Piotr Labenz
Does Frege’s Defnition of Existence Invalidate
the Ontological Argument?
Affirmation of existence is nothing but the
denial of number nought.
Because existence is a property of concepts,
the ontological argument for the existence
of God breaks down[1].
1. The purpose and arrangement of the paper
In his attempt to build a basis for mathematics,
Frege proposed definitions of some very basic
concepts that proved to be of import not
only to the Grundlagenproblem, but to other
areas of philosophy, ontology in particular,
as well. Frege himself seems to have noticed
that, as is indicated by his remark that
the definition of existence he gave in the
Grundlagen der Arithmetik[2] shows the invalidity
of Saint Anselm’s ontological argument for
the existence of God. As often is the case
with Anselmian problems, this claim seems—given
the definition—obvious at first glance, but
upon closer inquiry—quite philosophically
perplexing and not obvious at all. Hence,
in this paper we shall expound that definition
and investigate how can it be used to criticise
the ontological argument. In order to do
this, we will also need to give some attention
to the context of Frege’s system, possible
criticisms of his definition of existence,
and logical structure of Anselm’s argument.
To begin with, we will introduce some of
the basic ideas of Frege’s system, which
form the rudiments of modern predicate calculus
as well as of Fregean ontology. Of course
we shall not attempt any interpretation of
Frege’s system as a whole in this paper,
but rather adopt its standard account, and
only focus on our subject[3]. (It seems acceptable
to speak of ‘Fregean ontology’ as clearly
Frege did conceive of the results of his
linguistico-logical inquiries as applying
to the real world, that is, describing not
merely relations between expressions, but
between actual objects as well[4]. In any
case, only this allows using a definition
of existence stemming from pure language-analysis
in discussing an ontological issue.)
Then, we will present Frege’s definition
of existence itself, together with arguments
in support of adopting it; and next, its
criticisms. This discussion will conclude
with certain results that will be important
for further analysis of how the definition
relates to the ontological argument. Next,
we will proceed to exposing the structure
of the ontological argument for the existence
of God and considering several questions
significant in its analysis. Finally, we
will put the results of the analysis together
and look at what do they imply for the topic
question of the paper, and conclude with
an answer to it.
2. Frege’s definition of existence
2.1. Object and concept
The crucial, for us, idea in Frege’s system
is the distinction of objects and concepts,
and the latter of first and second order;
or, more precisely, distinction of names
of objects, names of first-order concepts
and names of second-order concepts. We will
follow Forgie[5] by abbreviating ‘names of
objects’ as ‘A-expressions’, ‘names of first-order
concepts’ as ‘B1-expressions’ and ‘names
of second-order concepts’ as ‘B2-expressions’[6].
An A-expression is such that can be the grammatical
subject of an utterance, but cannot be the
grammatical predicate; or which is a complete
declarative sentence. The latter is because
sentences, according to Frege, denote Truth
or Falsity, which are objects, hence sentences
denote objects, like other A-expressions,
rather than as, loosely speaking, expressing
facts (which is somewhat confusingly different
from the nowadays-standard name-sentence
distinction[7]. Examples of A-expressions:
‘horse’, ‘Socrates’, ‘the teacher of Plato’,
‘God’, ‘7’, ‘Socrates is mortal’, ‘5+7=12’.
Now, A-expressions can serve as arguments
for functions, that is, expressions having
argument-places, such as: ‘––– is mortal’,
‘the capital of –––’, ‘––– + ––– =12’ (where
‘–––’ is the argument-place) and so forth.
If an argument, i. e. an A-expression(s)
is (are) substituted for argument-place(s),
an A-expression is obtained from the function
(strictly speaking, from the name of the
function). If the A-expression resulting
from substituting an A-expression into a
given name of function denotes Truth or Falsity,
then the name of the function is a B1-expression,
that is, a name of a first-order concept;
e. g. ‘is mortal’, ‘…+…=12’, but not ‘is
the capital of. (By substituting an A-expression
into it, a saturated expression is obtained
from an unsaturated B-expression; ‘saturated’
meaning complete, self-standing, able of
being meaningful[8].)
Finally, B1-expressions can be the arguments
of second-order functions. If the A-expression
resulting from substituting a B1-expression
into a given name of second-order function
denotes Truth or Falsity, then the name of
this second-order function is a B2-expression.
Examples of B2-expressions are adjectives
of number: ‘there are 460 of’, ‘there are
as many…as’ and[9], as will be shown, existence.
They name second-order concepts, because
these concepts are being assigned to another
concepts, rather than no objects. Frege’s
illustration of this point is that whereas
‘is thoroughbred’ is a B1-expression, the
argument of which is the name of an object,
say ‘horse’, ‘there are four’ is a B2-expression,
the argument of which is a concept, say ‘thoroughbred
horses’[10] [11].
2.2. Definition of existence
Now, Frege defines existence as the negation
of number zero[12]. To say that x exists
is to say that there is a nonzero number
of x-s; that is, to say simply that there
are x-s. Therefore, ‘exists’ is a B2-expression,
meaning in fact ‘there are more than zero
of’. This is a key claim of Frege’s, denying
existence to be a first-order concept, which
it might at the first sight appear to be.
A single argument for the claim that existence
is a second-order predicate in Frege might
be the weak natural-language-analogy argumentation
for the claim that number in general is a
second-order predicate[13]. However, we have
proposed another argument on more ontological
lines[14]. Now, let ‘P’ stand for the concept
of ‘there are 460 of’; what falls under it
is, e. g. ‘the members of the Polish Diet’;
let ‘Q’ stand for the concept of ‘there are
0 of’. In case of a parliamentary crisis
leading to the dissolution of the Diet, Q
would apply to ‘the members of the Polish
Diet’; and there would be no such members.
If ‘P’, ‘Q’ were B1-expressions, then in
that case ‘Q’ would have a nonexistent argument.
This, however, would lead to the serious
ontological problem of so-called ‘Plato’s
Beard’—predication about nonexistent objects.
Although this is not a proper place for discussing
‘Plato’s Beard’, we can assert this difficulty
should be avoided[15]. And this can only
be done by treating ‘P’, ‘Q’, and hence also
‘exists’, as B2-expressions.
Moreover, let us remark that the second-order
concept of existence applies (semantically)
to the object, not to the first-order concept,
under which that object falls[16]. So, if
we substitute ‘thoroughbred horse’ into ‘——
exists’, we assert the existence of a horse
(and a thoroughbred one), and not of thoroughbredness.
This is significant, as otherwise this second-order
concept of existence would lead to difficulties
with fictional entities. Let the first-order
concept be that of a unicorn, the second-order
that of negation of existence. Then, what
is meant to be nonexistent is not the concept
of unicorn, but unicorns—the object, not
the first-order concept[17].
An opposite view is held by Munitz[18], supported
by some evidence from Frege’s writing. He
claims that existence, being a second-order
concept, applies to the first-order concept
rather than to object, and is to be read:
‘is instantiated’, rather than ‘exists’.
Hence ‘a thoroughbred horse exists’ would
be ‘thoroughbredness is instantiated’; similarly,
‘unicorns do not exist’ would be ‘being-a-unicorn
is not instantiated’. This is to be so, because,
according to Munitz[19], the B1-expression
‘exists’ is equivalent to quantifying existentially
an A-expression; and what Frege means is
to define existence in terms of the quantifier
rather than as a predicate. But even if this
interpretation is correct, still it is undeniable
that what is concerned is the existence of
the object, not merely non-emptiness of the
concept. When we say ‘a thoroughbred horse
exists’, we do not only say that ‘thoroughbredness
is instantiated’, but that it is instantiated
by a horse (rather than by, say, a hound,
or a language[20]) as well. Therefore, the
analysis holding the existential quantifier
equivalent to second-order concept being
generally correct[21], still the latter applies
to objects the previous quantifies, and not
to their properties—which are quite irrelevant
for the quantifier, and hence must be so
for the second-order concept of existence
too. Thus Munitz is right (and quite insightful,
perhaps) in that analysis, but he contradicts
himself saying that existence applies to
first-level concepts, not objects[22].
Finally, an issue that should be made clear
is whether on Frege’s definition existence
is a predicate or not. On the face of it,
Frege expressly and repeatedly says that
it is a property[23], hence a predicate.
However, it has been widely claimed—after
Kant—that in fact it is essentially not a
full-fledged, real predicate like ‘thoroughbred’,
‘mortal’ etc., but merely a logical (i. e.,
behaving like a predicate syntactically,
but not being one semantically) and non-determining
one (i. e., such that does not enlarge the
argument’s connotation[24])[25]. And this
has been put forward as the gist of both
Kant’s and Frege’s criticisms of the ontological
argument[26]. Moreover, the second-level
predicate has been interpreted as equivalent
to first-order quantifier[27], thus further
confusing the distinction between predicate
and quantifier definitions of existence.
The discussion on whether existence is a
predicate by far exceeds the scope of this
paper[28]; therefore, we should content ourselves
with a following view, perhaps not very satisfactory,
but seeming in accordance with most expositions.
For Frege, then, existence is a second-order
predicate—which may be understood as paraphrase
of first-level quantifier—but not a first-order
predicate. The latter claim will be discussed
critically in the following section.
2.3. Criticism
However, there are arguments against treating
existence as a second-order predicate. Firstly,
there is a problem with the existence of
individual objects—that is, such, that we
do not need to assert any additional concept
besides that of existence of them—such as,
say ‘Andrzej Golota’[29]. It seems obvious
that we can say ‘Golota exists’
(even if not much more exciting could be
said about him), and that there are not two
concepts there, but one, that of existence,
and it is a first-order concept applied to
the object of Golota. Perhaps, though, a
Fregean would answer that ‘Golota’ itself
is a concept (however preposterous might
that sound), and uttering ‘Golota exists’,
we actually say ‘something, which is Golota,
exists’, and existence remains a second-order
predicate quite well[30]. That sounds somewhat
odd, but might perhaps be agreed upon. However,
let us put forward a weaker claim than about
the existence of Golota, namely: ‘something
exists’. Now, this is true (and entailed
by the existence of Golota, or of whatever),
and cannot be analysed in terms of second-order
concept of existence[31]. This precludes
the attempt to eliminate the problem by the
means of a theory of descriptions. While
(Ex) P(x)—where ‘P’ is the property of being
a given individual, say, Golota—could be
treated as f(P(x)) or so, (Ex) x cannot—we
can only rephrase it to some F(x): a first-order
predicate, that is, concept of existence
is needed here (and similarly (Ep) p; rephrasing
into (Ex) x=p is of no help, and above-suggested
rephrasing into (Ex) P(x) is in fact doubtful,
as will be shown below).
Therefore, it is impossible to dispense with
a first-order concept of existence, because
some utterances about existence cannot be
formulated without it, but using only the
second-order one. Apparently, Frege himself
has noticed it, but was not able to account
for it[32]. Rather, he would say that it
is meaningless to talk about the existence
of individual objects; they can be ‘real’[33],
but cannot exist. To say that Golota exists
is, according to Frege, meaningless[34].
This is so, because ‘Golota’ is not a property
(i. e., ‘—— is Golota’ is not a B1-expression);
it does not make sense to talk about ‘falling
under “Golota”’ (meaning, of course, the
concept of Golota) in the same way as ‘falling
under “thoroughbredness”’. Therefore the
existence of individual objects cannot be
expressed by the means of a second-order
concept of existence.
However, an attempt to relieve that problem
has been made by C. J. F. Williams in his
interpretation of the Fregean doctrine[35].
He claims that the solution is to treat individual
objects as ‘unique instantiations’ of certain
properties, or sets of properties that unambiguously
point to these individuals[36] (for Golota,
these might be e. g. ‘—— lost against Lennox
Lewis’, ‘—— lost against Mike Tyson’ etc.).
Then, in accordance with our above analysis
of Frege, existence as second-order concept
can be asserted of such individual. Though,
we can easily see that this counter-argument
is not much more than a simple negation of
the claim that being an individual is not
a property; it is to be, in fact, a property
or conjunction of several properties that
determine it in the above manner. Now, it
seems that not much can be done about that:
these are two opposing views and apparently
there are no convincing arguments to falsify
either[37]. Thus we might conclude by only
recalling that, firstly, Frege himself held
that to assert existence of individuals is
meaningless and, secondly, there is no satisfactory
account for ‘something exists’ in Fregean
terms.
Moreover, even, if all propositions about
existence were expressible in terms of Frege’s
definition, it still would be not a sufficient
and compelling reason to accept it. As has
been shown above, some utterances are better—more
simply and intuitively—analysable using the
first-order concept of existence. Now, there
is apparently no good reason to discard first-order
formulations in favour of second-order ones,
which Frege would do. Different utterances
can express the same thought by the means
of quite different predicative structures[38].
Hence there is no good reason to restrict
ourselves to second-order concept of existence,
if some utterances have predicative structures
better explicable by other means.
Finally, let us consider one more apparent
objection. As we have already said, Frege’s
second-order concept of existence applies
to objects rather than first-order concepts.
But what, if we wanted to ascribe existence
to a concept—for instance to say: ‘the wisdom
of the people exists’, meaning the existence
of the wisdom, not of the people? We would
seemingly need to use a third-order concept
to account for it, as the second-order concept
ascribes existence to the object only, not
to the concept describing it. However, according
to the definition of A-expression, a saturated
name of a concept (e. g., ‘the wisdom of
the people’) can be perfectly well treated
as an A-expression, and hence an argument
of a B1-expression, to the referent of which
existence can apply. Thus no higher-that-second-order
concepts are needed here.
2.4. Conclusion
Therefore, we can say that Frege’s doctrine
of existence cannot be defended in its full
strength. It cannot account for the sentence
‘something exists’, and deals rather inadequately
with sentences about individuals. On the
other hand, it disallows the use of first-order
concept of existence, which would solve this
problem. Of course this does not mean that
Frege’s definition of existence is thoroughly
wrong. For instance, it might be enough to
follow Munitz’s approach and combine Fregean
definition with a quantifier definition of
existence in order to eliminate these difficulties.
However, to do that would require exact and
extensive analysis, being too ambitious task
to be attempted here. For our purposes it
will be enough to recapitulate the following
about the Fregean definition. Firstly, it
applies to objects characterized by some
first-order concept. Secondly, it could be
thus applied to individual objects, by the
means of a theory of descriptions, but in
a rather awkward manner and contrary to Frege’s
contention that is meaningless to do so.
Thirdly, it claims existence to be a second-order
predicate, which means it not to be a real
predicate.
3. The criticism of the ontological argument
3.1. Formulation of the argument
To begin with, there is not a single ontological
argument, but numerous versions of it, which
are not necessarily equivalent. The inventor
of the argument, St. Anselm of Aosta himself
gave (at least) two logically distinct formulations
of the argument[39], and other philosophers,
from Descartes and Leibniz to Alvin Plantinga
have put forward other formulations. Now,
Frege in his brief remark does not refer
to any particular version of the argument;
therefore we will consider here the most
well-known formulation, that is, that of
Proslogion, Chapter 2, which is what he most
probably meant when writing about ‘ontological
argument for the existence of God’.
Furthermore, even this single formulation
of the argument—originally formulated in
eleventh-century Latin—has indeed numerous
paraphrases in terms of more modern technical
philosophical vocabulary or various logical
systems[40]. We will not attempt to offer
an exact analysis of that sort; rather, as
Frege probably intended his remark against
the simple common explication of the argument,
we will present such elementary, but still
interesting and general enough, account.
As our end is analytical rather than historical,
we will generalize rather than follow Anselm’s
text exactly (this will be essential for
further discussion). Hence, the argument
has the following form:
(1) God is by definition the being than which
no greater being can be thought of (or, conceived).
(2) A being, which is thought of and exists
in reality[41], is greater than an otherwise
identical being, which is thought of but
does not exist in reality.
(3) I think of God, so God is thought of.
(4) If (a) He existed in reality besides
being thought of, he would be greater, than
if (b) he didn’t exist in reality.
(5) But He is the greatest being, so (a)
is the case rather than (b), that is: God
exists in reality. Q. E. D.
The key point as far as Fregean (and Kantian)
critique is concerned is (2), and more specifically
the claim that to exist is greater than not
to exist, entailed in (2). In Anselm’s text
the argument is formulated in terms of adding
another property, and is formulated more
like: ‘a being having all the God’s properties
except existence is less perfect than a being
having all the God’s properties including
existence’. Then the critics straightaway
claim that existence is not a perfection,
that is, in more modern terms, a real predicate,
that is, in Fregean terms, a first-order
concept[42], so this step is, according to
them, invalid and hence the argument fails.
However, we have seen, in the above formulation
of the argument, no resort to the notion
of God’s properties. Obviously in intuitive
terms, there is a notion of ‘adding’ or ‘subtracting’
existence from the set of properties of God
in the argument. But it is not indispensable
for the argument at all[43]. (It has been
argued that the existential use of quantifier
is excluded in the case of the ontological
arguments, for it involves quantification
over objects both existing really and only
thought of, so it is necessary to use the
existence predicate. Though, it is not a
valid objection, as it seems enough to use
predicates distinguishing real and intensional
objects, without an existence predicate[44].)
Therefore probably Frege was thoroughly incorrect
in his remark, basing it on the mistaken
belief that existence’s being a property,
and a first-order property, is essential
for the argument[45]. (Besides, not uncommonly
is the assertion that existence is not a
predicate deemed to prevent us from defining
God into existence[46]. However, it seems
equally inappropriate to define, say Golota
into being victorious, or a horse into thoroughbredness;
and it is not a reason to claim that being
victorious and thoroughbredness are not predicates.)
Still, it might be worthwhile inquiring whether
Frege’s critique would invalidate the argument
if it did rely on the assumption that existence
is a predicate; we will, then, try to apply
Frege’s definition to the argument and see
whether it would result in showing the latter’s
fallibility.
3.2. Is ‘God’ a name or a predicate?
Apart from the above-granted (for the purpose
of the inquiry only) proposition that the
ontological argument relies on existence’s
being a predicate (we have not granted it
must be first-order). Then, having in mind
the above discussion, there seems to be a
point worth consideration: is ‘God’ a proper
name (an A-expression) or a predicate (a
B1-expression)? In the first case we would
run into all the above-mentioned difficulties
with reconciling Frege’s definition and existence
of individuals; in the second, on the contrary,
we might use the second-order concept of
existence perfectly well.
It has been claimed problematic both to regard
‘God’ as a logically proper name, that is,
name with reference but without sense, and
as a predicate[47]. Of course the argument
that to treat it as a name would be defining
into existence misses the point, for there
obviously are proper names of nonexistent
objects as well, e. g. ‘Gandalf’. Nor is
the argument that existence cannot be asserted
of the referents of proper names, because
existence is a second-order property, sound—it
would be a petitio principii to accept it
here, of course.
However, it might be claimed[48] that ‘God’
could not be a property just because of what
the ontological proof seems to rest on: that
the concept of God involves necessary existence
and certain properties (omniscience, omnipotence,
benevolence etc.). Then, it seems unfeasible
to say: ‘x is God iff x exists necessarily
and is omniscient etc.’, because it suggests
that x has the properties of being omniscient
etc., and of being God, too, which is obviously
a misunderstanding. Thus, being God seems
not to be a property (just like being Golota,
as has been indicated above), which agrees
with the intuition rejecting predicates that
have, by definition, only one instantiation.
Though still, these are not conclusive, however
intuitively appealing, arguments either way.
From a logical point of view—which is the
decisive one—both solutions are possible;
therefore we will consider both in further
analysis.
3.3. Can second-order predicates be used
freely?
An argument has been put forward[49] that
second-order predicates are such that cannot
be ascribed freely to objects—as some first-order
predicates can—but are a matter of fact independent
of the language-users’ decisions. For instance,
we can define a ‘gavagai’ to have the first-order
property of being gray, but we cannot influence
the second-order property of whether it exists,
or of how many gavagais are there, etc. We
can conceive of an object’s having first-order
predicates, but not second-order predicates
(thus it is meaningless to say ‘x can be
conceived as existing’).
However, there are other accounts of second-order
properties quite different from this. For
instance, Cocchiarella[50], following medieval
logicians, has proposed a logic in which
first-order predicates (and relations) are
such that entail existence of the object
(so called ‘e-attributes’), and second-order
predicates are such that do not: for instance,
respectively, ‘is thoroughbred’ and ‘is thought
to be thoroughbred’. Now, of course, this
assumes a quite different interpretation,
or rather use of second-order predicates,
and entirely different approach (e. g., fictional
objects on this account have—and necessarily
so—only second-order properties, while on
the previous they have only first-order properties).
Therefore, we can see that the account claiming
second-order predicates not to be freely
ascribable to objects is not compelling.
It is possible to adopt a contrary position;
both can be well formulated in logical terms.
This points to a more general conclusion.
It is not unusually the case that there are
several distinct, and often mutually contradictory,
possibilities of formulating a philosophical
point. There might be a heated discussion
over these, involving arguments in favour
and against particular solutions of the problem.
However, as long as a solution is not proven
to be either self-contradictory or contradicting
something we otherwise hold—all arguments
can be at most suggestive, and not conclusive.
As long as a solution can be consistently
formulated, there is no compelling reason
for rejecting it. This conclusion obviously
applies to several above-discussed issues,
too; hence we will now not attempt defending
particular solutions of these, but rather
try to see how the various possibilities
influence the ontological argument.
3.4. Application of the results to the argument
In order to present the results of the above
analysis, we will collect them in the form
of the following table, showing possible
outcomes. It has to be noted that the table
has been constructed in a simplified way,
outright excluding some impossible combinations
(e. g. that existence cannot be a predicate
and first-order predicate can express existence).
(1) Is existence’s being a property essential
for the ontological argument? Yes. (2) Can
existence be a predicate (real or merely
logical—i. e., of any order)? No. The whole
critique is besides the point.
Yes. (3) Is ‘God’ a name or a first-order
predicate? No. The whole critique is besides
the point.
A name. (4) Can a first-order predicate express
existence? A first-order predicate. (5) Can
second-order predicates be used freely?
Yes. The argument holds, critique fails.
No. The proof is fallacious. (*) Yes. Then
the critique fails, the argument is not invalidated.
No. The critique is right, the argument is
invalidated.
Now, we have established that (1) is false,
but we have suspended that result so that
further inquiry would be possible. The answer
to (2) has been shown to be ‘yes’. The answers
to (3) and (5) have been shown to be possible
either way. It has been shown, too, that
answer to (4) is possible either way, and
that Frege’s answer was ‘No’. And, moreover,
Frege would also (need to) answer to (3)
that ‘God’ is a first-order predicate, for,
as it has been shown, existence, according
to his doctrine, applies to objects characterized
by a first-order concept (here: ‘—— is God’).
Needless to say, he would say ‘No’ to (5).
(A short comment should be made on the case
(*). If ‘God’ is a name, and first-order
predicates do not express existence, and
existence can be expressed by a predicate,
then existence must be expressed by a second-order
predicate. But then, the first-order predicate
is not the concept of God, for it would be
redundant to predicate it of an object named
‘God’—though, perhaps, sill possible, but
then the case would fall under ‘ “God” is
a predicate’. Then, it must express something
else; we do not want to engage in philosophical
fiction and speculate, what. In any case,
then the argument would aim at the existence
of that mysterious concept rather than God,
and be thus fallacious. Therefore we will
neglect the case (*).)
Therefore, we can conclude the following.
Granted the false assumption that existence’s
being a property is essential for the ontological
argument, and granted the assumptions, for
which no conclusive proof is offered, that
‘God’ is a first-order predicate and that
second-order predicates cannot be used freely—indeed,
then, Frege’s definition of existence invalidates
the ontological argument for the existence
of God. Hence in order to uphold this critique
one would have to make the case for all the
assumptions, which would be quite an ambitious
task, especially as far as the problem whether
it is indispensable for the ontological argument
that existence were a predicate is concerned.
The assumptions present interesting problems
in themselves, and imply several other problems,
e. g. whether a first-order predicate must
be a real predicate etc. However; if all
the mentioned assumptions are not granted—and,
let us repeat, seemingly no convincing reason
for doing so has been given—then the critique
falls short of its objective, and Frege’s
definition of existence is quite unharmful
to the ontological argument for the existence
of God.
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Soviet Studies on Frege, tr. and ed. I. Angelelli,
Reidel, Dordrecht 1964
7. I. M. Bochenski, A History of Formal Logic,
Univ. of Notre Dame Press, Notre Dame 1961
8. N. Cocchiarella, ‘A Second Order Logic
of Existence’, Journal of Symbolic Logic,
XXXIV, 1 (Mar., 1969)
9. ———, ‘Properties as Individuals in Formal
Ontology’, Nous, VI, 2 (May, 1972)
10. M. Dummett, Frege: Philosophy of Language,
2nd ed., Harvard Univ. Press, Cambridge 1983
11. ———, Frege: Philosophy of Mathematics,
Harvard Univ. Press, Cambridge 1991
12. ———, The Interpretation of Frege's Philosophy,
Harvard Univ. Press, Cambridge 1981
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Ontological Argument’, Nous, VI, 3 (Sept.,
1972)
14. G. Frege, The Foundations of Arithmetic,
tr. J. L. Austin, Blackwell, Oxford 1950
15. ———, The Translations from the Philosophical
Writings of Gottlob Frege, ed. and tr. P.
T. Geach, M. Black, Blackwell, Oxford 1960
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Philosophy, Northwestern Univ. Press, Evanston
1969
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seems not to be’, Dialectics and Humanism,
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in Some Anselmian Ontological Proofs: A Syllogistic
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Univ. of Illinois Press, Urbana 1968
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ontologicznym na istnienie Boga’ [Some Remarks
on the Ontological Proof for the Existence
of God], Filozofia Nauki,
23-24 (3-4/ 1998)
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J. Jadacki’s seminar at the Institute of
Philosophy, Univ. of Warsaw, 1999]
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The Philosophical Review, LXIX, 1 (Jan.,
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and the Ontological Argument’, The Journal
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[in: W. Kneale, G. E. Moore, ‘Symposium:
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Press, New York 1974
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Ontological Argument’, The Journal of Philosophy,
LXIII, 19 (Oct., 1966)
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[in:] idem, Ontological Relativity and Other
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--------------------------------------------------------------------------------
[1] Frege (1950: 65).
[2] i. e. Frege (1950).
[3] For a throughout interpretation of Frege,
see e. g. Dummett’s works (1981, 1983, 1991).
For Frege’s logic, see Bochenski (1961: 291-292,
320-322 et al.). In this paper we will focus
on the philosophical issue and not on exegesis
of Frege’s writings (which, even though clearly
written, are obsolete and sometimes ambiguous,
and hence prone to differing interpretations).
Nor will we touch upon any related issues,
such as whether Frege was a realist or a
nominalist—see papers in Klemke (1968), Biriukov
(1964), or for implications of the definition
of existence for traditional metaphysics—see
Angelelli (1967: 225-227).
[4] This is proven by Wells (1954: 537 ff).
It would suffice to support it by recalling
the vast literature devoted to Frege’s ontology,
notably Cocchiarella (1972: 181 ff), Klemke
(1968), and the monograph by Williams (1981).
[5] Forgie (1972: 254-256) gives an exceptionally
lucid account of this, which we will thus
follow in this section. Also see e. g. Walker
(1965: Ch. 2.), Grossmann (1969: Ch.
2.), Munitz (1981: 82-104). References to
relevant passages in Frege can be found there.
[6] Forgie (1972: 254). The following definitions
of A-, B1- and B2-expressions are due to
Forgie as well.
[7] Which has been introduced by Ajdukiewicz
(1935).
[8] See e. g. Biriukov (1964: 25 ff) on saturation.
[9] Munitz (1981:98-99) following Dummett
(1973: 262).
[10] Of course, ‘thoroughbred horses’ is
a saturated concept, and an A-expression,
obtained from the unsaturated first-order
concept, the function ‘——is thoroughbred’,
which is a B1-expression.
[11] Frege (1950: 59, 64).
[12] Frege (1950: 65). See also wider exposition
in Williams (1981: Ch. 3.) and comparison
with other definitions of existence in Labenz
(1999: 3-4).
[13] Frege (1950: 59, 64).
[14] Labenz (1999: 3).
[15] For a good analysis of ‘Plato’s Beard’
problem (which originates from Plato’s Parmenides),
see Jadacki (1981) and Williams (1981: 37-41).
[16] This ought to be reflected in the model
(of logic) of an ontology using this concept.
[17] Grossmann (1969: 69-70) recognizes the
problem, claiming that Frege confuses the
two possibilities: asserting the existence
of a concept and of an object.
[18] Munitz (1974: 78-80).
[19] Ibidem.
[20] Fowler in The Modern English Usage calls
German a thoroughbred language.
[21] An exception will be pointed at in the
next section.
[22] Munitz (1974: 78).
[23] Frege (1950); see Labenz (1999: 1-3).
[24] Like ‘identical with itself’, ‘thoroughbred
or not thoroughbred’, etc. In Polish there
is a proverbial expression for that: ‘buttery
butter’.
[25] See Williams (1981: 17-41), Shaffer
(1962: 309-311).
[26] Commonly at least; and while rightly
so about Kant, we will consider whether rightly
about Frege, too.
[27] Munitz (1974), Williams (1981).
[28] A classic text on this widely-discussed
issue is Moore (1936); a interesting treatment
is given by Sommers (1973).
[29] This objection has been raised by Forgie
(1972: 259-261) and Grossmann (1969: 67-69).
Walker (1965: 32) ignores it, apparently
not regarding it as a problem at all, quite
counterintuitively. Munitz (1974: 78-85)
soothes the problem away, on the ground of
quantifier interpretation of Frege’s definition
of existence, reducing it into the Russelian
theory of descriptions.
[30] Of course this would be an answer somewhat
in the manner of Russelian theory of descriptions;
Russell (1905). See Munitz (1974: 84-86).
[31] One might try some exquisite word-constructions
to save Fregean approach here, e. g. analyse
‘something’ as ‘whatever thing’, ‘whatever
——’ being a B1-expression. However, this
seems quite unfeasible.
[32] Forgie (1972: 260); the passage referred
to there is Frege (1960: 108).
[33] Wirklich.
[34] Frege (1960: 50); see Grossmann (1969:
64-68) for a discussion of this view in relation
to Russell and Moore.
[35] Williams (1981: 81-107).
[36] Williams (1981: 106).
[37] As Leszek Kolakowski has once noticed,
it is a common fate of all the philosophical
problems of past twenty five centuries than
no party is ever convinced by the opponents’
arguments.
[38] Forgie (1972: 259). Forgie names this
claim ‘principle P’.
[39] In Proslogion Ch. 2 and Ch. 3—Anselm
(1979); the distinctness of the proofs is
shown by Malcolm (1960).
[40] Some of these are Kolodziejczyk (1998),
Kelly (1994), Tichý (1979), Adams (1971),
Plantinga (1966).
[41] As opposed to ‘in thought’, so simply—which
exists.
[42] Thus Plantinga (1966: 538).
[43] This observation has been made by Forgie
(1972: 251).
[44] Then existence could be expressed by
the quantifier, in the manner of Quine (1969),
only the universe would have to include both
real and intensional objects.
[45] On the ground of the above-considered
Frege’s doctrine of existence, Plantinga’s
(1966: 54) ‘desultory gesture’ against Frege
in terms of problems with non-existents is
quite besides the point.
[46] E. g. Allen (1961: 59).
[47] Allen (1961: 61-63). Unfortunately,
most of his argumentation is based on premises
we have not granted, or otherwise doubtful.
[48] Somewhat inspired by Allen (1961: 62),
but we have changed the argument significantly.
[49] Mavrodes (1966).
[50] Cocchiarella (1969).
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