50, 117, 118).24 Grice’s maxim of quality says: Try to make your contribution one that is true; so do not convey what youbelieve false or unjustified. C.f.: Grice (1989[1975], 27).

19

contents that are more directly expressed by (19) or (20). So there are ways to meet the

second mentioned part of our challenge in plausible ways as well.

Let us now turn to the challenge provided by an example like (14). Defenders of the second

mentioned variant of the first-order discriminating view of existence seem to have an

argument in favour of the correctness of their proposed analysis of (14). They claim that it is

possible to derive the sentence ‘Some things do not exist’ from certain true premises that

everyone should accept as literally true. An argument of this sort is the following:

Unicorns do not exist.

[Every unicorn is a thing.]

Some things do not exist.

This argument seems to be intuitively valid and it seems to have premises that are literally

true. Therefore, it seems to be the case that we also have to accept that the conclusion of

the argument is literally true.

How should a defender of our proposed first-order non-discriminating property view react to

this argument? Firstly, there are good reasons to doubt the logical validity of the mentioned

argument. According to our opponent, the conclusion of the mentioned argument has the

logical form ‘x(Tx E!x)’. On this basis the only plausible way to account for the logical

validity of the given argument is by assigning the following logical form to it:

x(Ux E!x)

x(Ux Tx)

x(Tx E!x)

But now our proposed and defended analysis of bare plural sentences with ‘exist‘ comes into

play. If we make use of this analysis, the first premise of the mentioned argument has a

different logical form, namely: x(Ux E!x). Remember that ‘Unicorns do not exist’ is

semantically equivalent with ‘No unicorns exist’! And if we interpret the first premise in this

way the argument turns into an invalid argument. This also shows that this argument is not

as intuitive and acceptable to all parties as it might seem at first sight: it is only formally

valid, if we presuppose a questionable and incorrect analysis of the first premise.

Furthermore, the proposed logical form of the first premise of the argument given by our

opponent is itself problematic. It is only then true if there are non-existent things that are

20

unicorns. One might doubt that there are any non-existent things, because the notion of a

non-existent thing does not seem to be coherent. But even if we accept such kind of things it

is not clear whether they can have an ordinary sort of property like the property of being a

unicorn. If non-existent object are, for example, merely possible objects, those objects might

only be possibly a unicorn, but not actually.

There does not seem to be an uncontroversial argument to establish the truth of (14). A

defender of a first-order discriminating view may, for example, try to justify the truth of (14)

in the following alternative way:

Pegasus does not exist.

[Pegasus is a thing.]

Some things do not exist.

A defender of our two proposed versions of single-domain free logic might object to the

correctness of this argument in two different ways. Firstly, he may claim that the argument is

invalid, because ‘Pegasus does not exist’ does not have any of the required existential

implications to make this argument valid. Secondly, he may claim that a sentence like

‘Pegasus is a thing’ cannot literally be true if ‘Pegasus’ is an empty fictional name. So again

there is no plausible sound reading of this second argument that is generally acceptable.

In the absence of an uncontroversial justification of the truth of the semantic content of (14)

the best way to account for the proposed true reading of (14) seems to be again provided in

pragmatic terms.

According to our first-order non-discriminating property view, (14) has the logical form (13’)

and this form cannot be true. But we can make use of either of the two mentioned

pragmatic mechanisms and convey a true proposition by using (14). Such a true content can

be more explicitly expressed by a sentence like the following:

(21) Some kinds of things do not have instances that exist.

(22) Some examples of things are not examples of things that exist.

We can plausibly assign logical forms to both of this claims that have a true interpretation.

The concept of a witch or a unicorn represent kinds of things that do not have instances. And

examples of these two kinds of things provided in fictional works are examples of things that

do not exist.

21

There also seem to be independent reasons that favour our alternative way to account for a

true reading of (14). We intuitively accept that the following two arguments are valid, but

not really informative:

Some things are red.

There are red things.

Something is a unicorn.

There is a unicorn.

We can give a good and straight-forward explanation for both intuitions if we make use of

any of the proposed two versions of a single-domain free logic to formalize both arguments.

A defender of the second variant of the classical version of the first-order discriminating

property view of existence does not have the resources to account for both intuitions in a

way that is plausible and not ad hoc. It is not plausible to claim that these arguments are

enthymematic, because there does not seem to be a way to add a premise that makes the

argument logically valid and nevertheless assigns a significant logical role to the listed first

premises of both arguments. It would be ad hoc and implausible to claim that either ‘some’

and ‘something’ or ‘unicorn’ and ‘red thing’ are ambiguous and have in some cases an

existence entailing reading. It also seems to be ad hoc and implausible to make use of a

semantic or pragmatic mechanism of quantifier-domain restriction to existent things to get

the desired reading of our premises, because there aren’t any good reasons to restrict the

scope of an already true particular quantification.25 And even if this strategy would be

acceptable it would nevertheless be necessary to insist contrary to common wisdom that

there is a reading relative to which both arguments are clearly invalid. Therefore, when it

comes to the analysis of arguments like the mentioned two single-domain free logic seems

to be the better choice.

Therefore, it seems that our proposed pragmatic strategy to account for true readings of

claims like (13), (14) not only allows us to meet the mentioned challenge, but our proposed

analysis of the semantic content of (13) and (14) also seems to provide the far more

plausible account of the overall use of ‘some’ and ‘something’ in natural language than the

semantic strategy proposed by a defender of the second variant of the first-order

discriminating view of existence.

25 C.f.: Stanley and Gendler Szabó (2000, 241-242).

22

(6) The third challenge: universal generalisations with the existence predicate

This is the final challenge for our proposed first-order non-discriminating view of existence

that I want to discuss in this paper. It stems from the Fregean logical framework we use, and

it concerns the logical status of universal generalisations that contain ‘exist’ like the

following:

(23) Every unicorn exists.

(24) Every horse exists.

On the basis of the already mentioned claim that ‘E!y’ and ‘x(x=y)’ are semantically

equivalent, the sentences (32) and (33) are semantically equivalent to sentences that have

the following logical form:

(25) x(Hx y(y=x))

And a sentence of the form (25) is true relative to every interpretation and therefore, in

Tarskian terms, a logical truth. Therefore, (23) and (24) come out as true against this

background, although intuitively these sentences are contingent truths that have different

truth-values: (23) seems to be not true, while (24) seems to be true.

It has to be noticed that the mentioned equivalence is not the root of the problem, we can

formulate the problem also by directly making use of the logical form of (23) and (24):

(26) x(Hx E!x))

And we get the same result on the basis of our mentioned view that ‘E!’ is a logical predicate

with our proposed contribution to truth-conditions.26

How should we react to this problem? Intuitively, it seems to be the case that (23) is in some

sense odd. But it is not clear whether we should hold that this sentence is false or whether

we should regard it as neither true nor false, because a specific presupposition of (23) is not

true. So there seem to be different possible ways to react to this problem.

26 It has to be noticed that even for a defender of the classical version of the first-order discriminating propertyview of existence there is no straight-forward semantic way to account for the two different intuitive readingsof (23) and (24), although (23) and (24) are not logical truths according to this view. Therefore, our thirdchallenge is a problem that concerns any of our distinguished views on existence.

23

Firstly, one could claim that the given Fregean truth-conditions of sentences like (23) and

(24) are incorrect and that such sentences in fact have those truth-conditions that Aristotle

ascribed to them.27 On this basis such sentences would be equivalent to sentences of the

logical form ‘(xHx x(Hx y(y=x)))’ and would therefore have the desired truth-values.

This seems to be a relatively substantial change to solve a relatively small problem.

Secondly, one could alternatively claim that the Fregean truth-conditions provide the correct

analysis of the semantic content of sentences like (23) and (24), but sentences of the

grammatical form ‘Every F is a G’ can be used to convey contents that have equivalent truth-

conditions with a sentence of the logical form ‘(xHx x(Hx y(y=x)))’. Therefore, an

expression of the form ‘Every F’ may trigger an implicature that has existential import. This

reaction seems to be the more reasonable and moderate.

Thirdly, one might claim that sentences of the form ‘Every F’ have existential

presuppositions. That is, the truth and the falsity of such a claim presupposes the truth of

‘xFx’. According to this conception, (23) is not literally false, but rather neither true nor

false. But we can explain on this basis our intuition that (23) is not true, while (24) is true.

There is one reason that speaks against the first option and another one that speaks against

the first and the third option. Firstly, if we decide to accept an Aristotelian analysis of

universal generalisations, it seems to be natural to extend this analysis to all kinds of

quantifiers. For logical reasons it seems to be even necessary to do so in order to provide a

relatively plausible conception of inference between different kinds of generalisations. But

such a move would not only lead to a massive reinterpretation of our logical framework it

would also undermine the proposed analysis of bare plural sentences with ‘exist’. Our

analysis was justified by means of three semantic assumptions. Two remain true even on the

basis of the Aristotelian reinterpretation: that ‘Unicorns do not exist’ and ‘No unicorns exists’

are semantically equivalent; and that ‘Unicorns exist’ is the contradictory opposite of

‘Unicorns do not exist’ and vice versa. But the proposed analysis of the truth-conditions of

‘No unicorns exists’ is no longer correct. According to the Aristotelian analysis, ‘No unicorns

exist’ is true iff the extension of ‘is a unicorn’ is non-empty and the intersection of the

27 C.f.: Heim und Kratzer (1998, 159-162).

24

extension of ‘is a unicorn’ and ‘exists’ is empty. But such an analysis is clearly not compatible

with a first-order non-discriminating view of existence.28

Secondly and more importantly, the existential import of universal generalisations is not a

general feature of claims of the form ‘Every F is G’. It seems to depend on certain semantic

features of the predicates ‘F’ and ‘G’ whether such a claim has existential import or not. To

clarify this point let us compare the following universal generalisations:

(27) Every round square is a round square.

(28) Every bachelor is unmarried.

(29) Every teacher that I had was crazy.

(30) Every tree in my garden is higher than one meter.

The sentences (27) and (28) are clear-cut examples of universal generalisations without

existential import. The examples (29) and (30), on the other hand, intuitively seem to be

examples of universal generalisations with existential import. Certain intuitive tests confirm

this diagnosis: We can add ‘even if/but there are no round squares’ to (27) or ‘even if/but

there are no bachelors’ to (28) without altering the reading of these sentences or producing

any sort of oddity. But this cannot be done in case of (29) and (30). If we add ‘even if/but

there are no teachers that I had’ to (29) or ‘even if/but there are no trees in my garden’ to

(30) the resulting sentences sound odd and even in some sense contradictory. This seems to

show that only (29) and (30) are in some sense existentially loaded.

It is a difficult issue to name semantic features that account for the difference between the

mentioned two pairs of examples.29 There are several interesting features of the given

examples. An interesting feature of (27) and (28) is that they are non-contingently true. (27)

seems to be a logical truth, while (28) seems to be a conceptual truth. The other examples

are contingently true or false. An interesting feature of (29) and (30) is that they contain

28 At first sight this Aristotelian framework might seem attractive for a defender of the second discussed variantof the classical version of a first-order discriminating property view of existence. Against this background ‘Nounicorn exists’ is not only semantically equivalent with ‘Unicorns do not exist’, but this claim also logicallyentails ‘Some unicorns do not exist’. C.f.: Heim and Kratzer (1998, 160). So there is at least one alternativelogical interpretation of the first mentioned argument in section (5) to justify the truth of (14). However, themain obstacle of this framework is that it does not provide a plausible account of our overall use of quantifierexpressions in natural language.29 In Horn (1997, 166-167) it is, for example, suggested that the distinction between universal generalisationswith and those without existential import corresponds with a distinction between contingent and law-likegeneralisations.

25

predicates that are relativized to objects that exist in the actual world. These observations

might be relevant or irrelevant for a general explanation of the mentioned difference. But

one thing is especially important for our purpose: (23) and (24) are according to their

standard Fregean formal interpretation logical truths like (27) and they do not contain any

relativized predicates like (29) or (30). So, (23) and (24) at least do not seem to be

prototypical examples of universal generalisations with existential import.

It might turn out that an ambiguity view concerning ‘every’ provides the best explanation to

distinguish universal generalisations with existential import from those without, but then

sentences like (23) and (24) would rather be treated as universal generalisations without

existential import according to such an account. Against this background, the best

alternative explanation of our intuitions concerning the truth-values of (23) and (24) seems

to be an explanation in pragmatic terms. If (23) and (24) are literally logical truths, then their

literal contents seem to be of no use for communicative purposes. But we can, nevertheless,

use logical truths to convey informative contents. Grice’s account of conversational

implicatures can be used to explain how this is possible: If we use a sentence like (23) for

communicative purposes, we may exploit the conversational maxim of quantity30 and on this

basis convey the same content that the more elaborated expression ‘Unicorns exist and

every unicorn does so’ literally expresses. Alternatively, we could also classify (23) as an

example of loose talk and hold that a sentence like ‘Unicorns exist and every unicorn does

so’ is a more accurate expression of the content that can be conveyed by (23).

I do not want to defend any of these two possible responses to our problem here in detail,

but the mentioned data shows that this problem is independent from the question of which

logical status ‘exist’ has. It is a problem that concerns the semantic analysis of sentences of

the form ‘Every F is G’ in general and the sketched accounts seem to show that it is possible

to solve this problem in a plausible way even on the basis of our proposed view on existence.

This observation concludes my defence of existence as a first-order non-discriminating

property.31

Burge, T. (1974): “Truth and Singular Terms”, in: Noûs, 8, 309-325.

30 Grice’s maxim of quantity essentially says: Be as informative as required. C.f.: Grice (1989[1975], 26).31 I would like to thank Gail Leckie, Wilfried Keller, Peter Ridley, Peter Sutton and Thomas Spitzley for

interesting and helpful discussions. Specials thanks go to two anonymous referees of this journal for their

detailed and very helpful comments.

26

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