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Frege's Objection to the Ontological ArgumentAuthor(s): J. William ForgieSource: Noûs, Vol. 6, No. 3 (Sep., 1972), pp. 251-265Published by: WileyStable URL: http://www.jstor.org/stable/2214773 .

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Frege's bjectionto the OntologicalArgument1

J. WILLIAM FORGIE

UNIVERSITY OF CALIFORNIA, SANTA BARBARA

God possesses all perfections; existence is a perfection;

therefore, God possesses existence, i.e., God exists. Many

philosophers have claimed that:

1) Descartes' ontological proof of the existence of God rests on

the assumption that existence is a first-level, or first-order,

property (attribute, quality), a property of individuals such

as horses or buildings or you and me.

Some of these same philosophers have then gone on to argue

that:

2) Existence is actually a second-level property, a property

of concepts, or of properties, or even of propositional

functions. (A similar claim is made by saying that the

concept of existence is of second-level.)

3) Therefore, existence is not a first-level property.

4) Therefore, Descartes' argument fails.

Although I do not believe claim 1) has ever adequately been

made out, I shall, for the purpose of this paper, assume it to be true.

I shall also assume that the inference from claim 2) to claim 3) isvalid.

To my knowledge the most elaborate defense of the idea that

existence is a second-level property has been put forth by Frege.

In this paper I will discuss that defense and argue that it fails.

I

In The Foundations of Arithmetic, Frege writes:

.the propositionthat there exists no rectangularequilateralrecti-linear triangle does state a property of the concept rectangular

equilateralrectilinear riangle; it assignsto it the numbernought.

251








252 NOUS

In this respectexistence is analogous o number.Affirmation f exist-

ence is in fact nothing but denial of the number nought. Because

existence is a propertyof concepts the ontologicalargumentfor theexistence of God breaksdown. ([6], section 53).

According to Frege when we make an affirmative existence assertion

we deny of a concept the number nought. Similarly to make a

negative existence assertion is to ascribe to a concept the number

nought. From other passages it is clear that to assign or deny the

number nought to a concept is to say, respectively, that nothing,

or something, falls under that concept:

... the number 0 belongs to a concept, if the propositionthat a doesnot fall under that concept is true universally,whatevera may be.

([6], section 55).

In On Concept and Object Frege makes a similar point.

When we make an affirmative existence assertion we say of a

concept that it is not empty; or, we subsume that concept under a

higher, a second-level concept:

I have called existence a propertyof a concept. . . In the sentencethere s at leastone squareroot of 4 , we havean assertion,not about

(say)the definitenumber2, nor about -2, but abouta concept, square

rootof 4; viz., that it is not empty... our sentence assertssomething

about it [theconcept,square ootof4]; it can be regardedas expressing

the fact that a concept falls under a higherone. ([8], pp. 48-9).

I believe Frege's position can be put as follows. (In order to

simplify matters, I shall hereafter mean by 'existence assertion'

or 'existence sentence' simply affirmative existence assertion or

sentence.) When we make various existence assertions we are

subsuming a first-level concept under a second-level concept; or,

put differently, we are ascribing a second-level property to a

first-level concept.2 This second-level concept (this second-level

property), which we can describe as the concept, not being empty,3

(the property not being empty ), is evidently thought to be the

concept of existence (the property of existence). Consequently,

the concept of existence is of second-level (existence is a second-

level property).4 The ontological argument, which assumes that

existence is a first-level property, must be rejected.

Let us consider Frege's first claim. How does he try to show

that in making an existence assertion we are subsuming a first-level

concept under a second-level concept ?5As a start toward answering








FREGE S OBJECTION TO THE ONTOLOGICAL ARGUMENT 253

this question we will do well to consider the following extended

passage from On the Foundations of Geometry :

Let us takethe proposition 2 is a primenumber. Linguisticallywe

distinguish two parts: a subject two and predicative part is a

primenumber .. The firstpart, two ,is a propernameof a certain

number, designates an object, something complete that does not

require a complement. The predicativepart, is a prime number ,

on the otherhand, does requirea complementand does not designate

an object. I shall call the first part saturatedand the second part

unsaturated.To this distinction among the symbols there naturally

correspondsan analogousdistinction in the realm of references:to

a propername correspondsan object and to the predicativepartcor-respondswhatI will callaconcept.This is not meantto be a definition.

For the decompositionin saturatedand unsaturatedparts must be

regardedas a primitivefeatureof logicalstructure,which must simply

be recognizedandacceptedbut whichcannotbe reduced o something

more primitive . . .... Let us now considerthe proposition thereis a squareroot of 4.

Obviouslywe cannotbe talkingabouta particular quareroot of 4; we

are ratherdealingwith the concept. And here too it has preserved ts

predicativenature.That this is so can be seen from the fact that we

can rewritethe proposition n the followingway: thereis something

which is a squareroot of 4, or it is false that, whatevera may be,

a is not a square root of 4. But in this case we obviously cannot

split the propositionup so that the unsaturatedpart is a concept and

the saturatedpart an object.If we comparethe proposition thereis

somethingwhich is a prime number with the proposition there is

somethingwhich is a squareroot of 4, we recognizethat what they

havein commonis there s somethingwhich containingwhatwould

genuinelybe called the logicalpredicate,whereasthe partsthat differ,

despite their predicative,unsaturatednature,play a role analogous othat of the subject in other cases. Here there is something being

predicatedof a concept. But obviouslythere is a very greatdifference

betweenthe logicalplaceof the number2, if we predicateof it that it

is aprimenumber,andthe conceptprimenumber,if we saythatthere

is somethingwhich is a prime number. The first place can be filled

only by objects,the second only by concepts.Not only is it linguisti-

cally improper to say there is Africa or there is Charlemagne,

but it is nonsensical.We may well say there is somethingwhich is

called Africa, and the words is called Africa signify a concept.

Thus the expressionthere s somethingwhich s also unsaturated,butin a totally differentway from is a primenumber. n the first case we

cansaturate he expressiononly with a conceptandin the secondonly

with anobject.We takeaccountof the similarityanddisparityof these

cases by means of the following terminologicaldistinction. In the








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proposition 2 is a prime number we say that an object 2 - fallsundera first-levelconcept prime number whereas n the proposi-tion there is a prime number we say that a first-level conceptprime number - falls within a certain second-level concept. Thusfirst-level concepts can stand to second-level concepts in a similarrelation to that of objects to first-level concepts. ([7], pp. 569-572.See also [8], pp. 48-50).

In order to understand the import of this passage, we shall

have to have some grasp of Frege's terminology. As is well known,

Frege divides the world into objects and functions. ([8], p. 32;

[5], pp. 35-6). Conceptsare a

speciesof functions.

( .. . a

conceptis a function whose value is always a truth-value. [8], p. 30). There

are at least two types of concepts those of first-level (under

which objects fall) and those of second-level (within which first-

level concepts fall). ([8], p. 50). Corresponding to these ontological

distinctions are certain linguistic distinctions. Frege distinguishes

proper names and function-names. The former are names of

objects, the latter of functions. ([5], p. 81). Proper names are said

to be saturated , function-names unsaturated . ([5], pp. 34, 36).

Some function-names are names of concepts, and some of those are

names of first-level concepts, others of second-level concepts.

We shall be interested in the distinctions between those expres-

sions which Frege claims are names of objects, those which he

claims are names of first-level concepts, and those which allegedly

are names of second-level concepts. For the sake of brevity, let us

call these A-expressions, B1-expressions, and B2-expressions,

respectively. How can we tell whether a given expression is an A,

a B1, or a B2 expression?

A-expressions: Certain expressions (e.g., 'Nixon', '2', 'the present

King of France') can be used as the grammatical subject of a

sentence, but can never be used as the grammatical predicate of a

sentence. Frege regards such expressions as A-expressions. (See

[8], pp. 43, 47-48). Although many expressions which satisfy this

criterion (e.g., 'the least rapidly convergent series') will not actually

have a denotation, we are still to regard them as A-expressions.

([8], p. 58). Frege also regards any complete declarative sentence

(hereafter simply: 'sentence') as an A-expression. ([8], p. 63). Any

such sentence which has a truth-value denotes its truth-value; and

truth-values, the True and the False , are objects. ([8], pp.

63-4). Sentences which are neither true nor false do not have a

denotation, but are A-expressions nonetheless.6 Let us say that an

expression is an A-expression if either a) it is a sentence, or b) it can









serve as the grammatical subject of a sentence, but never as a

grammatical predicate. Among A-expressions it is necessary to

distinguish those which have a denotation ( denoting A-expres-

sions ) and those which do not ( non-denoting A-expressions ).

BJ-expressions: If from an A-expression we remove an A-

expression (e.g., remove '2' from '2 is a prime number' and 'Eng-

land' from 'the capital of England'), we are left with expressions

(' is a prime number' and 'the capital of -') which Frege calls

names of first-level functions.7 Such expressions are to be regarded

as carrying argument places indicating where an A-expression

can be.placed so that the resulting expression is itself an A-expres-

sion. If the A-expression thus put into the argument place is a

denoting A-expression, then what it denotes is called the ar-

gument of that function for that particular case, and the resulting

A-expression is said to be the name of the value of the function

for that argument. ([5], p. 34). (For example, in the complex ex-

pression, 'the capital of England', the term 'England' is the name

of the argument (for this case) of the function named by 'the

capital of ', and the entire complex expression is the name of the

value of that function for that argument. In this example the

argument, of course, is England, and the value London.)

Now there is, for Frege, a special sub-class of expressions

which are names of first-level functions. The expression, 'is a

prime number', (unlike 'the capital of') is such that whenever an

A-expression is formed from it by filling its argument place with a

denoting A-expression, the resulting A-expression denotes either

the True or the False. Put differently, the expression, 'is a prime

number', is the name of a first-level function whose value for any

argument is always a truth-value. Now concepts, for Frege, are

functions whose value for any argument is always a truth-value.

Consequently 'is a prime number' is the name of a special kind of

first-level function, viz., a first-level concept. In general it is fair to

say that for Frege an expression is a B1-expression if it satisfies the

following conditions: 1) it is not an A-expression; but 2) an

A-expression can be formed from it by combining it with an A-ex-

pression; and 3) whenever an A-expression is formed from it by

combining it with a denoting A-expression, the resulting A-

expression is a sentence with a truth-value.8 (It is apparent that

'exists' satisfies these three conditions and hence counts as the

name of a first-level concept. This creates obvious difficulties for

Frege see below.)

B2-expressions: In the passage quoted from On the Founda-








256 NOUS

tions of Geometry, Frege distinguishes the expression 'is a prime

number' from the expression 'there is something which'. They are

both unsaturated expressions, but they are unsaturated in

totally different ways. We have seen that 'is a prime number'

qualifies as a B 1-expression. We can form an A-expression (a

sentence) from it by combining it with an A-expression. But we

cannot in this way form an A-expression from 'there is something

which'. 'There is something which 2,' or 'there is something which

Africa,' are not sentences; they are nonsensical expressions. It

seems that if we want to form an A-expression from 'there is

something which' we must add a B 1-expression like 'is a square

root of 4,' or 'is called Africa,' or 'is a prime number.' We should

regard 'there is something which' as carrying an argument place

indicating where we must add a B1-expression so as to form an

A-expression. Any such expression, Frege says, is the name of a

second-level function. ([5], p. 81). The BI-expression we put into

the argument place is the name of the argument of that second-level

function for that case, and the resulting A-expression is the name

of the value of that function for that argument.

Now there is a special sub-class of those expressions which are

names of second-level functions, and 'there is something which'

belongs to that sub-class. For whenever we form an A-expression

from it by filling its argument place with a B1-expression, the

resulting A-expression denotes either the True or the False. That is,

the expression 'there is something which' is the name of a second-

level function whose value for any argument is always a truth-value.

Since concepts are functions whose value for any argument is a

truth-value, 'there is something which' is the name of a special kind

of second-level function, viz., a second-level concept. ([5], p. 74).

In general for Frege, an expression is a B2-expression if it meets the

following conditions: 1) it is not an A-expression; but 2) an A-

expression can be formed from it by combining it with a BI-

expression; and 3) whenever an A-expression is formed from it in

this way, the resulting A-expression is a sentence with a truth-

value.9

Let us now return to the passage from On the Foundations

of Geometry in hopes of discovering why Frege holds that in

making an existence assertion we are subsuming a first-level concept

under a second-level concept. I believe the argument in that passage

can be put as follows. Frege asks us to consider the sentence, 'there

is a square root of 4.' (The translator of On the Foundations of

Geometry translates 'Satz' as 'proposition.' But since Frege









discusses the linguistic parts of this or that Satz, and since these

parts are groups of words, 'sentence' seems a preferable translation.)

He tells us that we can rewrite this sentence as 'there is something

which is a square root of 4.' In this rewritten sentence, 'is a square

root of 4' is a BI-expression and 'there is something which' is a

B2-expression. So the rewritten sentence contains a name of a

first-level concept and a name of a second-level concept.10 Frege

seems to take this as showing that the rewritten sentence subsumes

a first-level concept under a second-level concept. (I believe this

is the point of the two sentences ending with: Here there is

something being predicated of a concept. ) And he appears to

conclude from this that the original sentence also subsumes a first-

level concept under a second-level concept, for ostensibly the sole

purpose of rewriting the original sentence was to bring out more

clearly exactly what is being subsumed under what. But what

first-level concept is being subsumed under what second-level

concept ? The passages cited earlier suggest that it is the first-level

concept, square root of 4, which is being subsumed under the

second-level concept, not being empty.

Letting S represent the sentence 'there is a square root of 4' and

T represent the sentence 'there is something which is a square root

of 4,' we may set out Frege's argument in the following way:

1) S can be rewritten as T.

2) 'is a square root of 4' is a B1-expression, and 'there is some-

thing which' is a B2-expression.

3) Therefore, T contains a name of a first-level concept and

a name of a second-level concept.

4) Therefore, T subsumes a first-level concept under a

second-level concept.5) Therefore, S subsumes a first-level concept under a second-

level concept; or, more specifically, S subsumes the first-

level concept, square root of 4, under the second-level

concept, not being empty.

The argument Frege here uses with sentence S obviously can,

with appropriate changes, be applied to any existence sentence of

the form 'there is (exists) a i.'

I believe Frege's argument is circular. To see this, let us

suppose that sentence T does subsume a first-level concept under asecond-level concept. Will it follow that the same is true of sentence

S ? In other words, suppose step 4 in the above argument is true-

will step 5 follow? This will depend on how we understand step 1,








258 NOUS

for evidently it is this step that is supposed to license the move from

step 4 to step 5.

In On Concept and Object, Frege claims that all three of

the following sentences:

a) there is at least one square root of 4

b) the concept, square root of 4, is realized

c) the number 4 has the property that there is something of

which it is the square

express the same thought. ([8], p. 49. Frege equates the thought

expressed by a sentence with the sense of that sentence. See [5],p. 35, and [8], p. 62). But he also claims they have what we might

call different predicative structures. That is, a) subsumes the

concept, square root of 4, under the concept, not being empty. But

b) predicates something of an object, not a concept, the special

kind of object denoted by the expression, 'the concept, square root

of 4.' Sentence c) also predicates something of an object, but a

different one-the number 4. These observations:

. .will be surprisingonly to somebodywho fails to see that a thought

can be split up in many ways so that now one thing, now another,appearsas subjector predicate.The thought itself does not yet deter-

mine what is to be regardedas the subject.If we say the subjectof

this judgment we do not designateanything definite unless at the

same time we indicatea definitekind of analysis;as a rule,we do this

in connexionwith a definitewording.But we must never forget that

differentsentences may express the same thought. ([8], p. 49).

Frege would certainly, and I believe rightly, regard as misguided

any attempt to argue that because a), b), and c) express the same

thought, they therefore have the same predicative structure.

Consequently, we cannot infer that our sentence S has the same

predicative structure as sentence T if step 1 merely claims that S

and T express the same thought. The inference from step 4 to

step 5 in Frege's argument will not be valid unless step 1 is taken

as maintaining, at least in part, that S and T have the same predica-

tive structure.

But now the circularity in Frege's argument is apparent. For

he can infer the predicative structure of S from that of T only if he

first assumes that the two have the same predicative structure.

Since T allegedly subsumes the first-level concept, squareroot of 4,

under the second-level concept, not being empty, this will require

that he assume that S subsumes the same first-level concept under









the same second-level concept. But that S has just that predicative

structure is what his entire argument purports to show.

Frege's argument is defective because at bottom it ignores the

principle that different sentences may express the same thought

and yet have quite different predicative structures (principle P).

This principle places restrictions on the way one may legitimately

proceed in trying to reveal the predicative structure of this or that

sentence. It makes illegitimate any procedure by which one attempts

to show that one sentence has such-and-such predicative structure

by showing that it expresses the same thought as another sentence

which does have that predicative structure.

Principle P will eventually be useful in making what seems to

be a very elementary objection against Frege. Let us suppose he

has made out his claim concerning the predicative structure of

sentences of the form 'there is (exists) a f.' Still there is a class of

existence sentences which would appear, at least at first glance, to

create difficulty for his view that all existence sentences subsume a

first-level concept under the concept, not being empty. Consider

singular existence sentences like 'God exists,' or 'the Taj Mahal

exists.' The word 'exists' is a BI-expression and the expressions

'God' and 'the Taj Mahal' are A-expressions. Both sentences

then contain an A-expression and a B1-expression. It would seem

to follow that these existence sentences do not subsume first-level

concepts under a second-level concept. Rather they subsume

objects under the first-level concept denoted by 'exists.' But isn't

this simply to say that these sentences subsume an object under the

first-level concept of existence (or that they ascribe a first-level

property of existence to objects) ? Frege's own principles seem to

imply that the concept of existence is of first-level (that existence is

a first-level property) after all.

In raising this objection we have not used principle P. This

principle will be of use, however, in suggesting why that objection

cannot easily be overcome. Surely it is of no use to argue that

'God exists' expresses the same thought as 'there is something

which is called God' or 'there is something which is identical with

God.' For even if these latter sentences subsume first-level concepts

under a second-level concept, principle P can be invoked to show

that nothing follows concerning the predicative structure of 'God

exists.'

Nor is it to the point to show how the thought expressed by

'God exists' can be expressed in the symbolism of Frege's Begriff-

schrift. It might be said, for example, that the thought expressed by








260 NOUS

'God exists' can be translated into the Begriffschrift as 'TU ITa is

called God,' or '(aT a is identical with God.' Each of these

sentences is of the form 'my f(a),' in which 'f' represents a

BI-expression and 'aUT . . (a)' is a B2-expression. (See [8],

pp. 37-38. Frege's expression 'aUT f(a)' is equivalent to the

Principia Mathematica expressions' -. (x) -. (Fx)' and' (3x) (Fx)'.)This move is useless, for, again, whatever the predicative structure

of the Begriffschrift sentence, principle P shows that nothing

follows concerning the predicative structure of 'God exists.'

Although we no doubt gain many insights about sentences of

ordinary language by translating them into a formal symbolism, we

surely do not gain any insight into their predicative structures.

There is at least one passage in which Frege appears to concede

an exceptional character to singular existence sentences:

We must here keep well apart two wholly different cases that areeasily confused, because we speak of existence in both cases.In onecase the question is whether a proper name designates, names,something; in the other, whethera concept takesobjectsunder tself.If we use the words there s a. . . we have the lattercase.([8], p. 104).

These remarks suggest that Frege wants to make a distinction

between two kinds of existence sentences. In making assertions

with sentences of the form 'there is a b'we say of a concept that it

takes objects under itself, that is, we subsume a first-level concept

under the second-level concept, not being empty. But there is

another kind of case in which we speak of existence. Here, I

take it, Frege is thinking of singular existence sentences. In making

assertions with sentences of the form 'A exists' or 'the S exists' we

are not subsuming a first-level concept under a second-level

concept. Rather, we are saying of a proper name (in Frege's senseof 'proper name,' i.e., an A-expression - this includes both 'God'

and 'the Taj Mahal.') that it designates something, we are sub-

suming a proper name under the concept, having a designation. 1

If this is Frege's view about the predicative structure of singular

existence sentences, then three comments are in order. First, such

a view represents a departure from Frege's claim, in The Foundations

of Arithmetic, that affirmation of existence is a denial of the number

nought. A sentence of the form 'A exists' does not subsume a

first-level concept under the second-level concept, not being empty.Second, such a view also represents a departure from Frege's

syntactic-semantic principles (i.e., 1) the criteria for picking out

A, B1, and B2-expressions; together with 2) the rough procedural









rules Frege seems to follow in determining the predicative structure

of a given sentence - see note 10.) For those principles seem to

require the view that singular existence sentences subsume objects

under first-level concepts. It must be noted, however, that Frege

did not regard his syntactic-semantic principles as infallible guides

to determining the predicative structure of any given sentence, and

on occasion held that the predicative structure of a particular

sentence was other than what those principles would suggest. (See,

for example, [8], pp. 13-14, 45-48). But third, and most important,

Frege surely now owes us an argument to support this view about

singular existence sentences. All along we have been trying to see

why Frege thinks existence sentences have the predicative structure

he claims they have. We can understand why he thinks that a

sentence like 'there is something which is a square root of 4'

subsumes a first-level concept under a second-level concept because

we can see how such a view is a consequence of his syntactic-

semantic principles. And if Frege had held that singular existence

sentences subsume objects under first-level concepts we could also

understand that view as a consequence of those same principles.

But his view, evidently, is that singular existence sentences subsume

proper names under the concept, having a designation. Because

such a view is not a consequence of Frege's syntactic-semantic

principles, he owes us some other reasons for subscribing to it.

II

Frege is unsuccessful in his attempt to show that existence

sentences subsume first-level concepts under the second-level

concept, not being empty. Indeed nothing he has shown prevents

us from holding that at least some existence sentences subsume

objects under the first-level concept of existence. And if we can

hold this we can hold that the concept of existence is of first-level.

For purposes of argument, however, let us suppose that the fol-

lowing two propositions have been established:

1) No existence sentence subsumes an object under a first-level

concept of existence.

2) Instead, all existence sentences subsume a first-level concept

under the second-level concept, not being empty.

We must now ask what bearing such views about the predicative

structure of existence sentences have on the issue of whether

existence is a (first-level) property.








262 NOUS

Consider the sentence 'there is (exists) a dragon' or 'there is

(exists) at least one dragon.' According to Frege this sentencesubsumes the concept, dragon, under the second-level concept,

not being empty. But suppose one were to make the following

objection. Fafner is a dragon. Consequently, because of Fafner, the

concept, dragon, is not empty. So the sentence 'there exists a

dragon' cannot subsume the concept, dragon, under the concept,

not being empty. If it did, that sentence would express a true propo-

sition. And, of course, it does not.

Frege could meet this objection simply by saying that he

understands the notion of a concept's not being empty in such away that the concept, dragon, is not made non-empty by any

mythological, and hence non-existent, thing such as Fafner.12 But

we can distinguish two senses of a concept's not being empty. In

sense A a concept is non-empty if something which exists falls under

it. In sense B a concept is non-empty if something existing or

something non-existing (e.g., something in mythology or in fiction)

falls under it. If Frege's second-level concept is the concept, not

being empty, in sense B, then the sentence 'there exists a dragon'

will express a true proposition. The concept, dragon, is not empty in

sense B. So in order to avoid holding that 'there exists a dragon'

expresses a true proposition, we must say that the second-level

concept in question is the concept, not being empty, in sense A.

But consider the concept, not being empty (sense A). Being a

second-level concept its instances will be various first-level con-

cepts. But which first-level concepts? Only those which themselves

have instances which exist. But the expression 'instances which

exist' in the previous sentence would appear to be synonymous with

the expression 'instances which fall under the concept of existence.'

So it seems that Frege's concept, not being empty (sense A), is one

which applies only to first-level concepts which themselves have

instances which fall under the concept of existence. Or, more

briefly, it applies only to those first-level concepts which themselves

are partially coextensive with the concept of existence. But this

means that the concept of existence is of first-level after all, for if

it is partially coextensive with first-level concepts then both it and

those first-level concepts will have objectsas instances.

The upshot of this is that the following three claims are

perfectly compatible:

1) No existence sentence subsumes an object under a first-

level concept of existence.









2) Instead, all existence sentences subsume a first-level

concept under the second-level concept, not being empty

(sense A).

3) This second-level concept applies to those first-level

concepts which are partially coextensive with the equally

first-level concept of existence.

Whenever we speak of something falling under the concept,

red, or the concept, horse, we can also speak of a thing's having the

property, being red, or being a horse. Nothing in Frege

prevents us from maintaining that when we speak of things falling

under the concept of existence we can also speak of things havingthe property of existence. Indeed, for Frege, talking about the

concepts under which some object falls is just another way of

talking about the properties that object has. (See [8], p. 51).

Accordingly, I do not see why anyone holding claim 3) just listed

cannot also hold:

4) This second-level concept applies to those first-level

concepts which apply to objects having the first-level

property of existence.

If, as I believe, it is possible to hold claims 1) through 3), it is also

possible to hold claims 1) through 4).

It appears, then, that Frege's view about the predicative

structure of existence sentences does not show that existence

is anything but a first-level property. A defender of Descartes'

ontological proof could agree that all existence sentences subsume

a first-level concept under a second-level concept, but go on to

claim that there is a first-level property of existence in terms of

which that second-level concept can be explicated.13

REFERENCES

[1] Cartwright, R., Negative Existentials, in C. Caton (ed.), Philosophy and

Ordinary Language (Urbana, 1963): 55-66.

[2] Cocchiarella, N. B., Existence Entailing Attributes, Modes of Copulation

and Modes of Being in Second Order Logic, THIS JOURNAL, III, 1 (1969):

33-48.[3] , A Second Order Logic of Existence, JSL, XXXIV (1969): 57-69.

[4] , Some Remarks on Second Order Logic with Existence Attributes,THIS JOURNAL, II, 2 (1968): 165-175.

[5] Frege, G., The Basic Laws of Arithmetic, trans., in part, by M. Furth (Berkeley

and Los Angeles, 1964).

[6] , The Foundations of Arithmetic, trans. J. L. Austin (New York, 1960).








264 NOUS

[7] , On the Foundations of Geometry, trans. M. E. Szabo, in E. D.

Klemke (ed.), Essays on Frege (Urbana, Chicago, and London, 1968).

[8] , Translations from the Philosophical Writings of Gottlob Frege, ed.,and trans., by P. T. Geach and M. Black (Oxford, 1960).[9] Kant, I., Der einzig m6glich Beweisgrund zu einer Demonstration des

Daseins Gottes, in E. Cassirer (ed.) Werke (Berlin, 1912), Vol. II: 76-79.[10] Plantinga, A., Kant's Objection to the Ontological Argument, Journal of

Philosophy, LXIII (1966): 537-546.[11] Shaffer, J., Existence, Predication, and the Ontological Argument, Mind,

LXXI (NS) (1962): 307-325.

[12] Wolterstorff, N., Referring and Existing, Philosophical Quarterly, XI (1961):

335-349.

NOTES

1 I am grateful to Nelson Pike and William Rowe for helpful criticisms of

earlier versions of this paper.2 I have here written in terms of what speakers do, not what sentences do.

Frege normally speaks of what sentences do. In this paper I will employ both

idioms interchangeably.3 For Frege, of course, one cannot identify a concept by means of the

expression 'the concept O'.One succeeds only in picking out or referring to what he

calls an object. See, for example, [8], pp. 46-47. For our purposes, however, this

difficulty can safely be ignored.

4 It is worth noting that Frege was not the first to argue in this way.His

argument, at least in broad outline, was put forth by Kant in [9]. This historical

parallel is not often noticed. There Kant claims that when we make an existence

assertion we are ascribing a property not to things, but thoughts, or concepts, or a

collection of properties. He concludes from this that existence is a property of

thoughts, or concepts, or a collection of properties - a second-level property.

5 It is obvious that if one holds that the concept of existence is not of first-level

he must also hold that when we make an existence assertion we are not subsuming

things (objects) under a first-level concept of existence. Frege evidently thinks his

claim about what we are doing in making any existence assertion is relevant in

showing that the concept of existence is of second-level, not of first-level. I take

him, then, to be making the following assumption: if an existence sentence sub-

sumes a first-level concept under a second-level concept, then it does not subsumean object under a first-level concept. For the purposes of this paper we need not

quarrel with this assumption.6 [8], pp. 62-63. Frege suggests that sentences of the form 'A is i,' or 'the S is

P.' will have no truth-value if 'A' or 'the S' have no denotation. It is for this reason,

he says, that the sentence 'Odysseus was set ashore at Ithaca while sound asleep'

may not have a truth-value.

7 [5], p. 81. More specifically, they are names of first-level functions of one

argument. For our purposes the distinction between names of first-level functions

of one and two arguments (ibid.) will not be important. In this paper all examples

of names of first-level functions will be names of first-level functions of one argu-

ment.8

For our purposes we will assume that all Bl-expressions have denotation.A name of a first-level function has a denotation if the A-expression which results

from its argument place being filled with an A-expression always has a denotation

if the A-expression in the argument place has a denotation. ([5], p. 84). Conse-

quently, a name of a first-level concept has a denotation if the sentence which results

from its argument place being filled with an A-expression always has a truth-value









if the A-expression in the argument place has a denotation. Elsewhere, Fregesuggests that if a Bl-expression is to have a sense at all it must satisfy this condition.Otherwise, we should have to abandon the Law of Excluded Middle. See, forexample, [8], p. 159. That is, Frege seems to suggest that if a Bl-expression is to

have a sense it must have a denotation.9 We will assume that all B2-expressions have denotation. A name of a second-

level function has a denotation if the A-expression which results from its argumentplace being filled with a Bl-expression always has a denotation if the Bl-expressionin the argument place has a denotation ([5], p. 84). Consequently a name of asecond-level concept has a denotation if the sentence which results from its argu-ment place being filled with a Bl-expression always has a truth-value if the Bi-expression has a denotation. Since we are assuming that all B1-expressions havedenotation (see note 8), our assumption that all B2-expressions have denotation isthe assumption that, for any B2-expression, the sentence which results from itsargument place being filled with a Bl-expression always has a truth-value.

10Frege's general procedure here can be described as follows. First, we find

out what kind of expressions, whether A, Bi, or B2, make up a given sentence.This will tell us of what the various elements of the sentence are names. And thenwe can determine what is being subsumed under what. If the sentence contains anA and a Bi-expression, then that sentence subsumes an object (if there is one, i.e.,if the A-expression in fact has a denotation) under a first-level concept. If thesentence contains a Bi and a B2 expression, the sentence subsumes a first-levelconcept under a second-level concept.

11I am assuming that the distinction between the two kinds of existencesentences is to be drawn in terms of their predicative structures. It may be thatFrege wishes only to make the point that sentences of the form 'A exists' or 'the Sexists' express the same thought as corresponding sentences of the form 'the name

A (or the S ) has a designation.' For what are by now familiar reasons, such a

point will have no bearing on questions about the predicative structure of singularexistence sentences.

12 He might, of course, try to argue that there is no sense in which non-existententities can be said to have properties or fall under concepts. Recent work, however,has suggested that this line of response is not very promising. See, for example,Cartwright [1], Plantinga [10], Schaffer [11], and Wolterstorff [12]. Plantinga'spaper also contains a brief discussion (though differently motivated than the presentpaper) of Frege. See also Cocchiarella, [3] and [4], for a formulation and discussionof a logic of existence centering around a distinction between those attributes whichentail existence and those which do not. Although Cocchiarella agrees with thewriters just cited in holding that objects which do not exist may nevertheless haveattributes, the class of attributes which he claims do not entail existence appears to

be more restricted than it apparently is for these others - see [2].13Strictly speaking, all that has been argued is that Frege's view about the

predicative structure of affirmative existence sentences is compatible with holdingthat existence is a first-level property. For it was stipulated that, in this as well asin most of the preceding section, 'existence sentence' would mean simply 'affir-mative existence sentence'. It should be clear, however, that a similar line of argu-ment will show that Frege's view about negative existence sentences is also compat-ible with the claim that existence is a first-level property.

5

forgie w frege s objection to the ontol arg

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