\documentclass[12pt]{article}

\begin{document}

\vskip 10pt
\noindent{\bf George BOOLE}\\
b. 2 November 1815 - d. 8 December 1864
\vspace{.5cm}

\noindent {\bf Summary}. Boole's name in probability theory and statistical
inference is preserved by Boole's Inequality, an important tool for dealing with
the presence of statistical dependence. In the design of electronic circuitry,
Boolean algebra is widely used. 
\vspace{.5 cm}

\noindent Boole's name is well known in connection with the founding of
symbolic logic, and entries on him may be found, primarily in this
context, in major reference sources such as the {\it Encyclopaedia
Britannica}, {\it Encyclopedia of Philosophy, Dictionary of Scientific
Biography} as well the well-known {\it Men of Mathematics} of E.T. Bell.
Symbolic logic, simplistically speaking, uses for class-terms, propositions 
and relations between them, algebraic symbols and relational symbols much
in the way that is done in set theory.  This permits a mechanistic
solution, as in algebra, of logical problems.  Commensurately named
{\it Boolean algebra} is now widely used often in connection with
 binary codes and electronic switching in computer science.  Since
modern probability theory according to the Kolmogorov axiomatization
is based on set theory, it is not in retrospect surprising that Boole
had a strong interest in probability theory as an adjunct to formal
logic, and was indeeed part of a British tradition of this kind, with
his near contemporaries De Morgan (q.v.), Jevons (q.v.), and Venn (q.v.).

Boole's (1854) major study {\it An Investigation into the Laws of
Thought, on Which Are Founded the Mathematical Theories of Logic and
Probabilities}, has remained in print.  In this British tradition, too, was
Lewis Carroll (Charles Lutwidge Dodgson, 1832-1898) whose paradoxes and
books in a lighter genre, {\it Symbolic Logic} and {\it The Game of
Logic}, have continued also to attract interest.

It is worth mentioning that, specifically on the probabilistic side of
the tradition, of which De Morgan stands as head, the books of Isaac
Todhunter (1820-1884) {\it A History of the
Mathematical Theory of Probability from the Time of Pascal to That of
Laplace} (1865) and William Allen Whitworth (1840-1905)
{\it Choice and Chance} (1867) have also continued to be in print and in
use, a remarkable record of longevity. Boole's own books in their own 
time did not enjoy popularity; they were said, by a superficial
commentator ({\it Athenaeum}, December 17, 1864) given to the {\it bon
mot} to have:

\begin{quote}
`` ... sought a very limited audience, and we believe, found it."
\end{quote}

Boole was born in humble circumstances in the city of Lincoln in
England, where his father was a shoemaker, and a mathematical amateur given
to the construction and adaptation of optical instruments, a taste 
which was passed onto George, who revered him.  George's schooling was
ordinary, although he showed prodigious talent, and his early ambition
ran in the direction of attaining proficiency in classical Greek and
Latin.  He was essentially self-taught in mathematics from the age of
about 17 when he began as teacher in small schools, eventually opening his own
day-school for youth of both sexes in Lincoln, then going on to run a
school at which he had formerly taught, in Waddington, to support his
 parents and other members of his family.  From Waddington came his
earliest mathematical papers with the support and encouragement from
1839 of one of the editors of the newly founded (1837) {\it Cambridge
Mathematical Journal}, Duncan Farquharson Gregory (1813 - 1844).  With
Gregory's encouragement, in 1844 Boole communicated a paper to the Royal
Society of London. This was awarded the Royal Medal in that year as the
most important paper communicated to its {\it Transactions}. \ In 1849
Boole was rescued from schoolmastering by being selected to fill (even
though he had no formal university degree) the office of Professor of
Mathematics in the newly formed Queen's College, in Cork, where he was
prolific in his research and active in the scientific context of Ireland
till his death.

In 1855, Boole married Mary Everest, niece of Colonel Everest after whom
Mount Everest was named, an intellectually active woman with whom he had
five daughters.  He was elected Fellow of the Royal Society in 1857.
The Queen's Colleges of Belfast, Galway and Cork were united to form the
Queen's University of Ireland at about this time and Boole was appointed
one of the public examiners, a position which he filled with eminence.
His premature death, at Ballintemple, County Cork, came about as a
consequence of lecturing in clothes wet from the rain.

The burden of supporting his young family fell on his wife, Mary Everest
Boole, who moved back to London.  The youngest daughter, Ethel Lilian
eventually married a Polish revolutionary,
and as Ethel Lilian Voynich
(1864 - 1960) achieved fame in the Communist bloc as author of {\it The
 Gadfly}, (in Russian: {\it Ovod}), whose hero was held
 to be a model for Communist youth.  It appears, paradoxically, that the
book was based on her
association with Sidney Reilly, English spy and enemy of the Bolsheviks
(MacHale, 1985, pp.269-276).  Karl Pearson (q.v.) through his
interest in socialism and women's emancipation, has a cameo
part in her history.  Several of George Boole's great grandchildren,
from another daughter, attained scientific eminence in Britain and the
U.S. in our own time.

In probability theory, and therefore by extension in statistics, Boole's
name is attached to {\it Boole's Inequality}: for events \ $\{A_{i}\} ,
\quad 1 \leq i \leq n$ ,
$$P \left({\cup_{i=1}^n} A_{i}\right) \leq \sum_{i=1}^n P (A_i)$$
which can be written equivalently, through use of complementation, as
$$P \left({\cap_{i=1}^n} B_{i}\right) \geq 1 - \sum_{i=1}^n (1 - P (B_i))$$
where \ $B_i = {\bar A}_i$ . \ For Boole's various inequalities as they
actually occur in his {\it An Investigation ...}, see Seneta (1992).  The
strength of the above inequality is that it holds without events being
mutually exclusive or independent.  Dependence of events defined by
successive dependent sample means ${\bar X}_{(n)} = \sum_{i=1}^n X_i / n , \quad
 n \geq 1$ , \ where the \ $X_i$'s \ are independent random variables,
was the obstacle Francesco Paolo Cantelli (1875-1966) was able to
overcome with the aid of Boole's (``forgotten") Inequality, to give in
1917 a  proof of the Strong Law of Large Numbers for general
random variables, after an attempt in 1909 by Borel (q.v.) for
Bernoulli random variables.  Boole's Inequality is the first of a
sequence of inequalities of increasing complexity (``degree") which have
found use in simultaneous statistical inference because they can
cope with probabilities of unions of arbitrary dependent events.  
The second inequality of the sequence is:
$$P \left({\cup_{i=1}^n} A_{i}\right) \geq \sum_{i=1}^n P(A_i) -
 \sum_{i<j} P(A_i \cap A_j) ;$$
and the inequalities for \ $P (\cup_{i=1}^n A_i)$ \ continue in this way,
with alternation of inequality sign from one inequality to the next in
sequence.  The sequence is known as the Bonferroni
Inequalities, after Carlo Emilio Bonferroni (q.v.) who explicitly, in
1936, attributes the first inequality to Boole.  However usage of the
first inequality in statistical inference is generally mistitled the
Bonferroni adjustment, presumably through the more musical quality of
the name.  Nothing is ever named after its discoverer: Stigler's Law of
Eponymy.
\vspace{.5cm}

\begin{thebibliography}{3}
\bibitem{1}   Boole, G. (1952). {\it Studies in Logic and Probability.} (Ed. Rush
Rhees) Watts and Co., London.

\bibitem{2}  Hailperin, T. (1986). {\it Boole's Logic and Probability.} (2nd edn.) North
Holland, Amsterdam.

\bibitem{3}  Harley, R. (1866). George Boole, F.R.S. {\it British Quarterly
 Review}, July, 141-181. 
[Also as pp. 425-472 of Boole (1952)].

\bibitem{4}  Heath, P. (1967). Boole, George (1815-1864) {\it Encyclopedia
of Philosophy} (Paul Edwards, Ed.)
Macmillan Free Press, London, pp.346-347.

\bibitem{5}  Kneale, W. (1948). Boole and the Revival of Logic. {\it Mind},
{\bf 57}, 149-175.

\bibitem{6}  MacHale, D. (1985). {\it George Boole. His Life and Work}. 
Boole Press, Dublin.

\bibitem{7}  Seneta, E. (1992). On the History of the Strong Law of Large
Numbers and Boole's Inequality.
{\it Historia Mathematica} {\bf 19}, 24-39.
\vspace{1cm}

\hfill{Peter Heath and Eugene Seneta}
\end{thebibliography}


\end{document}

