Stephan Käufer
Canonical histories of logic, such as that of I. Bochenski or William and Martha Kneale, ignore the work of nineteenth-century philosophers working on traditional logic. This disregard is somewhat justified. Mainstream philosophers working on logic in the nineteenth century make no enduring or significant changes to the old scholastic syllogistic logic; and their concerns do not seem to be properly logical at all, but metaphysical, methodological, or epistemological. Kant’s work illustrates this approach to logic. In his logic lectures Kant adheres to the Aristotelian syllogistic and states that “from Aristotle’s time on, logic has not gained much in content, nor can it by its nature do so” (1800: 534). This echoes his assessment of the syllogistic in the Critique of Pure Reason, that logic is correct and complete, and further improvements are neither necessary nor possible (1781: Bviii). At the same time, however, Kant develops his astoundingly complex and innovative “transcendental logic,” which radically changes metaphysics and epistemology, but whose title seems misleading from the point of view of modern logic.
Recently historians of logic and philosophy have begun to re-examine the logical writings of the post-Kantian philosophers. The early and middle decades of the nineteenth century are unprecedented in the intensity and breadth of challenges to precisely the traditional logic that Kant had proclaimed complete. Immediately following Kant’s work, and indeed because of Kant’s work, philosophers call for a complete overthrow of logic. In 1812 Fichte demands that “general logic be destroyed to its very foundation” (Martin 2003: 36). Hegel writes in his Science of Logic that “if logic has not undergone any changes since Aristotle … then we should rather conclude that it requires a total reworking” (1831: 46). With such provocations Fichte and Hegel invigorate contemporaneous logical thought. They initiate a three-way debate between Kantian, Hegelian, and other logicians about what comes to be known as “the logic question,” a debate that dominates the middle decades of the century. In Wayne Martin’s phrase, this is a period of deep and pervasive “logical radicalism,” of attempts to rework the entire edifice of logic from the ground up.
The importance and core interests of the logical radicalism of the nineteenth century do not consist of technical advances in formal or symbolic logic; before Frege such advances are few and modest and, like Frege’s work, they take place at the margins of post-Kantian philosophy. The same philosophers who want to overthrow logic in debating the logic question, reproduce much of the entrenched logic of terms and syllogisms in their lectures and textbooks. This is also true for philosophers with a greater penchant for symbolic logic. For example, William Stanley Jevons, who works on extending Boole’s and Hamilton’s innovations in symbolic logic, publishes a short and elegant vademecum called Lessons in Logic, in which he exhorts students to recite the “ingenious, yet wholly unscientific” medieval mnemonic verse “Barbara, Celarent, Darii, Ferioque …” and practice reducing valid syllogisms to those of the first figure (1881: 145). Logical radicalism runs alongside continued adherence to the traditional syllogistic. Its historical and philosophical importance lies in its attempts to understand the nature and bounds of logic and to provide it with metaphysical grounding and epistemological import.
In the nineteenth century a standard exposition of logic comprises three parts: the theories of concepts, judgments, and inferences. They reproduce the ancient stock of logical analyses whose core goes back to the syllogistic logic first developed by Aristotle in the books of his Organon, especially the Prior Analytics. The theory of inferences (Schlusslehre) consists of the Aristotelian theory of syllogisms. It contains the list of valid syllogisms divided into moods and figures and examples of how to reduce some syllogisms to others. It usually also discusses hypothetical syllogisms, induction, and modal syllogisms. From a modern perspective the theory of inferences looks hopelessly outdated. But nineteenth-century philosophers consider it complete and do not devote much effort to overhauling it.
The theory of judgments (Urteilslehre) analyzes the logical form of judgments. Since syllogisms require premises and conclusions to be in subject–predicate form, much of the theory of judgments consists of proposals for shoehorning the grammatical variety of sentences into this logical mould. Some such proposals are fairly complicated, especially in the case of existential judgments (“God exists”), and impersonal judgments, which abound in German (“it rains”). Among the better known views is that of Sigwart, who claims that there is only a single form of judgment, “S is P,” and that all variation derives from the content of its component concepts; or the view proposed by Trendelenburg that precisely impersonal judgments reveal the original unified judgment-form from which the subject–predicate distinction arises.
The theory of judgments also contains the analysis of negation and quantification. It divides judgments into universal (“all S are P”), particular (“some S are P”), and singular (“this S is P”), in both affirmative and negative modes. Like the syllogisms, this analysis derives from Aristotle, who also first points out the inferences that obtain between some pairs of judgments. “All As are Bs” implies that “some As are Bs” and that “no As are not Bs” and so forth, and Aristotle uses these operations to reduce all valid syllogisms to the ones he considers basic. These “immediate” or “direct” inferences, i.e. inferences which do not require a third term beyond the two contained in a given judgment, are also expounded in the theory of judgments. A common mode of presenting them is the traditional square of oppositions, which organizes contraries, contradictories, subcontraries, and subaltern inferences in a diagram. Unlike syllogistic inferences the immediate inferences are the cause of some disagreement among logicians. Stoic and medieval logicians already know that some immediate inferences are problematic and among the logical radicals of the nineteenth century these problems provide a touchstone for rejecting old views of logic and proposing new ones.
The third part of traditional logic, the theory of concepts, analyzes the logical structure of concepts, or terms, that can take the place of a logical subject or predicate. Syllogistic logic is often referred to as “term logic,” because much of the inferential significance modeled in a syllogism derives from the structure and content of its major, minor, and middle terms. Hegel’s counter-intuitive claim that “the inference is the completely posited concept” (1816: 351) amounts to this: a syllogism is nothing but an articulation of one aspect of the content of its terms. This part of logic has no specific predecessor in ancient and medieval logics and there is limited overlap between its many expositions. Most authors agree that there is not much to be said in general about the structure of concepts per se. Conservative authors focus on the idea that concepts are aggregates of attributes or marks (Merkmale) and that the conceptual relations articulated in logic are grounded in the inclusion or exclusion of these attributes. They adduce illustrative examples from the hierarchies of genera and species being developed by botanists and zoologists, who, in turn, find their systems of classification validated by its “logical” structure. Most of the logical radicals of the nineteenth century find this attribute analysis of concepts either true, but empty (Herbart 1813: 88; Drobisch 1836: 16); or misleading (Lotze 1874a: 47); or false (Hegel 1816: 295; Trendelenburg 1870: 250; cf. Käufer 2005: 273). Many logicians supplement this analysis with accounts of the formation of concepts, or the a priori status of some concepts. Here they find themselves in a gray area in which they commonly stray into psychology, epistemology, or transcendental philosophy.
Virtually all logic books of the nineteenth century consist of these three parts. Normally they begin with the theory of concepts, followed by the theory of judgments, and end with the theory of inferences. This order indicates a commitment to a type of Aufbau view, according to which concepts are the most elemental components of logic, which are then combined into judgments, which, in turn, make up syllogistic inferences. This view itself also comes to be challenged. Hegel thinks that none of the three parts is logically more elemental; instead they constitute a system each part of which articulates an aspect of the others. Others argue that judgments are logically more fundamental than concepts, because they, not concepts, are the unit of cognition. Regardless, the division into the three theories persists even in texts that challenge the presumptions inherent in it. Famously, the transcendental logic, which makes up the vast majority of Kant’s Critique of Pure Reason, is organized by a plan that reproduces this same division. Hegel’s Science of Logic also includes the theories of concepts, judgments, and inferences in its second book, the Subjective Logic, even though they only make up a part of the book and he makes radical changes to the content of each.
The most important authors and most influential positions of logical radicalism fall into three main camps: Kantians, Hegelians, and a group of philosophers who sought to develop positions on the middle ground between these two.
The Kantian school consists of philosophers who claim to teach general logic in his sense within the framework of his critical philosophy. They do not take themselves to be radicals and do not advocate an overthrow of logic. Much of the work of the Kantian school of logic consists of logic textbooks, such as Herbart’s 1813 Lehrbuch zur Einleitung in die Philosophie or Moritz Drobisch’s 1836 Neue Darstellung der Logik nach ihren einfachsten Verhältnissen. In the course of the evolving debate these Kantians are forced to defend their overall position against challenges from the other camps. Often these defenses are integrated into successive editions of textbooks. Drobisch’s book is a prime example of this, both because of its longevity and the diligence of its author. His explanation of the nature and unity of a concept, for instance, changes in each edition, from the second in 1851 to the fifth in 1887. The view of the Kantian school is also called “subjective logic” or “formal” logic.
Hegel scholars today debate how to interpret the role of Hegel’s logic in his overall system. The book is clearly concerned with more than logic, and it may not have much to say about logic at all, especially given our contemporary understanding of that discipline. Philosophers in the early nineteenth century, by contrast, do not question the status of Hegel’s Wissenschaft der Logik as logic. From the moment Hegel publishes it until the decline of the first wave of Hegelianism, philosophers agree that this work is at least supposed to be a logic and that it should be interpreted as such. Of course, these philosophers presume that it is more than an amendment to the technical apparatus of syllogistic logic. They see its scope and ambition as on par with Kant’s transcendental logic. Indeed, roughly speaking Hegel’s overall goal is to unify general and transcendental logic, and in doing so to overcome a number of Kantian distinctions that Hegel deems untenable.
Hegel inspires a lot of devoted followers and acolytes, such as Karl Ludwig Michelet, Karl Rosenkranz and Johann Eduard Erdmann, many of whom are enthralled by his style and language. A few philosophers defend and refine what they take to be Hegel’s central insights into logic by engaging in a reasoned and reasonably clear debate with other logical radicals. Most important among these is Kuno Fischer, whom Windelband credits with the “art of translating Hegel into German” (1904: 180). In two very different editions of his Logik und Metaphysik, Fischer defends Hegel’s view that the structures of logic are the structure of being. These works constitute a Hegelian school of logic and their position is also called the “metaphysical logic.”
A third, somewhat looser group of philosophers explores middle ground between the Kantian and Hegelian schools. They agree with Hegel’s basic insight, that Kant’s transcendental logic must be reconciled with general logic into a single, coherent discipline. They also agree that this unification of the two logics requires a fundamental rethinking of the metaphysics and epistemology of logic. However, this third group of philosophers strictly opposes the style and much of the substance of the Hegelian attempts. Hegel, they claim, goes too far and in effect abandons logic as such by conflating it with his ambitious metaphysics. His “metaphysical logic” is no longer a logic at all. Trendelenburg first articulated a forceful and detailed version of this opposition to Hegel’s logic in his 1840 Logische Untersuchungen. In response to the pugnacious and disdainful replies this book evokes from Hegelian devotees he reiterates the main outlines of his criticisms in his 1843 essay Die Logische Frage in Hegel’s System. This essay initiates the three-way debate on the nature of logic and gives this debate its name. Trendelenburg’s work is enormously important for its detailed criticisms of Hegel’s logic and for bringing the “logic question” into focus. However, Trendelenburg’s positive proposals, which he develops in great detail and at length in the two volumes of his Logical Investigations, do not gain many adherents. A more influential middle view is that of Hermann Lotze, who defines logical reality as its own realm. While metaphysics is concerned with being, logic articulates the realm of Geltung, validity. Truth and falsity, the coherence, unity, and structure of a proposition or syllogism are all structures of the realm of validity. We should not say that they “are” but that they “are valid.” Other authors in this middle group are Friedrich Ueberweg and Hermann Ulrici.
These authors are not neo-Kantians, strictly speaking. The neo-Kantian movement, which has been the topic of some exciting historical scholarship lately, starts later. It is rallied by Otto Liebman’s slogan “back to Kant” in his 1865 Kant und die Epigonen and produces its first serious Kant interpretation in Hermann Cohen’s influential 1871 Kants Theorie der Erfahrung. It grows, ramifies, and blossoms in the writings of Heinrich Rickert and the Southwest school and Ernst Cassirer and the Marburg school. The work Lotze, Trendelenburg and others do on logic in the 1840s and 1850s prefigures but predates such an explicit and programmatic return to Kant. Trendelenburg’s Kant-criticisms are part of an attempt to develop his own middle view of metaphysics that he associates with a return to Aristotle. More than a proto-neo-Kantian, he is a product of the classical scholarship in nineteenth-century German philosophy. He studies classical languages at the Gymnasium and at the Universities of Kiel and Berlin and his first scholarly work is an attempt to reconcile Aristotle and Plato on ideas and numbers.
Trendelenburg (1846 and 1867), Kuno Fischer (1852 and 1865), and Ueberweg (1857) write histories of logic in addition to their contributions on the logic question. Scholarly work on the continent has always emphasized the history of an issue; even today a good part of a typical philosophy book published in Germany is dedicated to a critical exposition of the history of its topic. And the nineteenth century witnesses an especially broad and important increase in historical awareness and historicist approaches. But in Ueberweg’s Geschichte der Logischen Lehren or Trendelenburg’s Geschichte der Kategorienlehre more is at stake than scholarly diligence. Their historical overviews are intended to show the range of different approaches to logic since Aristotle and hence position their own reform efforts in the grand tradition of innovations in logic.
“General logic” is Kant’s name for the traditional syllogistic logic. He frequently lectures on general logic, elaborating on available textbooks. While Kant never writes a logic textbook, his student Christian Jäsche edits and publishes Kant’s lecture notes in 1800. According to Kant, general logic contains the principles of the correct use of the understanding and reason, “without which no use of the understanding takes place.” It studies the laws of thought without any regard for the objects it is about, or even whether it is about objects at all. A more specific task is taken up by types of logic that study “the rules for correctly thinking about a certain kind of objects.” So a particular logic might study the rules for thinking about motion or plants and hence be the logic of physics, or the logic of botany. As Kant puts it, such a particular logic is always the “organon of this or that science” (1781: A52/B76). It presumes prior knowledge of the objects studied in that science and consists of rules for ordering the body of knowledge about those objects. General logic is not bound by any such orientation towards objects and hence governs more widely than just this or that science.
With his critical philosophy, Kant develops the idea of transcendental logic, which he contrasts to general logic. Object cognition requires twin contributions of sensibility and the understanding. The former provides intuitions of the object and the latter provides concepts under which to cognize the object. In the Critique of Pure Reason Kant argues that there are pure intuitions that are not received by the senses and pure concepts that are not derived from empirical experience. Pure concepts applied to pure intuitions yield cognition of a pure object, one that is devoid of specific content and that does not belong to “this or that science.” The rules governing the understanding’s contribution to such cognition of pure objects do not fall into any particular logic bound to an empirical science. Nor do they fall into general logic which studies thinking irrespective of its directedness towards objects. Kant therefore develops the “idea of a science of pure understanding and of the pure cognition of reason, by means of which we think objects completely a priori. Such a science, which would determine the origin, the domain, and the objective validity of such cognitions, would have to be called transcendental logic” (1781: A57/B81). Kant’s transcendental logic is superficially similar to general logic, as it, too, is divided into chapters on concepts (the Analytic of Concepts), judgments (the Analytic of Principles), and inferences (the Transcendental Dialectic). But the content of these chapters differs radically from general logic. Instead of the “brief and dry science” required by the “scholastically correct presentation” of syllogisms and forms of judgments (A54/B78), the transcendental logic makes substantial claims about what we can experience objectively.
The difference between these two logics lies in the pure content of cognition. General logic “abstracts from all contents of the cognition of the understanding” (1781: A54/B78). Kant expresses the same point by saying that general logic abstracts from the “difference of its objects” (A54/B78), or from “any relation of [cognition] to the object” (A55/B79), or from the “matter” of cognition (A59/B83). He treats this criterion as the definition of generality. A science, a statement, or a criterion is general with respect to cognition precisely if it applies to “all cognitions without any distinction among their objects” (A58/B83). Transcendental logic, on the other hand, does “not abstract from all content of cognition” (A56/B80, emphasis added). It excludes the empirical content of cognition, but focuses precisely on the pure content. It is thus sensitive to certain kinds of distinctions among the objects of pure cognitions. For instance, it distinguishes between pure objects that are possible or impossible, or between objects that are something and those that are nothing (A290/ B346).
However, transcendental logic shares two crucial features with general logic. First, both are normative for thought, or, as Kant puts it, both are a “canon of the understanding.” A canon is “the sum total of the a priori principles of the correct use of certain cognitive faculties in general” (1781: A796/B824). One can think, or “use the understanding,” correctly or incorrectly. Whether a given act of thinking is correct as thinking is determined by its adherence to the principles contained in the canon of the understanding. These principles are thus normative; they prescribe features to which a mental act must conform if it is to count as thought. General logic is “a canon for understanding and reason in general,” i.e. it contains norms for all of thought, and it is a canon “only as far as form is concerned, since it abstracts from all content” (A796/ B824). It is silent on cognitions whose content is false, or even nonsensical. “For although a cognition may be in complete accord with logical form, i.e., not contradict itself, yet it can still always contradict the object” (A59/B84). Transcendental logic is “a canon for the assessment of empirical use [of the understanding]” (A63/B88) and “the canon of the pure understanding” (A796/B824). Cognition of objects is subject to the normative constraints of transcendental logic. These constraints concern the content of empirical judgments, insofar as this content includes the general features of an object. “For no cognition can contradict it without at the same time losing all content, i.e., all relation to any object, hence all truth” (A62/B87).
Second, transcendental logic, too, is formal, though it does not abstract from all content. Kant uses “form” in two different senses. First, he contrasts it to the sensible matter of cognition. “I call that in appearance which corresponds to sensation its matter, but that which allows the manifold of appearance to be intuited as ordered in certain relations I call the form of appearance” (1781: A20/B34). Transcendental logic is formal in this first sense, since it abstracts from objects as they are given in intuition. Second, in his definition of general logic, Kant contrasts form to the content of cognition, where this content itself is constituted by form (in the first sense) and matter. General logic is formal in this second sense. As regards transcendental logic, Kant warns against “the danger of making a material use of the merely formal principles of pure understanding” (A63/B88) and points out that pure concepts contain “only the form of thinking of an object in general” (A51/B75). This warning is directed at an undisciplined use of the transcendental logic that would aim to spin it into results beyond possible experience, a tendency that the Transcendental Dialectic is meant to reign in. All intuition is particular, so what transcendental logic abstracts from is particular content, or material content. It is directed at a pure object, or an object in general, and outlines the form of possible objectivity, which needs to be filled in by intuition in order to yield cognition.
Kant’s distinction between general and transcendental logic rests on the idea that the cognitions treated in transcendental logic are a proper subset of cognitions in general. In other words, Kant countenances a class of thoughts that is wider than object-directed thoughts, whether this object is possible or impossible, something or nothing. For Kant the idea of thought without directedness to an object makes sense. Thoughts in this class, sanctioned by the norms of general logic but beyond the scope of the norms of transcendental logic, have no content on grounds of their failure to be object-related. They are formally nonself-contradictory and they are not about anything.
In his survey of the century’s developments on logic, Wilhelm Windelband writes that “the relationship between [general and transcendental logic], these two logical doctrines that Kant regarded as entirely separate, was the agent that brought enormous ferment into the investigations about the essence of scientific thought, and it triggered a wealth of new movements whose tendencies still are not nearly reconciled and define our position to this day” (1904: 163). As Wayne Martin shows, Fichte may be the first to voice a radical challenge to logic explicitly, and he does so in precisely these terms. After Kant rejects his Wissenschaftslehre, claiming that it aims to derive substantial claims from purely formal reflection, Fichte argues that Kant here is mistaken because he confuses the formality of general logic with the method of transcendental philosophy. Indeed, he claims that Kant’s “own philosophy requires that general logic be destroyed to its very foundation” (Martin 2003: 36). Put less dramatically, the proposal is to dissolve general logic in transcendental logic. This is what Hegel’s logic intends to do.
Hegel rejects the idea of a logic that is more general than a properly conceived transcendental logic. Unlike Kant, he thinks that the idea of non-objective thoughts is incoherent. All thought is at least minimally object-directed. Rules governing objectdirected thoughts, found in a transcendental logic, therefore constitute precisely the “canon for understanding and reason in general” that Kant reserves for general logic. It amounts to the same thing that Hegel also rejects Kant’s second distinction between the form and content of thought, according to which general logic is concerned only with the form of thought. Hegel denies that it is coherent to conceive of thought as opposed to its content in this sense at all, and he criticizes Kant sharply and persistently on this distinction. He says it belongs among the presumptions “on which the conception of logic has rested so far, but that either have already gone under, or whose time has come to disappear completely” (1831: 36). Consequently it makes no sense to think of logic as studying the abstract forms of thought, the way traditional general logic does. Instead, Hegel claims, logic studies the content that all thought necessarily has insofar as it is about objects. It analyzes the most general content of thoughts, not their abstract form. In other words, logic is transcendental logic.
Hegel’s overall conception of logic corresponds exactly to Kant’s idea of a transcendental logic, in that it analyzes the concepts and principles that make objective thought possible. The details of Hegel’s logic, however, are very different. Kant’s transcendental logic establishes a table of pure concepts that function jointly and are logically independent from one another. Hegel argues that these concepts are not independent, but imply one another in a sequence of increasingly specific articulation. His logic traces the development of the entire system of concepts and principles through these articulations. The beginning of this process lies in the idea of pure objective thought itself, whose simplest and most immediate content is the idea of being. Logic then proceeds dialectically, i.e. by way of a series of logical arguments that show how every partial concept implies distinctions that go beyond itself; this means that every concept implies a further concept that is distinct from it and a third concept that comprehends this distinction itself. This dialectical sequence covers all pure concepts, types of judgments, and types of inferences and establishes their interrelation. It culminates in a system of the elements of pure thought that Hegel simply calls “the idea.”
Kant derives his table of categories from the forms of judgments of traditional general logic. Since Kant conceives of transcendental logic as a restriction of traditional general logic, it does seem consistent that the categories should be based on the traditional forms of judgment. The laws of general logic govern all thought, and therefore also hold for the objective cognitions analyzed in transcendental logic. Hegel, however, finds this approach deeply flawed. “Kant’s philosophy is inconsistent when its transcendental logic borrows the categories from subjective logic, which collected them empirically. It may as well help itself empirically to the categories right away” (1816: 289). He points out that traditional logic simply organizes kinds of concepts and judgments as it finds them in experience. It does not derive them logically and therefore remains an “empirical logic,” based on experience. This may be acceptable for a mere presentation of forms of thought for the didactic purposes of a school logic. But transcendental logic must establish the possibility of objective experience in the first place, and therefore must provide a derivation of the structures that constitute this possibility. Hegel claims that Fichte’s philosophy “has the deep merit of reminding us that the determinations of thought have to be essentially derived and revealed as necessary” (1830: 68). Logic must derive its own structures, as the dialectic aims to do.
Hegel argues that logic is not formal. Kant claims that general logic is formal insofar as it abstracts from all content of thought. This, Hegel claims, is incoherent; there is no thought without content. General logic, at best, amounts to a limited, uncritical survey of structures of thought found in experience. These structures are absorbed in a properly philosophical logic where they have their place in a dialectical derivation that shows their interrelation and demonstrates their necessity. Kant also claims that transcendental logic is formal, in the sense that it does not concern itself with the matter of sensations given in intuition. Hegel agrees that transcendental logic, which for him is the only kind of logic, has this feature. Like Kant, he prefers to call it its “purity.” “Logic is the science of the pure idea” (1830: 53), just as Kant’s transcendental logic studies the pure concepts and principles of pure reason. Purity is just the formality that Kant attributes to transcendental logic: objective thought shorn of intuitive material content.
Since logic is pure and abstracts from sensations, there is a sense in which it can be called formal. Hegel calls it the “science of the absolute form.” However, he points out that “this form is of a completely different nature from what logical form is commonly taken to be” because “it has its own content or reality in itself” (1816: 265). This specific content consists of the structures that enable thought to be about objects. The principles of logic govern all thought not because they abstract from its content, but because they partake in it and constitute it. The logical constitution of objective content, Hegel claims, is ubiquitous and substantial and logical content is therefore not abstract or empty. “This [logical] form must be conceived as being much richer in determinations and content and as having infinitely greater effect on the concrete than is commonly thought” (1816: 267). Hegel says that Kant himself realized that pure concepts are the seed of necessary content. “Kant introduced this point of view through the highly important thought, that there are synthetic judgments a priori. This original synthesis of apperception … contains the beginning of the true grasp of the nature of the concept and is entirely opposed to any empty identity or abstract generality” (1816: 260). In Hegel’s interpretation, Kant’s transcendental logic shows that logical concepts in themselves have concrete content and hence undermines the basic idea of a formal, general logic.
Regardless of its importance to metaphysics and other subjects beyond logic, Hegel’s radically new version of logic does not attract philosophical followers beyond the generation of his devotees. His sweeping criticisms of traditional logic and his broad conception of the foundation and scope of logic are influential in less direct ways. They have a liberating effect on the thought of philosophers of logic in the following decades. Trendelenburg begins his logical investigations with a careful and detailed response to Hegel’s logic. Lotze follows both Trendelenburg and Hegel in arguing for an epistemological conception of logic. Fischer starts out as a faithful Hegelian, but later departs from his initial strict Hegelianism and develops a looser version that incorporates elements of traditional syllogistic logic more strictly. Due to their Hegelian beginnings these logical radicals share the principle that logic is not formal, although they go on to develop this claim in different ways. Trendelenburg wants to supplement formal logic with a study of the material content of the basic concepts of the sciences. Lotze argues for a conception of logic as the formal structure of cognitive content. Fischer, finally, defends the Hegelian position that logic is transcendental and as such also specifies the basic features of objects.
In one way or another, all major logical radicals after Hegel reject the claim that logic is, or should be, formal. MacFarlane (2000, 2002) has disentangled different senses in which the philosophical tradition claims that logic is a formal discipline, and he has shown that one of these senses originates with Kant. Prior to Kant’s critical philosophy, metaphysicians in the Wolffian tradition argue that logical principles can yield substantial insight about some objects, such as God or the soul. Kant rejects such views and begins to maintain that logic is formal in the sense that it must abstract from all content of thought. Although Kant does not produce an explicit argument to this effect, MacFarlane discerns such an argument in the combination of two views that Kant is committed to. First, he claims in his logic lectures that logic is general, i.e. that it covers the rules of all thinking. Second, Kant’s critical epistemology implies that if logic is to be general, it must abstract from all relation of thought to sensibility and, therefore, from all content. (MacFarlane 2002: 49ff.). This is a claim about general logic, not transcendental logic. When Kant claims that transcendental logic is formal, he does not use this word in the same sense; for Kant transcendental logic is not general.
In its historical context Kant’s claim is both original and controversial. Asserting the formality of logic, he rejects the position of Wolffian metaphysics and is in turn rejected by the logical radicals. However, in challenging the formality of logic, the post-Kantian logical radicals reject neither Kant’s critical epistemology, nor the view that logic is general. Both Fichte and Hegel argue that these two views can be reconciled in a conception of logic that is not formal. Indeed, they claim that Kant himself undermines the idea of the formality of logic through his discovery of transcendental logic, which, they argue, implies that the most general concept of thinking is precisely the transcendental notion of thinking an “object in general.” In rejecting the formality of logic they both claim to develop Kant’s own philosophy to its consistent end. Similarly later philosophers from Trendelenburg to Windelband, who articulate and motivate arguments against the formality of logic, think that Kant’s philosophy is amenable to their views. They do not think that they are attacking Kant, even though they occasionally use the label “Kantian logic” as equivalent to “formal logic.” Instead they target the logic of Herbart and his followers Twesten and Drobisch.
For the logical radicals the so-called Kantian school of formal logic is not Kantian in any straightforward or privileged sense. The debate about the nature of logic implies a debate about the proper interpretation of Kant’s critical philosophy. Herbart and Drobisch also base their views about the nature of logic on an interpretation of Kant that rejects some of his most fundamental claims and emphasizes others. Herbart calls his Kant interpretation “realism” in explicit opposition to Fichte and Hegel’s idealism. His view insists on the givenness of experience as the starting point of critical analysis and rejects the idea that the transcendental subject can constitute this givenness. The purpose of transcendental philosophy, particularly transcendental logic, is to show how the subject cognizes according to principles that inhere in the given objects themselves and, in particular, to explain how the natural sciences can discern such principles. Consequently Herbart rejects the basic claim of Kant’s transcendental deduction, that self-consciousness is grounded in a synthesis that also makes cognition of objects possible. He writes that Kant here provides “the first occasion for the manifold errors in recent philosophy,” pointing out that Fichte is particularly indebted to this point (1813: 75).
It is more accurate to draw the distinctions between the three camps of logical radicals in terms of the aspects of Kant’s critical philosophy that they most emphasize. Herbart insists on the givenness of experience and maintains that the primary task of philosophy is to analyze and explain this givenness. Fichte and Hegel think that the most important and fruitful insights are contained in the transcendental deduction; and with their analyses of subject-object identity they take themselves to be faithful to the idea of critical philosophy in drawing out the consequences of Kant’s argument that the same synthesis constitutes the possibility of subjective self-consciousness and the cognition of objects. Logic, they argue, analyzes the transcendental concepts that constitute object cognition. Trendelenburg’s work undermines Kant’s strict division between the two sources of cognition. He thinks that the transcendental aesthetic points the way to the foundations of cognition. Kant argues according to a false dilemma when he infers from his arguments that the forms of intuition must belong to a subjective faculty that they could not also be in the objects themselves. Both subjective and objective aspects of space and time, Trendelenburg argues, derive from the metaphysically fundamental principle of Aristotelian original movement. In his view, a complete logic needs to take account of the common origin of both thought and objects in original movement.
Herbart’s conception of logic as a formal general discipline follows from his realist interpretation of critical philosophy. Logic is faced with given concepts and should restrict itself to the analysis of the structure and interrelations of these concepts without admixture of sensible content or the forms of sensibility. “Logic posits concepts as known and does not deal with their particular content. Hence it is not a tool of inquiry in which something new is to be discovered, but a guide for expounding what one already knows” (1813: 45). In other words, he insists on the formality that Kant claims for general logic. Drobisch similarly states that the proper subject matter of logic includes only those relations among concepts that “apply to all concepts, independently of what is thought in them” (1836: 3). He underscores this minimal conception of the role and content of logic by suggesting an improvement to Kant’s own definition of general logic. Where Kant writes that logic studies the “necessary laws of thinking, but not with respect to specific objects, but all objects in general” (Kant 1800: 9), Drobisch comments that “the only mistake in this explanation is that it relates thinking to objects” (1836: 7). Kant’s “object in general” imports more content than formal logic in the Herbartian tradition is willing to admit.
A common argument against the formality of logic is that by restricting itself to the mere form of thought, logic is unable to explain some inferences that it should be able to account for. For example, traditional logic licenses the conversion inference from “all As are Bs” to “some Bs are As.” This is correct if we consider examples such as “all dogs are mammals” which implies precisely that “some mammals are dogs,” no more and no less, presuming, as logicians before the twentieth century do, that “all As are Bs” means that there are at least some As. However, when we infer from “all figures with the Pythagorean property are right triangles” to “some right triangles are figures with the Pythagorean property,” the licensed inference is misleading. There are many such cases having to do with conversion inferences and the immediate inferences of the square of oppositions, and logicians from antiquity onwards discuss how best to handle these examples.
One solution in the nineteenth century, developed in detail by Hamilton, is that propositions implicitly quantify both subject and predicate. So the propositions in this example differ logically; the first states that “all dogs are some mammals,” while the second states that “all right triangles are all Pythagorean figures.” Drobisch proposes that the correct conversion of universal affirmative judgments can only be justified by an analysis of the object itself and hence falls beyond the reach of logic into the domain of a specific science. It is a question of geometry, not of logic, whether right triangles are in fact all the Pythagorean figures (1836: 46). Lotze holds the broader view that all conversion inferences and immediate inferences belong to applied logic (1874a: 101).
Trendelenburg does not look for a particular solution to the problems of conversion inferences. Instead he cites this example as evidence for the vastly more general point that a purely formal analysis of inferences, which abstracts from all content of the concepts involved, is impossible: “They found it significant that nothing more follows from the universal form of judgments. We believe that we see in the insufficiency of this whole result an indication that the entire standpoint of the science, in which the form is detached from the content, is itself insufficient” (1870: 336n). Formal logic excludes logically salient information by abstracting from the material content of concepts. Moreover, according to Trendelenburg, the very idea of a form of the concept without consideration of its material content denatures the concept and hence undermines itself. Formal logic, he claims, is not only overly restrictive; it also forces logicians to accept a false view of the structure of concepts.
Traditional logic converges on a quantitative account of concepts, according to which they are aggregates of attributes. Judgments express inclusion and exclusion relations between the sets of attributes of two different concepts, and syllogisms do the same for three or more concepts. According to a purely formal conception of this analysis, any set of attributes forms a concept. Trendelenburg points out that some pairs of attributes, such as “square” and “circular” necessarily exclude one another, for reasons that lie in the objects themselves. Since such exclusion relations between attributes are necessary, they lie within the proper purview of logic and the theory of concepts ought to be able to express them. Similarly, the formal quantitative account does not distinguish between essential and accidental attributes of a given concept. So, in an example Trendelenburg borrows from Hegel, “plant” is an essential attribute of “rose” while “red” is not. Once again, this difference has necessary inferential import, but formal logic does not have the analytic tools for representing it. Trendelenburg generalizes the point and faults formal logic for its inability to conceptualize the “particular connection” that forms the unified whole of a concept and instead reduces it to a senseless aggregate (1862: 21).
Trendelenburg’s approach here can be compared to one of the basic principles that Frege establishes for his Begriffsschrift, that a logical notation must “fully express everything needed for a correct inference.” Like Frege, Trendelenburg and other logical radicals are disposed to revise logic in its entirety. Nevertheless, unlike Frege, Trendelenburg’s criticisms issue in proposals for extensions of traditional logic that keep its core intact. Detaching the form of thought from its content is “insufficient,” not because the very distinction between form and content is incoherent, but because in some respects the content of thought has logical import. Trendelenburg argues that there is a logical task that goes beyond a purely formal analysis. Logic includes a formal part, but this formal part must be supplemented by some analysis of the content of concepts. In essence, then, Trendelenburg makes the same point that Drobisch and the defenders of formal logic also make: the puzzle about conversion inferences points to material questions about the objects denoted by the concepts in the examples. But where Drobisch draws the limits of logic at formality, Trendelenburg wants to extend these limits to include those aspects of the sciences that contribute to the correct analysis of an inference. The question is motivated by a reconsideration of the nature of logic, not by new discoveries about the structure of inferences.
To make the scope of his proposal clear, Trendelenburg distinguishes between a narrow and a wide sense of logic in his Logical Investigations. In the narrow sense logic maintains its formality, but is limited to a mere analysis of contradictions and incapable of making fruitful contributions to the sciences. In the wider sense the domain of logic is defined by necessity, not merely by formal contradictions. Necessity in thoughts and in facts can have logical and metaphysical origins. Therefore Trendelenburg aims for a single theory which must comprise “both domains to understand the inner possibility of knowledge and conceive thought in its striving toward knowledge” (1862: 11). Logic in the wider sense aims to analyze the necessity of scientific knowledge by explaining the metaphysical ground of the objects of scientific knowledge and the structure of thought about such objects. The culminating element of Trendelenburg’s theory is that thought and objects have the same metaphysical principle. They both derive from the original “movement” that constitutes the ground of being in space and time as well as the possibility of combining discrete concepts in judgments.
The theory Trendelenburg develops in great detail in his Logical Investigations has a fate similar to Hegel’s Logic, which it criticizes so sharply. The work is influential in stimulating the debates about the status of logic, but no philosopher is persuaded by Trendelenburg’s view about movement as a metaphysical ground of thought and being.
Lotze writes two editions of his influential logic. The 1843 “small” logic contains a lengthy preface in which he stakes out a position between formal and Hegelian approaches and largely agrees with Trendelenburg’s view that the proper boundaries of logic include the conceptual groundwork of the sciences. Lotze’s justification for this proposed extension of logic differs from Trendelenburg’s. Trendelenburg argues that basic material content belongs to logic because thought and objects share a single metaphysical root and hence cannot be analytically divided without falsifying the structure of concepts and inferences. Lotze, on the other hand, argues that logic can only study the forms of thought insofar as these forms tend to the cognition of real objects. The main body of Lotze’s Logic consists of a detailed exposition of the theories of concepts, judgments, and inferences with significant changes of emphasis from traditional presentations. In the 1874 “big” logic Lotze completely rewrites the preface, produces a revised version of the body of the small logic in a first volume called “On Thinking” and adds two further volumes, “On Inquiry” and “On Cognition.” With its focus on this third volume, the later logic develops the epistemological conception of the small logic explicitly. Unlike Trendelenburg’s Investigations, Lotze’s Logic becomes one of the standard logic texts of the nineteenth century, along with Drobisch and Sigwart’s books. He is a logical radical who captures the mainstream.
Lotze spreads his revisionist arguments about the nature of logic throughout the presentation of the theories of concepts, judgments, and inferences. He echoes Trendelenburg’s arguments that the quantitative analysis of concepts as aggregates of attributes misses the essential nature of concepts (1874a: 47). We only fully grasp a concept when besides its attributes we also grasp why precisely these attributes are connected in precisely this way, to the exclusion of others (1874a: 39). Lotze also agrees that a purely formal treatment of the immediate inferences is useless unless it is merged with an analysis of the basic concepts of the actual sciences in applied logic. He includes a discussion of these immediate inferences only “for the sake of tradition” (1874a: 101).
Lotze’s overall conception of logic preserves features of both Herbartian formal logic and Hegelian metaphysical logic. He writes that the discipline is neither formal, nor real. “Logic should be formal in the sense that it is a theory of the operations of thinking by which the subject prepares his thoughts for cognition; but not in the sense that these forms are something factically given without explicit reference to the task of cognizing the real” (1843: 13). The main fault of Herbartian formal logic, according to Lotze, is that it simply takes its tables of forms and laws as given. It is for this reason that logic has become a dry and largely useless discipline. It is precisely such uncritical deference to existing collections of doctrine that leads to the misleading analyses of the structure of concepts and inferences. While Herbart defends this character of logic as the essential expression of its proper boundaries, Lotze finds that such a logic does not fulfill its purpose. “Logic is not supposed to list the laws of thinking, but explain the origin of these laws and their relation to other activities of the mind; this way it should attain more influence on the development of actual knowledge than is possible for an abstract formalism” (1843: 5).
Hegel makes virtually the same criticism of the “dead bones” of formal logic and argues that the task of logic is to systematically derive the forms and laws of logic from the very idea of pure thought. But, Lotze claims, in doing so Hegel’s dialectic conflates logical and metaphysical structures and reduces to nothing but a “play on words, which is not far from a logical error” (1843: 11). Nevertheless, much like Hegel, Lotze conceives of the laws of logic as having real significance and as overlapping with the structure of real objects. Only such overlap can ground the cognitive import of logical structures. Lotze indicates his middle position with the claim that “there are motives in the nature of things that force the cognizing mind to produce precisely these shapes of perception and combinations of objects” (1843: 13). On the face of it, this seems to be a naturalistic claim, in that it attributes the laws of logic to the impact of non-logical laws on the mind. Lotze, however, strongly opposes psychological construals of the origins of logical laws. At pains to avoid the appearance of psychologism in his own case, he turns to Kant, and specifically to the schematism of the categories.
According to Kant, the schemata are rules for producing intuition according to patterns of time-determination. The transcendental imagination applies these rules and brings intuited content into patterns that make such content understandable according to the categories. In other words, the transcendental imagination produces patterns in which the mind intuits material as categorically formed. Lotze proposes that in addition to this intuition oriented schematism there is an analogous schematism for logical form. Just as patterns of time determination enable the categorical cognition of intuitable content, so the laws of logic are patterns that enable the cognition of thinkable content. Laws and structures of logic are rules according to which the mind can cognize categorically formed and thinkable, yet intuitively empty, content. “Forming schemata, in this sense, means nothing other than thinking, as opposed to cognizing” (1843: 29). So, for example, Lotze suggests that the logical schema of the category of substance is the concept of the logical subject and the logical schema of the category of inherence is the logical copula (1843: 30).
Of course full cognition requires both intuitive and logical schematization of the categories. In this sense Lotze maintains that logic is a formal discipline. Its structures do not apply to actual objects and do not extend to material content. But logic is not formal in Herbart’s sense, because it must contain an account of the origin of logical laws, and this account must be grounded in the role that logical laws play in cognition. And it is not real, or metaphysical, in Hegel’s sense, because while the laws of logic are necessary and binding on the thinking subject, they are not themselves constitutive of real objects. Nor, finally, is Lotze’s conception that of a transcendental logic, since he distinguishes logical forms from the categories. Transcendental logic is the doctrine of the categories. Logical concepts are not categories, but they presume the categories and explicate how it is possible that the categories apply to thinkable content.
Kuno Fischer is one of the few prominent nineteenth-century Hegelians who is not a student of Hegel himself. He comes to Hegel’s philosophy after Hegel’s death through the influence of his dissertation adviser, Johann Eduard Erdmann. Perhaps it is precisely this separation that enables Fischer to present and defend Hegel’s logic in a clear and sober manner, with such distinction that Windelband calls his work the “by far most interesting presentation of Hegel’s logic” (1904: 180). Fischer’s exposition of Hegel’s logic is the rare such work that gains adherents and promotes continuous study of Hegel’s philosophy throughout the nineteenth and into the twentieth century. Fischer writes two editions of a logic textbook. The earlier, 1852 edition is a faithful Hegelian work. He prefaces the main body of this text with a survey of the history of logic, which culminates in the development of Kant’s critical philosophy into the philosophy of identity in Fichte, Schelling and Hegel. The body of the text is a concise version of Hegel’s own logic and covers the entire dialectical development of the categories of thought from pure being to the absolute idea. A small part of this is the exposition of the “subjective logic,” which contains the doctrines of concepts, judgments, and inferences. Like Hegel’s own exposition of these doctrines, Fischer abstracts entirely from their traditional content in favor of a dialectical demonstration how they arise from one another. In the second, 1865 edition Fischer distances his view from the details of Hegel’s logic because, he writes, “even though it set itself the right task, its execution was all wrong” (1865: vi). Even though in its main outlines the exposition still follows Hegel’s logic, Fischer now includes a more traditional version of the doctrines contained in the subjective logic, and he frames his conception of the relation between logic and metaphysics in more recognizably Kantian terms. He also defends his Hegelian conception of logic against Herbart and Trendelenburg’s alternative proposals.
According to Fischer, logic analyzes concepts that transcend the domains of specific sciences and make scientific cognition possible. In a nod to Kant, Fischer calls these concepts the categories. Fischer thus develops Hegel’s argument against the formality of logic, itself derived from Kant’s idea of a transcendental logic, that logic is concerned with the constitution of objects in pure thought. Logical concepts and structures are not the form of thought without content, but the function of thought in constituting its objects. While formal logic limits itself to empty concepts, logic ought to aim at necessary concepts (1865: 204). As Fischer puts it, the concepts of logic “do not generalize, but combine; they do not represent, but judge” (1865: 9). Since categories such as being, quantity, or force are also the basic determinations of objects, logic is synonymous with metaphysics. This conception of logic is grounded in Hegel’s philosophy, for “the task of logic is solved in the transcendental standpoint of the philosophy of identity” (1852: 46). According to this Hegelian argument, logic is not formal at all. It is transcendental, and its necessity and generality derive from this.
Fischer’s logic is similar to Hegel’s and thus deviates more radically from the details of traditional logic than Trendelenburg and Lotze. For instance, Fischer adduces an example used by virtually all authors who aim to demonstrate the limits of traditional or formal logic. Traditional logic wrongly finds that the two judgments “the square is white” and “the square is a parallelogram” have the same form (1865: 432). Trendelenburg solves this problem by extending the theory of concepts to include a distinction between accidental and essential attributes. Lotze argues that in this case the different connections imply different interpretations of the copula in the theory of judgments. Fischer mounts a broader criticism that incorporates both of these. He uses this example to challenge the whole theory of judgments that posits a table of quantities, qualities, relations and modalities. These forms of judgments, says Fischer, must first be developed dialectically as determinations of concepts. Such a dialectic development of the forms of judgments yields the logical distinctions between accidental and necessary attributes. Further, this distinction between accidental and necessary attributes of a concept also constitutes a distinction between different forms of judgments.
Indeed, in the later version of his logic, Fischer pretends to go further than Hegel himself. He rejects Hegel’s way of integrating the structure of traditional formal logic in his dialectical progression as overly beholden to the received tables of logical forms. “Just as for Kant the judgment forms of ordinary logic served as the guiding thread for the discovery of the categories, so Hegel and his school used it as the guiding thread for the development of judgments” (1865: 433). Instead of merely repeating the table of forms, Hegel makes a dialectic series of arguments that produces a sequential order of the same forms. In effect he, too, uses the existing tables as a guiding thread. But it is an open question, to be resolved by logic itself, whether these forms of judgment appear in the dialectic. The dialectic should unfold according to the necessary content of pure thought, according to the schema that Hegel had identified as the progression from the logic of pure being to pure essence and finally the pure concept. In the same edition, however, Fischer actually returns to a more conservative treatment of the syllogistic than Hegel’s. Instead of applying Hegel’s schema in the “subjective logic” and reducing all syllogistic figures to a single relation between universal, particular, and individual determinations of a concept, he reproduces an entire theory of inferences according to the traditional figures and types of inference.
In the early edition, Fischer ends his survey of the history of logic with Hegel. In the second edition he takes account of the debates among logical radicals in the intervening decades. Fischer repeats his claim that the task of logic is to explain the possibility of scientific object-cognition. However, he now measures the standpoint of the philosophy of identity against the criticisms leveled at it by Herbart and the “formal” logicians on the one hand, and by Trendelenburg on the other. This additional layer of critical examination of the claims of subject-object identity leads Fischer to formulate his basic conception of logic by appealing to Kant’s notion of the synthetic function of the categories, rather than the unity of apperception that he sees as the basic principle in Fichte and Hegel. In other words, for Fischer the nature of logic as transcendental logic is primarily determined by the function of the categories in grounding objective cognition, and only secondarily by the structure of self-consciousness. Although this is only a slight shift of emphasis, it reflects the antiHegelian dynamics of the arguments of the logical radicals. To defend his Hegelian logic, Fischer retreats to less controversial Kantian grounds.
Herbart, of course, disagrees with the entire standpoint of subject-object identity. Metaphysics and logic, he maintains, are fundamentally distinct sciences. Fischer’s main argument against Herbart’s realism consists of a Hegelian refutation of the notion of a mind-independent reality. This notion amounts to a point of view that attempts to “think being as independent of thinking; i.e. being is thought as not thought” (1865: 134). This amounts to a contradiction in the concept of “reality” that Herbart wants to reserve as the domain of metaphysics and oppose to the domain of logic. Self-contradictory concepts, however, need to be developed dialectically and hence Herbart’s realism is swallowed up by Hegelian logic.
Fischer’s criticisms of Trendelenburg’s view are more detailed. Trendelenburg’s idea of movement as a substrate that grounds both being and thought resembles Fischer’s standpoint of subject-object identity. Trendelenburg, too, aims to be able to explain the cognitive possibility and objective validity of logical structures and he uses his metaphysics of movement as the bridging principle. Fischer argues that in doing so Trendelenburg cannot preserve the unity of his idea of movement. At best he establishes an analogy between the material ground of objects and the logical ground of thought. More importantly, Fischer also criticizes Trendelenburg’s “loophole” objection to Kant. Kant’s Critique, Fischer argues, does not overlook the possibility that space could be objective; on the contrary, Kant proves that this is impossible by demonstrating the possibility of a priori knowledge of mathematics. For, Fischer writes, if space were objective, it would be given in experience and hence “mathematical insights would have to be judgments of experience and as such could be neither general nor necessary” (1865: 175). These criticisms of Trendelenburg’s loophole objection lead the latter, who has a penchant for such Auseinandersetzung, to write a disparaging denouncement of Fischer’s understanding of Kant. The resulting exchange plays a significant role in the development of the Kant interpretations of Hermann Cohen, and hence contributes to the views of the neo-Kantian schools, which focus on ways to integrate the transcendental aesthetic with the functions of the concepts of the understanding. Thus Trendelenburg for a second time becomes the catalyst of a debate that shapes nineteenth-century philosophy.
The post-Kantian use of “formal logic” is not universal. In England, where mathematical and symbolic approaches to logic have more influence on philosophy, the phrase is used to denote mathematical or symbolic logic. For example Augustus De Morgan titles his 1847 textbook Formal Logic. This is not entirely coincidental. Windelband speculates that “pure formal logic must follow a natural inclination to express its forms in mathematical formulae and to seek its grounding in mathematical relations” (1904: 166). Drobisch experiments with such notation in a “logico-mathematical appendix” to his Neue Darstellung. Following Herbart he conceives of judgments as the relation between the spheres of two concepts and writes that therefore “simple judgments can also be represented by algebraic equations” (1836: 132). Hamilton’s idea to quantify over both subject and predicate may come from Herbart’s discussion of conversion inferences in his Lehrbuch. Herbart accounts for the apparent oddness of the fact that “All As are B” converts to “Some Bs are A” by positing the logical principle that propositions never assert the “entire sphere” of the predicate (1813: 104), i.e. that the predicate “B” does not mean “all Bs.” Trendelenburg claims that Hamilton “essentially transplanted Kant’s German formal logic onto the ground of English philosophy” (1862: 15), giving further evidence of some influence of post-Kantian philosophers of logic on English symbolic logic.
Due to such perceived commonality of Herbartian formal logic and mathematical logic, and in the wake of the influential criticisms of formal logic by the logical radicals, the philosophical mainstream in Germany spurns the idea that mathematical logic can provide new insights, even as such approaches begin to yield results that eventually revolutionize the entire discipline. In a representative gesture, Windelband dismissively credits both the Herbartian and the later British mathematical logics with the same emphasis on mere “acumen” and notes that “in Germany this logical sport which, it cannot be denied, has the merit of exercising formal acumen, has met with little approval.” He likens it to attempts to ground logical relations in spatial or geometric relations and writes that in both cases “a successful tool of visualization is confused with the essence of the matter” (1904: 166f.). The upshot of the criticisms of formality in Trendelenburg, Lotze, and the Hegelians is that formal considerations do not get at the heart of logical issues. The thrust of philosophical thought on logic through the middle of the century is to conceptualize the content of judgments in pure and general ways, so as to reinvigorate logic and once again make it relevant to scientific cognition. This tendency is exemplified most dramatically in Sigwart’s logic, which is one of the most widely used logic textbooks in the latter decades of the century up into the early decades of the twentieth century. Sigwart in effect reduces the forms of logic to a bare minimum. He argues that there is only a single form of judgment, the categorical affirmative judgment. Other differences amount only to differences in the content of subject or predicate. Similarly, he argues, there is only a single form of inference and the entire array of syllogistic forms derives merely from variations in content.
Mathematical and symbolic logic does flourish in Germany, outside the circle of post-Kantian philosophers of logic and somewhat later than in England. So, for example, a number of mathematicians develop extensive algebraic treatments of logic. Chief among these are the brothers Hermann and Robert Grassmann and, following them, especially Ernst Schröder who produces a systematic synthesis between Boolean algebra and Peirce’s discoveries. Some philosophers show genuine understanding of these innovations. One of the earliest is Ulrici’s 1855 review of Boole’s An Investigation of the Laws of Thought. This is a positive and detailed review that displays a good deal of understanding of Boole’s technical innovations, but it goes mostly unnoticed by Ulrici’s philosophical colleagues. Two decades later the Austrian neo-Kantian philosopher Alois Riehl writes another detailed and appreciative review of Boole’s Investigation and Jevons’ The Principles of Science in which he points out the most significant implications of this work. Riehl remarks that a great merit of Boole’s system is that it makes clear that the syllogisms only cover a small part of all logically valid inferences, and he points out specifically that mathematical deductions are examples of non-syllogistic inferences (1877: 60). He also maintains that mathematical logic has the great advantage that it develops logical structures independently of grammatical or linguistic structures, which are the basis of traditional syllogistic logic (1877: 53).
Arguments that logic ought to include an analysis of the content of concepts in addition to its formal part are, in a sense, conservative. Trendelenburg and Lotze, for example, leave the main structure of traditional logic intact and seek to supplement it in various places to make up for its inadequacies. A more radical strain of logical thought aims to undermine that structure itself. Often the same philosophers deploy both the conservative and radical strains of argument and thus betray a fundamental ambivalence over the status of traditional logic.
Underlying this ambivalence is the fact that the established parts of traditional logic reinforce one another. The form of the syllogisms requires an analysis of judgments as the synthetic combination of two terms. This analysis of the subject–predicate form, in turn, bolsters the analysis of the structure of concepts as conglomerates of attributes that can be subjected to a quantitative analysis of inclusion and subsumption. The theories of concepts, judgments, and inference in traditional syllogistic logic form a unified whole. Consequently criticisms that aim at a single aspect of the entire edifice tend to be formulated within the overall structure of traditional logic. The challenges to the formality of logic are mainly bound up with proposals to amend the theory of concepts, since this is where critics locate the cognitive, material, or transcendental content of thought. But they focus on the logical structure of concepts in such a way as to leave their role with respect to judgments and syllogistic inferences untouched. Fischer and other Hegelians, of course, reject traditional logic in a more wholesale fashion. Fischer even criticizes Hegel’s own logic for being too beholden to the traditional table of judgment forms. The logical radicals of the other schools develop arguments of similarly sweeping implications, though these implications are not often made explicit. These arguments, which lead to challenges that question the entire edifice of logic, mostly derive from attempts to reject the traditional analysis of the structure of judgments. Besides the criticisms of the formality of traditional logic, revisionist thought about the logical structure of judgment makes up the second main strain of arguments within logical radicalism.
Post-Kantian philosophical arguments about the logic of judgments cluster around three main topics. First, philosophers question the structure of judgments itself. Second, they debate the priority of judgments over concepts. And third, a few philosophers begin to challenge the role judgments play in inference.
Investigations into the structure of judgments begin with widespread recognition that the distinction between subject and predicate owes its origins to grammar and therefore requires careful logical interpretation. Even Herbart and Drobisch, who maintain and defend the traditional analysis of judgments as the combination of two concepts, recognize this need. The main component of the logical interpretation of judgments is the copula, the logical structure that corresponds to the grammatical “is.” The different functions of the copula constitute various forms of judgments. To emphasize the independence of the logical notion of the copula from grammatical structures, almost all writers of logic texts consider examples of judgments that do not fit the common “S is P” schema. Standard examples are existential judgments (“S exists”) and impersonal judgments (“It rains”), which abound in German (particularly with respect to atmospheric conditions: “es blitzt,” “es regnet,” “es donnert,” “es friert” are the usual examples). Traditionalists argue that the logical structure of these judgments is identical to that of less recalcitrant cases, consisting of the combination of logical subject and predicate by means of a copula. Wayne Martin has shown how Kantian logicians struggle with this attempt to conflate judgment forms, and that this very problem motivates a radical restructuring of logical form in Brentano (2006: 58f).
Lotze also analyzes impersonal judgments as subject–predicate compounds. He regards impersonal judgments as “the first act of judging in thought; as it were a preliminary step to categorical judgments” (1874a: 70). But as an act of thought, these judgments must still have a logical subject, which is, however, completely undetermined and vaguely expressed in language by the “it.” Despite this conservative strain, he proposes substantial changes to the traditional table of judgment forms. The novelty of Lotze’s analysis of judgments lies mostly in his interpretation of the logical copula. It expresses the validity (Geltung) of the fact that is thought in the combination of subject and predicate terms. Validity, he argues, is the metaphysical mode specific to logical entities, as opposed to the being of real entities. Lotze argues in general for a robustness and uniformity of the copula; since it expresses validity, it is fundamentally the same in every judgment. Consequently, Lotze’s analysis reduces the traditional variety of the forms of judgment, attributing them to differences in terms or content, rather than form. For instance, he claims that difference in quantity between universal, particular, and singular judgments are due to the quantity of the subject term, while the asserted connection between subject and predicate is identical in all three cases.
A particularly interesting line of argument derives from Lotze’s consideration of negative judgments. Where the Kantian table of judgments distinguishes affirmative from negative judgments, Lotze maintains that the copula is exactly the same in both cases. In each case the copula joins the subject and predicate to yield a thinkable content. This content as a whole is affirmed in one case and negated in the other (1874a: 61). Affirmative and negative judgments, then, differ not in logical form, but in the content of a second judgment about the first one. This view, which Sigwart later develops even further, initiates a debate about the double nature of judgments. One logical act forms the judgable content and another act affirms, denies, or takes some other evaluative attitude towards this content. This view of the double nature of judgments proves influential in the neo-Kantian Wertphilosophie of Windelband and Heinrich Rickert as well as phenomenology through the work of Brentano and Husserl. It may also be at the root of Frege’s invention of the judgment stroke in his Begriffsschrift (Martin 2006: 75f.).
More radical proposals reject the subject–predicate structure as a basic analysis of judgments. This is Hegel’s approach. A judgment, he claims, is an articulated concept, and its synthetic structure is the expression of this articulation. “The judgment is the concept in its specificity, as the distinguishing relation of its moments” (1830: 155). In this view, the underlying concept is the warrant for the unity of the judgment, and consequently there is no need for the synthetic function of the copula. Rather than combining two terms, judgments divide a single concept. “The etymological meaning of judgment [Urteil] in our language is deeper and expresses the unity of the concept as the first thing, and its differentiation as the original division [ursprüngliche Teilung], which is what the judgment is in truth” (1830: 155). Though this etymology, which Hegel shares with Hölderlin, turns out to be mistaken, the point itself is clear.
Trendelenburg develops a view of judgments that inverts Hegel’s, but also resembles it. He, too, finds that rather than producing a unity by means of a logical copula, judgments of subject–predicate form differentiate a more original unity. He rejects the traditional view according to which “thought is only the means, as it were the vehicle, to bring concepts together” (1870: 232). Like Hegel, Trendelenburg claims that this traditional analysis arises from an external perspective that does not grasp the nature of judgment. According to Trendelenburg’s metaphysics, the subject–predicate form of thought mirrors the substance-activity form of reality, because both thought and being are constituted by the “original movement” which first makes it possible that thought can be about being. Activity, however, is more basic than substance. At the most fundamental level thought grasps pure activity without substance, reflected grammatically in subjectless predicates. “We think in predicates. This main concept originally arises alone” (1870: 232). Further articulation of thought attributes the activity to a substance. So the intellect fixes a substance “the lightning” to which we can attribute the activity “lightning” and which it then uses as the subject of further judgments, “lightning strikes,” etc. Trendelenburg claims that far from being recalcitrant cases, the impersonal judgments best represent this logical structure of judgments. “We judge when we think, and in every complete judgment we distinguish subject and predicate. But this differentiated form points us back to a unity. In language this is represented in the so-called impersonal verbs, such as es braust, es blitzt, es friert” (1870: 231). Trendelenburg further draws on etymological evidence from Jacob Grimm that many nouns derive from verbs. So, for instance, Hahn (cock) derives from the lost verb hanan (to crow) (1870: 236). The activity precedes the substance in thought.
However, the conclusion Trendelenburg draws from this analysis of the original unity of judgment is precisely the opposite of Hegel’s. Where Hegel grounds judgments in concepts, Trendelenburg argues that the original unities of thought are themselves judgments. This is the point illustrated by impersonal judgments. “It rains” expresses a full judgment in its structural simplicity. “The beginnings of language lie in verbs, but in such a way that these in themselves form judgments” (1870: 236). Similarly the first names and words uttered by children represent entire sentences, despite their grammatical, and logical, simplicity. Indeed it is the concepts that owe their beginnings to such simple, unified judgments. Concepts, both nominative and predicative, only arise from an original thought and only within the articulated structure of derivative judgments. Here Trendelenburg once again appeals to evidence from etymology, this time from Gruppe, who “has shown that every concept is based on a judgment” (1870: 234). To take the same example, the concept “lightning” is only a concept insofar as it is thought as something to which a predicate can be attached, that it “strikes,” “shines” or whatever. Concepts essentially are partial and incomplete judgments. It is noteworthy that Trendelenburg makes this claim for all concepts, not only for logical predicates.
The question about the logical structure of judgment has implications for the structure of logic as a whole. In traditional analyses, such as Herbart’s, concepts have priority in two senses. First, they have priority in the order of exposition. The presentation of logic proceeds from the most elemental to the most complex, from concepts to inferences. Secondly, concepts have a logical priority, insofar as they make judgments and inferences possible as their constituent parts. Judgments are combinations of concepts, and the validity of inference is explained by the interrelations of the three terms of a syllogism.
In Hegel’s logic, too, concepts have priority. This is evident from the structure of his exposition. In the Encyclopedia logic, his versions of the three theories that traditionally make up logic, the theories of the concept “as such,” of judgments, and of inferences, are all included in the chapter on the “subjective concept,” which itself is part of the “doctrine of the concept.” Similarly in the Science of Logic, the three theories are part of the division on “subjectivity” in the subjective logic, which is also the doctrine of the concept. In its general outline, the same exposition is maintained in Fischer and other Hegelians. However, the logical priority Hegel accords to concepts differs from the traditional account. Whereas the tradition claims that concepts are the component parts of judgments and inferences, Hegel claims that these latter structures are nothing but articulations of particular concepts. Further, these particular concepts are parts of a whole that itself has the structure of a single concept.
Trendelenburg’s view, on the other hand, accords logical priority to judgments, not concepts. On his analysis, there could be no concepts without judgments. This shift in understanding logical priority, prompted by his revision of the analysis of the structure of judgments, is representative of the overall epistemological turn of post-Kantian logic. As philosophers debate the cognitive and epistemological import of logic, they focus on judgments as the basic form of cognitive achievement and hence also as the basic logical act. This same shift is evident in Lotze’s three-volume logic. In the first volume he adheres to the traditional order of exposition and writes that “without doubt, pure logic must posit the form of the concept before that of the judgment” (1874a: 24). However, when he analyzes the cognitive import of logic in the third volume, the fundamental role judgments play in cognition leads him to posit a different kind of logical priority for them. Logical entities, he argues, are distinct from real entities in that their metaphysical realm is validity (Geltung) as opposed to being (Sein). However, one cannot say of concepts that they are valid, because they are only meaningful insofar as they are parts of judgments. “Of concepts we can at most say that they mean something; but they mean something, because sentences about them are valid” (1874b: 521). In other words, concepts only belong to the realm of the logical insofar as they are constituent parts of judgments. It is the judgment that confers logicality onto the concept. So from the point of view of “pure logic,” which disregards the application of logical structures to real cognition, concepts show themselves to be elemental constituent parts. But from the point of view of applied logic judgments are the basic logical unit.
This trend becomes pervasive and at the end of the century philosophical logic preponderantly emphasizes the epistemological import of logical structures and consequently focuses more on the theory of judgments. The goal is to develop a thorough classification of the logical forms of judgment as the basic unit of cognition. Windelband, who develops a theory of double judgments, sums up this state of the art: “The theories of concept and inference are only appendices of the theory of judgments, which is the main problem of logic. This we may regard clearly as the foundation for future developments of this science: logic is the theory of judgments” (1904: 169). At this point philosophers of logic entirely lose interest in the bottom-up approach of constructing logic from concepts to judgments to inferences.
As Windelband indicates, new work on the theory of judgments also displaces the theory of syllogistic inferences from the center of logical concerns. Traditional syllogisms stipulate that premises and conclusions of an inference must be in subject– predicate form. As new theories of judgment challenge this analysis of judgments, it becomes apparent that traditional syllogisms cover only a small part of scientific deductions and arguments. Indeed from the beginning of ancient logic, logicians recognize that, while it is often possible to state arguments as a series of such syllogisms, some valid arguments resist this form. This is why expositions of the theory of inference usually also contain discussions of hypothetical and disjunctive syllogisms, as well as inductive, modal, and probabilistic inferences.
Further, logicians are aware that in traditional syllogisms the subject–predicate form plays two different roles that can easily be analytically separated. For example, syllogisms with hypothetical premises of the form “If A is B, then A is C” are treated as syllogistic inferences with categorical premises in which the major, minor, and middle terms stand for judgments, not for concepts. In demonstrating the validity of the inference, logicians substitute the entire judgment “If A is B, then A is C” for a premise of the form “S is P,” where “If A is B” is substituted for S, “then A is C” is substituted for P, and the copula becomes irrelevant. In such cases “S is P” does not stand for the form of a judgment, but for the schema of a valid inference, whose substitution instances can have many different forms.
This ambiguity of the subject–predicate form in syllogistic logic gives philosophical logicians license to challenge the traditional analysis of judgments. It implies that the inferential significance of judgments is not tied to the subject–predicate form, since other forms of judgment can figure in perfectly valid inferences. Indeed, as philosophers begin to explore different theories of judgment, they also begin to think of inferential schemas along different, non-syllogistic lines. A telling example, here, is once again Sigwart’s system of logic, which reduces all syllogisms to a single form of inference and differentiates arguments according to the judgments they involve. While Sigwart and others fall well short of developing a substantial non-syllogistic theory of inference, they nevertheless uncouple the analysis of judgments from the dictates of syllogistic form, leaving the way open toward a radically novel conception of inferential significance.
It is tempting to think that this entire debate is made irrelevant by the discoveries of Frege’s Begriffsschrift and the development of modern mathematical logic. Judging by the chronological sequence of events, this is certainly not the case. In 1904 Windelband was writing as one of the foremost German philosophers about the work of other prominent philosophers that the central issues in logic were to develop a new theory of judgments and to reconcile the discipline with Kant’s notion of a transcendental logic. While he and other philosophers are to a limited extent cognizant of developments in mathematical logic, they regard them as refinements in mode of presentation and training of logical acumen, but not as solutions to central logical problems. As late as 1930 Carnap complains that “the majority of philosophers have even now taken little cognizance of the new logic” (1930: 134). He suggests that the neo-Kantian philosophers who continue to elaborate theories of judgment are laboring on an anachronism. However, such a sweeping dismissal mistakes the substance of the views that make up logical radicalism. They are concerned with questions about what Trendelenburg defines as the “wider sense” of logic, such as the relation between logic and metaphysics and the possibility of objective cognition. Once we survey this wider ambit, a lasting and profound influence is more evident.
To begin with, it has been argued that both Frege and Carnap themselves owe much of their philosophical stance more or less directly to these authors. Sluga (1980) and Gabriel (1986, 1989a, b) show how the intellectual context of Lotze, Trendelenburg, and the neo-Kantians who follow them shapes Frege’s work. Similarly, Richardson (1992) and Friedman (1997) trace the origin of Carnap’s logical empiricism to neo-Kantian transcendental logics.
Some of this influence derives from early rejections of psychologism. Due to its focus on the mind’s cognitive faculties, Kant’s transcendental logic prompts a number of early psychologistic interpretations. Jakob Friedrich Fries, for example, claims that Kant’s critique is based on the inner experience of cognitive processes. In keeping with Kant’s claim that logic is a canon of the understanding, both Herbart and Drobisch oppose such tendencies by insisting that general logic studies how one ought to think, not how one actually thinks. Nevertheless, Herbart claims elsewhere that transcendental arguments are psychological and hence proposes a scientific psychology in place of a critical epistemology. Consequently he, too, becomes a target of Trendelenburg’s broad anti-psychologism (1846: 349); the latter was particularly diligent in detecting elements of psychology in the arguments of Hegel’s dialectic. Lotze, too, is rigorously anti-psychologistic and he uses his notion of validity to mark a clean separation between the logical and the actual.
However, the central legacy of this period of logic-criticism derives from its novel conception of just what a philosophical logic is supposed to do. In rejecting the formality of logic, the logical radicals search for ways to capture its proper content. As they attempt to reconcile a notion of the cognitive import of logic with a Kantian understanding of the structure of cognition, they in effect redefine the scope and nature of theoretical philosophy. They create a discipline at the area of overlap between transcendental philosophy, critical epistemology, the logic of scientific cognition, and the theory of scientific inquiry. While post-Kantian logical radicalism does not give us modern mathematical logic, it does give rise to the conception of philosophy as the theory of the sciences, a conception that continues to shape theoretical philosophy today.
Carnap, Rudolf (1930) “The Old and the New Logic,” trans. Isaac Levi, in A. J. Ayer (ed.) (1959) Logical Positivism, New York: Free Press.
De Morgan, Augustus (1847) Formal Logic: The Calculus or Inference, Necessary and Probable, London: Taylor & Walton.
Drobisch, Moritz (1836) Neue Darstellung der Logik nach ihren einfachsten Verhältnissen. Leipzig: Leopold Voß.
Fischer, Kuno (1852) Logik und Metaphysik oder Wissenschaftslehre, Heidelberg; repr., edited by Hans-Georg Gadamer, Heidelberg: Manutius Verlag, 1997.
—— (1865) System der Logik und Metaphysik oder Wissenschaftslehre, 2nd rev. edn. Heidelberg.
Friedman, Michael (1997) “Overcoming Metaphysics: Carnap and Heidegger,” in Ronald Giere and Alan Richardson (eds) Origins of Logical Empiricism, Minnesota Studies in Philosophy of Science, vol. 16, Minneapolis, MN: University of Minnesota Press, pp. 45–79.
Gabriel, Gottfried (1986) “Frege als Neukantianer,” Kant-Studien 77: 84–101.
—— (1989a) “Lotze und die Entstehung der modernen Logik bei Frege,” introduction to reprint of Lotze (1874a).
—— (1989b) “Objektivität: Logik und Erkenntnistheorie bei Lotze und Frege,” introduction to reprint of Lotze (1874b).
Hegel, G. W. F. (1813) Wissenschaft der Logik: Die Lehre vom Wesen; repr. Werke, vol. 6, Frankfurt: Suhrkamp, 1986, pp. 13–240.
—— (1816) Wissenschaft der Logik: Die Subjektive Logik; repr. Werke, vol. 6, Frankfurt: Suhrkamp, 1986, pp. 241–573.
—— (1830) Enzyklopädie der Philosophischen Wissenschaften im Grundrisse; repr., edited by F. Nicolin and O. Pöggeler, Hamburg: Meiner, 1959.
—— (1831) Wissenschaft der Logik: Die Objective Logik; repr. Werke, vol. 5, Frankfurt: Suhrkamp, 1986.
Herbart, Johann Friedrich (1813) Lehrbuch zur Einleitung in die Philosophie, Königsberg: A. W. Unzer; repr. Hamburg: Meiner, 1993.
Jevons, William Stanley (1881) Elementary Lessons in Logic: Deductive and Inductive, London: Macmillan.
Kant, Immanuel (1781) Kritik der Reinen Vernunft, edited by R. Schmidt. Hamburg: Meiner, 1993; trans., P. Guyer and A. Wood, as Critique of Pure Reason, Cambridge: Cambridge University Press, 1997.
—— (1800) Logik, edited by C. Jäsche, in Wilhelm Weischedel (ed.) Schriften zur Metaphysik und Logik, Frankfurt: Suhrkamp, 1996; edited, trans. Young, J. Michael, as Lectures on Logic, Cambridge: Cambridge University Press, 1992.
Käufer, Stephan (2005) “Hegel to Frege: Concepts and Conceptual Content in Nineteenth-Century Logic,” History of Philosophy Quarterly 22: 259–80.
Lotze, Rudolf Hermann (1843) Logik, Leipzig: Weidmann’sche Buchhandlung.
—— (1874a) Logik, bk 1: Vom Denken, Leipzig: S. Hirzel; repr., edited by G. Gabriel, Hamburg: Meiner, 1989.
—— (1874b) Logik, bk 3: Vom Erkennen, Leipzig: S. Hirzel; repr., edited by G. Gabriel, Hamburg: Meiner, 1989.
MacFarlane, John (2000) What Does It Mean to Say That Logic Is Formal? PhD diss., University of Pittsburgh.
—— (2002) “Frege, Kant, and the Logic in Logicism,” Philosophical Review 111: 25–65.
Martin, Wayne (2003) “Nothing More or Less than Logic: General Logic, Transcendental Philosophy, and Kant’s Repudiation of Fichte’s Wissenschaftslehre,” Topoi 22: 29–39.
—— (2006) Theories of Judgment: Psychology, Logic, Phenomenology, Cambridge: Cambridge University Press.
Richardson, Alan (1992) “Logical Idealism and Carnap’s Construction of the World,” Synthese 93: 59–92.
Riehl, Alois (1877) “Die Englische Logik der Gegenwart,” Vierteljahrsschrift für wissenschaftliche Philosophie 1: 51–80.
Sigwart, Christoph (1873) Logik, vol. 1: Die Lehre vom Urteil, vom Begriff und vom Schluss. Tübingen: H. Laupp.
Sluga, Hans (1980) Gottlob Frege, London: Routledge.
Trendelenburg, Friedrich Adolf (1843) Die Logische Frage in Hegel’s System: Zwei Streitschriften, Leibzig: F. A. Brockhaus.
—— (1846) Geschichte der Kategorienlehre, Berlin: G. Bethge.
—— (1857) “Ueber Leibnizens Entwurf einer allgemeinen Charakteristik”; repr. Trendelenburg (1867), pp. 1–47.
—— (1862) Logische Untersuchungen, vol. 1, 3rd edn, Leipzig; repr. Hildesheim: Georg Olms, 1964.
—— (1867) Historische Beiträge zur Philosophie, vol. 3, Berlin: G. Bethge.
—— (1870) Logische Untersuchungen vol. 2, 3rd edn, Leipzig; repr. Hildesheim: Georg Olms, 1964.
Ueberweg, Friedrich (1857) System der Logik und Geschichte der Logischen Lehren, Bonn: Adolph Marcus.
Ulrici, Hermann (1855) A review of George Boole, An Investigation of the Laws of Thought, Zeitschrift für Philosophie und philosophische Kritik 27: 273–91.
Windelband, Wilhelm (1904) “Logik,” in Wilhelm Windelband (ed.) Die Philosophie im Beginn des zwanzigsten Jahrhunderts: Festschrift für Kuno Fischer, vol. 1, Heidelberg: Carl Winter.
Kuno Fischer, Anti-Trendelenburg: Eine Gegenschrift (Jena: Hermano Dabis, 1870), contains Fischer’s response to Trendelenburg’s “loophole” argument. Detailed treatment of mathematical and symbolic logic in Germany during this period, with a thorough bibliography can be found in Volker Peckhaus, Logik, Mathesis universalis und allgemeine Wissenschaft: Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert (Berlin: Akademie Verlag, 1997). Friedrich Adolf Trendelenburg, “Über eine Lücke in Kants Beweis von der ausschließenden Subjektivität des Raumes und der Zeit,” in Historische Beiträge zur Philosophie, vol. 3 (Berlin: G. Bethge, 1867), contains Trendelenburg’s “loophole” argument that space may be both objective and subjective; and Kuno Fischer und sein Kant (Leipzig: S. Hirzel, 1869), Trendelenburg’s vitriolic attack on Fischer’s challenge to his loophole argument.