Kant and Frege on Existence1

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Kant and Frege on Existence1

Forthcoming in Synthese

Toni Kannisto, University of Oslo, t.t.kannisto@ifikk.uio.no
1 – Introduction
Kant famously claims that being is “obviously not a real predicate” (KrV, A 598/B 626)2, i.e. a determination or a property of a thing. As Frege similarly states that existence is not a first-level predicate of objects but a second-level predicate of concepts, it is not surprising that the two philosophers have been compared on this point. Indeed, Jonathan Bennett speaks of the “Kant-Frege view”, according to which Frege first gave solid logical foundations for Kant’s claim (Bennett 1974, 62–5, 231).3 To my mind, although there is some truth to the Kant-Frege view, there is a fundamental disparity between Kant’s and Frege’s conceptions of existence that far outweighs their similarities.
I submit – similarly to Hans Sluga (1980, 88) but for different reasons – that although Kant and Frege agree on what existence is not, they agree neither on what existence is nor on the importance and justification of existential propositions. Furthermore, the two philosophers are deeply at odds in their treatment of truth about merely possible objects. This difference, I argue, goes to the heart of philosophy and especially of the question, whether and how metaphysics could be justified. Consequently it is not true that Frege gave logical clarity to Kant’s thesis about existence: they have fundamentally different and competing theories of existence.
I will spell out the disparity between the philosophers in three main sections. Sections 2 and 3 introduce and analyse Kant’s and Frege’s conceptions of existence, respectively, whereas section 4 identifies three of their key disagreements: First, whereas for Frege existence is a non-relational property of a concept, for Kant it is a relational property that pertains between the concept and intuition of an object. Second, whereas for Frege truth about individuals presupposes their existence, for Kant truth is in many cases (including judgments about individuals) independent of the (possible) existence of objects. Third, whereas Frege binds existence to logic and “eliminates alethic modalities from his logic” (Haaparanta 1985, 146; cf. BS, §4), for Kant existence is a modal category that is emphatically removed from the domain of (general) logic and set in the core of metaphysics. Consequently, whereas for Kant assertions about (merely) possibly existing objects or individuals are crucial to his critical metaphysics, for Frege they are nothing less than senseless. I will conclude in section 5 by explicating how this difference makes it impossible for Frege to conduct the kind of meta-metaphysical inquiry that is essential to Kant’s philosophy. Note that apart from pointing out some of the immediate virtues of Kant’s and Frege’s respective stances, it is not here my aim to thoroughly assess their relative advantages and disadvantages.
2 – Kant
Despite having been a major focus of Kant-scholarship, Kant’s view of existence is generally not well understood, and the number of competing interpretations is great.4 To my mind, there are three major reasons for this, only the latter two of which will be addressed here. First, Kant’s claims about existence have not been sufficiently related to those of his immediate predecessors – especially Alexander Baumgarten – in light of which, however, they should properly be interpreted (see Kannisto 2016). Second, the focus has mainly been on Kant’s negative claim that existence is not a real predicate – on which Kant and Frege fully agree – whereas his positive characterisation of it as “absolute positing” – that sets the two philosophers apart – has received little attention. Third, it has not been sufficiently recognised that Kant’s theory of existence is founded on his more general theory of modality, which in turn is opposed to that of Frege.
2. 1. Kant on Existence and Predication
Kant’s famous claim that existence is not a (real) predicate predates the Critique of Pure Reason by almost two decades. In The Only Possible Ground for a Demonstration of the Existence of God (OPG, 1763) Kant supports his claim that “[e]xistence is not a predicate or a determination of a thing” (OPG, 72) with the following explanation:
Take any subject you please, for example, Julius Caesar. Draw up a list of all the predicates which may be thought to belong to him, not excepting even those of space and time. You will quickly see that he can either exist with all these determinations, or not exist at all. The Being who gave existence to the world and to our hero within that world could know every single one of these predicates without exception, and yet still be able to regard him as a merely possible thing which, in the absence of that Being's decision to create him, would not exist. […] It cannot happen, therefore, that if they [merely possible things] were to exist they would contain an extra predicate; for, in the case of the possibility of a thing in its complete determination, no predicate at all can be missing. (Ibid.)
However many predicates we may attribute to a thing, it remains undecided whether the thing, along with these predicates, exists. In the Critique the same claim is presented concisely and without reference to God: “If the concept of a thing is already entirely complete, I can still ask about this object whether it is merely possible, or also actual,5 or, if it is the latter, whether it is also necessary? No further determinations in the object are hereby thought […].” (KrV, A 219/B 266.) It is these claims that are incorporated in the famous slogan: “Being is obviously not a real predicate, i.e., a concept of something that could add to the concept of a thing.” (KrV, A 598/B 626.)
Notably, however, although existence adds nothing to the thing or its concept, it does add something: “Through the actuality of a thing I certainly posit more than possibility, but not in the thing; for that can never contain more in actuality than what was contained in its complete possibility.” (KrV, A 234–5/B 287 n.) The reason why existence adds nothing to the thing is simple: it rather adds the thing itself:
[A] distinction must be drawn between what is posited and how it is posited. As far as the former is concerned: no more is posited in an actual thing than is posited in a merely possible thing, for all the determinations and predicates of the actual thing are also to be found in the mere possibility of that same thing. However, as far as the latter [the “how”] is concerned: more is posited through actuality […], for positing through an existent thing involves the absolute positing of the thing itself as well. (OPG, 75, my emphasis.)
For an actual thing to correspond to its concept, it must instantiate exactly the same predicates that are also thought in its concept, and so the “absolute positing” (a concept to be clarified shortly) cannot add any predicates or properties to the object:
A hundred actual dollars do not contain the least bit more than a hundred possible ones. For since the latter signifies the concept and the former its object and its positing in itself, then, in case the former contained more than the latter, my concept would not express the entire object and thus would not be the suitable concept of it. (KrV, A 599/B 627.)
But what does it mean for actual dollars to “contain” no more than possible ones? For Kant containment is a technical term, defined as follows:
Every concept, as partial concept, is contained in the representation of things; as ground of cognition, i.e., as mark, these things are contained under it. In the former respect every concept has a content [Inhalt], in the other an extension [Umfang]. (JL, 95.)
This distinction is similar to the contemporary one between intension and extension of concepts. The content of a concept is the set of partial concepts (conjunction of predicates) that constitute its thought-content (intension); its extension is the set of (actually) existing things to which it refers.6 E.g. the concept bachelor contains the concepts man and unmarried, whereas its extension are all actual things that instantiate the predicates of man and unmarried. The partial or sub-concepts of a concept Kant calls its (characteristic) marks (Merkmale) – a term Frege employs similarly (see 3.1).
Dollars “contain” those properties7 that they have in virtue of instantiating the predicates contained in the concept of dollar. Rather than enlarging the intension of A by adding a predicate to it, “A exists” posits a non-empty extension for A by positing the thing itself – the referent of A. Thus actual dollars must instantiate exactly the predicates expressed by their concept (i.e. possible dollars), and whatever difference actual and possible dollars may have, it does not consist in the former having an extra property: existence. If existence added to the content of A, “A exists” would change the intension of A rather than positing a non-empty extension that corresponds exactly to this intension.8 But what does “positing” mean?
2. 2. Existence as Absolute Positing
In The Only Possible Ground Kant defines positing as follows:
The concept of positing or setting [Setzung] is perfectly simple: it is identical with the concept of being in general. Now, something can be thought as posited merely relatively, or, to express the matter better, it can be thought merely as the relation (respectus logicus) of something as a characteristic mark to a thing. In this case, being, that is to say, the positing of this relation, is nothing other than the copula in a judgment. If what is considered is not merely this relation but the thing posited in and for itself, then this being is the same as existence. (OPG, 73, translation altered.)
Kant also distinguishes between relative and absolute positing (OPG, 73–4; AA 28: 554; AA 29: 822; Longuenesse 1998, 352). In the former predicates are posited to the subject-term, whereas the latter posits the subject itself. This distinction is manifest in the two uses of is, logical and existential, in “A is B” and “A is”:
In the logical use [being] is merely the copula of a judgment. The proposition God is omnipotent contains two concepts that have their objects: God and omnipotence; the little word “is” is not a predicate in it, but only that which posits the predicate in relation to the subject. Now if I take the subject (God) together with all his predicates […] and say God is, or there is a God, then I posit no new predicate to the concept of God, but only posit the subject in itself with all its predicates, and indeed posit the object in relation to my concept. Both must contain exactly the same, and hence when I think this object as given absolutely (through the expression, “it is”), nothing is thereby added to the concept that expresses merely its possibility. (KrV, A 598–9/B 626–7, translation altered.)
Being is not a predicate of a thing either in its relative or absolute use. The logical “is” of “A is B” states that the predicate B belongs to the subject A – it predicates B of A.9 The absolute “is” of existence in “A is” states that A itself is instantiated by an actual thing, i.e. that A, along with its content, exists. It does not posit anything additional in the thing but posits the thing itself: “[w]ith actuality, the object is added to a concept, but nothing is added to the object” (AA 29: 822). “Wombats (actually) exist,” or “There (actually) are wombats,” states that some actual things instantiate all the predicates contained in the concept of wombat.
Although even absolute positing in fact involves a relation (see the long quote above), this relation is essentially different from that of relative positing. Whereas the latter relates two concepts (subject and predicate terms), the former relates the subject concept to objects or things (as its referents), i.e. to entities of a wholly different kind.10 (Cf. 3.2 and 4.2.) Thus absolute and relative positing denotes two different and mutually independent relations. Relative positing adds the predicate B to the subject A but does not as such concern the (possible) existence of A. This existence is added via the absolute positing of A, which determines some thing a as instantiating the predicates of A.
Although the two positings can appear together, and often do (most notably always in empirical judgments), they are independent to the extent that one can predicate B of A without positing A itself absolutely, and one can posit A absolutely without adding any new predicate B to it. Since no consideration of existence is involved in relative positing, it can even be used to relate fictions (non-entities) to one another:
If I say: ‘God is omnipotent’ all that is being thought is the logical relation between God and omnipotence, for the latter is a characteristic mark of the former. […] Whether God is, that is to say, whether God is posited absolutely or exists, is not contained in the original assertion at all. For this reason, ‘being’ is also correctly employed even in the case of the relations which non-entities [Undinge] have to each other. For example: ‘The God of Spinoza is subject to continuous change.’ (OPG, 74, translation altered.)
2. 3. Kant on Existence and Truth
Since existence does not belong to the content of concepts, analytic judgments cannot decide anything about it. This is because, as Kant’s explanation of the analytic/synthetic distinction makes clear, an analytic judgment merely analyses or makes explicit this content:
In all judgments in which the relation of a subject to the predicate is thought […] this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case I call the judgment analytic, in the second synthetic. (KrV, A 6–7/B 10.)
If existence cannot be a part of the content of concepts, analytic judgments can neither express nor on their own involve existence of either the subject or the predicate. This fits Kant’s view that analytic judgments can be (necessarily) true without being true of any objects, namely if the objects it speaks of do not exist.11 (E.g. KrV, A 151–2/B 190–91.) For example, “Unicorns are one-horned” is a necessary analytic truth despite there (likely) being no unicorns. Kant employs this point extensively in the Transcendental Dialectic, e.g.:
However, that the I of the “I think” must always be regarded as subject […] is an apodictic and even identical proposition [i.e. necessarily true]; but it does not signify that I as object am for myself a self-subsisting being or substance. (KrV, B 407, translation altered.)
God is omnipotent; that is a necessary [i.e. necessarily true] judgment. Omnipotence cannot be cancelled if you posit a divinity, i.e., an infinite being, which is identical to this concept [i.e. the judgment is analytic]12. But if you say, God is not, then neither omnipotence nor any other of his predicates is given; for they are all cancelled together with the subject, and in this thought not the least contradiction shows itself. (KrV, A 595/B 623; cf. OPG, 74.)
Truth’s independence of existence might appear to conflict with Kant’s definition of truth as correspondence (e.g. KrV, A 58–60/B 82–5; JL, 50–3). But in fact, first, for Kant correspondence between concept and object is required only for what he calls “material truth,” namely truth about (possibly existing) objects. What he calls “formal” or “logical” truth does not concern objects but merely “the agreement of cognition with itself, in complete abstraction from all objects whatsoever” (JL, 51). Hence an analytic judgment can be formally true or false even when its material truth is impossible.13
Existential judgments concern the move from formal to material truth by determining whether and how the concept is instantiated. Ascribing a modality to a concept determines whether the concept can instantiated, is instantiated, or must be instantiated. In each case the concept and the predicates contained in it must be the same, which is why existence (or any other modal concept) cannot amplify this content.
While the truth of analytic judgments is for Kant independent of the existence of their objects, this is not so for all synthetic judgments. Knowing whether “Socrates is a man” is true requires experiential knowledge about him, which seems to depend on him existing or having existed. Thus, plausibly, the truth of empirical judgments of this sort implies the existence of their objects. This picture is substantially complicated by fictions or imaginary beings, however: the truth of “Gilgamesh is a man” should not imply his existence. One way to deal with this is to say that judgments about fictions indeed cannot be true – this is the Fregean view – but this seems intuitively unappealing: asserting that Gilgamesh is a woman will certainly earn as strong objections from historians as claiming that Socrates is a woman. (Cf. 3.3 and note 27.)
There is a bevy of complex problems associated with the existential status of fictions, and for the greater part they must be set aside here. What is important is that there is conceptual space in Kant for true synthetic judgments about non-existing things. Since synthetic judgments require intuitions, there would have to be intuitions of non-existing objects. This is indeed a plausible reading of how Kant sees imagination, the capacity for “representing an object even without its presence in intuition” (KrV, B 151) or “a faculty of intuition without the presence of the object” (AA 7: 167). In a lecture note, Kant specifically states that imagination “is the faculty for producing images from oneself, independent of the actuality of objects” (AA 28: 237).14 Indeed, among others, synthetic a priori judgments of geometry e.g. about a million-angle are true irrespective of whether there actually are any million-angles. Thus the view suggested here is that the judgments “Socrates is a man” and “Gilgamesh is a man” are both true synthetic judgments, but since Socrates exists and Gilgamesh not, only the former is true of an actually existing object while the latter is true only of a possibly existing object. In terms of correspondence, the judgment can correspond to actual states of affairs and objects or to merely possible ones (see 4.3). This allows Kant to keep truth separate from (actual) existence also for some synthetic judgments. (Cf. Vanzo 2014, 226–8.)
Explication of the status of such possibilia and the exact nature of existence, construed as a relative property, will have to wait until section 4. An analysis of Frege’s theory of existence provides us with powerful tools for developing the thus far merely preliminary analysis of Kant’s theory of existence further.
3 – Frege
According to the Kant-Frege view, Frege approved of Kant’s critique of the ontological argument and gave logical and formal clarity to Kant’s idea “that being is a property of thought […] in the view that existence is a property of a concept” (Haaparanta 1985, 144; cf. Bennett 1974, 62; GA, § 53). Frege grounds his formalisation of existence on an analysis of the ambiguity in the word “is.” He recognises four different meanings of “is”: identity, predication, existence, and subordination or class-inclusion (e.g. GA, § 57; BG, 194 ff.; Haaparanta 1986, 269–70). If I say “7 + 5 is 12” or “Hesperus is Phosphorus,” I mean that 7 + 5 denotes the same individual object as 12 and Hesperus the same individual as Phosphorus. Here meanings (Bedeutungen) of proper names are identified with each other (SB, 25). Subordination, in turn, relates classes or sets of objects: “Planets are celestial bodies.” Predication occurs when an individual is subsumed under a class, e.g. “Venus is a planet.” Finally, the existential “is” states e.g. that “Planets are” or “There are planets.”15
One of Frege’s great contributions to logic is that he assigns each sense of is a different logical form. In contemporary notation, identity is expressed by the identity sign a = b; predication by F(a);16 subordination by x(F(x) G(x)); and existence by xF(x) (Haaparanta 1985, 14–15; 1986, 269–70). Thus existence is denoted by the existential quantifier.17 Frege also adheres to the interdefinability of the existential and universal quantifiers (xF(x)xF(x)) proposed by George Boole and Augustus De Morgan a few decades earlier (Vilkko & Hintikka 2006, 370–1). In this view “all” is parsed as a lack of exception, which, when combined with Frege’s view of existence as denoted by the existential quantifier, introduces an extremely important asymmetry to the quantifiers: since lack of exception does not imply existence of positive instances, the universal quantifier “all” lacks existential force but the existential quantifier “some” has it. Hence in contemporary logic the classically valid inference from all to some becomes invalid, for the former can involve empty terms while the latter implies their non-emptiness.
3. 1. Existence as Second-Level Concept
What does it mean that existence is denoted by a quantifier? The answer lies in Frege’s view that all logical operations share the same basic form of a function that takes an argument. A function alone is “unsaturated” and has to be completed by an argument. (FB, 6.) In identity the function is expressed by the identity sign ( ) = ( ), where the arguments taking the empty places are individuals a and b – e.g. Hesperus and Phosphorus. In predication the function is F( ) (a concept) and the argument is an individual a: F(a), e.g. “Hesperus is a planet.” What is the function for existence? According to Frege, in contemporary notation it is ( ) so that the argument place is filled with a first order predicate F( ): x(F(x)). Here “x” denotes the variable object or argument of predication. In predication the concept F applies to an individual a; in existential propositions the existential quantifier is applied to the concept F. That the concept F rather than the object a is the argument of the existential quantifier means that existence is a second-level property of concepts rather than a first-level property of objects: it is a function of concepts, not of their objects.18 (GA, § 53; BG, 199.)19 Frege exemplifies his view of existence thus:
I have called existence a property of a concept. How I mean this to be taken is best made clear by an example. In the sentence ‘there is at least one square root of 4’, we have an assertion, not about (say) the definite number 2, nor about -2 [i.e. about the objects 2 or -2], but about a concept, square root of 4; viz., that it is not empty. (BG, 199.)
Thus “There are A’s” is true just in case the concept of A is instantiated by at least one object, which is to say that the concept A has the property of being non-empty (having a meaning) rather than the object a having a property of existing. In other words, for Frege existence is a concept’s property of being instantiated (NS, 73–4).
But what kind of a property is a second-level property? Notably, Frege rejects certain obvious candidates: in the sentence “the concept ‘horse’ is a concept easily attained” (BG, 195), “the concept ‘horse’” designates an object while “easily attained” designates a first-level concept. According to him, the sentence is an example of subordination that “must not be confused with” something falling under a higher-level concept (ibid.). Subordination is for Frege a second-level relation (not a property) between two (first-level) concepts (Macbeth 2005, 91–2, 104; Künne 2010, 222).
Leila Haaparanta clarifies Frege’s point via his distinction between properties and characteristics (Merkmale) (Haaparanta 1986, 271–2; cf. also Kluge 1980, 99; Kenny 1995, 75; GA, §53; BG, 201–2). By characteristics (or characteristic marks) Frege – like Kant – means the predicates or sub-concepts that make up the concept (GA, §53) – or, in Kant’s terms, its content (cf. 2.1). An object that falls under a first-level concept has the characteristics of that concept as its properties (Haaparanta 1986, 271–2). That existence is for Frege a second-level property of concepts means that it cannot be one of their characteristic marks (for those are properties of objects). Thus Frege’s claim is very reminiscent of Kant indeed: in Kantian terms, existence does not belong to the content of the concept as its predicate (characteristic), and hence the object falling under it does not have any existence-property; rather existence is a property of (not in) the concept itself. “A is” means that the predicates of the concept-word A – in Kantian terms, the content of the concept A – are instantiated by something, which is a property of the concept rather than of this “something.”
Now, what Frege does not mean is that existence were a property of the concept in the sense that it – the concept – exists. (Cf. note 31.) Although an obvious point, it is still surprising to an extent, for the view that Frege is taken to oppose is that “Wombats exist” means that (some) wombats have the property of existing. To say that it is rather the concept of wombat that has this property might seem to suggest that it is the concept that exists. That this is not the case shows that Frege makes two transformations: he not only moves the property of existence up one level but also makes existence a specific kind of property, the property of being instantiated.
This is important, for it suggests certain alternatives to Frege’s conception. Perhaps existence is not a property of a concept to be instantiated but a property of an object to instantiate: “a exists” would be true if the object denoted by “a” has the property of instantiating some concept. (That this point concerns individual objects will become important in 3.2, 4.1 & 4.2.) This is similar to Meinong’s view that (pace Frege) existence is a property of (some) objects.20 A third alternative also seems possible: perhaps existence denotes an instantiation-relation between the concepts and objects – a kind of correspondence or concurrence (Zusammenfallen). Although these differences may seem minute, they are highly central given that for Frege relations and concepts are mutually exclusive: “We called such functions of one argument concepts; we call such functions of two arguments relations.” (FB, 28.) This becomes crucial, for, as I will argue shortly, Kant holds a variant of such a relational interpretation of existence, rather than either the Fregean second-level property or Meinongian first-level property view.
3. 2. The Existential Quantifier
The question why Frege does not construe existence as a relation proper arises also with the (Fregean as well as contemporary) notation of existential sentences, in contemporary notation: x(F(x)). Although the -quantifier is to take F(x) as its sole argument, the formalisation cannot dispose of the x, i.e., the thing(s) that F in turn takes as its argument. Moreover, rather than F(x), x would appear to be the immediate argument of . It is an x that exists: “There is an x such that…” In this picture one might construe the -function as taking two arguments: x and F(x). Since a two-placed function is a relation (FB, 28), one might try to interpret existence (as existential quantifier) as a relation between an object x and its concept F(x). But according to Frege, F(x) is the sole argument of the -function, and the reference to x is just to “delimit[] the scope” (BS, §11) of the quantifier. For example, the universal quantifier states: “whatever we may take for its argument, the function is a fact” (BS, §11).
Indeed, on a closer look it is clear that the “x” in x(F(x)) does not stand for an argument at all. It expresses the scope of objects of which we state that at least one of them instantiates F. That the existential quantifier is not a relation – at least not in the sense of being a two-placed function – is evident also in following ways. First, completing the relation-word x > y e.g. as 3 > 2 retains all the elements – the >-function and the two arguments 3 and 2. But in the case of x(F(x)), substituting x with an object a (existential instantiation) makes the -function disappear: one does not write a(F(a)) as if replacing x with a, but simply F(a): a instantiates F. Second, whereas x > 2 has no truth value until x is determined, x(F(x)) clearly does: it is true if at least one member – determinate or not – of the scope x is F, otherwise false. The “x” stands for a class of objects that make up the scope – in the unlimited case it is the class of all objects. Thus one also commonly writes e.g. x
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