George Boole

George Boole (1815–1864) was an English mathematician and a
founder of the algebraic tradition in logic. He worked as a
schoolmaster in England and from 1849 until his death as professor of
mathematics at Queen’s University, Cork, Ireland. He revolutionized
logic by applying methods from the then-emerging field of symbolic
algebra to logic. Where traditional (Aristotelian) logic relied on
cataloging the valid syllogisms of various simple forms, Boole’s
method provided general algorithms in an algebraic language which
applied to an infinite variety of arguments of arbitrary
complexity. These results appeared in two major works,
The Mathematical Analysis of Logic (1847)
and
The Laws of Thought (1854).

• 1. Life and Work
• 2. The Context and Background of Boole’s Work In Logic
• 3. The Mathematical Analysis of Logic (1847)
• 4. The Laws of Thought (1854)
• 5. Later Developments
  • 5.1 Objections to Boole’s Algebra of Logic
  • 5.2. Modern Reconstruction of Boole’s System
• 6. Boole’s Methods
  • 6.1 The Three Methods of Argument Analysis Used by Boole in LT
  • 6.2. Boole’s General Method for Primary Propositions
  • 6.3. Boole’s General Method for Secondary Propositions
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Life and Work

George Boole was born November 2, 1815 in Lincoln, Lincolnshire,
England, into a family of modest means, with a father who was
evidently more of a good companion than a good breadwinner. His father
was a shoemaker whose real passion was being a devoted dilettante in
the realm of science and technology, one who enjoyed participating in
the Lincoln Mechanics’ Institution; this was essentially a community
social club promoting reading, discussions, and lectures regarding
science. It was founded in 1833, and in 1834 Boole’s father became the
curator of its library. This love of learning was clearly inherited by
Boole. Without the benefit of an elite schooling, but with a
supportive family and access to excellent books, in particular from
Sir Edward Bromhead, FRS, who lived only a few miles from Lincoln,
Boole was able to essentially teach himself foreign languages and
advanced mathematics.

Starting at the age of 16 it was necessary for Boole to find gainful
employment, since his father was no longer capable of providing for
the family. After 3 years working as a teacher in private schools,
Boole decided, at the age of 19, to open his own small school in
Lincoln. He would be a schoolmaster for the next 15 years, until 1849
when he became a professor at the newly opened Queen’s University in
Cork, Ireland. With heavy responsibilities for his parents and
siblings, it is remarkable that he nonetheless found time during the
years as a schoolmaster to continue his own education and to start a
program of research, primarily on differential equations and the
calculus of variations connected with the works of Laplace and
Lagrange (which he studied in the original French).

There is a widespread belief that Boole was primarily a
logician—in reality he became a recognized mathematician well
before he had penned a single word about logic, all the while running
his private school to care for his parents and siblings. Boole’s
ability to read French, German and Italian put him in a good position
to start serious mathematical studies when, at the age of 16, he read
Lacroix’s Calcul Différentiel, a gift from his friend
Reverend G.S. Dickson of Lincoln. Seven years later, in 1838, he would
write his first mathematical paper (although not the first to be
published), “On certain theorems in the calculus of
variations,” focusing on improving results he had read in
Lagrange’s Méchanique Analytique.

In early 1839 Boole travelled to Cambridge to meet with the young
mathematician Duncan F. Gregory (1813–1844) who was the editor
of the Cambridge Mathematical Journal
(CMJ)—Gregory had founded this journal in 1837 and
edited it until his health failed in 1843 (he died in early 1844, at
the age of 30). Gregory, though only 2 years beyond his degree in
1839, became an important mentor to Boole. With Gregory’s support,
which included coaching Boole on how to write a mathematical paper,
Boole entered the public arena of mathematical publication in
1841.

Boole’s mathematical publications span the 24 years from 1841 to 1864,
the year he died from pneumonia. If we break these 24 years into three
segments, the first 6 years (1841–1846), the second 8 years
(1847–1854), and the last 10 years (1855–1864), we find
that his work on logic was entirely in the middle 8 years.

In his first 6 career years, Boole published 15 mathematical papers,
all but two in the CMJ and its 1846 successor, The
Cambridge and Dublin Mathematical Journal. He wrote on standard
mathematical topics, mainly differential equations, integration and the
calculus of variations. Boole enjoyed early success in using the new
symbolical method in analysis, a method which took a differential
equation, say:

d²y/dx² − dy/dx − 2y = cos(x),

and wrote it in the form Operator(y) = cos(x).
This was (formally) achieved by letting:

D = d/dx, D² = d²/dx²,
etc.

leading to an expression of the differential equation as:

(D² − D − 2) y = cos(x).

Now symbolical algebra came into play by simply treating the operator
D² − D − 2 as though it were an ordinary
polynomial in algebra. Boole’s 1841 paper “On the Integration of
Linear Differential Equations with Constant Coefficients” gave a
nice improvement to Gregory’s method for solving such differential
equations, an improvement based on a standard tool in algebra, the use
of partial fractions.

In 1841 Boole also published his first paper on invariants, a paper
that would strongly influence Eisenstein, Cayley, and Sylvester to
develop the subject. Arthur Cayley (1821–1895), the future
Sadlerian Professor in Cambridge and one of the most prolific
mathematicians in history, wrote his first letter to Boole in 1844,
complimenting him on his excellent work on invariants. He became a
close personal friend, one who would go to Lincoln to visit and stay
with Boole in the years before Boole moved to Cork, Ireland. In 1842
Boole started a correspondence with Augustus De Morgan
(1806–1871) that initiated another lifetime friendship.

In 1843 the schoolmaster Boole finished a lengthy paper on
differential equations, combining an exponential substitution and
variation of parameters with the separation of symbols method. The
paper was too long for the CMJ—Gregory, and later De
Morgan, encouraged him to submit it to the Royal Society. The first
referee rejected Boole’s paper, but the second recommended it for the
Gold Medal for the best mathematical paper written in the years
1841–1844, and this recommendation was accepted. In 1844 the
Royal Society published Boole’s paper and awarded him the Gold
Medal—the first Gold Medal awarded by the Society to a mathematician.
The next year Boole read a paper at the annual meeting of the British
Association for the Advancement of Science at Cambridge in June 1845.
This led to new contacts and friends, in particular William Thomson
(1824–1907), the future Lord Kelvin.

Not long after starting to publish papers, Boole was eager to
find a way to become affiliated with an institution of higher learning.
He considered attending Cambridge University to obtain a degree, but
was counselled that fulfilling the various requirements would likely
seriously interfere with his research program, not to mention the
problems of obtaining financing. Finally, in 1849, he obtained a
professorship in a new university opening in Cork, Ireland. In the
years he was a professor in Cork (1849–1864) he would
occasionally inquire about the possibility of a position back in
England.

The 8 year stretch from 1847 to 1854 starts and ends with Boole’s
two books on mathematical logic. In addition Boole published 24 more
papers on traditional mathematics during this period, while only one
paper was written on logic, that being in 1848. He was awarded an
honorary LL.D. degree by the University of Dublin in 1851, and this was
the title that he used beside his name in his 1854 book on logic.
Boole’s 1847 book, Mathematical Analysis of Logic, will be
referred to as MAL; the 1854 book, Laws of Thought,
as LT.

During the last 10 years of his career, from 1855 to 1864, Boole
published 17 papers on mathematics and two mathematics books, one on
differential equations and one on difference equations. Both books were
considered state of the art and used for instruction at Cambridge. Also
during this time significant honors came in:

1857	Fellowship of the Royal Society
1858	Honorary Member of the Cambridge Philosophical Society
1859	Degree of DCL, honoris causa from Oxford

Unfortunately his keen sense of duty led to his walking through a
rainstorm in late 1864, and then lecturing in wet clothes. Not long
afterwards, on December 8, 1864 in Ballintemple, County Cork, Ireland,
he died of pneumonia, at the age of 49. Another paper on mathematics
and a revised book on differential equations, giving considerable
attention to singular solutions, were published post mortem.

The reader interested in an excellent and thorough account of
Boole’s personal life is referred to Desmond MacHale’s George
Boole, His Life and Work, 1985, a source to which this article is
indebted.

• 1815 — Birth in Lincoln, England
• 1830 — His translation of a Greek poem printed in a local paper
• 1831 — Reads Lacroix’s Calcul Différentiel
• Schoolmaster
• 1834 — Opens his own school
• 1835 — Gives public address on Newton’s achievements
• 1838 — Writes first mathematics paper
• 1839 — Visits Cambridge to meet Duncan Gregory, editor of the
  Cambridge Mathematical Journal (CMJ)
• 1841 — First four mathematical publications (all in the
  CMJ)
• 1842 — Initiates correspondence with Augustus De Morgan
  — they become lifelong friends
• 1844 — Correspondence with Cayley starts (initiated by
  Cayley) — they become lifelong friends
• 1844 — Gold Medal from the Royal Society for a paper on
  differential equations
• 1845 — Gives talk at the Annual Meeting of the British
  Association for the Advancement of Science, and meets William Thompson
  (later Lord Kelvin) — they become lifelong friends
• 1847 — Publishes Mathematical Analysis of Logic
• 1848 — Publishes his only paper on the algebra of logic
• Professor of Mathematics
• 1849 — Accepts position as Professor of Mathematics at the new Queen’s University in Cork, Ireland
• 1851 — Honorary Degree, LL.D., from Trinity College, Dublin
• 1854 — Publishes Laws of Thought
• 1855 — Marriage to Mary Everest, niece of George Everest,
  Surveyor-General of India after whom Mt. Everest is named
• 1856 — Birth of Mary Ellen Boole
• 1857 — Elected to the Royal Society
• 1858 — Birth of Margaret Boole
• 1859 — Publishes Differential Equations; used as a
  textbook at Cambridge
• 1860 — Birth of Alicia Boole, who will coin the word
  “polytope”
• 1860 — Publishes Difference Equations ; used as a
  textbook at Cambridge
• 1862 — Birth of Lucy Everest Boole
• 1864 — Birth of daughter Ethel Lilian Boole, who would write
  The Gadfly, an extraordinarily popular book in Russia after
  the 1917 revolution
• 1864 — Death from pneumonia, Cork, Ireland

2. The Context and Background of Boole’s Work In Logic

To understand how Boole developed, in such a short time, his
impressive algebra of logic, it is useful to understand the broad
outlines of the work on the foundations of algebra that had been
undertaken by mathematicians affiliated with Cambridge University in
the 1800s prior to the beginning of Boole’s mathematical publishing
career. An excellent reference for further reading connected to this
section is the annotated sourcebook From Kant to Hilbert by
Ewald (1996).

The 19th century opened in England with mathematics in the doldrums.
The English mathematicians had feuded with the continental
mathematicians over the issues of priority in the development of the
calculus, resulting in the English following Newton’s notation, and
those on the continent following that of Leibniz. One of the obstacles
to overcome in updating English mathematics was the fact that the great
developments of algebra and analysis had been built on dubious
foundations, and there were English mathematicians who were quite vocal
about these shortcomings. In ordinary algebra, it was the use of
negative numbers and imaginary numbers that caused concern. The first
major attempt among the English to clear up the foundation problems of
algebra was George Peacock’s Treatise on Algebra, 1830 (a
second edition appeared as two volumes, 1842/1845). He divided the
subject into two parts, the first part being arithmetical
algebra, the algebra of the positive numbers (which did not permit
operations like subtraction in cases where the answer would not be a
positive number). The second part was symbolical algebra,
which was governed not by a specific interpretation, as was the case
for arithmetical algebra, but by laws. In symbolical algebra there were
no restrictions on using subtraction, etc.

The terminology of algebra was somewhat different in the 19th
century from what is used today. In particular they did not use the
word “variable”; the letter x in an expression like 2x + 5
was called a symbol, hence the name “symbolical
algebra”. In this article a prefix will sometimes be added, as in
number symbol or class symbol, to emphasize the
intended interpretation of a symbol.

Peacock believed that in order for symbolical algebra to be a useful
subject its laws had to be closely related to those of arithmetical
algebra. For this purpose he introduced his principle of the
permanence of equivalent forms, a principle connecting results in
arithmetical algebra to those in symbolical algebra. This principle has
two parts:

(1) General results in arithmetical algebra belong to the laws
of symbolical algebra.

(2) Whenever an interpretation of a result of symbolical algebra
made sense in the setting of arithmetical algebra, the result would
give a correct result in arithmetic.

A fascinating use of algebra was introduced in 1814 by
François-Joseph Servois (1776–1847) when he
tackled differential equations by separating the differential operator
part from the subject function part, as described in an example given
above. This application of algebra captured the interest of Duncan
Gregory who published a number of papers on the method of the
separation of symbols, that is, the separation into operators
and objects, in the CMJ. He also wrote on the foundation of
algebra, and it was Gregory’s foundation that Boole embraced, almost
verbatim. Gregory had abandoned Peacock’s principle of the permanence
of equivalent forms in favor of two simple laws. Unfortunately these
laws fell far short of what is required to justify even some of the
most elementary results in algebra. In “On the foundation of
algebra,” 1839, the first of four papers on this topic by De
Morgan that appeared in the Transactions of the Cambridge
Philosophical Society, one finds a tribute to the separation of
symbols in algebra, and the claim that modern algebraists usually
regard the symbols as denoting operators (e.g., the derivative
operation) instead of objects like numbers. The footnote:

Professor Peacock is the first, I believe, who
distinctly set forth the difference between what I have called the
technical and the logical branches of algebra.

credits Peacock with being the first to separate (what are now called)
the syntactic and the semantic aspects of algebra. In the second
foundations paper (in 1841) De Morgan proposed what he considered to be
a complete set of eight rules for working with symbolical algebra.

3. The Mathematical Analysis of Logic (1847)

Boole’s path to logic fame started in a curious way. In early 1847 he
was stimulated to launch his investigations into logic by a trivial
but very public dispute between De Morgan and the Scottish philosopher
Sir William Hamilton (not to be confused with the Irish mathematician
Sir William Rowan Hamilton). This dispute revolved around who deserved
credit for the idea of quantifying the predicate (e.g.,
“All A is all B,” “All A
is some B,” etc.). Within a few months Boole had
written his 82 page monograph, Mathematical Analysis of
Logic, giving an algebraic approach to Aristotelian logic. (Some
say that this monograph and De Morgan’s book Formal Logic
appeared on the same day in November 1847. )

The Introduction chapter starts with Boole reviewing the symbolical
method. The second chapter, First Principles, lets the symbol 1
represent the universe which “comprehends every conceivable class
of objects, whether existing or not.” Capital letters X, Y, Z,
… denoted classes. Then, no doubt heavily influenced by his very
successful work using algebraic techniques on differential operators,
and consistent with De Morgan’s 1839 assertion that algebraists
preferred interpreting symbols as operators, Boole introduced the
elective symbol x corresponding to the class X, the elective symbol y
corresponding to Y, etc. The elective symbols denoted election
operators—for example the election operator red when applied to
a class would elect (select) the red items in the class. (One can
simply replace the elective symbols by their corresponding class
symbols and have the interpretation used in LT in 1854.)

Then Boole introduced the first operation, the
multiplication xy of elective symbols. The standard notation
xy for multiplication also had a standard meaning for operators (for
example, differential operators), namely one applied y to an object and
then x is applied to the result. (In modern terminology, this is the
composition of the two operators.) Thus, as pointed out by
Hailperin (1986), it seems likely that this established notation
convention handed Boole his definition of multiplication of elective
symbols as composition of operators. When one switches to using classes
instead of elective operators, as in LT, the corresponding
multiplication of two classes results in their intersection.

The first law in MAL was the distributive law
x(u+v) = xu + xv,
where Boole said that u+v corresponded to dividing
a class into two parts. This was the first mention of addition. On
p. 17 Boole added the commutative law xy
= yx and the idempotent
law x² = x (which Boole called
the index law). Once these two laws of Gregory were secured,
Boole believed he was entitled to fully employ the ordinary algebra of
his time, and indeed one sees Taylor series and Lagrange multipliers
in MAL. The law of idempotent class symbols,
x² = x, was different from the two
fundamental laws of symbolical algebra—it only applied to the
individual elective symbols, not in general to compound terms that one
could build from these symbols. For example, one does not in general
have (x+y)² = x+y in
Boole’s system since, by ordinary algebra with idempotent class
symbols, this would imply 2xy = 0, and
then xy = 0, which would force x
and y to represent disjoint classes. But it is not the case
that every pair of classes is disjoint.

Boole focused on Aristotelian logic in MAL, with its 4 types
of categorical propositions and an open-ended collection of
hypothetical propositions. In the chapter Of Expression and
Interpretation, Boole said that necessarily the class not-X
is expressed by 1−x. This is the first appearance of
subtraction. Then he gave equations to express the categorical
propositions (see in
Section 6.2
below). The first to be expressed was All X is Y,
for which he used xy = x, which he then
converted into x(1−y) = 0. This was the first
appearance of 0 in MAL—it was not introduced as the
symbol for the empty class. Indeed the empty class did not appear
in MAL. Evidently an equation E = 0 performed the
role of a predicate in MAL, asserting that the class denoted
by E simply did not exist. (In LT, the empty class
would be denoted by 0.) Boole went beyond the foundations of
symbolical algebra that Gregory had used in 1844—he added De
Morgan’s 1841 single rule of inference, that equivalent operations
performed upon equivalent subjects produce equivalent results.

In the chapter on conversions, such as Conversion by Limitation—All X is Y, therefore Some Y is X—Boole found the
Aristotelian classification defective in that it did not treat
complements, such as not-X, on the same footing as the named classes X,
Y, Z, etc. With his extended version of Aristotelian logic in mind
(giving not-X equal billing), he gave (p. 30) a set of three
transformation rules which allowed one to construct all valid
two-line categorical arguments (providing you accepted the
unwritten convention that simple names like X and not-X denoted
non-empty classes).

Regarding syllogisms, Boole did not care for the Aristotelian
classification into Figures and Moods as they seemed rather arbitrary
and not particularly suited to the algebraic setting. His first
observation was that syllogistic reasoning was just an exercise in
elimination, namely the middle term was eliminated to give the
conclusion. Elimination was well known in the ordinary algebraic theory
of equations, so Boole simply borrowed a standard result to use in his
algebra of logic. If the premises of a syllogism involved the classes
X, Y, and Z, and one wanted to
eliminate y, then Boole put the equations for the two
premises in the form:

ay + b = 0

a′y + b′ = 0.

The result of eliminating y in ordinary algebra gave the
equation

ab′ − a′b   =  0,

and this is what Boole used in his algebra of logic to derive the
conclusion equation. Although the conclusion is indeed correct,
unfortunately this elimination result would be too weak for his algebra
of logic if he only used his primary translations into equations. In
the cases where both premises were translated as equations of the form
ay = 0, the elimination conclusion turned out to be 0 = 0, even though
Aristotelian logic might demand a non-trivial conclusion. This
was the reason Boole introduced the alternative equational translations
of categorical propositions, to be able to derive all of the valid
Aristotelian syllogisms (see p. 32). With this convention, of using
secondary translations when needed, it turned out that the only cases
that led to 0 = 0 were those for which the premises did not belong to a
valid syllogism.

Boole emphasized that when a premise about X and Y
is translated into an equation involving x, y and v,
the understanding was that v was to be used to express
“some”, but only in the context in which it appeared in
the premiss. For example, “Some X is
Y” has the primary translation v
= xy, which implied the secondary translation
vx = vy. This could also be read as
“Some X is Y”. Another consequence of v
= xy is v(1−x) =
v(1−y). However it was not permitted to read
this as “Some not-X is not-Y”
since v did not appear with 1−x in the
premiss. Boole’s use of v in the translation of propositions
into equations, as well as its use in solving equations, has been a
long-standing bone of contention.

Boole analyzed the seven forms of hypothetical syllogisms that were
in Aristotelian logic, from the Disjunctive Syllogism to the Complex
Destructive Dilemma, and pointed out that it would be easy to create
many more such forms. In the Postscript to MAL, Boole
recognized that propositional logic used a two-valued system, but
he did not offer a propositional logic to deal with this.

Beginning with the chapter Properties of Elective Functions, Boole
developed general theorems for working with equations in his algebra of
logic—the Expansion Theorem and the properties of constituents
are discussed in this chapter. Up to this point his sole focus was to
show that Aristotelian logic could be handled by simple algebraic
methods, mainly through the use of an elimination theorem borrowed from
ordinary algebra.

It was natural for Boole to want to solve equations in his algebra of
logic since this had been a main goal of ordinary algebra, and had led
to many difficult questions (e.g., how to solve a 5th degree
equation). Fortunately for Boole, the situation in his algebra of
logic was much simpler—he could always solve an equation, and
finding the solution was important to applications of his system, to
derive conclusions in logic. An equation was solved in part by using
expansion after performing division. This
method of solution
was the result of which he was the most proud—it described how
to solve an elective equation for one of its symbols in terms of the
others, and it is this that Boole claimed (in the Introduction chapter
of MAL) would offer “the means of a perfect analysis of
any conceivable set of propositions, …”. In LT
Boole would continue to regard this tool as the highlight of his
work.

Boole’s final example (p. 78) in MAL used a well known
technique for handling constraint conditions in analysis called
Lagrange Multipliers—this method, like his use of Taylor series,
was evidently considered overkill, if not somewhat dubious, and did not
appear in LT (Taylor series did appear in a footnote in
LT—Boole had not completely given up on them).

4. The Laws of Thought (1854)

Boole’s second logic book, An Investigation of The Laws of Thought
on which are founded the Mathematical Theories of Logic and
Probabilities, published in 1854, was an effort to correct and
perfect his 1847 book on logic. The second half of this 424 page book
presented probability theory as an excellent topic to illustrate the
power of his algebra of logic. Boole discussed the theoretical
possibility of using probability theory (enhanced by his algebra of
logic) to uncover fundamental laws governing society by analyzing large
quantities of social data.

Boole said that he would use simple letters like x to represent
classes, although later he would also use capital letters like V. The
universe was a class; and there was a class described as
having “no beings” which we call the empty class.
The operation of multiplication was defined to be
intersection, and this led to his first law, xy = yx. Next (some pages
later) he gave the idempotent law x² = x. Addition
was introduced as aggregation when the classes were disjoint. He stated
the commutative law for addition, x + y = y + x, and the distributive
law z(x + y) = zx + zy. Then followed x − y = − y + x and
z(x − y) = zx − zy.

One might expect that Boole was building toward an axiomatic
foundation for his algebra of logic, just as in MAL, evidently
having realized that the three laws in MAL were not enough.
Indeed he did discuss the rules of inference, that adding or
subtracting equals from equals gives equals, and multiplying equals by
equals gives equals. But then the development of an axiomatic approach
came to an abrupt halt. There was no discussion as to whether these
axioms and rules were sufficient to build his algebra of logic. Instead
he simply and briefly, with remarkably little fanfare, presented a
radically new foundation for his algebra of logic.

He said that since the only idempotent numbers were 0 and 1, this
suggested that the correct algebra to use for logic would be the
common algebra of the ordinary numbers modified by restricting the
symbols to the values 0 and 1. He stated what, in this article, is
called The Rule of 0 and 1, that a law or argument held in
logic iff after being translated into equational form it held in
common algebra with this 0,1-restriction on the possible
interpretations (i.e., values) of the symbols. Boole would use this
Rule to justify his main theorems (
Expansion,
Reduction,
Elimination
), and for no other purpose. The
main theorems in turn yielded Boole’s
General Method
for analyzing the consequences of propositional premises.

In Chapter V he discussed the role of uninterpretables in
his work; as a (partial) justification for the use of uninterpretable
steps in symbolic algebra he pointed to the well known use of
√−1. In succeeding chapters he gave the Expansion Theorem,
the new full-strength Elimination Theorem, a Reduction Theorem, and the
use of division to solve an equation.

After many examples and results for special cases of solving
equations, Boole turned to the topic of the interpretability of a
logical function. Boole had already stated that every equation is
interpretable (by converting it into a collection of constituent
equations). However terms need not be interpretable, e.g., 1+1 is not
interpretable.

Boole’s chapter on secondary propositions was essentially the same as
in MAL except that he changed from using “the cases
when X is true” to “the times when X is
true”. In Chapter XIII Boole selected some well-known arguments
of Clarke and Spinoza, on the nature of an eternal being, to put under
the magnifying glass of his algebra of logic, starting with the
comment:

2. The chief practical difficulty of this inquiry will
consist, not in the application of the method to the premises once
determined, but in ascertaining what the premises
are.

One conclusion was:

19. It is not possible, I think, to rise from the
perusal of the arguments of Clarke and Spinoza without a deep
conviction of the futility of all endeavours to establish, entirely a
priori, the existence of an Infinite Being, His attributes, and His
relation to the universe.

In the final chapter on logic, chapter XV, Boole presented his
analysis of the conversions and syllogisms of Aristotelian logic. He
considered this ancient logic to be a weak, fragmented attempt at a
logical system. This much neglected chapter is quite interesting
because it is the only chapter where he analyzed particular
propositions, making essential use of additional letters like
“v” to encode “some”. This is also the chapter
where he detailed (unfortunately incompletely) the rules for working
with “some”.

Briefly stated, Boole gave the reader a summary of traditional
Aristotelian categorical logic, and analyzed some simple examples
using ad hoc techniques with his algebra of logic. Then he launched
into proving a comprehensive result by applying his General Method to
the pair of equations:

vx  =  v′y
wz  =  w′y,

noting that the premises of many categorical syllogisms can be put in
this form. His goal was to eliminate y and find expressions
for x, 1−x and vx in terms
of z, v, v′, w,
w′. This led to three equations involving large
algebraic expressions. Boole omitted almost all details of his
derivation, but summarized the results in terms of the established
results of Aristotelian logic. Then he noted that the remaining
categorical syllogisms are such that their premises can be put in the
form:

vx  =  v′y

wz  =  w′(1−y),

and this led to another triple of large equations.

5. Later Developments

5.1 Objections to Boole’s Algebra of Logic

Many objections to Boole’s system have been published over the years;
three among the most important concern:

• the use of uninterpretable steps in derivations,
• the treatment of particular propositions by equations, and
• the method of dealing with division.

We look at a different objection, namely at the Boole/Jevons dispute
over adding X + X = X as a law. In Laws
of Thought, p. 66, Boole said:

The expression x + y seems indeed uninterpretable,
unless it be assumed that the things represented by x and the things
represented by y are entirely separate; that they embrace no
individuals in common.

[The following details are from “The development of the theories
of mathematical logic and the principles of mathematics, William
Stanley Jevons,” by Philip Jourdain, 1914.]

In an 1863 letter to Boole regarding a draft of a commentary on
Boole’s system that Jevons was considering for his forthcoming book
(Pure Logic, 1864), Jevons said:

It is surely obvious, however, that x+x is equivalent
only to x, …

Professor Boole’s notation [process of subtraction] is
inconsistent with a self-evident law.

If my view be right, his system will come to be regarded as a
most remarkable combination of truth and error.

Boole replied:

Thus the equation x + x = 0 is equivalent to the
equation x = 0; but the expression x + x is not equivalent to the
expression x.

Jevons responded by asking if Boole could deny the truth of x + x = x.

Boole, clearly exasperated, replies:

To be explicit, I now, however, reply that it is not
true that in Logic x + x = x, though it is true that x + x = 0 is
equivalent to x = 0. If I do not write more it is not from any
unwillingness to discuss the subject with you, but simply because if we
differ on this fundamental point it is impossible that we should agree
in others.

Jevons’s final effort to get Boole to understand the issue was:

I do not doubt that it is open to you to hold …[that x + x = x
is not true] according to the laws of your system, and with this
explanation your system probably is perfectly consistent with itself
… But the question then becomes a wider one—does your
system correspond to the Logic of common thought?

Jevons’s new law, X + X = X, resulted from his conviction that
“+” should denote what we now call union, where the
membership of X + Y is given by an inclusive “or”. Boole
simply did not see any way to define X + Y as a class unless X and Y
were disjoint, as already noted.

Various explanations have been given as to why Boole could not
comprehend the possibility of Jevons’s suggestion. Boole clearly had
the semantic concept of union—he expressed the union of X and Y
as x + (y−x), a union of two disjoint classes, and pointed out
that the elements of this class are the ones that belong to either X or
Y or both. So how could he so completely fail to see the possibility of
taking union for his fundamental operation + instead of his curious
partial union operation?

The answer is simple: the law x + x = x would have destroyed his
ability to use ordinary algebra: from x + x = x one has, by
ordinary algebra, x = 0. This would force every class symbol to denote
the empty class. Jevons’s proposed law x + x = x was simply not true if
one was committed to making ordinary algebra function as the algebra of
logic.

5.2. Modern Reconstruction of Boole’s System

Given the enormous degree of sophistication achieved in modern algebra
in the 20th century, it is rather surprising that a law-preserving
total algebra extension of Boole’s partial algebra of classes did not
appear until Theodore Hailperin’s book of 1976—the delay was
likely caused by readers not believing that Boole was using ordinary
algebra. Hailperin’s extension was to look at labelings of the universe
with integers, that is, each element of the universe is labeled with an
integer. Each labeling of the universe creates a multi-set
(perhaps one should say multi-class) consisting of those
labeled elements where the label is non-zero—one can think of
the label of an element as describing how many copies of the element
are in the multi-set. Boole’s classes correspond to the multi-sets
where all the labels are 1 (the elements not in the class have the
label 0). The uninterpretable elements of Boole become interpretable
when viewed as multi-sets—they are given by labelings of the
universe where some label is not 0 or 1.

To add two multi-sets one simply adds the labels on each element of
the universe. Likewise for subtraction and multiplication. (For the
reader familiar with modern abstract algebra, one can take the
extension of Boole’s partial algebra to be Z^U where Z is the
ring of integers, and U is the universe of discourse.) The multi-sets
corresponding to classes are precisely the idempotent multi-sets. It
turns out that the laws and principles Boole was using in his algebra
of logic hold for this system. By this means Boole’s methods are proved
to be correct for the algebra of logic of universal
propositions. Hailperin’s analysis did not apply to particular
propositions. F.W. Brown’s 2009 paper proposes that one can avoid multi-sets by working with the ring of polynomials Z[X] modulo a certain ideal.

Boole could not find a translation that worked as cleanly for the
particular propositions as for the universal propositions. In 1847
Boole used the following two translations, the second one being a
consequence of the first:

Some Xs are Ys
…………. v = xy
and vx = vy.

He initially used the symbol v to capture the essence of
“some”. Later he used other symbols as well, and also he
used v with other meanings (such as for the coefficients in
an expansion). One of the problems with his translation scheme
with v was that at times one needed “margin
notes,” to keep track of which class(es) the v was attached to
when it was introduced. The rules for translating from equations with
v‘s back to particular statements were never clearly formulated. For
example in Chapter XV one sees a derivation of x =
vv′y which is then translated as
Some X is Y. But he had no rules for when a product
of v‘s carries the import of “some”. Such problems detract
from Boole’s system; his explanations leave doubts as to which
procedures are legitimate in his system when dealing with particular
statements.

There is one point on which even Hailperin was not faithful to Boole’s
work, namely he used modern semantics, where the simple
symbols x, y, etc., can refer to the empty class as
well as to a non-empty class. With modern semantics one cannot have
the Conversion by Limitation which held in Aristotelian logic: from
All X is Y follows Some Y is X. In
his Formal Logic of 1847, De Morgan pointed out that all
writers on logic had assumed that the classes referred to in a
categorical proposition were non-empty. This restriction of the class
symbols to non-empty classes, and dually to non-universe classes, will
be called Aristotelian semantics. Boole had evidently
followed this Aristotelian convention because he derived all the
Aristotelian results, such as Conversion by Limitation. A proper
interpretation (faithful to Boole’s work) of Boole’s system requires
Aristotelian semantics for the class
symbols x, y, z, … ; unfortunately
it seems that the published literature on Boole’s system has failed to
note this.

6. Boole’s Methods

While reading through this section, on the technical details of
Boole’s methods, the reader may find it useful to consult the

supplement of examples from Boole’s two books.

These examples have been augmented with comments explaining, in
each step of a derivation by Boole, which aspect of his methods is
being employed.

6.1 The Three Methods of Argument Analysis Used by Boole in LT

Boole used three methods to analyze arguments in LT:

(1) The first was the purely ad hoc algebraic manipulations that
were used (in conjunction with a weak version of the Elimination
Theorem) on the Aristotelian arguments in MAL.

(2) Secondly, in section 15 of Chapter II of LT, one finds
the method that, in this article, is called
the Rule of 0 and 1.

The theorems of LT combine to yield the master result,

(3) Boole’s General Method (in this article it will always be referred
to using capitalized first letters—Boole just called it
“a method”).

When applying the ad hoc method, he used parts of ordinary algebra
along with the idempotent law x² = x to
manipulate equations. There was no pre-established procedure to
follow—success with this method depended on intuitive skills
developed through experience.

The second method, the Rule of 0 and 1, is very powerful, but it
depends on being given a collection of premiss equations and a
conclusion equation. It is a truth-table like method (but Boole never
drew a table when applying the method) to determine if the argument is
correct. Boole only used this method to establish the theorems that
justified his General Method, even though it is an excellent tool for
simple arguments like syllogisms. The Rule of 0 and 1 is a somewhat
shadowy figure in LT—it has no name, and is never
referred to by section or page number. A precise version of Boole’s
Rule of 0 and 1 that yields Boole’s results is given in Burris and
Sankappanavar 2013.

The third method to analyze arguments was the highlight of Boole’s
work in logic, his General Method (discussed immediately after this).
This is the one he used for all but the simplest examples in
LT; for the simplest examples he resorted to the first method
of ad hoc algebraic techniques because, for one skilled in algebraic
manipulations, using them is usually far more efficient than going
through the General Method.

The final version (from LT) of his General Method for
analyzing arguments is, briefly stated, to:

(1) convert (or translate) the propositions into equations,

(2) apply a prescribed sequence of algebraic processes to the
equations, processes which yield desired conclusion equations, and
then

(3) convert the equational conclusions into propositional
conclusions, yielding the desired consequences of the original
collection of propositions.

With this method Boole had replaced the art of reasoning from
premiss propositions to conclusion propositions by a routine mechanical
algebraic procedure.

In LT Boole divided propositions into two kinds, primary
and secondary. These correspond to, but are not exactly the same as,
the Aristotelian division into categorical and hypothetical
propositions. First we discuss his General Method applied to primary
propositions.

6.2. Boole’s General Method for Primary Propositions

Boole recognized three forms of primary propositions:

• All X is Y
• All X is all Y
• Some X is Y

These were his version of the Aristotelian categorical propositions,
where X is the subject term and Y the predicate term. The terms X and
Y could be complex names, for example, X could be X₁ or
X₂.

STEP 1: Names are converted into algebraic terms as follows:

We will call the letters x, y, … class symbols (as
noted earlier, the algebra of the 1800s did not use the word
variables).

STEP 2: Having converted names for the terms into algebraic terms,
one then converts the propositions into equations using the
following:

Boole used the four categorical propositions as his primary forms in
1847, but in 1854 he eliminated the negative propositional forms,
noting that one could change “not Y” to
“not-Y”. Thus in 1854 he would express “No X is
Y” by “All X is not-Y”, with the translation

x(1 − (1 − y))  =  0,

which simplifies to xy  =  0.

STEP 3: After converting the premises into algebraic form one has a
collection of equations, say

p1  =  q1,

p2  =  q2,

…,

pn  =  qn.

Express these as equations with 0 on the right side, that is, as

r1  =  0,

r2  =  0,

…,

rn  =  0,

with

r1  :=  p1 − q1,

r2  :=  p2 − q2,

…,

rn  :=  pn − qn.

STEP 4: (REDUCTION) [LT (p. 121) ]

Reduce the system of equations

r1  =  0,

r2  =  0,

…,

rn  =  0,

to a single equation r  =  0. Boole had three different
methods for doing this—he seemed to have a preference for
summing the squares:

r  :=  r₁² + · · · +
r_n²  =  0.

Steps 1 through 4 are mandatory in Boole’s General Method. After
executing these steps there are various options for continuing,
depending on the goal.

STEP 5: (ELIMINATION) [LT (p. 101) ]

Suppose one wants the most general equational conclusion derived
from r  =  0 that involves some, but not all, of the class
symbols in r. Then one wants to eliminate certain symbols. Suppose r
involves the class symbols

x₁, … , x_j and y₁,
… , y_k.

Then one can write r as r(x₁, … , x_j,
y₁, … , y_k).

Boole’s procedure to eliminate the symbols x₁, …,
x_j from

r(x₁, … , x_j, y₁,
… , y_k)  =  0

to obtain

s(y₁, … , y_k)  =
0

was as follows:

1. form all possible expressions r(a₁, … ,
a_j, y₁, … , y_k) where
a₁, … , a_j are each either 0 or 1,
then

2. multiply all of these expressions together to obtain
s(y₁, … , y_k).

For example, eliminating x₁, x₂ from

r(x₁, x₂, y)  =
0

gives

s(y)  =  0

where

s(y)  :=  r(0, 0, y) · r(0, 1, y) ·
r(1, 0, y) · r(1, 1, y).

STEP 6: (DEVELOPMENT, or EXPANSION) [ MAL (pp. 60), LT (pp. 72, 73) ]

Given a term, say r(x₁, … , x_j,
y₁, … , y_k), one can expand the term with
respect to a subset of the class symbols. To expand with respect to
x₁, … , x_j gives

r  =  sum of the terms

r(a₁, … , a_j, y₁,
… , y_k) · C(a₁, x₁)
· · · C(a_j,
x_j),

where a₁, … , a_j range over all sequences
of 0s and 1s of length j, and where the C(a_i, x_i)
are defined by:

C(1, x_i)  :=  x_i,
and  C(0, x_i)  :=  1−
x_i.

Boole said the products:

C(a₁, x₁) · · ·
C(a_j, x_j)

were the constituents of x₁, … ,
x_j. There are 2^j different constituents for j
symbols. The regions of a Venn diagram give a popular way to visualize
constituents.

STEP 7: (DIVISION: SOLVING FOR A CLASS SYMBOL)
[MAL (p. 73), LT (pp.
86, 87)] ]

Given an equation r  =  0, suppose one wants to solve this
equation for one of the class symbols, say x, in terms of the other
class symbols, say they are y₁, … , y_k. To
solve:

r(x, y₁, …, y_k)  =
0

for x, first let:

N(y₁, … , y_k)  =
− r(0, y₁, … , y_k)

D(y₁, … , y_k)  =  r(1,
y₁, … , y_k) − r(0, y₁,
… , y_k).

Then:

x  =
s(y₁,…, y_k)
(*)

where s(y₁,…, y_k) is:

(1) the sum of all constituents

C(a₁, y₁) · · ·
C(a_k, y_k),

where a₁, … , a_k range over all sequences
of 0s and 1s for which:

N(a₁, … , a_k)  =
D(a₁, … , a_k)  ≠
0,

plus

(2) the sum of all the terms of the form

V_{a1 … ak} ·
C(a₁, y₁) · · ·
C(a_k, y_k)

for which:

N(a₁, … , a_k)  =
D(a₁, … , a_k)  =
0.

The V_{a1 … ak} are parameters, denoting
arbitrary classes (similar to what one sees in the study of linear
differential equations, a subject in which Boole was an expert).

To the equation (*) for x adjoin the side-conditions (that we
will call constituent equations)

C(a₁, y₁) · · ·
C(a_k, y_k)  =  0

whenever

D(a₁, … , a_k)  ≠
N(a₁, … , a_k)  ≠
0.

Note that one is to evaluate the terms:

D(a₁, … , a_k)   and
N(a₁, … , a_k)

using ordinary arithmetic. Thus solving an equation r  =  0
for a class symbol x gives an equation

x  =
s(y₁,…, y_k),

perhaps with side-condition constituent equations.

STEP 8: (INTERPRETATION) [MAL pp. 64–65, LT
(Chap. VI, esp. pp. 82–83)]

Suppose the equation r(y₁, … , y_k)
=  0 has been obtained by Boole’s method from a given
collection of premiss equations. Then this equation is equivalent to
the collection of constituent equations

C(a₁, y₁) · · ·
C(a_k, y_k)  =  0

for which r(a₁, … , a_k) is not 0. A
constituent equation merely asserts that a certain intersection of the
original classes and their complements is empty. For example,

y₁(1−y₂)(1−y₃)
=  0

expresses the proposition “All Y₁ is Y₂ or
Y₃,” or equivalently, “All Y₁ and not
Y₂ is Y₃.” It is routine to convert
constituent equations into propositions.

6.3. Boole’s General Method for Secondary Propositions

Secondary propositions were Boole’s version of the propositions that
one encounters in the study of hypothetical syllogisms in Aristotelian
logic, statements like “If X or Y then Z.” The symbols X,
Y, Z, etc. of secondary propositions did not refer to classes, but
rather they referred to (primary) propositions. In keeping with the
incomplete nature of the Aristotelian treatment of hypothetical
propositions, Boole did not give a precise description of possible
forms for his secondary propositions.

The key (but not original) observation that Boole used was simply
that one can convert secondary propositions into primary propositions.
In MAL he adopted the convention found in Whately (1826), that
given a propositional symbol X, the symbol x will denote “the
cases in which X is true”, whereas in LT Boole let x
denote “the times for which X is true”. With this the
secondary proposition “If X or Y then Z” becomes simply
“All x or y is z”. The equation x  =  1 is the
equational translation of “X is true” (in all cases, or for
all times), and x  =  0 says “X is false” (in all
cases, or for all times).

With this translation scheme it is clear that Boole’s treatment of
secondary propositions can be analyzed by the methods he had developed
for primary propositions. This was Boole’s propositional logic.

Boole worked only with Aristotelian propositions in MAL,
using the traditional division into categoricals and hypotheticals. One
does not consider “X and Y,” “X or Y,” etc., in
categorial propositions, only in hypothetical propositions. In
LT this division was replaced by the similar but more general
primary versus secondary classification, where the subject and
predicate were allowed to become complex names, and the number of
propositions in an argument became unrestricted. With this the
parallels between the logic of primary propositions and that of
secondary propositions became clear, with one notable difference,
namely it seems that the secondary propositions always translate into
universal primary propositions.

Bibliography

Primary Literature

• Boole, G., 1841, “Researches on the Theory of Analytical
  Transformations, with a special application to the Reduction of the
  General Equation of the Second Order,” The Cambridge
  Mathematical Journal, 2: 64–73.
• –––, 1841, “On Certain Theorems in the
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• –––, 1841, “On the Integration of Linear
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  Cambridge Mathematical Journal, 2: 114–119.
• –––, 1847, The Mathematical Analysis of
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• –––, 1848, “The Calculus of Logic,”
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• –––, 1854, An Investigation of The Laws of
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• –––, 1859, A Treatise on Differential
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• –––, 1860, A Treatise on the Calculus of
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• De Morgan, A., 1839, “On the foundation of algebra,”
  Transactions of the Cambridge Philosophical Society, VII,
  174–187.
• –––, 1841, “On the foundation of algebra,
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• –––, 1847, Formal Logic: or, the Calculus of
  Inference, Necessary and Probable, Originally published in London
  by Taylor and Walton. Reprinted in London by The Open Court Company,
  1926.
• –––, 1966, On the Syllogism, and Other
  Logical Writings, P. Heath (ed.), New Haven: Yale University Press. (A
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• Gregory, D.F. 1839, “Demonstrations in the differential
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• –––, 1839, “I.–On the elementary
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• –––, 1840, “On the real nature of
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• –––, 1865, The Mathematical Writings of
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• Jevons, W.S., 1864, Pure Logic, or the Logic of Quality apart
  from Quantity: with Remarks on Boole’s System and on the Relation of
  Logic and Mathematics, London: Edward Stanford. Reprinted 1971 in
  Pure Logic and Other Minor Works, R. Adamson and H.A.
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• Lacroix, S.F, 1797/1798, Traité du calcul
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• Lagrange, J.L., 1797, Théorie des fonctions
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• –––, 1788, Méchanique Analytique,
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• Peacock, G., 1830, Treatise on Algebra, 2nd ed., 2 vols.,
  Cambridge: J.&J.J. Deighton, 1842/1845.
• –––, 1833, “Report on the Recent Progress
  and Present State of certain Branches of Analysis”,
  In Report of the Third Meeting of the British Association for the
  Advancement of Science held at Cambridge in 1833,
  pp. 185-352. London: John Murray.
• Schröder, E., 1890–1910, Algebra der Logik, Vols.
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Secondary Literature

Cited Works

• Brown, F.W, 2009, “ George Boole’s deductive system”,
  Notre Dame Journal of Logic, 50: 303–330.
• Burris, S. and Sankappanavar, H.P., 2013, “The Horn theory
  of Boole’s partial algebras”, The Bulletin of Symbolic
  Logic, 19: 97–105.
• Ewald, W. (ed.), 1996, From Kant to Hilbert. A Source Book in the
  History of Mathematics, 2 Vols, Oxford: Oxford University Press.
• Grattan-Guiness, I., 2001, The Search for Mathematical
  Roots, Princeton, NJ: Princeton University Press.
• Hailperin, T., 1976, Boole’s Logic and Probability,
  (Series: Studies in Logic and the Foundations of Mathematics, 85),
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  Revised and enlarged, 1986.
• –––, 1981, “Boole’s algebra isn’t Boolean
  algebra”, Mathematics Magazine, 54: 172–184.
• Jourdain, P.E.B., 1914, “The development of the theories of
  mathematical logic and the principles of mathematics. William Stanley
  Jevons”, Quarterly Journal of Pure and Applied
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• MacHale, D., 1985, George Boole, His Life and Work,
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Other Important Literature

• Couturat, L., 1905, L’algèbre de la Logique, 2d edition,
  Librairie Scientifique et Technique Albert Blanchard, Paris. English
  translation by Lydia G. Robinson: Open Court Publishing Co., Chicago
  & London, 1914. Reprinted by Dover Publications, Mineola,
  2006.
• Dummett, M., 1959, “Review of Studies in Logic and
  Probability by George Boole”, Watts & Co., London, 1952,
  edited by R. Rhees. The Journal of Symbolic Logic, 24:
  203–209.
• Frege, G., 1880, “Boole’s logical calculus and the
  concept-script”, in Gottlob Frege: Posthumous Writings,
  Basil Blackwell, Oxford, 1979. English translation of
  Nachgelassene Schriften, vol. 1, edited by H. Hermes,
  F. Kambartel, and F. Kaulbach, Felix Meiner, Hamburg, 1969.
• Kneale, W., and M. Kneale, 1962, The Development of
  Logic, The Clarendon Press, Oxford.
• Lewis, C. I., 1918, A Survey of Symbolic Logic,
  University of California Press, Berkeley. Reprinted by Dover
  Publications, Inc., New York, 1960. Chap. II, “The Classic, or
  Boole-Schröder Algebra of Logic.”
• Peirce, C. S., 1880, “On the Algebra of Logic”,
  American Journal of Mathematics, 3: 15–57.
• Smith, G. C., 1983, “Boole’s annotations on The
  Mathematical Analysis of Logic”, History and Philosophy
  of Logic, 4: 27–39.
• Styazhkin, N. I., 1969, Concise History of Mathematical Logic
  from Leibniz to Peano, Cambridge, MA: The MIT Press.
• van Evra, J. W., 1977, “A reassessment of George Boole’s
  theory of logic”, Notre Dame Journal of Formal Logic,
  18: 363–77.
• Venn, J., 1894, Symbolic Logic, 2d edition, Macmillan, London.
  Reprinted, revised and rewritten. Bronx: Chelsea Publishing Co., 1971.
• Whitney, H., 1933, “Characteristic functions and the algebra
  of logic”, Annals of Mathematics, Second Series, 34:
  405–414.

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Other Internet Resources

• George Boole,
  The MacTutor History of Mathematics
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• Augustus De Morgan,
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  William Jevons,
  George Peacock,
  Ernst Schröder,
  The MacTutor History of
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• Algebraic Logic Group,
  Alfred Reyni Institute of
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• George Boole 200,
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