The British mathematician George Boole is best known for his work on Boolean logic which was named after him and is very important in computing and electronics. November 2nd 2015 is Boole’s 200th birthday if such a thing makes any sense. We like to celebrate famous people from tech history so it’s a good enough excuse for us!

Boolean Logic is a foundational concept in computing and is definitely important to learn about today. But as well as learning about the ideas we think it’s important to learn about where they came from.

For such a significant topic, Boolean logic is very accessible to everyone.

I’ve put together some resources about George Boole and Boolean logic that are suitable for use with older children and teenagers. I’ll be talking to my kids about George Boole after school today and sharing these resources with them.

## What is Boolean Algebra?

Boolean logic uses AND, OR, NOT and related operators to evaluation whether statements are TRUE or FALSE. It’s simple, but really powerful. It’s used in electronics, databases and in computer programming languages. It’s really not an exaggeration to same that Boolean logic is one of the foundational concepts of the tech age.

Google supports Boolean logic for searches and in many countries they have a Google Doodle in honour of George Boole.

## Who was George Boole?

## Why Learn About George Boole?

Boolean Algebra is central to modern electronics, computing and data processing. I think it’s really important to understand where ideas came from as well as the ideas themselves.

## Boolean Resources

- Boole2School is an initiative to get school to teach Boolean logic on Boole’s 200th birthday. Definitely worth a look even if you’ve missed the date. There are resources for students aged 8-18 and tap into topics that kids are interested in such as Minecraft.
- The George Boole Timeline is an interactive resource which provides a very approachable way to learn about his life.
- University College Cork (where Boole was based for much of his academic career) has produced a short video that explains Boole’s legacy:
- BBC Bitesize Boolean Algebra – An overview of Boolean Algebra with colourful cartoons and clear explanations.
- littleBits Logic Expansion Pack – If you’ve got a littleBits set then the Logic expansion pack is a great way to learn Boolean logic in a hands-on way.
- As a child I loved logic puzzles like these grid-based ones and they use Boolean logic.

**Puzzle Baron’s Logic Puzzles**? @ Amazon

(1815–64). For centuries philosophers have studied logic, which is orderly and precise reasoning. George Boole, an English mathematician, argued in 1847 that logic should be allied with mathematics rather than with philosophy. Demonstrating logical principles with mathematical symbols instead of words, he founded symbolic logic, a field of mathematical/philosophical study.

In the new discipline he developed, known as Boolean algebra, all objects are divided into separate classes, each with a given property; each class…

Boolean algebra

In 1847 Boole published the pamphlet *Mathematical Analysis of Logic*. He later regarded it as a flawed exposition of his logical system, and wanted *An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities* to be seen as the mature statement of his views. Contrary to widespread belief, Boole never intended to criticise or disagree with the main principles of Aristotle’s logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability.^{[29]} Boole’s initial involvement in logic was prompted by a current debate on quantification, between Sir William Hamilton who supported the theory of “quantification of the predicate”, and Boole’s supporter Augustus De Morgan who advanced a version of De Morgan duality, as it is now called. Boole’s approach was ultimately much further reaching than either sides’ in the controversy.^{[30]} It founded what was first known as the “algebra of logic” tradition.^{[31]}

Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic. A 2003 article^{[32]} provides a systematic comparison and critical evaluation of Aristotelian logic and Boolean logic; it also reveals the centrality of wholistic reference in Boole’s philosophy of logic.

#### 1854 definition of universe of discourse[edit]

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilised men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.

^{[33]}

# George Boole

*First published Wed Apr 21, 2010; substantive revision Mon Apr 14, 2014*

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
- 6. Boole’s Methods
- 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:

d^{2}y/dx^{2}−dy/dx− 2y= cos(x),

and wrote it in the form Operator(*y*) = cos(*x*).

This was (formally) achieved by letting:

D=d/dx,D^{2}=d^{2}/dx^{2},

etc.

leading to an expression of the differential equation as:

(

D^{2}−D− 2)y= cos(x).

Now symbolical algebra came into play by simply treating the operator

*D*^{2} − *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 2*x* + 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* *x**y* of elective symbols. The standard notation

*x**y* 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*) = *x**u* + *x**v*,

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* *x**y*

= *y**x* and the *idempotent law*

*x*

^{2}=

*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,

*x*

^{2}=

*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 2

*x*

*y*= 0, and

then

*x*

*y*= 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 *x**y* = *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*

= *x**y*, which implied the secondary translation

*v**x* = *v**y*. This could also be read as

“Some *X* is *Y*”. Another consequence of *v*

= *x**y* 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, *x**y* = *y**x*. Next (some pages

later) he gave the idempotent law *x*^{2} = *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*) = *z**x* + *z**y*. Then followed *x* − *y* = − *y* + *x* and

*z*(*x* − *y*) = *z**x* − *z**y*.

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 *v**x* 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+yseems indeed uninterpretable,

unless it be assumed that the things represented byxand the things

represented byyare 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+xis equivalent

only tox, …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

equationx= 0; but the expressionx+xis not equivalent to the

expressionx.

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 Logicx+x=x, though it is true thatx+x= 0 is

equivalent tox= 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 areYs

………….v=xy

andvx=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* =

*v**v*′*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 *x*^{2} = *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 *X*_{1} or

*X*_{2}.

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

Terms | MAL | LT | ||||
---|---|---|---|---|---|---|

universe | 1 | p. 15 | 1 | p. 48 | ||

empty class | 0 | p. 47 | ||||

not X | 1 − x | p. 20 | 1 − x | p. 48 | ||

X and Y | xy | p. 16 | xy | p. 28 | ||

X or Y (inclusive) |
| p. 56 | ||||

X or Y (exclusive) | x(1 − y) + y(1 − x) | p. 56 |

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:

Primary Propositions | MAL (1847) | LT (1854) | ||
---|---|---|---|---|

All X is Y | x(1−y) = 0 | p. 26 | x = vy | p. 64, 152 |

No X is Y | xy = 0 | (not primary) | ||

All X is all Y | (not primary) | x = y | ||

Some X is Y | v = xy | vx = vy | ||

Some X is not Y | v = x(1−y) | (not primary) |

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 *x**y* = 0.

STEP 3: After converting the premises into algebraic form one has a

collection of equations, say

p_{1}=q_{1},p_{2}=q_{2},…, p_{n}=q_{n}.

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

r_{1}= 0,r_{2}= 0,…, r_{n}= 0,

with

r_{1}:=p_{1}−

q_{1},r_{2}:=p_{2}−

q_{2},…, r_{n}:=p_{n}−

q_{n}.

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

Reduce the system of equations

r_{1}= 0,r_{2}= 0,…, r_{n}= 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:=r_{1}^{2}+ · · · +

r_{n}^{2}= 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

x_{1}, … ,x_{j}andy_{1},

… ,y_{k}.

Then one can write *r* as *r*(*x*_{1}, … , *x*_{j},

*y*_{1}, … , *y*_{k}).

Boole’s procedure to eliminate the symbols *x*_{1}, …,

*x*_{j} from

r(x_{1}, … ,x_{j},y_{1},

… ,y_{k}) = 0

to obtain

s(y_{1}, … ,y_{k}) =

0

was as follows:

1. form all possible expressions *r*(*a*_{1}, … ,

*a*_{j}, *y*_{1}, … , *y*_{k}) where

*a*_{1}, … , *a*_{j} are each either 0 or 1,

then

2. multiply all of these expressions together to obtain

*s*(*y*_{1}, … , *y*_{k}).

For example, eliminating *x*_{1}, *x*_{2} from

r(x_{1},x_{2},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*(*x*_{1}, … , *x*_{j},

*y*_{1}, … , *y*_{k}), one can expand the term with

respect to a subset of the class symbols. To expand with respect to

*x*_{1}, … , *x*_{j} gives

r= sum of the terms

r(a_{1}, … ,a_{j},y_{1},

… ,y_{k}) ·C(a_{1},x_{1})

· · ·C(a_{j},

x_{j}),

where *a*_{1}, … , *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},

andC(0,x_{i}) := 1−

x_{i}.

Boole said the products:

C(a_{1},x_{1}) · · ·

C(a_{j},x_{j})

were the *constituents* of *x*_{1}, … ,

*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 *y*_{1}, … , *y*_{k}. To

solve:

r(x,y_{1}, …,y_{k}) =

0

for *x*, first let:

N(y_{1}, … ,y_{k}) =

−r(0,y_{1}, … ,y_{k})

D(y_{1}, … ,y_{k}) =r(1,

y_{1}, … ,y_{k}) −r(0,y_{1},

… ,y_{k}).

Then:

x=

s(y_{1},…,y_{k})

(*)

where *s*(*y*_{1},…, *y*_{k}) is:

(1) the sum of all constituents

C(a_{1},y_{1}) · · ·

C(a_{k},y_{k}),

where *a*_{1}, … , *a*_{k} range over all sequences

of 0s and 1s for which:

N(a_{1}, … ,a_{k}) =

D(a_{1}, … ,a_{k}) ≠

0,

plus

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

V_{a1 … ak}·

C(a_{1},y_{1}) · · ·

C(a_{k},y_{k})

for which:

N(a_{1}, … ,a_{k}) =

D(a_{1}, … ,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(a_{1},y_{1}) · · ·

C(a_{k},y_{k}) = 0

whenever

D(a_{1}, … ,a_{k}) ≠

N(a_{1}, … ,a_{k}) ≠

0.

Note that one is to evaluate the terms:

D(a_{1}, … ,a_{k}) and

N(a_{1}, … ,a_{k})

using ordinary arithmetic. Thus solving an equation *r* = 0

for a class symbol *x* gives an equation

x=

s(y_{1},…,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*(*y*_{1}, … , *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(a_{1},y_{1}) · · ·

C(a_{k},y_{k}) = 0

for which *r*(*a*_{1}, … , *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,

y_{1}(1−y_{2})(1−y_{3})

= 0

expresses the proposition “All *Y*_{1} is *Y*_{2} or

*Y*_{3},” or equivalently, “All *Y*_{1} and not

*Y*_{2} is *Y*_{3}.” 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.

Secondary Propositions | MAL (1847) | LT (1854) | ||
---|---|---|---|---|

X is true | x = 1 | p. 51 | x = 1 | p. 172 |

X is false | x = 0 | “ | x = 0 | “ |

X and Y | xy = 1 | “ | xy = 1 | “ |

X or Y (inclusive) | x + y −xy = 1 | p. 52 | ||

X or Y (exclusive) | x −2xy+ y = 1 | p. 53 | x(1 − y) + y(1 − x)= 1 | p. 173 |

If X then Y | x(1−y) = 0 | p. 54 | x = vy | p. 173 |

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*Pure Logic, or the Logic of Quality apart*, London: Edward Stanford. Reprinted 1971 in

from Quantity: with Remarks on Boole’s System and on the Relation of

Logic and Mathematics

*Pure Logic and Other Minor Works*, R. Adamson and H.A.

Jevons (eds.), New York: Lennox Hill Pub. & Dist. Co. - Lacroix, S.F, 1797/1798,
*Traité du calcul*, Paris: Chez Courcier.

différentiel et du calcul integral - Lagrange, J.L., 1797,
*Théorie des fonctions*, Paris: Imprimerie de la Republique.

analytique - –––, 1788,
*Méchanique Analytique*,

Paris: Desaint. - 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*held at Cambridge in 1833,

Advancement of Science

pp. 185-352. London: John Murray. - Schröder, E., 1890–1910,
*Algebra der Logik, Vols.*. Leipzig, B.G. Teubner; reprint Chelsea 1966.

I–III

### 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*, 19: 97–105.

Logic - Ewald, W. (ed.), 1996,
*From Kant to Hilbert. A Source Book in the*, 2 Vols, Oxford: Oxford University Press.

History of Mathematics - Grattan-Guiness, I., 2001,
*The Search for Mathematical*, Princeton, NJ: Princeton University Press.

Roots - Hailperin, T., 1976,
*Boole’s Logic and Probability*,

(Series: Studies in Logic and the Foundations of Mathematics, 85),

Amsterdam, New York, Oxford: Elsevier North-Holland. 2nd edition,

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*, 44: 113–128.

Mathematics - MacHale, D., 1985,
*George Boole, His Life and Work*,

Dublin: Boole Press.

#### 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*, The Clarendon Press, Oxford.

Logic - 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*, 4: 27–39.

of Logic - Styazhkin, N. I., 1969,
*Concise History of Mathematical Logic*, Cambridge, MA: The MIT Press.

from Leibniz to Peano - 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

Archive - Augustus De Morgan,

Duncan Farquharson Gregory,

William Jevons,

George Peacock,

Ernst Schröder,

The MacTutor History of

Mathematics Archive - Algebraic Logic Group,

Alfred Reyni Institute of

Mathematics, Hungarian Academy of Sciences - George Boole 200,

maintained at University College Cork.