Logic from Kant to Russell

Laying the Foundations for Analytic Philosophy

Sandra Lapointe

• Publisher: Routledge
• Year: 2019
• eBook ISBN: 9781351182225
• Edition: 1
• Topic: Philosophy
• Subtopic: Logic in Philosophy

The scope and method of logic as we know it today eminently reflect the ground-breaking developments of set theory and the logical foundations of mathematics at the turn of the 20th century. Unfortunately, little effort has been made to understand the idiosyncrasies of the philosophical context that led to these tremendous innovations in the 19thcentury beyond what is found in the works of mathematicians such as Frege, Hilbert, and Russell. This constitutes a monumental gap in our understanding of the central influences that shaped 19th-century thought, from Kant to Russell, and that helped to create the conditions in which analytic philosophy could emerge.

The aim of Logic from Kant to Russell is to document the development of logic in the works of 19th-century philosophers. It contains thirteen original essays written by authors from a broad range of backgrounds—intellectual historians, historians of idealism, philosophers of science, and historians of logic and analytic philosophy. These essays question the standard narratives of analytic philosophy's past and address concerns that are relevant to the contemporary philosophical study of language, mind, and cognition. The book covers a broad range of influential thinkers in 19th-century philosophy and analytic philosophy, including Kant, Bolzano, Hegel, Herbart, Lotze, the British Algebraists and Idealists, Moore, Russell, the Neo-Kantians, and Frege.

1 The Logicians of Kant’s School

(Or, If Logic Has Been Complete Since Aristotle, What’s Left For a Logician To Do?)

Jeremy Heis

One of the most infamous claims in Kant’s Critique of Pure Reason concerns the completeness of formal logic:

[S]ince the time of Aristotle [formal logic] has not had to go a single step backwards, unless we count the abolition of a few dispensable subtleties or the more distinct determination of its presentation, which improvements belong more to the elegance than to the security of that science. What is further remarkable about logic is that until now it has also been unable to take a single step forward, and therefore seems to all appearance to be finished and complete.¹

(Bviii)

This infamous claim was subject to severe criticism in the early part of the 19th century from figures as diverse as Bolzano, De Morgan, Hegel, and Fries.² Of course, it is not surprising that figures such as these would be critical of Kant’s claim, since their conception of the scope and content of logic differed fundamentally from his. What’s more surprising is that there arose starting already in the 1790s and into the 19th century a group of logicians who self-consciously thought of themselves as orthodox Kantians and who wrote extensive and original works in formal logic. Not only did these logicians show an affinity with Kant’s conception of logic, but their contemporaries and later logicians thought of them as forming a kind of school – what Friedrich Ueberweg, in his history of logic in the 19th century, calls the “logic of Kant’s school.”³ Ueberweg himself, who was very critical of the school, listed as members “Jakob, Kiesewetter, Hoffbauer, Maas, Krug, etc.”

What made these logicians ‘Kantian’ is that they held to Kant’s conception of logic as formal. This distinctive conception of logic had four parts. First, formal logic was distinguished from transcendental logic. Second, formal logic was considered the science of thinking, not of cognizing. Third, it was asserted that formal logic abstracts from all relation to an object. Fourth, formal logic was held to be independent of psychology and metaphysics.

Logicians in this school included Kant’s students, such as Schultz and Kiesewetter, but also later logicians who wrote after Kant’s death, such as Krug and Esser. Here is an undoubtedly incomplete list of some of the logicians in this Kantian school, along with some of their major works.⁴

1. Schultz, Johann. 1789. Prüfung der Kantischen Critik der reinen Vernunft. Vol. 1.
2. Jakob, Ludwig Heinrich. 1791. Grundriss der allgemeinen Logik. 2nd ed.
3. Kiesewetter, J.G.C. 1791. Grundriss einer allgemeinen Logik nach Kantischen Grundsätzen.
4. Hoffbauer, J.C. 1792. Analytik der Urtheile und Schlüße.
5. Maass, J.G.E. 1793. Grundriss der Logik.
6. Kant, I. 1800. Logik. Ed. Jäsche.
7. Krug, Wilhelm Traugott. 1806. Denklehre oder Logik.
8. Herbart, Johann Friedrich. 1813. Lehrbuch zur Einleitung in die Philosophie. Zweiter Abschnitt. Die Logik.
9. Esser, Wilhelm. 1823. System der Logik.
10. Drobisch, Moritz. 1836. Neue Darstellung der Logik. ²1851, ³1863.
11. Hamilton, William. 1860. Lectures on Logic. (Delivered 1837–8.)
12. Mansel, Henry Longueville. 1851. Prolegomena Logica: An Inquiry into the Psychological Character of Logical Processes.

Not surprisingly, most of these logicians were German and wrote in the few decades after the publication of the first Critique. However, the last two logicians on the list were British, and that requires some explanation. But first some background about the history of logic in Britain in the early 19th century. In 1826, Archbishop Whatley ([1826] 1866) restored the fortunes of logic in Britain by arguing that logic is a science, not an art. As a science, logic contains a law, namely the dictum de omni et nullo, that explains the validity of the figures of the syllogism. And since logic is not an art, it’s not meant to be a tool of discovery, nor a “medicine of the mind.” Whately used this conception of logic to defend logic against its detractors, who pointed to the alleged sterility and lack of utility of the study of logic. A few years later, William Hamilton defended Whately’s view of logic, but argued that in fact Whately’s point of view had already been expressed 50 years earlier by Kant in the Critique of Pure Reason. Hamilton ([1833] 1861) argued that Kant had already maintained that formal logic is a science; that the logical laws explain the validity of forms of inference; that, as a canon and not an organon, it is not a tool of discovery; and lastly, that it was not designed to cure errors in reasoning. These points about Kant’s conception of formal logic were captured in Kant’s distinguishing formal logic from special logic and applied logic, respectively. Hamilton, having defended Kant’s conception of logic, in the following years lectured extensively on formal logic using the texts of the German Kantian logicians Krug and Esser. These lectures were formally published many decades later, after a wide circulation, in 1860 (cf. Hamilton [1860] 1874). Among Hamilton’s students was Henry Mansel, who wrote Prolegomena Logica in 1851, defending a kind of Kantian conception of formal logic.

The very existence of a Kantian school seems surprising, since it’s not clear, from our point of view, what exactly the Kantian school took itself to be doing. After all, if logic has been complete since Aristotle, what’s left for a logician to do? In this chapter I want to answer this question. I’m going to identify four representative questions or issues that the logicians in this school considered and debated. In each case I’ll argue that this was a question that a reasonable reader of Kant might think Kant had left unsettled, and then I’ll give a representative sample of the kinds of answers that logicians in this school gave to these questions. My goal in this chapter is not to be exhaustive, but to introduce the reader to the kinds of questions, debates, and philosophy that were done by logicians in the Kantian school – a school that at least in the last century or so has been more or less absent from the historiography of logic. Since my goal in this chapter is to give the reader a sense of the questions asked by the Kantian logicians and to give the reader a sense of the kinds of answers that they gave, my goal in this chapter will not be to evaluate the answers given, either as interpretations of Kant or as philosophically defensible positions in their own right. Though I will occasionally editorialize, my goal here is simply to present the various views and not to evaluate them. I’ll identify four questions that the Kantian logicians attempted to answer.

1. 1 What is the relationship between formal logic and analyticity?
2. 2 What are the logical laws and how are they related?
3. 3 What does it mean to say that logic is formal?
4. 4 Are all concepts formed through comparison, reflection, and abstraction?

I’ll address each of these questions in turn in the following four sections of this chapter.

What Is the Relationship between Formal Logic and Analyticity?

It’s clear from Kant’s characterizations of analytic judgments that every analytic judgment is meant to be knowable through logical laws alone. At various points he asserts that analytic judgments are “thought through identity” (A7/B10), that “their certainty rests on identity of concepts” (JL, §36), and that they are knowable through the principle of contradiction alone.

For, if the judgment is analytic, whether negative or affirmative, its truth can always be adequately known in accordance with the principle of contradiction. The reverse of that which as concept is contained and is thought in the knowledge of the object, is always rightly denied. But since the opposite of the concept would contradict the object, the concept itself must necessarily be affirmed of it. The principle of contradiction must therefore be recognized as being the universal and completely sufficient principle of all analytic knowledge.

(A151/B190–1)

Since the principles of identity and non-contradiction are logical laws, and are sufficient for knowing all analytic judgments, it follows of course that, on Kant’s conception, all analytic judgments are derivable from logical laws alone. But is the converse true? Is it true that every judgment knowable through logical laws alone is an analytic judgment?

Kant does not say so, at least not directly. This may seem surprising. At least since Frege, analytic judgments have often been characterized as those that are provable from logical laws plus definitions. From this conception of analyticity, it of course follows that all logical propositions are themselves analytic.

The problem becomes, in fact, that of finding the proof of the proposition, and of following it up right back to the primitive truths. If, in carrying out this process, we come only on general logical laws and on definitions, then the truth is an analytic one.

(Frege [1884] 1950, §3)

However, this conception of analyticity not only differs from Kant’s, but was presented a full century after the Critique of Pure Reason. What’s more, this conception of analyticity seems to have been novel with Frege. I have been unable to find any source before Frege who asserts that analytic truths are those provable from logical laws plus definitions.

Of course, absence of evidence is not evidence of absence. But there is a further piece of convincing evidence here. Couturat in his Les Principes des Mathématiques, in 1905, includes an appendix on Kant’s philosophy of mathematics and his conception of analyticity. Couturat surveys various conceptions of analyticity and asserts that Frege’s definition was original with him. He writes:

One should therefore say, in order to remain true to the spirit, if not the letter of Kantian doctrine: a judgment is analytic, if it can be proved from definitions and laws of logic alone.

(Couturat 1905, 246)

In a footnote Couturat cites Frege’s Grundlagen and a paper by Gerardus Heymans from later in the 1880s (cf. Heymans 1889; cf. also Heymans 1886). Concerning Frege’s Grundlagen, he writes:

This latter work is one of the few in which the Kantian theory of arithmetic is handled with the greatest energy and depth…But it is also the only one which gets no mention in the bibliographies related to Kant. How useless are bibliographies!

(Couturat 1905, 246)

Frege’s characterization of analyticity, he admits, is not true to the letter of the Kantian definition. He gives two arguments for why post-Kantian developments were necessary in order to make this friendly amendment possible. First, it had to be clearly recognized that there are many other logical laws besides the principle of non-contradiction (PNC), and that there are other logical relations among concepts besides conceptual containment. Second, “what is thought in a concept” needed to be de-psychologized. Understood psychologically, this is a notion that is relative to a subject and a time; but, he argues, the definition of a concept is an objective matter, not relative to a subject. It is only these later developments in logic and philosophy, Couturat argues, that allow Frege’s definition to seem like the friendly amendment to Kant’s definition that it is.

One can dispute whether or not Frege’s definition really is an improvement over Kant’s, but I think the historical point is convincing: the conception of analyticity that makes it immediate that analytic truths are logical truths and vice versa did not arise until almost a century after Kant’s own writing. Given the fact, then, that Kant never asserts that all logical truths are analytic, the question then becomes whether or not his followers took him that way. Was it obvious to those in the Kantian school that all logical truths are analytic? Now, it’s true that some logical laws are clearly analytic in Kant’s view. In particular, he argues that the PNC itself is analytic (cf. A152). But is it the case that all of the logical laws are analytic? Or are all the propositions of pure general logic analytic?

Among the first two generations of Kantian logicians, which included Kant’s students and those writing in the first few decades of the 19th century, I found only one philosopher who asserts that all of formal logic is analytic. This is not in a logic text but in a passing remark in Schultz’s Prüfung from the 1780s. Schultz writes:

In general logic there is a pure part, which consists in propositions that are obviously a priori, but which are not synthetic, but rather are all analytic.

(Schultz 1789, 45)

This is as clear a statement as you’re going to find. However, Schultz’s argument for this claim unfortunately shows a serious misinterpretation of Kant’s view. The passage continues

Insofar as it determines the necessary rules of all thinking theoretically, without regard to its actual execution or application in particular cases, the understanding is concerned neither with any objects, nor with our sensibility, but merely only with itself, and the analysis of its form; consequently, the theoretical part of general logic as mere analysis of our form of understanding, is an entirely pure science, to which nothing empirical must be mixed, and whose rules are obviously a priori, but not synthetic, but rather analytic.

(Schultz 1789, 45–6, emph. added)

Schultz is certainly picking up on a real theme in Kant’s writings on logic. Kant believes that formal logic results from an analysis, namely, of the actions [Handlungen] of our faculty of the understanding.

Since merely formal logic, so conceived, abstracts from all content of cognition (whether it be pure or empirical), and concerns itself merely with the form of thinking (of discursive cognition) in general, it can also include in its analytic part, a canon for reason, the form of which has its secure precept, into which there can be a priori insight through mere analys...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright
• Table of Contents
• Introduction
• 1 The Logicians of Kant’s School
• 2 Kant’s Excessive Tenderness for Things in the World, and Hegel’s Diatheism
• 3 Hegel’s Conception of Thinking in His Logics
• 4 Bolzano on Logic in Mathematics and Beyond
• 5 Laws of Thought and Laws of Logic after Kant
• 6 Platonism in Lotze and Frege
• 7 Demystifying Cohen’s Logik
• 8 The Logic in Dedekind’s Logicism
• 9 What Russell Meant When He Called Moore a Logician
• 10 Sigwart, Russell and the Emergence of Scientific Philosophy
• 11 Kant and Formalism: Hilbert, Russell and Whitehead
• Contributors
• Index