THOMAS MORMANN THE DEBATE ON BEGRIFFSTHEORIE BETWEEN CASSIRER, MARC-WOGAU – AND SCHLICK 0. INTRODUCTION The aim of this paper is to reconstruct a peculiar debate between Ernst Cassirer and the Swedish philosopher Konrad Marc-Wogau on Begriffstheorie that took place in the late thirties of the 20th century. This debate may be conceived as sort of ersatz of the discussion between Cas sirer's Neokantian Begriffstheorie on the one hand, and logical empiricist accounts on the other, in particular Schlick's Begriffstheorie as presented in his Allgemeine Erkenntnislehre.1 Although Schlick did not participate in personam in the discussion that took place between Cassirer and Marc-Wogau, one may consider him as a "virtual" participant of the debate, since his Begriffstheorie played an important role in the background, in particular for Cassirer. More precisely, I'd like to show that the debate can be read as a dispute on the feasibility of a "rich" account of Begriffstheorie, favored by Cassirer, and the necessity of restricting Begriffstheorie to an "austere" approach whose protagonists were Marc-Wogau and Schlick, although in a quite different ways. More generally, the debate on Begriffstheorie exemplifi es the complex interactions – and non-interactions – between three important currents of scientifi c philosophy, namely, the Marburg Neokantianism of Cassirer, the scientifi cally minded philo so phers of the Uppsala School, and, indirectly, the Logical Empiricism of the Vienna Circle. To set the stage, fi rst let us recall briefl y some bi o gra phical details of the protagonists. After National Socialism had come to power in January 1933, Cassirer left Germany in April of the same year. First he went to England, in 1934 he settled down in Uppsala. When in 1941 a German invasion of Sweden seemed imminent, he went to the U.S. where he lived until his death in April 1945. Konrad Marc-Wogau (1902 – 1991) was Professor of Philo sophy in Uppsala from 1946 till his retire ment in 1968. During Cassirer's stay in Sweden he and Cassirer were engaged in a lively debate that mainly took place in the then newly founded journal Theoria. From 1936 to 1940 their exchange in Theoria comprises at least seven items. Moreover, already in 1936 Marc-Wogau had published the monograph Inhalt und Umfang des Begriffs in which he dealt with a variety of Begriffstheorien, 1 Another important current of Begriffstheorie fl ourishing in the Vienna Circle was the one put forward by Car nap in the Aufbau. For reasons of space I cannot deal with it here. 168 Thomas Mormann among them Cassirer's. He found all of them wanting, since they all led to "dialectical", i.e., inconsistent concepts of concepts. For Cassirer Begriffstheorie, i.e., the philosophical theory of the formation of scientifi c concepts, was not just one philosophical topic among others. Rather, he considered Begriffstheorie as a truly central point of philosophy überhaupt (cf. Cassirer 1928, 163). Marc-Wogau agreed with Cassirer on the importance of the Begriffsproblem. He was well aware of the fact that he did not attack some minor point of Cassirer's approach, but launched an assault against the very center of Cassirer's philosophy. Moreover, he conceived his attack not only as directed against Cassirer's theory, but against the traditional philosophical account of concepts as a whole. In the background of the debate on Begriffstheorie between Cassirer and Marc-Wogau, Schlick and his Allgemeine Erkenntnislehre played an important role. Marc-Wogau claimed that Cassirer's Begriffstheorie could not survive logical analysis in that it led to an inconsistent notion of the concept. Although in Inhalt und Umfang he did not deal with Schlick's account of Begriffstheorie as elaborated in Allgemeine Erkenntnislehre it transpires from his criticisms of the other theories of concepts treated that he would have judged Schlick's account as "dialectical", i.e., as inconsistent, as well. On the other hand, Cassirer held Schlick's Allgemeine Erkenntnislehre in high esteem as a step in the right direction (cf. Cassirer 1927), but he criticized Schlick in that he took Begriffe as merely conventional symbols, laying "stress only on the negative moment of the function of "denoting" and its "conventional" character." According to him, "a sharper analysis of this function discovers immediately another, more positive aspect. (Cassirer 1927, 136). This "more positive aspect" of the symbolic character of scientifi c concepts he claimed to have unfolded in his constitutive account of Begriffstheorie as pre sented in Substance and Function (Cassirer 1910) and later in The Philosophy of Symbolic Forms (Cassirer 1923 – 1929). Complementarily, Schlick, in Allgemeine Erkenntnislehre, criticized (Ne o-) Kan tian accounts of Begriffstheorie as overstating the power of thinking without men ti o ning Cassirer by name. He pithily asserted: Thinking does not create the relations of reality... [R]eality does not obtain form and regularity fi rst from con sci ous ness; on the contrary, consciousness is only a section cut out of reality. ... There are no synthetic judgments a priori. (GTK, §40) For Schlick, Begriffe were merely conventional symbols. In this sense, he subscribed to a rather austere version of Begriffstheorie. But even this kind of modest theory Marc-Wogau would have blamed as untenable and logically fl awed since it got involved in the pernicious dialectics of Inhalt and Umfang. The outline of this paper is as follows. In the next two sections we recall the basics of the accounts of Begriffstheorie of Schlick and Cassirer. This requires dealing with Helmholtz's theory of concepts in some detail. In section 3 we will The Debate on Begriffstheorie 169 deal with Marc-Wogau's critique of Cassirer's Begriffstheorie and Cassirer's counter-critique in some detail. In section 4 I put forward some arguments from modern Formal Theory of Concepts which show that Marc-Wogau's objections to Cassirer's Begriffstheorie are untenable. This does not mean that Cassirer's rich constitutive account of concepts was without problems, but at least it shows that there is no reason to suspect that every theory of concepts that subscribes to some kind of relation between Inhalt and Umfang is per se inconsistent. In section 5 we conclude with some general remarks on the complex relations between the Logical Empiricism of the Vienna Circle, the Marburg Neokantianism, and the Uppsala School as they show up in the debate on Begriffstheorie. 1. KNOWLEDGE AS COORDINATION: HELMHOLTZ AND SCHLICK The term Begriff is probably one of the most vague terms ever-used in philosophy, psychology, and other disciplines (cf. Weitz 1984, Marc-Wogau 1936). One cannot start with a neat and comprehensive defi nition. In this paper I propose to conceive Begriffstheorie as a result of two complementary infl uences: On the one hand, it may be understood as a result of post-Kantian epistemology, which no longer accepted Kantian "pure intuitions" as an important apriori ingredient for scientifi c knowledge. On the other hand, Begriffstheorie may be seen as a philosophical reaction of the conceptual evolution of the sciences, i.e., it was an attempt of philosophy to come to terms with the new conceptual developments of the sciences, in particular with those of logic, mathematics, the mathematized empirical sciences. Also insights of physiology and psychology that concerned the ways of human conceptualization required the attention of philosophy. A convenient starting point is Helmholtz's "semiotic" theory of knowledge (cf. Helmholtz 1921). Helmholtz considered himself as a (Neo)Kantian, moreover he was a fi rst-class scientist with an immense expertise in physics, physiology and other disciplines. Helmholtz's epistemology may be characterized as a rather special version of a "scientifi cally corrected" Kantianism. According to it, on one side there is the world W of Kantian things-in-themselves, on the other side there is the domain S of one's sensations. Things and sensations are correlated to each other in a 1-1-way in such that sensations are to be interpreted as signs of objects: Our sensations are precisely effects produced by external causes in our organs, and the manner in which one such effect expresses itself depends, of course, essentially on the type of apparatus which is affected. Insofar as the quality of our sensation gives us information about the peculiarity of the external infl uence stimulating it, it can pass for a sign – but not for an image. For one requires from an image some sort of similarity with the object imaged: ... A sign, however, need not have any type of similarity with which it is a sign for. The relations between the two are so restricted that the same object, taking effect under equal cir cum stances, produces the same sign, and hence unequal signs always cor res pond to unequal effects." (Helmholtz 1878, 347) 170 Thomas Mormann Knowledge, then, is based on a mapping W f S of the world W of things-in-themselves into the domain S of sensations satisfying the requirement f(x) ≠ f(y) x ≠ y. According to Helmholtz, this weakly "structure-preserving" relation between an outer world and a domain of inner sensations is suffi cient to ensure that we are able to know the lawful structure of reality (cf. Helmholtz 1878, 348). In his Allgemeine Erkenntnislehre Schlick, in general, faithfully followed Helmholtz's semiotic approach.2 A point where he deviated from Helmholtz was that he replaced "sensations" by "concepts". The reason was that he considered "sensations" or "mental images" as too vague and unde ter mined as that they could fulfi l the symbolic role that Helmholtz had provided for them. In order to ensure stability and determinateness of our thought, he proposed to replace Helmholtz's sensory images (Empfi ndungen) by "concepts". Concepts were distinguished from images by the fact that they were completely determined and had nothing uncertain about them (cf. GTK, §5, 20). One may ask, how natural minds like ours with their continuously changing sensory images can handle such ideal entities as concepts as Schlick defi ned them. Schlick offered an answer apparently inspired by Vaihinger's Philosophie des Als Ob: Strictly speaking, concepts do not exist, what is important is their functional role: We operate with concepts as if they were (sensory) images (Vorstellungen) with exa c tly delineated pro per ties that can always be re-cognized with absolute certainty. Their pro perties are called the characteristics or features (Merkmale) of the concept, and are laid down by means of specifi c stipulations which in their totality constitute the defi nition of the con cept. In logic, the totality of the characteristics of a concept is called its "in tension" (or "content"); the set of objects denoted by the concept is called its "ex tension. ... Accordingly, a concept plays the role of a sign for all those objects whose properties include all defi ning characteristics of that concept." (GTK, §5, 20) For later use it will be expedient to comment briefl y on this piece of traditional concept logic to which Schlick subscribes here. In traditional logic a concept has two complementary components: on the one hand, its Inhalt (intension), given as the set of its defi ning characteristics, and on the other hand its Umfang ("extension"), given as the set of all objects whose properties include all its defi ning characteristics. This duality suggests the so-called "law of reciprocity" (cf. MarcWogau 1936, 10ff) according to which the following "reciprocity" between the Inhalt and the Umfang of a concept holds: the larger the Inhalt of a concept, the smaller its Umfang, and vice versa. This time-honoured "law" of traditional logic 2 In an approving comment on Helmholtz's The Facts of Perception Schlick explicitly characterized his Ge n eral Theory of Knowledge as an attempt "to show that forming such a mapping of what is lawlike in the actual, with the help of a sign system, altogether constitutes the essence of all knowledge, and that therefore our cog ni tive process can only in this way fulfi l its task and needs no other method for doing so." (Schlick and Hertz 1921, 166, endnote 15) The Debate on Begriffstheorie 171 appears in various forms in virtually every logical treatise of the 19th and early 20th century. As will be discussed later in more detail, it is the target of Marc-Wogau's incisive criticism put forward in Inhalt und Umfang. More precisely, he contended that all accounts of Begriffstheorie that hold some version of the reciprocity law were doomed to be inconsistent. Before we come to this, let us note that the project of defi ning concepts by characteristic features is threatened by two complementary dangers, either by infi nite regress or ending up in some features that lack exact defi nitions but instead are grounded in some murky empirical intuition that undermined the exact character of concepts so defi ned. According to Schlick, it was Hilbert's account of implicitly defi ned concepts that provided a way out of this impasse. According to it the basic concepts of mathematical theories are just defi ned as entities that satisfy the axioms specifi ed for them. Hence, there seem to be at least some concepts that can be defi ned in a completely precise and unambiguous way. Let us assume, for the sake of the argument, that we possess concepts in Schlick's sense that are coordinated in a 1-1-way with objects. What is the purpose in coordinating concepts to objects? Schlick's answer in Allgemeine Erkenntnislehre is that coordinations enable us to make judgments about objects, and only 1-1-coordinations enable us to make true judgments (cf. GTK, § 10). This answer is un satisfying in that it still allows a "Lagadonian coordination"3 of concepts and objects that co or dinates each object with one concept in a 1-1-way in some arbitrary fashion. Such a Lagadonian con cep tual system would allow us to make true judgments in a trivial manner. In order to exclude such undesired conceptual systems, Schlick hastened to add that the real aim of coordinating objects and concepts is not simply to enable us to for mulate true judgments but to get knowledge, which depends on very special coordinations: Knowledge is more – much more – than mere truth. Truth requires nothing but uniqueness of coordination; as far as truth is concerned, it does not matter what sign is used for that purpose. Knowledge, on the other hand, means unique coor di na tion with the help of certain defi nite sym bols, namely, those that have already found applications elsewhere. ... Hence if we were to coordinate a special sign to each fact and object in the world, we should have nothing but isolated truths, each of which would have to be learned se parately. ... Our truths would be nothing but discrete points, so to speak; they would not form a coherent system. Yet it is only in such a system that knowledge is possible, since the fi nding anew of one thing in another presupposes a pervasive in ter connection. (GTK, 66, 67, dt. 97) Thus, an essential point of a Schlickian Begriffstheorie would have been to distinguish bet ween "good" and "not so good" conceptual coordinations. This issue, however, remained underdeveloped in Allgemeine Erkenntnislehre. Schlick was content to give some vague hints pointing at a sort of Machian thought economy 3 On the philosophical appeal of Lagadonian languages see D. Lewis On the Plurality of Worlds (Lewis 1986, p. 145). 172 Thomas Mormann by which we could single out "good" parsimonious from "bad" Lagadonian concepts. Summarizing we may say that Schlick's coordinative account of knowledge is characterized by two complementary features: on the one hand it was based on a rich notion of structured re ality that did not only recognize "simple" objects as real, but even the most "theoretical" re la tions; on the other hand, it ascribed a rather austere role to the conceptualizing activity of the subject: for Schlick, concepts were nothing but conventional 1-1 coor di nations that allow easy "syntactical" manipulations. Thereby his account claimed a neat separation between factual and conventional components of knowledge. As we shall see, Cassirer's Begriffstheorie pulled in the opposite direction: although based on co or di na tion as well, it emphasized the active role of the conceptualizing subject against that of reality "out there". 2. CONCEPTUAL CONSTITUTION: CASSIRER'S BEGRIFFSTHEORIE Cassirer's philosophy of science is concept-orien ted par excellence: "The theory of the concept becomes a cardinal problem of systematic philosophy. It be comes the nub around which logic, epistemology, philosophy of language and cog nitive psychology are rotating" (Cassirer 1928, 163). Since Substance and Function (Cassirer 1910) he conceived phi losophy of science as a theory of the formation of scientifi c concepts. His theory was natu ra lis tic in the sense that according to him philosophy should not decree what scientifi c concepts were and how they worked. Rather, since scientifi c concepts evolved in the history of science, it was the task of philosophy of science to study this conceptual development of science and to make philosophical sense of it, not to legislate it according to some preconceived philosophical ideas. The role concepts played in the evolution of scientifi c knowledge science according to Cassirer's Begriffstheorie may be described in telegram style as follows. Scientifi c knowledge does not cognize objects as ready-made entities. Rather, knowledge is organized objectually in the sense that in the continuous stream of experience invariant relations are fi xated. The unity of a concept is not to be found in a fi xed group of pro perties, but in a rule, which lawfully represents the mere di ver sity of experiences as a sequence of ele ments. The meaning of a concept depends on the system of concepts in which it oc curs. It is not completely determined by one single system, but rather by the con tinuous series of systems unfolding in the course of history. Scientifi c concepts and conceptual systems do not yield pictures of reality, rather, they provide guide lines for the conceptualisation of the world. The fun damental concepts of theoretical physics are blueprints for possible ex pe ri ences. Factual and theoretical components of scientifi c knowledge cannot be neatly se pa rated. In a scientifi c theory „real" and „non-real" components The Debate on Begriffstheorie 173 are inex tri cab ly interwoven. Not a single concept is confronted with re a li ty but a whole sys tem of con cepts. Our experience is always conceptually structured. There is no non-con cep tually structured „given". The „given" is an artifact of a bad meta physics. The concepts of mathematics and the concepts of the empirical sciences are essentially of the same kind. In a similar way, as the objects of a mathematical theory are constituted by a system of concepts, the objects of an empirical theory are constituted by theory's concepts. With this account of scientifi c concepts and their role in the ongoing evolution of science Cassirer goes beyond Helmholtz's and Schlick's structural realist accounts of knowledge and coordination. He emphasized the "constitutive" character of symbolic representation: ... we do not know "objects" as if they were already independently determined and given as objects, – but we know objectively, by producing certain limitations and by fi xating certain permanent elements and connections within the uniform fl ow of experience. The concept of the object in this sense constitutes no ultimate limit of knowledge, but is rather the fundamental in stru ment, by which all that has become its permanent possession is expressed and established. The object marks the logical possession of know ledge, and not a dark beyond forever removed from knowledge. "(SF, 303f) Instead of conceiving knowledge as a structure-preserving map between a world of trans cen dent things on the one hand and a domain of sensory images (Helmholtz) or con ventional sym bols (Schlick), for Cassirer knowledge as coordination meant the co or di nation of dif fe rent areas or stages of knowledge. This led to a new "internal" account of coordination or re presentation that described the coordination between thought and reality not as a relation be tween two ontologically different spheres but as a relation between different areas of know ledge. Thus, if one graphically represents Schlick's and Helmholtz's account by a simple relation W S between a world W of transcendent things and a domain of internal symbols S, Cassirer's account could perhaps be represented by an unending chain of conceptualizations evolving in the history of science: ... Ci Ci+1 Ci+2 ... Here, the Ci should not be interpreted as "mere conceptua li za tions". Rather, the Ci are always thought to aim at empirical confi rmation and corroboration. In modern terms, they may perhaps be conceived as interpreted models of reality. Thus, a scientifi c object is never "given as such", independently of all the Ci, it always appears in a lawful conceptual context by which it is constituted. In a similar way as a ma the matical object such as a geometrical point cannot be thought outside a geometrical system, an object of physics or of any other science cannot be thought outside its theoretical context to which it belongs. Indeed, Cassirer considered it as the essential task of critical idealist phi lo so phy of science to make clear "that the same foun da ti o nal syntheses (Grund syn thesen) on which logic and ma thematics rest also govern the scientifi c con struction of exp e ri en tial know ledge ..." (Cassirer 174 Thomas Mormann 1907, p. 44). This contention did not imply that empirical and mathe ma ti cal ob jects and concepts are one and the same thing. In Substance and Function he pointed out there was an important difference between mathematics and empirical concepts: In contrast to the mathematical concept, however, in empirical science the cha rac te ris tic difference emerges that the construction which within ma the ma tics arrives at a fi xed end, remains in prin ciple incompleteable with in ex pe rience. But no mat ter, how many „strata" of relations we may super im pose on each other, and how ever close we may come to all particular cir cum stances of the real process, never the less there is always the pos si bility that some co-op e rative factor in the total re sult has not been calculated and will only dis covered with the further pro gress of ex perimental analysis. Cas si rer (1910/1953, p. 254) In a nutshell, then, for Cassirer the difference between mathematical and empirical concepts re sided in the fact that the latter are open ("incompleteable") while the former are closed: the implicit defi nition of a point in Euclidean geometry fi xes the meaning of this concept once and for all. In contrast, the meaning of a concept such as "atom" is never fi xed by a single conceptual system. Cas sirer claimed that the key concepts of empirical science had a "serial form" ("Reihen form") in that their meaning was not fi xed once and for all by a single theoretical frame work. Rather, it emerged in a series of theoretical stages in the ongoing evolution of scien ti fi c know ledge. Thus concepts comprise two complementary moments: on the one hand they are rules for further investigations, on the other hand they are devices for determining the objects of scientifi c knowledge. The feasibility of this complex relation of the two com po nents is at stake in the debate between Cassirer and Marc-Wogau. 3. MARC-WOGAU'S CRITICISM AND CASSIRER'S DEFENSE The aim of Marc-Wogau's treatise Inhalt und Umfang des Begriffs. Beitrag zur Theorie des Begriffs (1936) was to clarify the es sence of the concept of concept. According to him, such a clarifi cation was urgently needed, since vir tu ally all extant accounts of Begriffstheorie were fatally fl awed. As an expe di ent starting point for such a clarifi cation he considered the problem of the relation between the Inhalt and the Umfang of a concept: "It seems to me that the nature of concept can best be clarifi ed at this problem." (Marc-Wogau 1936, 5). He pointed out that in the theory of con cepts one easily runs into logical diffi culties. For instance, the "concept of concept" (der Be griff des Begriffs) immediately leads to well-known paradoxes of a class that contains itself as an ele ment. Hence, in order to avoid such pitfalls one had to be extremely careful in the choice of the basic assumptions on which to build a consistent Begriffstheorie. Con se quent ly, Marc-Wo gau was prepared to recognize only those "determinations" (Bestimmt heiten) as concepts The Debate on Begriffstheorie 175 which were non-contradictory (widerspruchslos) or unequivocal (eindeutig) and could be grasped by a uniform (einheitlich) thought (ibid., 7). Thereby he hoped to exclude inconsistent expressions such as "round square" from the realm of Begriffstheorie. Although it is rather plausible not to admit openly contradictory concepts such as "being red and non-red all over at the same time" Marc-Wogau's requirement of uniformity is more tricky, in particular, since he considered nonuniformity as the main source of "dialectical", i.e., incon sis tent concepts. According to him, virtually all theories of concepts sinned against the com mand of uniformity and fell prey to inconsistency. The main entrance door for non-uniformity (and thence inconsistency) was that virtually all theories of concepts subscribed to a correlation between Inhalt and Umfang of a concept. Marc-Wogau claimed that this correlation could not be grasped in a "uniform thought" and therefore led to an inconsistent "double thought" (Doppelgedanken). He attempted to show that all accounts of Begriffstheorie endorsed a very strong version of the reciprocity law according to which the Inhalt uniquely determined the Umfang and the Umfang uniquely determined the Inhalt. This claim may well be doubted. For instance, a non-extensional Begriffstheorie readily allows for the existence of concepts having the same Umfang but different Inhalte. Fortunately, we need not go into the details of these quibbles when we wish to grasp the essence of Cassirer's and Marc-Wogau's dispute. Cassirer intended to refute Marc-Wogau principally, i.e., he readily admitted that he did subscribe to a "double thought" approach of concept. But he denied that this led to contradiction. Hence he argued that even if Inhalt and Umfang determined each other in the strict way that Marc-Wogau assumed, even then this fact would not lead to inconsistency. In the following I want to show that Cassirer was right, even if the argument he presented for this thesis, was less than convincing. In some sense, Marc-Wogau's arguments against the "double thought" hidden in the standard approaches of Begriffstheorie resemble those of the British idealists such as Bradley who claimed that the concept of relation was "unintelligible" and even "contradictory". As Marc-Wogau put it: If one relatum of this relation (between Inhalt and Umfang) is thought, thereby the other is thought as well. Consequently, the relata coincide. If A is to be related to B in such a way that A obtains its determination (or determinateness) only through B, then it is impossible to distinguish between A and B. They coincide." (Theoria 2, 291ff) Let us call this thesis Marc-Wogau's identity thesis. Cassirer's counter-argument against the identity thesis was to give a counter-example, i.e., he presented a decent, scientifi cally recognized relation whose relata strictly determined each other but nevertheless were not identical. Thus his strategy was based on the naturalist assumption that it is not the task of philosophy to decree what is possible and what is not pos sible but to understand the conceptual evolution of the sciences. According to him, there was no reason to assume that "thinking together" Inhalt and 176 Thomas Mormann Umfang led to contradiction since the conceptual evolution of science had shown that relations whose relata are different but nevertheless strictly determine each other, do not lead to contradictions. In other words, he accused Marc-Wogau of being caught in the trap of some unfounded philosophical pre ju dice refuted by the conceptual evolution of science. In order to refute Marc-Wogau's identity thesis Cassirer relied on Schlick's Allgemeine Erkenntnislehre, and pointed out that systems of implicitly defi ned concepts as considered by Schlick refute Marc-Wogau's thesis: In an implicitly defi ned conceptual system there is given a to ta li ty of concepts that stand in strict correlation to each other and have no in de pen dent content outside this correlation. None of them is meaningful "for itself", each is defi ned only with respect to the other, or, better said, with respect to the whole sys tem. Nevertheless this mutual dependence cannot be considered as a fl aw; rather it lays the foundation for a certain highly characteristic advan tage. One can not say that, due to the fact that none of the system's basic concepts can be exp lained or used meaningfully outside the system, their meaning dis ap pears or becomes ambiguous. Each has its well-determined place in the system and there by it distinguishes itself from any other concept of the system. (Cassirer 1938, 226) He concluded that thereby Marc-Wogau's thesis was "directly refuted" (ibid.). Even if from a formal point of view Cassirer's argument against Marc-Wogau seems fl awless, one may consider it not as fully con vincing: fi rstly, it is an abstract argument in the sense in that it has no thing to do with the specifi cs of the reciprocity law that correlates Inhalt and Umfang. It sim ply gives an example showing that there exist relata that strictly determine each other with out being identical. Secondly, the argument based on implicitly defi ned concepts does not provide any positive evidence for Cassirer's own version of a constitutive Begriffs theorie. In the next section I want to show that today we have powerful formal tools that allow one to refute Marc-Wogau's thesis on his own ground. That is to say, there are consistent theories of concepts that satisfy a strong version of the reciprocity law. 4. FORMAL THEORY OF CONCEPTS AND ADJOINT SITUATIONS Cassirer's refutation of Marc-Wogau's identity thesis by invoking implicit defi nitions may not be considered as fully adequate, since the implicit defi nition of concepts has nothing to do with the problematic of the relation between Inhalt and Umfang that occupies centre stage in Marc-Wogau's Begriffstheorie. In other words, Cassirer's argument is too general than to be really convincing. Fortunately, today better and more specifi c arguments are available to back up Cassirer's arguments against Marc-Wogau. I'd like to mention two different approaches. First, the so-called Formal Concept Analysis (FCA) inaugurated in the 1980s by the German mathematician Rudolf Wille and his collaborators. SecThe Debate on Begriffstheorie 177 ondly, on a more general level, the theory of adjoint situations that belongs to the core of the foundational discipline of category theory founded in the late 1940s by the American mathematicians Saunders Mac Lane and Samuel Eilen berg. Both approaches offer mathematical models of (generalized) concepts that are better suited to refute Marc-Wogau's identity thesis than Cassirer's vague allusion to Schlick's equally vague theory of implicit defi nitions in Allgemeine Erkenntnislehre. FCA starts with the reciprocity law. A concept is determined by its extent ("Umfang") and its intent ("Inhalt"). The extent consists of all objects belonging to the concepts, while the intent is the collection of all attributes shared by the objects. As it is often diffi cult to list all the objects and usually impossible to list all its attributes, it is natural to work within a specifi c context in which the sets of objects and attributes are fi xed.4 Then a context is defi ned as a triple (G, M, F) where G and M are sets and F G × M. The elements of G are called objects, and the elements of M are called attributes. If (g, m) F this is to be interpreted as the fact that in M the object g has the attribute m, or, put it dif fe rently that the attribute m is instantiated by g. For A G and B M defi ne I(A) := {m M; for all g A (g, m) F} U(B) := {g G; for all m B (g, m) F} Informally, I(A) is the set of attributes common to all the objects in A, and U(B) is the set of objects having all the attributes in B. Denoting the power set of A and B by PA and PB, the operators I and U just defi ned above may be conceived as mappings PA I PB and PB U PA These mappings have some interesting properties. For instance, they satisfy the following requirements (cf. Ganter and Wille 1999, chapter 0.4, Defi nition 16, 11): (1) A1 A2 I(A1) I(A2) (2) B1 B2 U(A1) U(B2) (3) A U(I(A)) and B I(U(B)) The pair (I, U) is called a Galois connection, and the maps I and U are called dually adjoint to each other. As is well known a pair of maps PA I PB and PB U PA is a Galois connection if and only if it satisfi es the equivalence (4) A U(B) B I(A) Now we are ready to defi ne concepts of a context (G, M, F) as pairs (A, B) PG x PM that are "balanced" in the sense that I(A) = B and U(B) = A. The Umfang 4 If this is not done, one runs into diffi culties, as is discussed in detail by Marc-Wogau (1936). 178 Thomas Mormann of the concept (A, B) is A while its Inhalt is B. The set of concepts C(G, M, F) := {(A, B); I(A) = B} has the structure of a complete lattice (cf. Theorem 3, p. 20, Ganter and Wille 1999). By defi nition, Umfang and Inhalt of a concept strictly determine each other. Nevertheless they are different. But given the Inhalt I one can calculate the Umfang U, and, vice versa, given the Umfang U, one can calculate the Inhalt I. For A PG and B PM) one obtains: U(B) = {A; B ≤ I(A)} and I(A) = {B; A ≤ U(B)} In sum, the Galois connection (I, U) neatly disproves Marc-Wogau's identity thesis according to which strict mutual determination im plies identity. Moreover, the refuting example is directly concerned with Inhalt and Umfang as key concepts of Begriffstheorie. By briefl y mentioning FCA and the theory of Galois con nections I only scratched at the surface of what may be charac te rized as a modern version of traditional Begriffstheorie. In this direction much more has to be done in order to fi nd out if traditional Begriffstheorie could indeed be fruitfully related to contemporary strands of research in category theory, computer science and cognitive science. It would be a gross underestimation of the theory of Galois connections to take it just as an abstruse calculus that is useful for some special theory such as FCA. Rather, Galois connections are a very special case of so called Adjoint Situations. Adjoint situations are, according to the assessment of Saunders Mac Lane, one of the founding fathers of category theory, THE fundamental concept of category theory. There is no time to explain this contention in any detail. Be it suffi cient just to state that in adjoint situations the rather austere structures PG and PM are replaced by appropriate, much more richly structures cate go ries, and the role of the mappings I and U is taken over by appropriate functors these categories. Then one of the fundamental theorems of category theory, the so-called Adjoint Functor Theorem, ensures that under certain conditions something like a generalization of the reciprocity law holds. Painting it with a broad brush we may contend that in this way the allegedly obsolete Begriffs theorie of the early 20th century, centering on the notorious "law of reciprocity", has found an unexpected come back in the guise of category theory. The fact that adjoint situations are one of the core concepts of category theory, and the fact that category theory is one of the most successful contemporary foundational theories suggest that even today Begriffstheorie may deserve more than mere philosophico-historical interest. 5. CONCLUDING REMARKS Begriffstheorie may be considered as hidden meeting point for a variety of philosophical currents more or less closely related to some sort of "scientifi c" philosoThe Debate on Begriffstheorie 179 phy, in particular Logical Empiricism, among them Schlick's empirio-criticism of the early 20s, Cassirer's critical idealism, and the logical philosophy of the Uppsala School. More generally, as Marc-Wogau's treatise Inhalt und Umfang shows the issue of Be griffs theorie was a common ground for the various currents of analytic and continental philosophy that in the following decades became neatly separated. Begriffs theorie was a topic where philosophers of quite different orientations met. It exemplifi es that once upon a time philosophers, who today are classifi ed as belonging to allegedly quite different traditions, were engaged in discussing similar problems. Begriffstheorie shows in particular that it would be a serious distorsion to characterize the continental tradition as anti-logical, and the analytical tradition as pro-logical. From Marc-Wogau's perspective the theories of concepts put forward by philosophers such as Cassirer, Frege, Husserl, Kant, Rickert, Russell, and others, all appeared to be rather similar, since they all suffered from similar defects. He treated them as united in the com mon endeavor of elucidating the nature of (scientifi c) concepts. Implicitly he thereby defi ed the sharp distinction between continental and analytic philosophy that later be came current. This feature of Begriffstheorie would have deserved more attention as I could give to it in a short paper like this. Rather, I concentrated on Begriffstheorie as a common ground of the more closely related currents of Cassirer's critical idealism and Schlick's early logical Empiricism as presented in his Allgemeine Erkenntnislehre. From Marc-Wogau's rigid conception of logic that did not al low one "to think together" the complementary aspects Inhalt and Umfang, both Cassirer's and Schlick's accounts of Begriffstheorie were untenable. Lo gically, Marc-Wogau's criticism is refuted by the existence of concepts that mutually de ter mine each other without being identical. Pragmatically, Marc-Wogau's account of Begriffstheorie is unacceptable, since it is hard to see how scientifi c concepts could do the work they are designed to do without assuming the existence of some kind of relation be t ween two components of concepts that more or less resemble the classical aspects of Inhalt and Umfang. Summing up one may say that Cassirer's Begriffstheorie survives Marc-Wogau's assault since his argument against "dual" accounts of concepts is fatally fl awed. This is not to say that Cassirer's rich "constitutive" account of Begriffstheorie did not suffer from its own problems. But that is another story. REFERENCES Ernst Cassirer, "Kant und die moderne Mathematik", in: Kant-Studien 12, 1907, pp. 1-49. Ernst Cassirer, Substanzbegriff und Funktionsbegriff, Darmstadt: Wissen schaft liche Buch ge sellschaft 1910 (1985). Ernst Cassirer, Erkenntnistheorie nebst den Grenzfragen der Logik und Denkpsy180 Thomas Mormann chologie, in ders., Erkenntnis, Begriff, Kultur, herausgegeben von R.A. Bast, Hamburg: Meiner, 1927 (1993), pp. 77-153. Ernst Cassirer, "Zur Theorie des Begriffs", in ders., Erkenntnis, Begriff, Kultur, herausgegeben von R.A. Bast, Hamburg: Meiner, 1928 (1993), pp. 155-164. Ernst Cassirer, "Inhalt und Umfang des Begriffs", in ders., Erkenntnis, Begriff, Kultur, herausgegeben von R.A. Bast, Hamburg: Meiner 1936 (1993), pp. 165-197. First published in Theoria 2, 207-232. Ernst Cassirer, Philosophie der symbolischen Formen, Darmstadt: Wissenschaftliche Buchgesellschaft 1923–1929 (1985). Ernst Cassirer, Zur Logik des Symbolbegriffs, in ders., Wesen und Wirkung des Symbolbegriffs, Darmstadt: Wissenschaftliche Buchgesellschaft, 1938 (1983), pp. 203-230. First published in Theoria 4, 145-175. Michael Friedman, Coordination, Constitution, Convention. The Evolution of the Apriori in Lo gical Empiricism, in A. Richardson, T. Uebel (eds.), The Cambridge Companion to Lo gi cal Empiricism, Cambridge: Cambridge University Press 2007, pp. 91-116. Bernhard Ganter, Rudolf Wille, Formal Concept Analysis, Berlin, Heidelberg: Springer 1999. Heinrich von Helmholtz, Schriften zur Erkenntnistheorie, herausgegeben und erläutert von Paul Hertz und Moritz Schlick, Wien: Springer 1921 (1998). David Lewis, On the Plurality of Worlds, Oxford: Basil Blackwell, 1986. Konrad Marc-Wogau, Inhalt und Umfang. Beitrag zur Theorie des Begriffs, Uppsala: Almquvist and Wiksell, and Leipzig: Otto Harrassowitz 1936. Thomas Mormann, Der begriffl iche Aufbau der wissenschaftlichen Wirklichkeit bei Cas sirer, Logos 4, pp. 268-293, 1997. Moritz Schlick, Allgemeine Erkenntnislehre, Translated by A. E. Blumberg as General Theory of Knowledge, Chicago and LaSalle: Open Court 1925 (1985). (GTK) Friedrich Stadler, The Vienna Circle, Studies in the Origins, Development, and Infl uence of Logical Empiricism, Wien and New York: Springer 2001. Maurice Weitz, Theories of Concepts, London: Routledge 1988. Department of Logic and Philosophy of Science University of the Basque Country (UPV/EHU) P.O. Box 1249 20080 Donostia-San Sebastián Spain ylxmomot@sf.ehu.es