Carnap on Categorial Concepts
Carsten Klein
(University of Bonn, Germany)

The term “category” is seldom used by Carnap himself, and his treatment of categories is not a topic that is frequently addressed by the literature on Carnap. Therefore, in what follows I shall first spend some time explaining what I have in mind when speaking about categorial concepts. In order to to this, I shall give a survey of the kind of problems I am interested in as they appear in the writings of Gottlob Frege, Bertrand Russell and Ludwig Wittgenstein. This will lead directly to Carnap’s discussion in the Logical Syntax of the problems that Wittgenstein’s Tractatus presented to him and to his efforts to solve them. By making Carnap part of the tradition that deals with categorial questions, I also hope to highlight some features of his general philosophical position.

1. Introduction 
The concept of a category goes back at least as far as Aristotle and is used throughout the history of philosophy in a number of different ways. The basic idea behind the type of category I shall focus on is that a certain domain is divided into a number of areas which are fundamentally different – so different, in fact, that they don’t have anything in common at all. Depending on where the specific interests lie, the categories under consideration can be of an ontological, an epistemological, a semantical, a grammatical or a logical kind, and perhaps there are many other types of categories.
In modern analytical philosophy, the concept of a category was re-introduced by Gilbert Ryle, whose definition of categorial difference is still echoed in the recent literature: 
Two proposition-factors are of different categories or types, if there are sentence-frames such that when the expressions for those factors are imported as alternative complements to the same gap-signs, the resultant sentences are significant in the one case and absurd in the other (Ryle 1938, 181).
In Ryles terminology, sentence-factors are just parts of sentences, sentence-frames are whatever is left of a sentence if a factor is taken away from it, and proposition-factors are significata of sentence-factors. Hence, categories are applied not to linguistic expressions but to their meaning, making them semantical rather then syntactical or linguistic categories. Unfortunately, Ryle does not specify the nature of these significata in his category-paper. In the broader context of Ryles philosophical view, however, it seems to be obvious that an ontological reading in the sense of a realistic semantical is out of the question. Since this is not the place for a detailed Ryle-exegesis, I would just like to point out that Ryle later identifies the meaning of linguistic expressions relevant for philosophical (in contrast to, e.g. philological) purposes with their “logic of functioning” (Ryle 1957, 371). For the purpose of this paper it is more important, however, to point to another feature of his definition, which is common to all categorial concepts which are of interest in the present context: Although not always defined by recourse to the meaninglessness or absurdity of sentences, categorial difference is always connected with it in an intimate way. This will be made clear in the following section. 

2. Categorial Concepts in Frege, Russell and Wittgenstein 
2.1 Frege
I will begin the discussion of the tradition of the use of categorial concepts in modern analytical philosophy with Gottlob Frege. Although Frege does not use the term, two categories are of fundamental importance in his logic and semantics, and they lie, in fact, at the very heart of his functional approach to predication: object (‘Gegenstand’) and concept or, more general, function (Frege 1891; 1893, §§1-2). A function is characterized by the property of being incomplete or “unsaturated”; it is taken not to be able to stand alone. It needs to be supplemented by an argument in order to yield a complete whole. An object, by contrast, is everything that is not a function, that can stand alone, that is complete. Linguistic expressions for functions, so-called functional signs, hint at the need of supplementation of the functions they designate by at least one placeholder, whereas the linguistic expressions for objects, i.e., names, do not have such an empty gap. Functions are further specified as to different levels: A first-order function can only have an object for its argument, a second-order function can only have a first-order function for its argument, and so on.
The distinction between objects and functions or functions of different order is a categorial distinction: no incomplete part or “frame” (to use the Rylean terminology) of a sentence which can be supplemented by a name to yield a complete and significant sentence can ever be supplemented by a concept word or functional sign, and vice versa. Any attempt to insert a concept word where a name is required would lead to nonsensical expressions. There is, however, one important difference between Frege and Ryle: Whereas Ryle’s categorial distinctions of the significata of linguistic expressions are founded on the meaningfulness of the expresions in question, the founding relation in Frege’s conception works the other way round: The categorial difference obtains between objects and concepts in the first place, allowing only items of different categories to be joined to form a complete whole. This categorial structure of the constituents of reality (in the broadest sense) is reflected on the level of language, allowing only a combination of concept words or predicates and names to yield well-formed sentences. Frege’s categorial distinction is primarily an ontological one, which is reflected by a syntactical distinction on the linguistic level. Hence, we need a new definition of categorial difference to capture the spirit of Frege’s approach. If we go back to the idea that categorially different items are such that they have no common property, and if we remember that concepts are functions according to Frege’s approach, the following definition suggests itself as a reconstruction of the idea of categorial difference inherent to Frege’s logic:
Two entitites belong to different categories, if there is at least one concept such that it can be combined with (saturated by) one of these two entities, but not with (by) the other one.
Concepts like “object” or “function”, which are associated with categories, shall be called categorial concepts.
This definition yields the result desired since every concept that can be saturated by an object is a first order concept and as such it therefore cannot be saturated by a function. And every concept that can be saturated by a first-order function is a second-order concept and as such it cannot be saturated by an object. On the other hand, every first order concept can be saturated by any object, and the same applies to concepts of a higher order. Frege’s distinction between objects and functions satisfies an even stronger condition as put forward by the following definition:
Two entitites belong to different categories, if it is true for all concepts that if the concept can be combined with (saturated by) one of these entities, it cannot be combined with (saturated by) the other one, and vice versa.
Any violation of these rules would result in categorial nonsense that manifests itself on the level of linguistic expressions in the form of syntactically ill-formed expressions.
An alternative to the ontological reading of the categorial difference between functions and objects would Frege take as having in mind an epistemic concept of a category, since a categorial distinction, analogous to the one just mentioned, applies to the level of Fregean senses. This distinction could be taken as constitutive for the definition on the level of Fregean “Bedeutungen”, i.e. the distinction between functions and objects. Whatever interpretation we choose, it is true that categorial differences are not based upon properties of linguistic expressions but upon a presumed categorial framework of the world (in the widest sense possible) or of human reason taken as the capacity of making jugdements about the world. In Frege’s universalistic conception of logic, the logical structure of judgements and the categorial distinction between objects and concepts is assumed to be apriori, absolute and universally valid. Whether the categories are given independently of our judgements and determine the structure of these judgements or whether the logical structure of our judgements should be regarded as fundamental and constitutive for the categorial structure – this is a question that goes beyond the scope of this paper. There is an ongoing debate in the literature on Frege as to whether he should be taken as a Platonist or rather as a Kantian philosopher.
However, according to Frege, explanations of and distinctions between categories are the genuine task of epistemology that result in statements like “F is a concept” or “the concept F has such and such properties”. We may call them category-propositions following Ryle or categorial elucidations borrowing a term used by Frege and Wittgenstein. Unfortunately, it is exactly this kind of statements that leads to serious problems since such elucidations result in either meaningless or paradoxical sentences. Since Frege addresses these issues by discussing the concept horse, the problems involved became known as the concept-horse-problem.
On the surface, the sentence 
(1)	(The concept) horse is a concept.
shares the same grammatical structure as
(2)	(The city of) Jena is a city.
Hence, it should be taken as a meaningful and true sentence in the same way that (2) is taken as meaningful and true. But when examined more closely (1) turns out to be false. According to Fregean semantics, the expression “the concept horse” or even “horse” by itself must be a name that denotes an object, as does any expression of the form “the concept F”. And it is indeed quite obvious that this expression can never replace a concept-word, because a concept-word must have at least one placeholder. Therefore, one is forced to accept the paradoxical sentences
(3)	(The concept) horse is not a concept.
or
(4)	(The concept) horse is an object.
as true sentences.
In his later works, Frege seems to have abandoned the idea that an expression like “the concept horse” is a name because there is no object that could possibly be designated by such an expression (1924/25, 369-370). This view helps to avoid the paradoxical result that sentence (1) is false whereas sentences (3) and (4) are true. But now it becomes impossible to assign any truth-value whatsoever to these sentences, because in Fregean semantics a sentence cannot have a truth-value when one of its parts has no Bedeutung. On both interpretations it is thus impossible to say anything meaningful at all about concepts because all efforts that make them the subject of discourse miss the mark as Frege himself explicitly recognized:
I admit that there is a quite peculiar obstacle in the way of an understanding with my reader. By a kind of necessity of language, my expressions, taken literally, sometimes miss my thought; I mention an object, when what I intend is a concept. I fully realize that in such cases I was relying upon a reader who would be ready to meet me halfway – who does not begrudge a pinch of salt. (Frege 1892, 192)
And this problem does not only concern sentences like “The concept horse is a concept”, but also sentences like “(The concept) horse is a one-place concept” or “(The concept) horse is unsaturated” which are either false or without truth-value and thus meaningless. It seems that categorial elucidations of this kind can only be expressed by paradoxical or meaningless sentences. The functional approach to predication reaches a limit when it comes to those concepts which lie at the heart of this very approach. Thus, the knowledge of the fundamental categorial difference lies outside the realm of conceptual knowledge and meaningful discourse.

2.2 Russell
The second philosopher whose work is crucial for any understanding of the notion of categorial difference as it occurs in modern philosophy is Bertrand Russell. We shall soon find out that his famous theory of types is afflicted with problems which are very similar to Frege’s concept-horse-problem. Unlike Frege, however, Russell did not pay much attention to these problems.
The theory of types, which has the advantage of avoiding the paradoxes that infected Frege’s system of logic, is a more general approach than Frege’s categorial distinction between functions of different order. It has become customary to distinguish two versions of this theory: the simple theory of types, which avoids the logical or set-theoretical paradoxes, and the ramified theory of types, which avoids the semantical paradoxes. In what follows, the discussion will be restricted to the simple theory, which is of primary importance for the questions addressed in this paper.
In his earliest explanation, Russell defines a type by considering ranges of significance of propositional functions (x), i.e. “a range within which x must lie if (x) is to be a proposition at all, whether true or false”. Ranges of significance form types: “If x belongs to the range of significance of (x), then there is a a class of objects, the type of x, all of which must also belong to the range of significance of (x), however  may ve varied” (ibid.). According to this theory, the lowest type is the class of individuals (type 0), i.e., any object which is not a class. The next type consists of classes of individuals (type 1), the type following classes of individuals consists of classes of classes of individuals (type 2) etc. Generally speaking, type m  1 consists of all classes of entities of type m (Russell 1903, 517, 523f.).
Russell’s own view with respect to an interpretation of his theory of types is ambiguous. One may agree with the view that it was not meant as description of any natural language or as prescription for reforming any such language, but rather as a recommendation for constructing a special artificial language which serves the purpose of mathematical logic and is at the same time free from contradictions (Copi 1971, 27). But it is still an open question whether the theory of types is to be understood as a formal theory only, i.e., as a syntactical meta-theory stating rules for the construction of permissible formulas. Although in 1918 Russell stated that “the theory of types is really a theory of symbols, not of things” (Russell 1918-19, 267), he very often uses a thing-language when explaining his theory by speaking of objects, individuals, constituents of facts or terms instead of names or words (e.g. Russell 1903, 517, 523-524, 1908, 75-80, 1924, 332.). Hence, an ontological reading cannot be dismissed at first glance. If the hierarchy of types is indeed meant as a hierarchy of different categories of entities, Russell’s theory runs into the same kind of problems that troubled Frege. They have been put forward forcefully by no less a logician than Kurt Gödel. In his contribution to the Schilpp-volume dedicated to Russell, Gödel notes that the theory of types is based on the following assumption: “Whenever an object x can replace another object y in one meaningful proposition, it can do so in every meaningful proposition.” But, Gödel continues, its very assumption makes the formulation of this principle as a meaningful proposition impossible, “because x and y must then be confined to definite ranges of significance which are either the same or different, and in both cases the statement does not express the principle or even part of it. Another consequence is that the fact that an object x is (or is not) of a given type also cannot be expressed by a meaningful proposition” (Gödel 1946, p. 149-150). To formulate the principle which lies at the heart of the theory of types, one has to violate the type-distinctions that are laid down by this very theory. This is illustrated by another definition of type-sameness given by Russell in 1924: “A and B are of the same logical type if, and only if, given any fact of which A is a constituent, there is a corresponding fact which has B as a constituent, which either results by substituting B for A, or is the negation of what so results” (1924, 332). But, as Russell himself points out a little later, the word “fact” (like “attribute”, “relation”, “complex”, “truth” among others) by its very nature sins against the theory because its application is not restricted to a specific type (ibid., 334). The similarity to Frege’s concept-horse-problem is obvious: just as Frege was unable to meet the requirements for significant sentences which are based on the categorial distinction between objects and concepts in order to introduce this very distinction and to classify items accordingly, Russell has to use concepts which do not fulfil the requirements for significant sentences stated by the theory of types in order to formulate the principles of this very theory and classify items with respect to their type-status. It seems that any attempt to express the categorial structure of reality necessarily trespasses the boundaries of meaningful discourse.
Russell’s way out of this problem is the move of semantical ascent: in his reply to Max Black, who discussed a similar problem within the theory of types in his contribution to the aforementioned Schilpp-volume (Black 1946, 233-239), he follows Black’s suggestion to apply type-distinctions not to entities, but only to words. They are taken to acquire their type-status neither as physical occurrences nor by categorial properties of the entities they might signify but by their syntactical functions (Russell 1946, 692). Russell thus goes back to the view put forward in 1918. It should be noted, however, that he himself admits in his reply to Black that this kind of problem has troubled him “a good deal”, which indicates that the purely formal reading of the theory of types was by no means the matter of course for him. The idea of a syntactical interpretation of the theory of types is very similar to Carnap’s treatment of Wittgenstein’s problems in the Tractatus as put forth in The Logical Syntax of Language in 1934; it will be discussed in the second part of this paper. But at first we will have to take a closer look at the Tractatus itself in order to complete the pre-Carnap history of categorial concepts.

2.3 Wittgenstein
The concepts we have called ‘categorial concepts’ are labelled ‘formal concepts’ by Wittgenstein. Among the examples he offers are object (“Gegenstand”), complex, fact, function, relation and number. Marks of formal concepts are formal or internal properties, i.e., properties of which it is impossible that an object does not possess them (Wittgenstein 1921, 4.122, 4.123, 4.126, 4.1272). Hence, we may capture Wittgenstein’s notion of a category by the following definition:
Properties are categorial if and only if every object that possesses them necessarily possesses them.	
Concepts that have categorial properties as characteristic marks are called categorial concepts. An example of formal properties is given at a very early stage in the Tractatus: “Space, time, and colour (being coloured) are forms of objects (2.0251)”. It is impossible that an object (in the sense of a physical object) is not situated in space and time and does not have a colour. And we must already know that physical objects but not numbers are in space and time; otherwise we could not make an assertion like “Christiane was present at the Carnap conference held in Jena in September 2001” or avoid an assertion like “2 + 2 at 3 o’clock equals 4” (4.1272).
Hence, formal properties constitute the categorial structure of reality; a structure which we must presuppose before we can formulate meaningful statements about the world. According to Wittgenstein’s picture-theory of language, the logical form of the world must be matched by the logical form of thought and language to make meaningful statements possible (3.12). Hence, the correct categorial structure of language is a necessary condition for any meaningful statement. 
Philosophy is in Wittgenstein’s view primarily concerned with explaining this categorial structure. While Frege admitted that hints or elucidations are an unavoidable part of philosophical (e.g., meta-logical or semantical) explanations, Wittgenstein goes even further in identifying philosophical statements with categorial elucidations: “A philosophical work consist essentially of elucidations” (4.112). This, however, will prove to have devastating consequences for the possibility of philosophy.
Unfortunately, it is exactly sentences of the kind of 2.0251, i.e., those that state the formal properties of objects and thus their categorial structure, that are meaningless or nonsensical (“unsinnig”) according to Wittgenstein. The reason for this is that formal concept-words cannot function as proper concept-words (4.1272). The difference between those two kinds of concepts can be described in the following way: To determine whether a given object falls under a concept, we must have a way of identifying the object independently of the properties which are the characteristic marks of the concept. In order to give an answer to the question whether Black Beauty is a horse, we must have a way of picking out Black Beauty and determining whether it actually possesses the properties we associate with the concept horse. And we can do so because in this case we have more general concepts than just horse, e.g., animal, and we can identify Black Beauty as an animal and determine in a second step whether it possesses the additional properties that make it a horse. Hence, we need not to presuppose that Black Beauty is a horse in order to speak about it: the argument is given independently of the function we are going to apply to it. To put it in a different way: The sentence “Black Beauty is a horse” is meaningful because, even if it is true, it could have been false. Otherwise, the sentence would not convey any information and, hence, would carry no sense:
It belongs to the essence of a proposition (“Satz”) that it should be able to communicate a new sense to us. (4.027)
However, when we consider a formal concept like object (in the sense of spatio-temporal object), it is exactly this condition which is not satisfied. For it makes no sense to look at a piece of paper and ask: “Is this object situated in space and time?”, because we must presuppose that it belongs to the category of spatio-temporal objects before we can make any meaningful statement about it. There is simply no way of talking about any spatio-temporal object without talking about it as a spatio-temporal object. The reason for this lies in the categorial character of formal concepts: there is no higher concept that can be used to refer to the object in question. The idea of an entity which is categorially indetermined is simply nonsense for Wittgenstein. For it is the very essence of formal concepts that their characteristic marks correspond to the properties that the objects in question necessarily possess. Therefore, sentences that contain formal concept words must be meaningless (nonsensical). They try to give a description of the categorial stucture of the world or of language that cannot be described because it has to be already presupposed in any description. We can make the same point by observing that a categorial proposition does not comply with the following demand:
A proposition states something only insofar as it is a picture. (4.03)
Categorial propositions, however, are not pictures of the world but rather try to express the formal properties a language must have in common with the world of facts in order to make pictures possible at all. But:
A picture cannot, however, depict its pictorial form: it displays it. (2.2172)
Formal concepts cannot be expressed by functions like proper (“eigentliche”) concepts. Rather, they are expressed by variables:
Every variable is the sign for a formal concept. (4.1271)
This is to be taken as saying that the form of a variable expresses the formal properties of its values. And these formal properties cannot be expressed in language, but they show themselves in the logical type of the variable. In a perfect symbolism, a variable immediately shows what kind of objects it can have as values. Formal concepts are used in the attempt to specify the type or category of those entities that are in the value-range of a variable. But according to Wittgenstein, that “what can be shown cannot be said” (4.1212).
It should have become clear that Wittgenstein encountered the same problem that troubled Frege and Russell: in order to speak about the categorial structure of the world it is necessary to use statements that are meaningless according to the rules of meaningful discourse that are required by the very structure which is to be explained. His reaction to this problem is similar to Frege’s and very different from Russell’s. When discussing Carnap’s treatment of the Tractatus, I will try to show that a syntactical reading of the Tractatus similar to Russell’s proposal for his own theory of types is in no way compatible with Wittgenstein’s intentions. This should be taken as evidence that Wittgenstein’s philosophical attitude in general and his view of the problems associated with categorial distinctions in particular are much more akin to Frege’s philosophy than to Russell’s. It is true, of course, that Wittgenstein developed his conception of saying versus showing during his long occupation with Russell’s logic and the theory of types. Wittgenstein’s idea that a symbol shows its logical type within an adequate symbolism (and it is thus neither possible nor necessary to say something about it) is meant as a way of preserving what he regarded to be the correct elements of Russell’s conception of type-differences while avoiding to ascribe to the possibility of a meta-theory of logic which deals with these differences. According to Wittgenstein, there is no theory of types outside logic: “Logic must look after itself” (5.473). Only in a second step Wittgenstein developed these ideas into a general philosophical position by including ontology, epistemology, ethics and aesthetics in the realm of what shows itself, but cannot be said. This fact granted, it is not surprising that Wittgenstein’s examples of categorial concepts are almost the same as Russell’s, while he differs from Frege’s ‘coarse-grained’ distinction between only objects and functions. But even if the core subject of Wittgenstein’s doctrine of showing may be Russell’s logic and theory of types, I want to advocate the thesis that only the study of Fregean semantic allowed him to develop a sensibility for the far-reaching consequences of the problems associated with categorial discourse. Unlike Russell, who was optimistic even in 1946 that a satisfactory theory of logical hierarchy would be developed eventually (1946, 692), Frege had taken these problems very seriously and acknowledged the fundamental impact they have on logic and semantics. Wittgenstein, rather than sharing Russell’s insouciance, took these problems even more seriously: Frege thought that despite the unavoidable obstacles of language he encountered, the properties of the categorial framework of our knowledge are accessible through language. This access, however, is not one of asserting, but of circumscribing, hinting at or of showing, where the speaker has to count on the readiness of the audience or the reader to meet him halfway. A philosopher of logic would thus have to use a mode of speaking that is quite different from that of a logician or a scientist, one that circles around knowledge which itself is non-propositional. In this way, a meta-theory of logic – or philosophy in general –would at least be possible. Wittgenstein too came to the conclusion that the knowledge of the categorial structure of language or of the world – its “logical form” (2.18) – cannot be expressed in meaningful sentences. By the requirements for meaningful discourse laid down in the Tractatus the categorial elucidations of the Tractatus itself turn out to be meaningless or “unsensical” in exactly the same way as are the statements of speculative metaphysics. But although Wittgenstein would have admitted that the statements of the Tractatus have a practical value in leading the reader to the insight of their own meaninglessness, unlike Frege he would not have give in to accepting the idea of a special kind of philosophical discourse which does not correspond to the truth-functional paradigm of logic and the natural sciences. Wittgenstein declares philosophy, even if understood as a meta-theory of logic and the sciences, to be impossible, since the features of the categorial framework cannot be expressed by language but only show themselves. If Frege is less rigorous than Wittgenstein in this respect, this holds for Russell to a much greater extent.

3. Carnap on Categorial Concepts
We have seen that categories are of fundamental importance in the writings of Frege, Russell and Wittgenstein. Since Carnap was heavily influenced by these three philosophers, it would be surprising if we could not find any trace of their treatment of categories in his works.
An explicit mention of the concept of a category can be found in the Aufbau (§ 83), although it appears only as part of a historical remark. (The paragraph bears the titel “Überschlagbar”, “can be skipped”.) It is the Kantian notion of a category for which Carnap seeks a place in his constitution system. But Carnap also uses a concept of a category in the Rylean sense in the Aufbau, although under a different heading: the concept of “Sphärenverwandtschaft”. Two objects are called “sphärenverwandt” if there is an argument-position in a propositional function such that the names of both of the objects are permissible arguments. Two objects which are not “sphärenverwandt” are called “sphärenfremd”. The class of those objects which are “sphärenverwandt” with a given object is called its “Gegenstandssphäre” (Carnap 1928, §29). The similarity of a “Gegenstandssphäre” to a Russellian type is obvious, and Carnap indeed explicitly mentions this. The use he makes of this concept, however, rather shows an affinity to the kind of meaning analysis which was developed by Ryle a little later: the uncovering of categorial nonsense hidden by the ambiguities of ordinary language (§§ 30, 31). Even the method of examining as many different examples as possible in order to find “category mistakes” (Carnap’s label is “Sphärenvermengungen”) anticipates Ryle’s method.
While Carnap in the Aufbau is mainly interested in the relevance of the concept of a “Gegenstandssphäre” for his constitution system, in his his famous paper “The Elimination of Metaphysics Through Logical Analysis of Language”, published in 1932, the main focus shifts to a critical examination of categorial nonsense as it appears in traditional philosophy. Let us take a look at his well known examples
(5)	Caesar is a prime number.
and
(6)	Caesar is and.
Carnap noticed that (6) is meaningless because it is formed countersyntactically, whereas (5) is meaningless for a quite different reason: it is syntactically correct, but it is meaningless because “prime number” is a predicate of numbers and can neither be affirmed nor denied of a person. Thus, (5) is a pseudo-statement. His argument for the meaninglessness of (5) already shows the influence of Wittgenstein: 
The fact that the rules of grammatical syntax are not violated easily seduces one at first glance into the erroneous opinion that one still has to do with a statement, albeit a false one. But “a is a prime number” is false if and only if a is divisible by a natural number different from a and from 1; evidently it is illicit to put here “Caesar” for “a”. (1931, 68)
Carnap tries to eliminate “Caesar” from the range of possible names that can be inserted for “a” by giving a definition for prime number in which the restriction to natural numbers is made explicit. The categorial difference between Caesar and numbers belongs to what he calls a logical syntax. And then he puts forward the Wittgensteinian idea that it is a defect of our language that the grammatical syntax does not correspond to the logical syntax. In a correctly constructed language category mistakes would already be ruled out by grammar, and all nonsensical sequences of words would be of the kind of (6) (“Caesar is and”). Hence, it would be possible to rule out all pseudo-statements, including the pseudo-statements of metaphysics, just by considerations of grammar. It would be unnecessary to pay attention to the meanings of the words over and above their syntactical category.
Thus, Carnap is following Wittgenstein’s idea of an ideal symbolism here in which syntax rules out philosophical nonsense. This idea, of course, is further developed in the Logical Syntax. Let us now take a closer look at this book, where Carnap most prominently discusses categorial concepts and the problems associated with them. This is not surprising, since it is in this book that he tries to prove that, despite the alleged meaninglessness of metaphysics, philosophy is still possible when restricted to a logic of science, i.e. a metatheory of scientific knowledge. Carnap is forced to find a solution to Wittgenstein’s problem of the inexpressibility of the formal properties of language, because otherwise a logic of science would be as meaningless as metaphysics. His argument comes in two steps: First, he tries to show that logical syntax is possible despite Wittgenstein’s claims to the contrary. (At least, this is Carnap’s reading of Wittgenstein’s claims.) And in a second step, he argues that a logic of science is actually the logical syntax of the language of science and, hence, a respectable discipline. With respect to the first point, Carnap tries to prove that the syntactical rules of a given object language can be formulated in a syntax language which itself obeys the rules of syntax and which is, thus, perfectly respectable. We may take it for granted that Carnap succeeds in doing so. But is logic of science nothing more than logical syntax? To prove this, Carnap would have to show that every statement that belongs to the logic of science can be translated into a syntactical sentence. In Carnap’s terminology, this would amount to the claim that every statement about the logic of science, though it seems to make claims about the properties of objects, is not an object-sentence proper. Rather, these sentences are pseudo-object sentences only, making assertions not about objects but about the syntactical properties of symbols, although this is disguised by the fact that they are formulated in the material mode of speech. Hence, they are quasi-syntactical sentences, belonging not to the object-language but to the meta-language in which logical syntax must be formulated. They can be transformed into syntactical sentences proper by translating them into the formal mode of speech. And that must be true especially for the kind of sentences we named categorial sentences, sentences which, as Carnap acknowledges, are central to the logic of science. So we have to take a look at Carnap’s treatment of categorial concepts in the Logical Syntax in order to decide whether his argument succeeds.
Words like “thing”, “object”, “property”, “relation”, “fact” etc. are called “universal words” (in German: “Allwörter”) by Carnap. They are defined in the following way:
A word is called a universal word if it expresses a property (or relation) which belongs analytically to all the objects of a genus [in German: “Gattung”], any two objects being assigned to the same genus if their designations belong to the same syntactical genus. (1934, 293)
This informal definition in the material mode of speech of course is only an elucidation of a syntactical definition proper in which there is no reference to objects but only to the properties of symbols. The notion of genus is also defined in a purely syntactical way:
Two expressions U1 and U2 are said to be syntactically related [in German: “verwandt”] when there existes an sentence S1 such that U1 occurs in S1 and S1 [U1/U2] is a sentence. 
Two related expressions U1 and U2 are called isogenous [in German: “gattungsgleich”] if for any S1, S1 [U1/U2] and S1 [U2/U2] are sentences. (1934, 169f.)
S1 [U1/U2] is the expression which results from S1 by replacing U1 by U2.
This is more or less a syntactical version of Ryle’s definition: two expressions belong to different categories if they are not syntactically related. And Carnap’s definition of universal words takes up Wittgenstein’s idea that formal concepts have internal properties as characteristic marks which necessarily belong to an object. Carnap gives this idea of necessity a precise meaning by using the term “analytic”, a term he also defines in a purely syntactical way (1934, 39).
According to Carnap, universal words can be employed in two different ways. First, they can be genuine parts of the object language. Here a universal word serves to point out the syntactical category of an expression. Sometimes, Carnap says, this is unnecessary because the category is determined by the form of the expression alone. His examples are expressions like “the number five” and “the process of crystalization”. We may note in passing that this is wrong because we need to look at the meaning of the words “five” and “crystalization” to determine their syntactical category, it is not given to us by their form alone. Carnap’s claim is only true for a language were the form of the symbols reflects the logical syntax. 
But sometimes expressions are ambigous with respect to their category, i.e. “the natural number 7” in contrast to “the real number 7”. But even here we could render the universal words “natural number” and “real number” superfluous by introducing different expressions for different kinds of number words or use an index for the same purpose. The same is true for variables, for which in the word languages universal words are of special importance. The reason is that the words used for the formulation of quantified sentences  – “every”, “a” etc. – are categorially indetermined. We need a universal word in order to specify the genus or the category of the values of the variables that are to be allowed. The universal word here becomes a designator for a range of variables – an idea we already encountered in Wittgenstein.
In a language that is not as deficient as our word language, however, universal words would not be needed because in such a language different kinds of names and variables are used for different categories. This is exactly Wittgenstein’s idea that the formal or internal properties of language show themselves in language. But it is the other claim made by Wittgenstein, namely, that the properties that show themselves cannot be said, that is rejected by Carnap. He claims that apart from the superfluous usage of universal words as categorial qualifiers in the object language, there is a way of using them as proper predicates in sentences like 
(7)	Five is not a thing, but a number.
Such a sentence, though it looks like an object sentence, is really a quasi-syntactical sentence in the material mode of speech. It is not meaningless as Wittgenstein thought, but only ill phrased, because it should be formulated in the formal mode of speech:
(7’)	“Five” is not a thing-word, but a number-word.
Such an expression, though it contains universal predicates, is syntactically well formed and therefore a proper sentence, contrary to what Wittgenstein claimed.
Note that the translation into the formal mode of speech does not only solve Wittgenstein's problem, but also Frege’s concept-horse-problem (a suggestion made by Michael Dummett). The sentence 
(4)	(The concept) horse is a concept.
when interpreted as a quasi-syntactical sentence in the material mode of speech becomes
(4’)	(The concept-word) “... is a horse” is a concept-word.
when translated into the  formal mode of speech. The expression “the concept word ‘... is a horse’” again is a name that refers to an object, according to Frege’s semantics. But the object it designates is the concept-word “... is a horse” which as an linguistic expression is an object in Frege’s bivalent ontology, but does not loose its syntactical property of being a concept-word by naming it. The concept-horse problem does not occur on the level of syntax, because all linguistic expressions belong to the onological category object, even if as symbols they refer to concepts.

But this seems to be to good to be true. So let us recall Wittgenstein’s argument in order to see how Carnap achieves his solution: sentences that contain formal concept words must be meaningless because they try to give a description of the categorial structure of language that cannot be described because it is already presupposed in any description. Hence, these sentences do not convey a new sense and are therefore not proper sentences. This no longer holds for Carnap’s sentences of logical syntax. Because they are formulated not in the object language but in a different meta-language, they lack the self-referential structure of Wittgenstein’s sentences. But what about the syntax of the meta-language? Do we not need a second meta-language to formulate it? There seems to be the danger of the need for an infinite hierarchy. Carnap, however, proved a very remarkable result: it is possible to formulate the syntax of any language in the language itself to an extent which is limited only by the richness in means of expression of the language (1934, 53ff.). This result seems to sweep away Wittgenstein’s inexpressibility-claim completely. But this seems to be amazing. Is it really possible that Carnap found such an elegant solution without paying any price for it? To answer this, we shall take a look at a simple example again:
(8)	“Five” is a number-word.
In this example, we use the ordinary word language in order to speak about syntactical properties of this very language. Why is this sentence not doomed to meaninglessness by Wittgenstein’s argument? Why is it not necessary to know already that “five” is a number word in order to speak about it? The reason is that sentence (8) speaks about contingent features only. That the predicate is called “number-word” does easily deceive us into thinking that it is a number-word because it is meant to refer to numbers. But in fact the assertion expressed by (8) refers to properties of the expression “five” only, stating that a certain expression is used in a certain way in our word-language. But this is a contingent feature of our language only, not a necessary feature of language as such in general. Wittgenstein makes this difference very explicit:
A proposition possesses essential and accidental features. Accidental features are those that result from the particular way in which the propositional sign is produced. Essential features are those without which the proposition could not express its sense. (1921, 3.34)
And the same applies to symbols in general. The claim as to the inexpressability of the features of language is meant to apply only to the essential features, because only essential features are necessary conditions for the possibility of the formulation of meaningful sentences about the world. Carnap’s syntactical sentences are not concerned with necessary properties of language. It is not an intrinsic or “internal” property of the word “five” to belong to the syntactical category of number-words, it is just the consequence of either an established usage when our ordinary word-language is concerned or a consequence of a stipulated convention with respect to the vocabulary of an invented symbolic language.
But what about syntactical propositions that are claimed to be valid for all possible languages? Carnap considers this as one possibility in which a syntactical sentence is meant to hold true. Of course sentence (8) – “Five” is a number-word – can hardly be interpreted in such a way, but we could imagine that someone wanted to claim that number-words and thing-words belong to different syntactical categories in every possible language:
(9)	For all languages: No number-word is a thing-word.
But even this statement does not express a necessary feature of language in the Wittgensteinian sense of “necessary” if it is interpreted in the spirit of Carnap. We would have to read it either as a claim about all existing languages or as a claim about the impossibility of inventing a syntax according to which certain symbols are both number-words and thing-words. On the first reading, it is an empirical statement and, thus, does not express any necessary feature of langauge. On the second reading, sentence (9) is simply false  because there are well known examples of languages which do not distinguish between number-words and thing-words (e.g., Frege’s language). If one would add the claim that those languages are “wrong” in some sense, necessity may sneak in again. But in order to do that we would have appeal to an absolute extra-linguistic standard and, hence, we should have to give up Carnap’s conventionalism with respect to the choice of a linguistic framework, culminating in the famous principle of tolerance which lies at the very heart of Carnap’s philosophical world-view (1934, 51).
So it is not just the trick of using a meta-language that does the job for Carnap, but the complete abandonment of the absolutistic conception of language. And he is very clear about this:
Since the rules of syntax of the word-language are not exactly established, and since linguistic usage varies considerably on just this point of the generic classification of words, our examples of universal words must always be given with the reservation that they are valid only for one particular use of language. (1934, 293)
The use of the material mode of speech leads ... to a disregard of the relativity to language of philosophical sentences; it is responsible for an erroneous conception of philosophical sentences as absolute. (1934, 299)
But he fails to notice that to “improve” Wittgenstein along these lines would mean nothing less than to abandon the whole conception of philosophy which is prominent in both Wittgenstein and Frege. The idea of the absolute validity of the philosophical sentences in question, viz. sentences about the categorial structure of language, is founded on the belief that this structure has to correspond to the categorial structure of the world in order to render meaningful discourse possible: “Logic ... is a mirror-image of the world” (Wittgenstein 1921, 6.13). Only by giving up this assumption, Carnap is able to get rid of the character of necessity the categorial features of language otherwise possess. In this respect, Carnap’s philosophical view is much closer to Russell, who willingly accepted Black’s suggestion to apply the theory of types only to words (understood as symbols subject to syntactical rules). This is accompanied by a loss of the transcendental character of philosophical investigations. Philosophy no longer is concerned with determining the necessary conditions and, hence, the limits of the possibility of human knowledge, which is not only the task of the Critique of Pure Reason but also of the Tractatus. Instead it becomes “Logic of Science”, but not in the sense of logic which Frege and Wittgenstein assumed. Rather, philosophy has to be content with formulating conventional languages and analyzing and assessing them with respect to their practical values for certain usages. I.e., philosophy is concerned with external questions in the sense of the late Carnap.
What I have said so far might give rise to the suspicion that it is Carnap’s restriction to syntactical questions only which is responsible for this different perspective. Does the situation not change when semantics comes in, when we talk about the objects linguistic expressions refer to? But this depends on what is meant by semantics, and certainly this is not true for Carnap. That this is so can be seen by considering his reaction to Quine’s famous principle of ontic commitment: To be is to be the value of a variable (Quine 1948). In Meaning and Necessity Carnap reacts with only subdued enthusiasm to Quine's suggestion (Carnap 1947, 42ff.). He agrees with Quine insofar as the use of a certain category of variables implies that one is willing not only to speak about particular objects belonging to this category, e.g. numbers, but also to speak about numbers in general and, hence, is willing to recognize the universal concept “number”:
It is important to emphasize the point just made that, once you admit certain variables, you are bound to admit the corresponding universal concept. (ibid.)
Carnap is anxious, however, to emphazise that this does not imply any ontological statement about the existence or reality of the entities in question and, hence, of the existence of a certain category in reality. That is why he dislikes Quine’s usage of the word “ontology”. What the so-called ontology is all about is nothing more than a decision to include a certain category of expressions in the language, thereby making it possible to use these expressions in formulating internal statements. This idea, of course, is further developed in his classic paper “Empiricism, Semantics, and Ontology” (Carnap 1950). So even in semantics the categorial features of language are still conventional, and the status of the sentences formulated in a now semantical meta-language remains the same as in the syntax. Though Carnap speaks about the categories of objects, these are actually categories of language. If, on the other hand, in a realistic semantic one would like to follow Frege in basing the categories of language on the categories of objects, the problems would reappear.
Carnap’s treatment of ontological questions, however, shows that there is a Kantian dimension in him left after all. This is the constitutional role of linguistic frameworks. Access to empirical and mathematical knowledge is only possible within a linguistic framework, a conceptual scheme which provides not only a vocabulary, but also the syntactical or semantical categories which are the formal conditions for making statements about the world. The choice of the framework, however, is conventional, governed only by pragmatic considerations. The shift of the constitutional element from reason to language is accompanied by the loss of necessity. Though the categories are still considered to be valid a priori, it is a relativized and revisable a priori.

To conclude: I considered the use of categorial concepts in Carnap’s Syntax and Semantics in order to place him in a certain philosophical tradition of treating categorial distinctions. I tried to show that Carnap’s transformation of ontological or epistemological categories into syntactical or semantical properties of linguistic frameworks succeeds in avoiding certain problems as to the inexpressability of categorial properties, problems that infect philosophical statements in general insofar as philosophy is concerned with categorial questions. I tried to show, however, that Carnap was successful only because he abandoned features of categorial concepts that were fundamental for the tradition to which, even if not Russell, then Frege and Wittgenstein still belonged: The view that categorial concepts represent formal properties that are essential for the objects of our knowledge. For my part, I think the price that Carnap had to pay is too high – a thoroughgoing conventionalism with respect to the formal aspects of human knowledge sacrifices too much of the traditional goals of epistemology which cannot be addressed from the standpoint Carnap adopted in the Logical Syntax. I do not want to imply that the program of a “Logic of Science” as it is formulated in this book is useless – on the contrary. But I think that it is no appropriate replacement for the issues Frege, Wittgenstein and maybe even Russell were interested in. Thus, it is no fully adequate substitute for philosophy in the traditional sense.

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