19th Century Logic between Philosophy and
Mathematics
Volker Peckhaus
Institut für Philosophie
der
Universität Erlangen-Nürnberg
Bismarckstr. 1, D-91054 Erlangen
E-mail: vrpeckha@phil.uni-erlangen.de
1 Introduction
Doubt could be expressed that a special section on late 19th century
mathematics, or, more specifically, on Victorian mathematics, was an appropriate
place for a lecture on 19th century logic. Most 19th century scholars would have
been of the opinion that philosophers are responsible for research on logic. On
the other hand, the history of late 19th century logic indicates clearly a very
dynamic development instigated not by philosophers, but by mathematicians. The
central feature of this development was the emergence of what has been called
the "new logic'', "mathematical logic'', "symbolic logic'', or, since 1904,
"logistics''. This new logic came from Great Britain, and was created by
mathematicians in the second half of the 19th century, finally becoming a
mathematical subdiscipline in the early 20th century. This development is, thus,
at the heart of Victorian mathematics.
The 100th anniversary of the death of Charles L. Dodgson, i. e., Lewis
Carroll (1832-1898) was the motivation behind this special section on late 19th
century mathematics given at the Commonwealth University in Ottawa which has
just commemorated its 150th anniversary. Carroll's well-known books on logic,
The Game of Logic of 1887 and Symbolic Logic of
1896 of which a fourth edition had already appeared in 1897, were
written "to be of real service to the young, and to be taken up, in
High Schools and in private families, as a valuable addition of their stock of
healthful mental recreations'' (Carroll 1896, xiv). They were meant "to
popularize this fascinating subject,'' as Carroll wrote in the preface
of the fourth edition of Symbolic Logic (ibid.). But, astonishingly
enough, in both books there is no definition of the term "logic''. Given the
broad scope of these books the title "Symbolic Logic'' of the second book should
at least have been explained.
Maybe the idea of symbolic logic was so widely spread at the end of the 19th
century in Great Britain that Carroll regarded a definition as simply
unnecessary. Some further observations support this thesis. They concern a
remarkable interest by the general public in symbolic logic, after the death of
the creator of the algebra of logic, George Boole, in 1864.
Recalling some standard 19th century definitions of logic as, e.g., the art
and science of reasoning (Whately) or the doctrine giving the normative rules of
correct reasoning (Herbart), it should not be forgotten that mathematical or
symbolic logic was not set up from nothing. It arose from the old
philosophical collective discipline logic. The standard presentations
of the history of logic ignore the relationship between the philosophical and
mathematical side of its development; they sometimes even deny that there has
been any development of philosophical logic at all. Take for example William and
Martha Kneale's programme in their eminent The Development of Logic.
They wrote (1962, iii): "But our primary purpose has been to record the
first appearances of these ideas which seem to us most important in the logic of
our own day,'' and these are the ideas leading to mathematical logic.
Another example is J. M. Bochenski's assessment of "modern classical logic''
which he scheduled between the 16th and the 19th century. It was for him a
noncreative period in logic which can therefore justly be ignored in the problem
history of logic (1956, 14). According to Bochenski classical logic was
only a decadent form of this science, a dead period in its development (ibid.,
20).
Such assessments show that the authors adhered to the predominant views on
logic of our time, i. e. actual systems of mathematical or symbolic logic. As a
consequence, they have not been able to give reasons for the final divorce
between philosophical and mathematical logic, because they have ignored the seed
from which mathematical logic has emerged. Following Bochenski's view Carl B.
Boyer presented a consistent periodization of the development of logic (Boyer
1968, 633): "The history of logic may be divided, with some slight
degree of oversimplification, into three stages: (1) Greek logic, (2) Scholastic
logic, and (3) mathematical logic.'' Note Boyer's "slight degree of
oversimplification'' which enabled him to skip 400 years of logical development
and ignore the fact that Kant's transcendental logic, Hegel's metaphysics and
Mill's inductive logic were called "logic'', too.
In discussing the relationship between the philosophical and the mathematical
development of logic, at least the following questions will be answered:
- What were the reasons for the philosophers' lack of interest in formal
logic?
- What were the reasons for the mathematicians' interest in logic?
- What did "logic reform'' mean in the 19th century? Were the systems of
mathematical logic initially regarded as contributions to a reform of logic?
- Was mathematical logic regarded as art, as science or as both?
This paper focuses not only on the situation in Britain, but also on the
development in Germany. This needs some justification in a symposium on
Victorian mathematics. British logicians regarded Germany as the logical
paragon. John Venn can be regarded as a chief witness. He deplored, in the
second edition of his Symbolic Logic of 1894, the lack of a traditions
in logic in Great Britain which caused problems in creating the collection of
books on logic for the Cambridge University Library (1894, 533):
At the time when I commenced the serious study of Symbolic Logic
many of the most important works which bore on the subject were not to be found
in any of those great libraries in this country to which one naturally refers in
the first place, and could therefore only be obtained by purchase from abroad.
[...] I suppose that the almost entire abandonment of Logic as a serious
academic study, for so many years in this country at least, had prevented the
formation of those private professorial libraries, the frequent appearance of
which in the market has kept the second-hand booksellers' shops in Germany so
well supplied with works on this subject.
It should be stressed, however, that when speaking of German logic Venn
wasn't referring to contemporary German logical sytems, but to the great 18th
century rationalistic precursors of the British algebra of logic beginning with
Gottfried Wilhelm Leibniz and ending with the Swiss, Johann Heinrich Lambert.
In the following sections surveys are given of the philosophical and
mathematical contexts in which the new logic emerged in Great Britain and
Germany. The strange collaboration of mathematics and philosophy in promoting
the new systems of logic will be discussed, and finally answers to the four
questions already posed will be given.
2 Contexts
2.1 The Philosophical Context in Great Britain
The development of the new logic started in 1847, completely independent of
earlier anticipations, e.g. by the German rationalist Gottfried Wilhelm Leibniz
(1646-1716) and his followers (cf. Peckhaus 1994a; 1997, ch.
5). In that year the British mathematician George Boole (1815-1864) published
his pamphlet The Mathematical Analysis of Logic (Boole 1847).
Boole mentioned that it was the struggle for priority concerning the
quantification of the predicate between the Edinburgh philosopher William
Hamilton (1788-1856) and the London mathematician Augustus De Morgan (1806-1871)
which encouraged this study. Hence, he referred to a startling philosophical
discussion which indicated a vivid interest in formal logic in Great Britain.
This interest was, however, a new interest, not even 20 years old. One can even
say that neglect of formal logic could be regarded as a characteristic feature
of British philosophy up to 1826 when Richard Whately (1787-1863) published his
Elements of Logic.1In his preface Whately added an extensive
report on the languishing research and education in formal logic in England. He
complained (1826, xv) that only very few students of the University of
Oxford became good logicians and that
by far the greater part pass through the University without
knowing any thing of all of it; I do not mean that they have not learned by rote
a string of technical terms; but that they understand absolutely nothing
whatever of the principles of the Science.
Thomas Lindsay, the translator of Friedrich Ueberweg's important System
der Logik und Geschichte der logischen Lehren (1857, translation
1871), was very critical of the scientific qualities of Whately's book,
but he, nevertheless, emphasized its outstanding contribution for the
renaissance of formal logic in Great Britain (Lindsay 1871, 557):
Before the appearance of this work, the study of the science had
fallen into universal neglect. It was scarcely taught in the universities, and
there was hardly a text-book of any value whatever to be put into the hands of
the students.
One year after the publication of Whately's book, George Bentham's An
Outline of a New System of Logic appeared (1827) which was to
serve as a commentary to Whately. Bentham's book was critically discussed by
William Hamilton in a review article published in the Edinburgh Review
(1833). With the help of this review Hamilton founded his
reputation as the "first logical name in Britain, it may be in the world.''2 Hamilton propagated a
revival of the Aristotelian scholastic formal logic without, however,
one-sidedly preferring the syllogism. His logical conception was focused on a
revision of the standard forms by quantifying the predicates of judgements.3 The controversy about
priority arose, when De Morgan, in a lecture "On the Structure of the
Syllogism'' (De Morgan 1846) given to the Cambridge Philosophical
Society on 9th November 1846, also proposed quantifying predicates. None had any
priority, of course. Application of the diagrammatic methods of the syllogism
proposed e. g., by the 18th century mathematicians and philosophers Leonard
Euler, Gottfried Ploucquet, and Johann Heinrich Lambert, presupposed
quantification of the predicate. The German psychologistic logician Friedrich
Eduard Beneke (1798-1854) suggested quantifying the predicate in his books on
logic of 1839 and 1842, the latter of which he sent to
Hamilton. In the context of this paper it is irrelevant to solve the priority
question. It is, however, important that a dispute of this extent arose at all.
It indicates there was new interest in research on formal logic.
This interest represented only one side of the effect released by Whately's
book. Another line of research stood in the direct tradition of Humean
empiricism and the philosophy of inductive sciences: the inductive logic of John
Stuart Mill (1806-1873), Alexander Bain (1818-1903) and others. Boole's logic
was in clear opposition to inductive logic. It was Boole's follower William
Stanley Jevons (1835-1882; cf. Jevons 1877-1878) who made this
opposition explicit.
Boole referred to the controversy between Hamilton and De Morgan, but this
influence should not be overemphasized. In his main work on the Laws of
Thought (1854) Boole went back to the logic of Aristotle by
quoting from the Greek original. This can be interpreted as indicating that the
influence of contemporary philosophical discussion was not as important as his
own words might suggest. In writing a book on logic he was doing philosophy, and
it was thus a matter of course that he related his results to the philosophical
discussion of his time. This does not mean, of course, that his thoughts were
really influenced by this discussion.
2.2 The Philosophical Context in Germany
It seems clear that, in regard to the 18th century dichotomy between German
and British philosophy represented by the philosophies of Kant and Hume,
Hamilton and Boole stood on the Kantian side. There are some analogies with the
situation in Germany, where philosophical discussion on logic after Hegel's
death was determined by the Kantian influence. In the preface to the second
edition of his Kritik der reinen Vernunft of 1787, Immanuel
Kant (1723-1804) wrote that logic has followed the safe course of a science
since earliest times. For Kant this was evident because of the fact that logic
had been prohibited from taking any step backwards from the time of Aristotle.
But he regarded it as curious that logic hadn't taken a step forward either (B
VIII). Thus, logic seemed to be closed and complete. Formal logic, in Kant's
terminology the analytical part of general logic, did not play a prominent rôle
in Kant's system of transcendental philosophy. In any case it was a negative
touchstone of truth, as he stressed (B 84). Georg Wilhelm Friedrich Hegel
(1770-1831) went further in denying any relevance of formal logic for philosophy
(Hegel 1812/13, I, Introduction, XV-XVII). Referring to Kant, he
maintained that from the fact that logic hadn't changed since Aristotle one
could infer that it needed a complete rebuilding (ibid., XV). Hegel created a
variant of logic as the foundational science of his philosophical system,
defining it as "the science of the pure idea, i.e., the idea in the abstract
element of reasoning '' (1830, 27). Hegelian logic thus coincides
with metaphysics (ibid., 34).
This was the situation when after Hegel's death philosophical discussion on
logic in Germany started. This discussion on logic reform stood under the label
of "the logical question'', a term coined by the Neo-Aristotelian Adolf
Trendelenburg (1802-1872). In 1842 he published a paper entitled "Zur Geschichte
von Hegel's Logik und dialektischer Methode'' with the subtitle "Die logische
Frage in Hegel's Systeme''. But what is the logical question according to
Trendelenburg? He formulated this question explicitly towards the end of his
article: "Is Hegel's dialectical method of pure reasoning a scientific
procedure?'' (1842, 414). In answering this question in the negative,
he provided the occasion of rethinking the status of formal logic within a
theory of human knowledge without, however, proposing a return to the old
(scholastic) formal logic. In consequence the term "the logical question'' was
subsequently used in a less specific way. Georg Leonard Rabus, the early
chronicler of the discussion on logic reform, wrote that the logical question
emerged from doubts concerning the justification of formal logic (1880,
1).
Although this discussion was clearly connected to formal logic, the
so-called reform did not concern formal logic. The reason was provided
by the Neo-Kantian Wilhelm Windelband who wrote in a brilliant survey on 19th
century logic (1904, 164):
It is in the nature of things that in this enterprise [i.e. the
reform of logic] the lower degree of fruitfulness and developability power was
on the side of formal logic. Reflection on the rules of the correct progress of
thinking, the technique of correct thinking, had indeed been brought to
perfection by former philosophy, presupposing a naive world view. What Aristotle
had created in a stroke of genius, was decorated with the finest filigree work
in Antiquity and the Middle Ages: an art of proving and disproving which
culminated in a theory of reasoning, and after this constructing the doctrines
of judgements and concepts. Once one has accepted the foundations, the safely
assembled building cannot be shaken: it can only be refined here and there and
perhaps adapted to new scientific requirements.
Windelband was very critical of English mathematical logic. Its
quantification of the predicate allows the correct presentation of extensions in
judgements, but it "drops hopelessly" the vivid sense of all judgements, which
tend to claim or deny a material relationship between subject or predicate. It
is "a logic of the conference table'', which cannot be used in the vivid life of
science, a "logical sport'' which has, however, its merits in exercising the
final acumen (ibid., 166-167).
The philosophical reform efforts concerned primarily two areas:
- the problem of a foundation of logic which itself was approached by
psychological and physiological means, leading to new discussion on the question
of priority between logic and psychology, and to various forms of psychologism
and anti-psychologism (cf. Rath 1994, Kusch 1995);
- the problem of logical applications focusing interest on the methodological
part of traditional logic. The reform of applied logic attempted to bring
philosophy in touch with the stormy development of mathematics and sciences of
the time.
Both reform procedures had a destructive effect on the shape of logic and
philosophy. The struggle with psychologism led to the departure of psychology
(especially in its new, experimental form) from the body of philosophy at the
beginning of the 20th century. Psychology became a new, autonomous scientific
discipline. The debate on methodology emerged with the creation of the
philosophy of science which was separated from the body of logic. The
philosopher's ignorance of the development of formal logic caused a third
departure: Part of formal logic was taken from the domain of the competence of
philosophy and incorporated into mathematics where it was instrumentalized for
foundational tasks.
2.3 The Mathematical Context in Great Britain
As mentioned earlier, the influence of the philosophical discussion on logic
in Great Britain on the emergence of the new logic should not be overemphasized.
Of greater importance were mathematical influences. Most of the new logicians
can be related to the so-called "Cambridge Network'' (Cannon 1978,
29-71), i. e. the movement which aimed at reforming British science and
mathematics which started at Cambridge. One of the roots of this movement was
the foundation of the Analytical Society in 1812 (cf. Enros 1983) by
Charles Babbage (1791-1871), George Peacock (1791-1858) and John Herschel
(1792-1871). In regard to mathematics Joan L. Richards called this act a
"convenient starting date for the nineteenth-century chapter of British
mathematical development'' (Richards 1988, 13). One of the first
achievements of the Analytical Society was a revision of the Cambridge Tripos by
adopting the Leibnitian notation for the calculus and abandoning the customary
Newtonian theory of fluxions: "the principles of pure D-ism in opposition to the
Dot-age of the University'' as Babbage wrote in his memoirs (Babbage
1864, 29). It may be assumed that this successful movement triggered
off by a change in notation might have stimulated a new or at least revived
interest in operating with symbols. This new research on the calculus had
parallels in innovative approaches to algebra which were motivated by the
reception of Laplacian analysis. Firstly the development of symbolical algebra
has to be mentioned. It was codified by George Peacock in his Treatise on
Algebra (1830) and further propagated in his famous report for the
British Association for the Advancement of Science (Peacock 1834,
especially 198-207). Peacock started by drawing a distinction between
arithmetical and symbolical algebra, which was, however, still based on the
common restrictive understanding of arithmetic as the doctrine of quantity. A
generalization of Peacock's concept can be seen in Duncan F. Gregory's
(1813-1844) "calculus of operations''. Gregory was most interested in
operations with symbols. He defined symbolical algebra as "the science
which treats of the combination of operations defined not by their nature, that
is by what they are or what they do, but by the laws of combinations to which
they are subject'' (1840, 208). In his much praised paper "On a General
Method in Analysis'' (1844) Boole made the calculus of operations the
basic methodological tool for analysis. However in following Gregory, he went
further, proposing more applications. He cited Gregory who wrote that a symbol
is defined algebraically "when its laws of combination are given; and that a
symbol represents a given operation when the laws of combination of the latter
are the same as those of the former'' (Gregory 1842, 153-154). It is
possible that a symbol for an arbitrary operation can be applied to the same
operation (ibid., 154). It is thus necessary to distinguish between arithmetical
algebra and symbolical algebra which has to take into account symbolical, but
non-arithmetical fields of application. As an example Gregory mentioned the
symbols a and +a. They are isomorphic in arithmetic, but in geometry they need
to be interpreted differently. a can refer to a point marked by a line whereas
the combination of the signs + and a additionally expresses the direction of the
line. Therefore symbolical algebra has to distinguish between the symbols a and
+a. Gregory deplored the fact that the unequivocity of notation didn't prevail
as a result of the persistence of mathematical practice. Clear notation was only
advantageous, and Gregory thought that our minds would be "more free from
prejudice, if we never used in the general science symbols to which definite
meanings had been appropriated in the particular science'' (ibid., 158).
Boole adopted this criticism almost word for word. In his Mathematical
Analysis of Logic of 1847 he claimed that the reception of symbolic algebra
and its principles was delayed by the fact that in most interpretations of
mathematical symbols the idea of quantity was involved. He felt that these
connotations of quantitative relationships were the result of the context of the
emergence of mathematical symbolism, and not of a universal principle of
mathematics (Boole 1847, 3-4). Boole read the principle of the
permanence of equivalent forms as a principle of independence from
interpretation in an "algebra of symbols''. In order to obtain further
affirmation, he tried to free the principle from the idea of quantity by
applying the algebra of symbols to another field, the field of logic. As far as
logic is concerned this implied that only the principles of a "true Calculus''
should be presupposed. This calculus is characterized as a "method resting upon
the employment of Symbols, whose laws of combination are known and general, and
whose results admit of a consistent interpretation'' (ibid., 4). He stressed
(ibid.):
It is upon the foundation of this general principle, that I
purpose to establish the Calculus of Logic, and that I claim for it a place
among the acknowledged forms of Mathematical Analysis, regardless that in its
objects and in its instruments it must at present stand alone.
Boole expressed logical propositions in symbols whose laws of combination are
based on the mental acts represented by them. Thus he attempted to establish a
psychological foundation of logic, mediated, however, by language. The central
mental act in Boole's early logic is the act of election used for building
classes. Man is able to separate objects from an arbitrary collection which
belong to given classes, in order to distinguish them from others. The symbolic
representation of these mental operations follows certain laws of combination
which are similar to those of symbolic algebra. Logical theorems can thus be
proven like mathematical theorems. Boole's opinion has of course consequences
for the place of logic in philosophy: "On the principle of a true
classification, we ought no longer to associate Logic and Metaphysics, but Logic
and Mathematics'' (ibid., 13).
Although Boole's logical considerations became increasingly philosophical
with time, aiming at the psychological and epistemological foundations of logic
itself, his initial interest was not to reform logic but to reform mathematics.
He wanted to establish an abstract view on mathematical operations without
regard to the objects of these operations. When claiming "a place among the
acknowledged forms of Mathematical Analysis'' (1847, 4) for the
calculus of logic, he didn't simply want to include logic in traditional
mathematics. The superordinate discipline was a new mathematics. This
is expressed in Boole's writing: "It is not of the essence of mathematics to be
conversant with the ideas of number and quantity'' (1854, 12).
2.4 The Mathematical Context in Germany
The results of this examination of the British situation at the time when the
new logic emerged-a reform of mathematics, with initially a lack of interest in
a reform of logic, by establishing an abstract view on mathematics which focused
not on mathematical objects, but on symbolic operations with arbitrary
objects-these results could be transferred to the situation in Germany without
any problem.
The most important representative of the German algebra of logic was the
mathematician Ernst Schröder (1841-1902) who was regarded as having completed
the Boolean period in logic (cf. Bochenski 1956, 314). In his first
pamphlet on logic, Der Operationskreis des Logikkalkuls
(1877), he presented a critical revision of Boole's logic of
classes, stressing the idea of the duality between logical addition and logical
multiplication introduced by William Stanley Jevons in 1864. In 1890 Schröder
started on the large project, his monumental Vorlesungen über die Algebra
der Logik (1890, 1891, 1895, 1905)
which remained unfinished although it increased to three volumes with four
parts, of which one appeared only posthumously. Contemporaries regarded the
first volume alone as completing the algebra of logic (cf. Wernicke
1891, 196).
Schröder's opinion concerning the question as to the end to which logic is
studied (cf. Peckhaus 1991, 1994b) can be drawn from an
autobiographical note, published in 1901 (and written in the third person), the
year before his death. It contains Schröder's own survey of his scientific aims
and results. Schröder divided his scientific production into three fields:
- A number of papers dealing with some of the current problems of his science.
- Studies concerned with creating an "absolute algebra,'' i. e., a general
theory of connections. Schröder stressed that such studies represent his "very
own object of research'' of which only little was published at that time.
- Work on the reform and development of logic.
Schröder wrote (1901) that his aim was
to design logic as a calculating discipline, especially to give
access to the exact handling of relative concepts, and, from then on, by
emancipation from the routine claims of spoken language, to withdraw any fertile
soil from "cliché'' in the field of philosophy as well. This should prepare the
ground for a scientific universal language that, widely differing from
linguistic efforts like Volapük [a universal language like Esperanto very
popular in Germany at that time], looks more like a sign language than like a
sound language.
Schröder's own division of his fields of research shows that he didn't
consider himself a logician: His "very own object of research'' was "absolute
algebra,'' and in respect to its basic problems and fundamental assumptions
similar to modern abstract or universal algebra. What was the connection between
logic and algebra in Schröder's research? From the passages quoted one could
assume that they belong to two separate fields of research, but this is not the
case. They were intertwined in the framework of his heuristic idea of a general
science. In his autobiographical note he stressed (1901):
The disposition for schematizing, and the aspiration to condense
practice to theory advised Schröder to prepare physics by perfecting
mathematics. This required deepening-as of mechanics and geometry-above all of
arithmetic, and subsequently he became by the time aware of the necessity for a
reform of the source of all these disciplines, logic.
Schröder's universal claim becomes obvious. His scientific efforts served to
provide the requirements to found physics as the science of material nature by
"deepening the foundations,'' to quote a famous metaphor later used by David
Hilbert (1918, 407) in order to illustrate the objectives of his
axiomatic programme. Schröder regarded the formal part of logic that can be
formed as a "calculating logic,'' using a symbolical notation, as a model
of formal algebra that is called "absolute'' in its last state of
development.
But what is "formal algebra''? The theory of formal algebra "in the narrowest
sense of the word'' includes "those investigations on the laws of algebraic
operations [ ...] that refer to nothing but general numbers in an unlimited
number field without making any presuppositions concerning its nature''
(1873, 233). Formal algebra therefore prepares "studies on the most
varied number systems and calculating operations that might be invented for
particular purposes'' (ibid.).
It has to be stressed that Schröder wrote his early considerations on formal
algebra and logic without any knowledge of the results of his British
predecessors. His sources were the textbooks of Martin Ohm, Hermann Günther Graß
mann, Hermann Hankel and Robert Graß mann. These sources show that Schröder was
a representative of the tradition of German combinatorial algebra and algebraic
analysis (cf. Peckhaus 1997, ch. 6).
Like the British tradition, but independent of it, the German algebra of
logic was connected to new trends in algebra. It differed from its British
counterpart in its combinatorial approach. In both traditions, algebra of logic
was invented within the enterprise to reform basic notions of mathematics which
led to the emergence of structural abstract mathematics. The algebraists wanted
to design algebra as "pan-mathematics'', i. e. as a general discipline embracing
all mathematical disciplines as special cases. The independent attempts in Great
Britain and Germany were combined when Schröder learned about the existence of
Boole's logic in late 1873, early 1874. Finally he enriched the Boolean class
logic by adopting Charles S. Peirce's theory of quantification and adding a
logic of relatives according to the model of Peirce and De Morgan.
The main interest of the new logicians was to utilize logic for mathematical
and scientific purposes, and it was only in a second step, but nevertheless an
indispensable consequence of the attempted applications, that the reform of
logic came into the view. What has been said of the representatives of the
algebra of logic also holds for the proponents of competing logical systems such
as Gottlob Frege or Giuseppe Peano. They wanted to use logic in their quest for
mathematical rigour, something questioned by the stormy development in
mathematics.
3 Accepting the New Logic
Although created by mathematicians, the new logic was widely ignored by
fellow mathematicians. In Germany Schröder was only known as the algebraist of
logic, and regarded as rather exotic. George Boole was respected by British
mathematicians, but his ideas concerning an algebraical representation of the
laws of thought received very little published reaction. He shared this fate
with Augustus De Morgan, the second major figure of symbolic logic at that time.
In 1864, Samuel Neil, the early chronicler of British mid 19th century logic,
expressed his thoughts about the reasons for this negligible reception: "De
Morgan is esteemed crotchety, and perhaps formalizes too much. Boole demands
high mathematic culture to follow and to profit from'' (1864, 161). One
should add that the ones who had this culture were usually not interested in
logic.
The situation changed after George Boole's death in 1864. In the following
comments only some ideas concerning the reasons for this new interest are hinted
at. In particular the rôles of William Stanley Jevons and Alexander Bain are
stressed which exemplify "the strange collaboration of mathematics and
philosophy in promoting the new systems of logic'' mentioned in the
introduction.
3.1 William Stanley Jevons
A broader international reception of Boole's logic began when William Stanley
Jevons made it the starting point for his influential Principles of Science
of 1874. He used his own version of the Boolean calculus
introduced in his Pure Logic of 1864. Among his revisions were the
introduction of a simple symbolical representation of negation and the
definition of logical addition as inclusive "or''. He also changed the
philosophy of symbolism (1864, 5):
The forms of my system may, in fact, be reached by divesting his
[Boole's] of a mathematical dress, which, to say the least, is not essential to
it. The system being restored to its proper simplicity, it may be inferred, not
that Logic is a part of Mathematics, as is almost implied in Professor Boole's
writings, but that the Mathematics are rather derivatives of Logic. All the
interesting analogies or samenesses of logical and mathematical reasoning which
may be pointed out, are surely reversed by making Logic the dependent of
Mathematics.
Jevons' interesting considerations on the relationship between mathematics
and logic representing an early logicistic attitude will not be discussed.
Similar ideas can be found not only in Gottlob Frege's work, but also in that of
Hermann Rudolf Lotze and Ernst Schröder. In the context of this paper, it is
relevant that Jevons abandoned mathematical symbolism in logic, an attitude
which was later taken up by John Venn. Jevons attempted to free logic from the
semblance of being a special mathematical discipline. He used the symbolic
notation only as a means of expressing general truths. Logic became a tool for
studying science, a new language providing symbols and structures. The change in
notation brought the new logic closer to the philosophical discourse of the
time. The reconciliation was supported by the fact that Jevons formulated his
Principles of Science as a rejoinder to John Stuart Mill's A System
of Logic of 1843, at that time the dominating work on logic and
the philosophy of science in Great Britain. Although Mill called his logic A
System of Logic Ratiocinative and Inductive, the deductive parts played
only a minor rôle, used only to show that all inferences, all proofs and the
discovery of truths consisted of inductions and their interpretations. Mill
claimed to have shown "that all our knowledge, not intuitive, comes to us
exclusively from that source'' (Mill 1843, Bk. II, ch. I, § 1). Mill
concluded that the question as to what induction is, is the most important
question of the science of logic, "the question which includes all others.'' As
a result the logic of induction covers by far the largest part of this work, a
subject which we would today regard as belonging to the philosophy of science.
Jevons defined induction as a simple inverse application of deduction. He
began a direct argument with Mill in a series of papers entitled "Mill's
Philosophy Tested'' (1877/78). This discourse proved that symbolic
logic could be of importance not only for mathematics, but also for philosophy.
Another effect of the attention caused by Jevons was that British algebra of
logic was able to cross the Channel. In 1877, Louis Liard (1846-1917), at that
time professor at the Faculté de lettres at Bordeaux and a friend of Jevons,
published two papers on the logical systems of Jevons and Boole (Liard
1877a, 1877b). In 1878 he added a booklet entitled
Les logiciens anglais contemporaines which ran into five editions until
1907, and was translated into German in 1880. Although Herman Ulrici
had published a first German review of Boole's Laws of Thought as early
as 1855, the knowledge of British symbolic logic was conveyed primarily
by Alois Riehl, then professor at the University of Graz, in Astria. He
published a widely read paper "Die englische Logik der Gegenwart'' ("English
contemporary logic'') in 1877 which reported mainly Jevons' logic and
utilized it in a current German controversy on the possibility of scientific
philosophy.
3.2 Alexander Bain
Finally a few words on Alexander Bain (1818-1903): This Scottish philosopher
was an adherent of Mill's logic. Bain's Logic, first published in 1870,
had two parts, the first on deduction and the second on induction. He made
explicit that "Mr Mill's view of the relation of Deduction and Induction is
fully adopted'' (1870, I, iii). Obviously he shared the "[ ...] general
conviction that the utility of the purely Formal Logic is but small; and that
the rules of Induction should be exemplified even in the most limited course of
logical discipline'' (ibid., v). The minor rôle of deduction showed up in Bain's
definition " Deduction is the application or extension of Induction to
new cases '' (40).
Despite his reservations about deduction, Bain's Logic was quite
important for the reception of symbolic logic because of a chapter of 30 pages
entitled "Recent Additions to the Syllogism.'' In this chapter the contributions
of William Hamilton, Augustus De Morgan and George Boole were introduced.
Presumably many more people became acquainted with Boole's algebra of logic
through Bain's report than through Boole's own writings. One example is Hugh
MacColl (1837-1909), the pioneer of the calculus of propositions (statements)
and of modal logic. He created his ideas independently of Boole, eventually
realizing the existence of the Boolean calculus by means of Bain's report. Even
in the early parts of his series of papers "The Calculus of Equivalent
Statements'' he quoted from Bain's presentation when discussing Boole's logic
(MacColl 1877/78). In 1875 Bain's logic was translated into French, in
1878 into Polish. Tadeusz Batóg and Roman Murawski (1996) have shown
that it was Bain's presentation which motivated the first Polish algebraist of
logic, Stanisaw Pi atkiewicz (1848-?) to begin his research on symbolic logic.
The remarkable collaboration of mathematics and philosophy can be seen in the
fact that a broader reception of symbolic logic commenced only when its
relevance for the philosophical discussion of the time came to the fore.
4 Conclusions
Finally, these are the answers to the initial questions:
- What were the reasons for the philosophers' lack of interest in formal
logic?
In Germany philosophers shared Kant's opinion that formal logic was a
completed field of knowledge. They were interested primarily in the foundations
and application of logic. In Great Britain there was hardly any vivid logical
tradition. Philosophy was predominated by empiricist conceptions. New systems of
formal logic therefore had difficulties in gaining a footing in the
philosophical discussion.
- What were the reasons for the mathematicians' interest in logic?
Foundational problems and problems in grasping new mathematical objects
forced some mathematicians to look intuitively at the logical foundations of
their subject. The interest in formal logic was thus a result of the dynamic
development of late 19th century mathematics. One should not assume, however,
that this was a general interest. Most mathematicians did not (and still do not)
care about foundations.
- How did the mathematicians' logical activities fit into the reform of
logic conceptions of the time?
In Germany in the second half of the 19th century, Logic reform meant
overcoming the Hegelian identification of logic and metaphysics. In Great
Britain it meant enlarging the scope of the syllogism or elaborating the
philosophy of science. Mathematicians were initially interested in utilizing
logic for mathematical means, or they used it as a language for structuring and
symbolizing extra-mathematical fields. Applications were e. g. the foundation of
mathematics (Boole, Schröder, Frege), the foundation of physics (Schröder), the
preservation of rigour in mathematics (Peano), the theory of probabilities
(Boole, Venn), the philosophy of science (Jevons), the theory of human
relationships (Alexander Macfarlane), and juridical questions. The
mathematicians' preference for the organon aspect of formal logic seems to be
the point of deviation between mathematicians and the philosophers who were not
interested in elaborating logic as a tool.
- Was mathematical logic regarded as art or as science?
From the applicational interest it follows that it was mainly regarded as an
art. The scientific aspect grew, however, with the insight into the power of
logical calculi. Nevertheless, in an institutional sense the new logic was
established only in the beginning of the 20th century as an academic subject,
i. e. as an institutionalized domain of science.
Bibliography
Babbage, Charles
1864 Passages from the Life of a Philosopher, Longman, Green, Longman, Roberts, & Green:
London; repr. Gregg: Westmead 1969.
Bain, Alexander
1870 Logic, 2 vols, pt. 1: Deduction, pt. 2: Induction, Longmans, Green, & Co.:
London
Batóg, Tadeusz/ Murawski, Roman
1996 "Stanisaw Pi atkiewicz and the Beginnings of Mathematical Logic in
Poland,'' Historia Mathematica 23, 68-73.
Beneke, Friedrich Eduard
1839 Syllogismorum analyticorum origines et ordinem naturalem, Mittler: Berlin.
-
1842 System der Logik als Kunstlehre des Denkens, 2 vols., F. Dümmler:
Berlin.
Bentham, George
1827 An Outline of a New System of Logic. With a Critical Examination of Dr.
Whately's "Elements of Logic'', Hunt and Clark: London; repr. Thoemmes:
Bristol 1990.
Bochenski, Joseph Maria
1956 Formale Logik, Alber: Freiburg/München (= Orbis Academicus, III,
2), 41978.
Boole, George
1844 "On a General Method in Analysis,'' Philosophical Transactions of
the Royal Society of London for the Year MDCCCXLIV, pt. 1, 225-282.
-
1847 The Mathematical Analysis of Logic. Being an Essay Towards a Calculus
of Deductive Reasoning, Macmillan, Barclay, and Macmillan: Cambridge/George
Bell: London; repr. Basil Blackwell: Oxford 1951.
-
1854 An Investigation of the Laws of Thought, on which are Founded the Mathematical
Theories of Logic and Probabilities, Walton & Maberly: London; repr. Dover:
New York [ca. 1951].
Boyer, Carl B.
1968 A History of Mathematics, John Wiley & Sons: New York/London/Sydney.
Cannon, Susan Faye
1978 Science in Culture: The Early Victorian Period, Dawson Scientific History
Publications: New York.
Carroll, Lewis
1887 The Game of Logic, MacMillan: London; reprinted in Carroll 1958.
-
1896 Symbolic Logic, MacMillan: London; reprinted in Carroll 1958.
-
1958 Mathematical Recreations of Lewis Carroll. Symbolic Logic and The Game
of Logic (both Books Bound as One), Dover Publications and Berkeley Enterprises:
New York.
De Morgan, Augustus
1846 "On the Syllogism I. On the Structure of the Syllogism, and on the
Application of the Theory of Probabilities to Questions of Argument and Authority,''
Transactions of the Cambridge Philosophical Society 8, 379-408.
Enros, Philip C.
1983 "The Analytical Society (1812-1813): Precursor of the Renewal of
Cambridge Mathematics,'' Historia Mathematica 10, 24-47.
Gregory, Duncan Farquharson
1840 "On the Real Nature of Symbolical Algebra,'' Transactions of the
Royal Society of Edinburgh 14, 208-216.
-
1842 "On a Difficulty in the Theory of Algebra,'' The Cambridge Mathematical
Journal 3, 153-159.
Hamilton, William
1833 "Logic. In Reference to the Recent English Treatises on that Science,''
Edinburgh Review 66 (April 1833), 194-238; again in Hamilton, Discussions on
Philosophy and Literature, Education and University Reform. Chiefly from the
Edinburgh Review; Corrected, Vindicated, Enlarged, in Notes and Appendices,
Longman, Brown, Green and Longmans: London/Maclachlan and Stewart: Edinburgh
1851, 16-174.
-
1859-1866 Lectures on Metaphysics and Logic, 4 vols., eds. H. L. Mansel/J.
Veitch, William Blackwood and Sons: Edinburgh/London.
Heath, Peter
1966 "Introduction,'' in Augustus De Morgan, On the Syllogism and Other
Logical Writings, ed. Peter Heath, Routledge & Kegan Paul: London (= Rare
Masterpieces of Philosophy and Science), vii-xxxi.
Hegel, Georg Wilhelm Friedrich
1812/1813 Wissenschaft der Logik, vol. 1: Die objective Logik, Johann Leonhard
Schrag: Nürnberg; critical edition: Hegel, Wissenschaft der Logik. Erster
Band. Die objektive Logik (1812/1813), eds. Friedrich Hogemann/Walter Jaeschke,
Felix Meiner: Hamburg 1978 (= Hegel, Gesammelte Werke, vol. 11).
-
1830 Encyclopädie der philosophischen Wissenschaften im Grundrisse. Zum
Gebrauch seiner Vorlesungen. Dritte Ausgabe, Oß wald'scher Verlag: Heidelberg.
Critical edition Hegel, Enzyklopädie der philosophischen Wissenschaften
im Grundrisse (1830), eds. Wolfgang Bonsiepen/Hans-Christian Lucas, Felix Meiner:
Hamburg 1992 (= Hegel, Gesammelte Werke, vol. 20).
Hilbert, David
1918 "Axiomatisches Denken,'' Mathematische Annalen 78, 405-415.
Jevons, William Stanley
1864 Pure Logic or the Logic of Quality apart from Quantity with Remarks on
Boole's System and the Relation of Logic and Mathematics, E. Stanford: London;
repr. in Jevons 1890, 3-77.
-
1874 The Principles of Science. A Treatise on Logic and Scientific Method,
2 vols., Macmillan and Co.: London [New York 1875].
-
1877-1878 "John Stuart Mill's Philosophy Tested,'' The Contemporary Review
31 (1877/78), 167-82, 256-275; 32 (1878), 88-99; again in Jevons 1890, 137-172.
-
1890 Pure Logic and Other Minor Works, eds. Robert Adamson/Harriet A. Jevons,
Macmillan and Co.: London/New York; repr. Thoemmes Press: Bristol 1991.
Kant, Immanuel
1787 Critik der reinen Vernunft, 2nd ed., Johann Friedrich Hartknoch: Riga;
Akademie-Ausgabe, vol. 3.
Kneale, William/ Kneale, Martha
1962 The Development of Logic, Clarendon Press: Oxford; Repr. 1986.
Kusch, Martin
1995 Psychologism. A Case Study in the Sociology of Philosophical Knowledge,
Routledge: London/New York (= Psychological Issues in Science).
Liard, Louis
1877a "Un nouveau système de logique formelle. M. Stanley Jevons''
Revue philosophique de la France et de l'Étranger 3, 277-293.
-
1877b "La logique algébrique de Boole'', Revue philosophique de
la France et de l'Étranger 4, 285-317.
-
1878 Les logiciens anglais contemporains, Germer Baillière: Paris, 51907.
-
1880 Die neuere englische Logik, ed. J[ohannes] Imelmann, Denicke's Verlag:
Berlin, 21883.
Lindsay, Thomas M
1871 "On Recent Logical Speculation in England," in Ueberweg 1871,
557-590.
MacColl, Hugh
1877/78 "The Calculus of Equivalent Statements (Second Paper),'' Proceedings
of the London Mathematical Society 9, 177-186.
Mill, John Stuart
1843 A System of Logic, Ratiocinative and Inductive. Being a Connected View
of the Principles of Evidence and the Methods of Scientific Investigation,
2 vols., J. W. Parker: London.
Neil, Samuel
1864 S. N., "John Stuart Mill,'' The British Controversialist and Literary
Magazine n. s. (1864), 161-173, 241-256.
Peacock, George
1830 A Treatise on Algebra, J. & J.J. Deighton: Cambridge/G.F. & J.
Rivington: London.
-
1834 "Report on the Recent Progress and Present State of Certain Branches
of Analysis,'' Report of the Third Meeting of the British Association for the
Advancement of Science held at Cambridge in 1833, John Murray: London, 185-352.
Peckhaus, Volker
1991 "Ernst Schröder und die `pasigraphischen Systeme' von Peano
und Peirce," Modern Logic 1, no. 2/3 (Winter 1990/91), 174-205.
-
1994a "Leibniz als Identifikationsfigur der britischen Logiker des 19.
Jahrhunderts," in VI. Internationaler Leibniz-Kongreß. Vorträge
I. Teil, Hannover, 18.-22.7.1994, Gottfried-Wilhelm-Leibniz-Gesellschaft: Hannover,
589-596.
-
1994b "Wozu Algebra der Logik? Ernst Schröders Suche nach einer universalen
Theorie der Verknüpfungen," Modern Logic 4, 357-381.
-
1997 Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die
Wiederentdeckung der formalen Logik im 19. Jahrhundert, Akademie Verlag: Berlin
(= Logica Nova).
Rabus, Georg Leonhard
1880 Die neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen
und Die logische Frage, Deichert: Erlangen.
Rath, Matthias
1994 Der Psychologismus in der deutschen Philosophie, Karl Alber Verlag: Freiburg/München.
Richards, Joan L.
1988 Mathematical Visions. The Pursuit of Geometry in Victorian England, Academic
Press: Boston etc.
Riehl, Alois
1877 "Die englische Logik der Gegenwart,'' Vierteljahrsschrift für
wissenschaftliche Philosophie 1, 51-80.
Risse, Wilhelm
1973 Bibliographia Logica. Verzeichnis der Druckschriften zur Logik mit Angabe
ihrer Fundorte, vol. 2: 1801-1969, Olms: Hildesheim/New York (=Studien und
Materialien zur Geschichte der Philosophie ; 1).
Schröder, Ernst
1873 Lehrbuch der Arithmetik und Algebra für Lehrer und Studirende, vol.
1 [no further volumes published]: Die sieben algebraischen Operationen, B.
G. Teubner: Leipzig.
-
1877 Der Operationskreis des Logikkalkuls, Teubner: Leipzig; repr. as "special
edition'', Wissenschaftliche Buchgesellschaft: Darmstadt 1966.
-
1890 Vorlesungen über die Algebra der Logik (exakte Logik), vol. 1, B.
G. Teubner: Leipzig; repr. Schröder 1966.
-
1891 Vorlesungen über die Algebra der Logik (exakte Logik), vol. 2, pt.
1, B. G. Teubner: Leipzig; repr. Schröder 1966.
-
1895 Vorlesungen über die Algebra der Logik (exakte Logik), vol. 3, pt.
1: Algebra und Logik der Relative, B. G. Teubner: Leipzig; repr. Schröder
1966.
-
1905 Vorlesungen über die Algebra der Logik (exakte Logik), vol. 2, pt.
2, ed. Karl Eugen Müller, B. G. Teubner: Leipzig; repr. Schröder
1966.
-
1901 "Grossherzoglich Badischer Hofrat Dr. phil. Ernst Schröder[,]
ord. Professor für Mathematik an der Technischen Hochschule in Karlsruhe
i. Baden,'' in Geistiges Deutschland. Deutsche Zeitgenossen auf dem Gebiete
der Literatur, Wissenschaften und Musik, Adolf Eckstein: Berlin-Charlottenburg.
-
1966 Vorlesungen über die Algebra der Logik (exakte Logik), ["second
edition''], 3 vols., Chelsea: Bronx, N.Y.
Trendelenburg, Friedrich Adolf
1842 "Zur Geschichte von Hegel's Logik und dialektischer Methode. Die
logische Frage in Hegel's Systeme. Eine Auffoderung [sic!] zu ihrer wissenschaftlichen
Erledigung'', Neue Jenaische Allgemeine Literatur-Zeitung 1, no. 97, 23 April
1842, 405-408; no. 98, 25 April 1842, 409-412; no. 99, 26 April 1842, 413-414;
separately published in Trendelenburg Die logische Frage in Hegel's System.
Zwei Streitschriften, Brockhaus: Leipzig 1843.
Ueberweg, Friedrich
1857 System der Logik und Geschichte der logischen Lehren, Adolph Marcus: Bonn.
-
1871 System of Logic and History of Logical Doctrines, translated from the
German, with notes and appendices by Thomas M. Lindsay, Longmans, Green, and
Co.: London; repr. Thoemmes Press: Bristol 1993.
Ulrici, Hermann
1855 Review of Boole 1854, Zeitschrift für Philosophie und philosophische
Kritik 27, 273-291.
Van Evra, James
1984 "Richard Whately and the Rise of Modern Logic'', History and Philosophy
of Logic 5, 1-18.
Venn, John
1894 Symbolic Logic, 2nd ed., "revised and rewritten,'' Macmillan & Co.:
London; repr. Chelsea Publishing: Bronx, New York 1971.
Wernicke, Alexander
1891 Review of Schröder 1891, Deutsche Litteraturzeitung 12, col. 196-197.
Whately, Richard
1826 Elements of Logic. Comprising the Substance of the Article in the Encyclopaedia
Metropolitana: with Additions, &c., J. Mawman: London.
Windelband, Wilhelm
1904 "Logik'', in Windelband (ed.), Die Philosophie im Beginn des zwanzigsten
Jahrhunderts. Festschrift für Kuno Fischer, vol. 1, Carl Winter: Heidelberg,
163-186.
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