SORITES ISSN 1135-1349
Issue #17 -- October 2006. Pp. 68-80
Does Frege's Definition of Existence Invalidate the Ontological Argument?
Copyright © by Piotr Labenz and SORITES
Does Frege's Definition of Existence Invalidate the Ontological Argument?
Piotr Labenz

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.Foot note 6_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 ArithmetikFoot note 6_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 subjectFoot note 6_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 wellFoot note 6_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 ForgieFoot note 6_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'Foot note 6_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 distinctionFoot note 6_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 meaningfulFoot note 6_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' andFoot note 6_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'Foot note 6_10 Foot note 6_11.

2.2. Definition of existence

Now, Frege defines existence as the negation of number zeroFoot note 6_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 predicateFoot note 6_13. However, we have proposed another argument on more ontological linesFoot note 6_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 avoidedFoot note 6_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 fallsFoot note 6_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 conceptFoot note 6_17.

An opposite view is held by MunitzFoot note 6_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 MunitzFoot note 6_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 languageFoot note 6_20) as well. Therefore, the analysis holding the existential quantifier equivalent to second-order concept being generally correctFoot note 6_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 objectsFoot note 6_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 propertyFoot note 6_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 connotationFoot note 6_24).Foot note 6_25 And this has been put forward as the gist of both Kant's and Frege's criticisms of the ontological argumentFoot note 6_26. Moreover, the second-level predicate has been interpreted as equivalent to first-order quantifierFoot note 6_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;Foot note 6_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'Foot note 6_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 wellFoot note 6_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 existenceFoot note 6_31. This precludes the attempt to eliminate the problem by the means of a theory of descriptions. While (∃x) P(x) -- where `P' is the property of being a given individual, say, Golota -- could be treated as φ(P(x)) or so, (∃x) 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 (?p) p; rephrasing into (∃x) x=p is of no help, and above-suggested rephrasing into (∃x) 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 itFoot note 6_32. Rather, he would say that it is meaningless to talk about the existence of individual objects; they can be `real'Foot note 6_33, but cannot exist. To say that Golota exists is, according to Frege, meaninglessFoot note 6_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 doctrineFoot note 6_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 individualsFoot note 6_36 (for Golota, these might be e.g. ` ---- lost against Lennox Lewis's, ` ---- 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 eitherFoot note 6_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 structuresFoot note 6_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 argumentFoot note 6_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 systemsFoot note 6_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:

God is by definition the being than which no greater being can be thought of (or, conceived).

A being, which is thought of and exists in realityFoot note 6_41, is greater than an otherwise identical being, which is thought of but does not exist in reality.

I think of God, so God is thought of.

If (a) He existed in reality besides being thought of, he would be greater, than if (b) he didn't exist in reality.

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 conceptFoot note 6_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 allFoot note 6_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 predicateFoot note 6_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 argumentFoot note 6_45. (Besides, not uncommonly is the assertion that existence is not a predicate deemed to prevent us from defining God into existenceFoot note 6_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 predicateFoot note 6_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 claimedFoot note 6_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 forwardFoot note 6_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, CocchiarellaFoot note 6_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.


Piotr Labenz
Institute of Logic, Language and Computation
University of Amsterdam
<piotr [at]>

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