Frege’s conceptual notation includes a significant revaluation of the grammatical role of the copula and, by extension, of the natural language intransitive verb, examples of which are “exists” or “there are.” Upon careful inspection the predicate position in any sentence is shown to be a function sign whose argument place could be filled by either a name or a compound expression for a concept and a variable. The value of the predicate is a concept and not a particular. "Exists" stands for an operator and not a predicate and so it is a mistake to say that it is a symbol of a concept. In Frege’s own example (Begriffsschrift p. 23), “There is a house” becomes “Something is a house (There is at least one house),” which can be understood as a function mapping (or subsuming) whatever satisfies its argument onto (or under) the concept, “…is a house.” Because this function is the truth functional equivalent to, “It is not the case that everything is not a house,” its expression can be replaced wherever it occurs by the latter without changing the meanings of the propositional functions involved.
This sharpened understanding of the “exists” or “is” verb is radically consequential for philosophical issues involving the existence of individual objects or classes of objects, and it is equally consequential for theological “proofs” that rely on the assumption that that verb is completely meaningful by itself, bereft of any supplementary meaning provided by a predicate concept.
Accordingly one wonders what Frege himself may have thought of non-logically motivated definitions of existence such as “To exist is to be perceived,” or proofs like the so-called ontological proof of the existence of god, or even propositions to the effect that god exists.
One text from the Nachlass entitled Dialog mit Pünjer Über Existenz contains a number of appropriate and enlightening remarks. Frege addresses the issue of whether existence can be defined as perceptibility or indeed as limited by any other predicate that means something more than the existential operator. In this dialogue Frege makes three points:
A) When we say that, as in the case of hallucinations or imaginary constructs, no object caused our mental idea (Vorstellung) (e.g. our idea of a centaur), the subject of our proposition is “mental idea of a centaur” and not “centaur.” This is a second-order proposition about ideas or representations; when we say something caused our idea of Peter, for example, our proposition predicates “corresponding to something” of ideas.
B) When we say, “Men exist” we are saying that it is not the case that (using standard notation) for any a whatsoever, “a is not a man” is true.
C) When we utter a statement including a referring term as its subject, we assume the object referred to exists, and our assumption is “self-evident” (selbstverständlich). At the same time, the words “is,” “exists,” “there is,” etc. have no meaning (keinen Inhalt). They are merely grammatical placeholders like punctuation marks. One way of understanding this is that they tell us no more about the subject of the sentence than that it is identical to itself. Any attempt to define existence terms of this sort with words that do have meaningful predicative content (such as “experientiable”) leads to contradiction.
(C) constitutes his objection to the doctrine that proposes that to exist means to be perceived or possibly perceived. For, (C) entails that certain statements about unperceived (or imperceptible) objects such as imaginary objects are self-contradictory. For example, by substitution of expressions with identical meanings, the following propositions should all mean the same thing, that is they should all have the same truth values.
1) Centaurs are imaginary.
2) Centaurs cannot be perceived.
3) There are centaurs that cannot be perceived.
4) There are centaurs that do not exist.
5) Some centaurs exist that do not exist.
So by (C) perceptibility, or indeed any predicate that similarly limits the meaning of existence, cannot be a valid definition of “exists.” Frege’s solution to this confusion is (A).
Frege’s conclusion has consequences for those philosophers, dating roughly from Locke onwards, for whom perceptibility is an important qualification on what can be known, and, in certain respects, also on what exists. One of the goals of modern philosophy has been to put a cap on what were viewed as wild conclusions about strange abstractions, conclusions drawn with virtually no justification beyond conformity to a few rules of deduction. And, because the abstractions involved were so poorly defined or ill conceived, they gave rise to the suspicion that they in fact meant nothing at all. The challenge was to give content to this suspicion. Bacon and Descartes first laid down the principle that real knowledge begins with what we sense; real knowledge is directly or indirectly knowledge about what we perceive or can perceive. Locke elaborated the Cartesian ground rule into a richer theory of ideas that presented a mechanism whereby sensations and other ideas constitute and provide the material for knowledge. Good Xtians that they were, Descartes and Locke excepted beliefs based on revelation as well as certain sorts of proof, mainly those having to do with god, from the Cartesian ground rule. Kant made at least two contributions to this critical function of modern philosophy. First, he lifted the exception of proofs about god from the Cartesian ground rule and subjected proofs of god’s existence, even those advocated by Descartes (an ontological proof that throws in causality for good measure) and Locke (the cosmological proof), to detailed criticism. Secondly Kant interpreted the Cartesian rule about what could be known as a set of putative rules about what exists. In Pünjer’s formulation, the principle that knowledge begins with sensation turns into a basic postulate that only things that directly or indirectly cause or could cause sensations do in fact exist. (Pünjer, of course, is not representative, of what Kant really said, for Kant began to distinguish between types of objects such that, in what comes down to a tautology, the perceptibility criterion was limited to objects of sense.)
Does Frege’s attack on Pünjer really target and thereby undo the critical conclusions of modern philosophers? Not really. Descartes and Locke limited their criticism to what could be known and did not in any clear way relate it to ontological issues. And Frege finds a way to formulate Kant’s insight that “existence is not a real predicate,” without embracing Kant’s view about faculties and ideas or indeed any theory of mind at all. He does this by creating a symbolic notation where the function of the copula and most of the natural language meanings of “exists’ are performed by the existential operator, an expression that can be read as stating, “This sentence is about….” (That is what Frege meant when he said that the imputation of existence in a well-formed proposition is selbstverständlich.)
Frege may not have had an explicit grasp of what the empiricists were trying to achieve when they linked knowledge (and sometimes existence) to what we perceive. (For example, Frege cites numerous philosophers’ views on the definition of number in the Grundlagen, but all his references come from one book, Baumann’s.) Empirically oriented philosophers want to prevent meaningless conclusions drawn by logical rules concerning meaningless concepts invented by the metaphysicians and schoolmen. Nevertheless, by not keeping an eye on the empiricist goal, Frege could produce a dispassionate analysis of what we really mean when we say some (mental) ideas are caused by external objects and some are not.
On p. 20 of the Nachlass Frege makes comments about what we mean when we argue about the existence of specific things that are not literally wrong but they do betray a lack of understanding of empiricist philosophies. He observes that the scientists who claimed that there exists a planet beyond Uranus did not mean that their idea of this planet was caused by an object acting on their senses (auf Grund einer Affektion des Ich), viz. that they actually perceived this planet. It is true that Leverrier did not say a ninth planet exists because he actually saw it (although Galileo said that Jupiter has moons because he actually saw them). However, Leverrier did rely on direct observation of the deviations in the orbit of Uranus that he felt were best explained by the existence of a ninth planet and its gravitational influence. And Leverrier meant (unless he was completely off the rails) that, were we to have telescopes strong enough, we would actually see the ninth planet, which indeed turned out to be the case. Leverrier’s hypothesis was what some would call an empirical claim. Namely it involved actual observations and perceptions of some sort and the prediction that, when certain conditions are met, people would actually perceive the ninth planet. What Leverrier probably meant when he said a ninth planet existed was not that it was perceived or perceivable, but that some body possessing mass was located in space and time such as to affect the orbit of Uranus. What empirically minded philosophers wanted to do was to distinguish what they considered legitimate issues such as the existence of Uranus from fanciful issues such as the Platonic chestnut that the Good as such exists. Their approach was to argue that legitimate questions about existence are related to what we perceive or what we can perceive. Despite the troubles with empiricist theories of mind, the need for making some such distinction remains relevant if we are to avoid uncontrolled speculation.
On the same page Frege uses exactly the same phrase concerning the existence of god. What those who affirm or deny the existence of god mean, he says, is not that their idea of god is or can be caused by the perception of an external object. Frege takes care to add that he means a direct perception (unmittelbare Affektion) of god. Indeed the Xtian definition of god is that thing whose essence is to exist. It is notably lacking in instructions as to where to find it. Likewise, the notion of existence as a predicate is critical to the proofs of the existence of god as proposed by Anselm, Descartes, Spinoza and Leibniz. So, while Xtian philosophers do not mean that god’s existence is tantamount to god’s perceivability, they do mean that god’s existence can be proved and that what is thereby proved says something more about god than the form of a proposition says about any argument that can occupy the subject position of that proposition.
The result is that one of the wild abstractions referred to above is god as defined by Xtian philosophers and theologians. And indeed Kant ties empiricist views and the ontological proof together nicely in the celebrated passage from his Critique of Pure Reason.
Frege endorses Kant’s observation that existence is not a real predicate with the attendant consequences that the propositions that make up the ontological proof of the existence of god are grammatically faulty and so meaningless. In fact Frege provides a basis for this observation that is much stronger than Kant’s thaler example. For, if existence is a real predicate, then a proposition like
6) God exists.
Should be grammatically indistinguishable from a simple predicate proposition like
7) God is stupid.
But they are not. (7) can be analyzed as,
8) There is a god who is stupid.
But (6) becomes,
9) There is a god that is.
Which sounds a bit like burning bush talk. Just as Frege coyly asserts that any proposition that denies existence to imperceptible objects contradicts itself, so (9) is redundant. Even this empty sense of existence as applying to anything you care to name or subsume under a predicate can be analyzed in such a way that existential propositions appear to be no more than a particular form of general proposition where the natural language words, “exist,” “is,” “there is” etc. don’t appear at all. In other words:
10) x(Fx) ≡ ¬(x)(¬Fx).
That is: An F exists if and only if it is not the case that everything is not an F. Under this analysis, to say that god exists is to say that it is not the case that everything is not god. That may or may not be true, just as it may or may not be true that it is not the case that everything is not a centaur. But the ontological proof purports to prove that (6) is not only true, it is necessarily true, whatever that means. The ontological proof incorporates propositions that require that existence be put back in the predicate position: “god’s essence is to exist,” “god has all the perfections including existence,” “Whatever other properties god may have, existence is one of them.” These quaint formulations all share a requirement for a stronger predicative sense of existence than the weak form that just serves to introduce any proposition whatsoever. Under Fregean analysis, any proposition that includes a distinct predicate “exists” that nevertheless means the same thing as the logical operator, “There is…,” dissolves into absurdity. But any proposition that includes a distinct predicate “exists” that means something more than or different from the logical operator both leads to contradictions as shown above and risks defining “god” in such a way as would be unacceptable to the proponents of the ontological proof.
The first thing is to see what happens when we try to insert the contextually defined version of “exists” into propositions in places where genuine first-order predicates (“is green,” “is stupid” etc. would normally be inserted). Begin with a simple assertion of existence:
11) There are centaurs.
Now let us replace “x is a centaur” with “Fx.” (11) now reads
By (10), therefore,
13) It is not the case that everything is not a centaur.
The working assumption is that “…exists” is a predicate just like “…is a centaur.” That is “x exists” can be substituted for “x is a centaur” in any well-formed proposition and the result remains a well-formed proposition subject to the same logical rules of substitution as propositions involving, “x is a centaur.” Let “Gx” stand for “x exists.” Now the proposition that x exists can be written,
By (10), (14) can be rewritten
By Frege’s analysis of “exists”,
16) “an F exists” =def “x(Fx)”
So, by (10)
17) “a G exists” =def ¬(x)(¬Gx).
Accordingly ¬(x)(¬Gx) can be substituted in (15) where Gx occurs. The result is
18) ¬(x)¬ (¬ (x)(¬Gx)).
But “Gx” appears in this proposition also and so we should be able to substitute (15) for “Gx” and so on ad infinitum. The result is not a proposition, but an infinite iteration of the incomplete propositional fragment, “It is not the case that, for anything whatsoever, it is not the case that….” The upshot is that by, using Fregean notation, which is in this respect no more than a clarification of our concepts, the superficially meaningful proposition (14), “Something exists,” is not a proposition at all.
All the ontological proofs involve as steps propositions that are identical to (14) or are like (14) or presuppose (14) or entail (14). Take the proposition that is a step in both Descartes’ and Anselm’s versions of the ontological proof: god’s essence is to exist (This statement can be found on p. 1137 of Descartes’ Letters). Restated without the concept of essence, this proposition states that, whatever other properties may be attributed to god, existence is one of them. Existing is incontrovertibly one of god’s properties. Anything that has all of god’s other properties but does not exist is not god. Let F1…F2 be a set of divine properties (It need not be complete, but it can be). Let G be the property “exists” as in our previous paragraph. So if a=god, then,
19) F1a &…&Fna &Ga.
Substituting for Ga in this proposition according to (10) and (17), we get,
20) F1a &…&Fna & ¬(x)(¬Gx).
So the superficially meaningful proposition “God exists” becomes the infinitely iterative propositional fragment,
21) ¬(x)¬( F1x &…&Fnx & ¬(x)(¬Gx))
22) ¬(x)¬( F1x &…&Fnx & ¬(x) ¬(¬(x)(¬Gx)))
and so on.
If we interpret “God’s essence is existence” to mean that “to be god” entails “to exist,” we get the same result.
23) (x)((x is god) Gx)
24) (x)((x is god) ¬(x)(¬(Gx))
and so on.
A foolish person might try to extricate himself from this unsatisfactory iteration by choosing to regard the logical operator “exists” as different from the predicate “exists” with the conclusion that the two can both be used in a proposition. The hoped for result would be well formed propositions that state that god exists. Frege shows us why such a person would be foolish when he points out that the negations of propositions asserting that there are objects that cannot be perceived are self-contradictory. For, if “perceptibility” means the same thing as “exists,” then such propositions come down to saying, “Something exists that does not exist.” In the same way every true statement containing only the existential operator and the foolish person’s existential predicate, viz. any proposition of the form, x(x exists), would be a tautology true of everything, and the negation of any such proposition would be a contradiction true of nothing.
That same foolish person, or perhaps someone like-minded, might try to assert that one can get by with just the existential operator. That is he might say that “There is a god” (and that god has all these F’s) is a true statement, or perhaps a “necessarily” true statement, and he might try to pimp the ontological proof to demonstrate that it is necessarily true. The problem is that Frege’s existential operator is no more than a grammatical placeholder introducing a proposition. It has no meaning, as Frege says. The negation of x(Fx), for example, does not state that x does not exist. It states that x is not F. The existential operator is just a live and let live kind of guy. As far as it is concerned, god exists, Zeus exists, Yahweh exists, Aphrodite exists and so does Moombalah the Terminator. And for my money, it’s far preferable for Aphrodite than for Yahweh to exist. After all, she’s just out of the bath, she’s got perky nipples and she’s ready for action. Much better than some smelly old bearded character who’s been hiding out in the desert.
Frege’s conceptual notation does not by itself refute the ontological proof of the existence of god. Nor is it meant to do so. The importance of the conceptual notation is that it allows us to express clearly and with perspicuous deductive consequences propositions in which we refer to things (It is important for Frege’s conceptual notation that every well formed sentence - except sentences where “exists,” as well as perhaps certain other “transcendental” terms from traditional metaphysics, is forced into the predicate position – can be fully and accurately transcribed) and, most importantly, propositions in which, using natural language, we assert that things do or do not exist. All of those situations in which we are concerned about what we had gathered together in the catch-all verb “exists” – situations involving our doubts or hopes about centaurs, or murder weapons or tenth planets – are much more perspicuously stated by replacing the verb “exists” by real predicates, such as “romping on the slopes of Olympus,” or “lying in the butler’s pantry,” or “describing orbit O.” Frege’s actual refutations come in his comments not only in his dialogue with the unfortunate Pünjer, but also in the Grundlagen. Indeed just stating what our real meaning when we use a word might be, is not in itself a complete argument. Someone might simply say they don’t agree that that is the real use, or else that they choose to use the word with some specialized sense. So when we clarify the use of a term from natural language or philosophical terminology, we need to bring the argument home, so to speak, by showing how the “unreal” use leads to contradictions, for example, or an inability to answer a simple question like, “What do you mean” (Uncle Ben’s inability to say what he means when he says, “Doo-li exists, all right,” is an example of this.)
On pp. 86-87 of the Grundlagen Frege gives a different formulation of the fallacies of the ontological proof. He begins by noting the similarity (Ähnlichkeit) between the concept of existence and he concept of number. Affirmation that and F exists is a denial that the number of things falling under the concept F is zero. Existence is a qualification of a concept and characteristics of concepts do not qualify the objects that fall under those concepts (although conclusions may be drawn about the objects from the fact that the concepts are characterized in a certain way). The fallacy behind the ontological proof is that it tries to apply a concept-specific quality to a particular, namely god. For the same reason the theological assertion that god is one is also a grammatical fallacy. The only thing troubling about this formulation is that to assert that existence, like number, is a qualification of concepts is not the same as to assert that existence in properly formulated propositions about particulars is either meaningless (keinen Inhalt) or self-evident (selbstverständlich). The most Frege can say is that the meaning of an existential proposition is contained in (10).
The other issue that the meaning of existence could have a bearing on is the one of the ontological status of various objects: mental objects, ideal objects etc. In one passage (p. 16) Frege seems to deny credence that existence is some sort of overarching concept (übergeordnete Begriff), a highest Aristotelian category. He equates existence with self-identity and states that it has no meaning. It functions as part of the form of a sentence. Derivative from this notion is the quasi-concept of entity (Seiendes) . No conclusions can be drawn about what this quasi-concept or its equivalents (e.g. Sein) could stand for. “Wenn die Philosophen von dem ‘absoluten Sein’ sprechen, so ist dies eigentlich eine Vergötterung der Kopula.” (p. 17) Certainly no one wants to deify the copula. That stirring issue deserves its own discussion.
Note. In other passages Frege asserts that existence is a kind of highest concept that subsumes (übergeordnet) other predicative concepts like being human (pp. 20-21). This position runs into Russell’s problems with universal sets which would be reflected in the failure of Basic Law V from Frege’s Grundgesetze and Heidegger’s frustration with coming up with a definition of being in general using “logical” thought. Instead of dealing with that complex topic in detail here, it is worthwhile to make two modest points. First, it is not clear that defining existence as a highest all-subsuming concept or category is compatible with saying that it is meaningless. For it could be argued that the meaningful content (Inhalt) of a concept is at least partially constituted by the meaningful content of the concepts it subsumes. For example, if the concept mammal subsumes the concept dog, then it could be argued that at least part of the meaning of “mammal” is “possibly being a dog.” Secondly the distinct doctrines that (a) the meaning of “exists” and related terms is determined by the form of the proposition and not because they are predicated of the proposition’s argument, and (b) existence is a concept that subsumes all other predicative concepts – these distinct doctrines do not entail each other. At least (a) does not entail (b). For example, the meaning of the propositional conjunction sign is part of the form of propositions containing as determined by truth functions. But conjunction is not a highest concept subsuming other concepts. So it is possible to retain Frege’s genuine insight into the meaning of existence terms and the consequences of that insight as described above without thereby being committed to treating existence as some sort of super category.