Boole was the son of a shoemaker. He received a primary school education and learnedLatin and modern languages through various means. At 16, he began teaching to support his family. He established his own school at 19 and later ran a boarding school in Lincoln. Boole was an active member of local societies and collaborated with fellow mathematicians. In 1849, he was appointed the first professor of mathematics at Queen's College, Cork (now University College Cork) in Ireland, where he met his future wife,Mary Everest. He continued his involvement in social causes and maintained connections with Lincoln. In 1864, Boole died due to fever-inducedpleural effusion after developingpneumonia.
Boole published around 50 articles and several separate publications in his lifetime. Some of his key works include a paper on earlyinvariant theory and "The Mathematical Analysis of Logic", which introduced symbolic logic. Boole also wrote two systematic treatises: "Treatise on Differential Equations" and "Treatise on the Calculus of Finite Differences". He contributed to the theory of linear differential equations and the study of the sum of residues of a rational function. In 1847, Boole developed Boolean algebra, a fundamental concept in binary logic, which laid the groundwork for the algebra of logic tradition and forms the foundation of digital circuit design and modern computer science. Boole also attempted to discover a general method in probabilities, focusing on determining the consequent probability of events logically connected to given probabilities.
Boole's work was expanded upon by various scholars, such as Charles Sanders Peirce and William Stanley Jevons. Boole's ideas later gained practical applications whenClaude Shannon andVictor Shestakov employed Boolean algebra to optimize the design of electromechanical relay systems, leading to the development of modern electronic digital computers. His contributions to mathematics earned him various honours, including the Royal Society's first gold prize for mathematics, the Keith Medal, and honorary degrees from the Universities of Dublin and Oxford. University College Cork celebrated the 200th anniversary of Boole's birth in 2015, highlighting his significant impact on the digital age.
Boole was born in 1815 inLincoln,Lincolnshire, England, the son of John Boole Snr (1779–1848), a shoemaker[7] and Mary Ann Joyce.[8] He had a primary school education, and received lessons from his father, but due to a serious decline in business, he had little further formal and academic teaching.[9] William Brooke, a bookseller in Lincoln, may have helped him with Latin, which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern languages.[2] In fact, when a local newspaper printed his translation of a Latin poem, a scholar accused him of plagiarism under the pretence that he was not capable of such achievements.[10] At age 16, Boole became the breadwinner for his parents and three younger siblings, taking up a junior teaching position inDoncaster at Heigham's School.[11] He taught briefly inLiverpool.[1]
Greyfriars, Lincoln, which housed the Mechanic's Institute
At age 19, Boole successfully established his own school in Lincoln: Free School Lane.[15] Four years later he took over Hall's Academy inWaddington, outside Lincoln, following the death of Robert Hall. In 1840, he moved back to Lincoln, where he ran a boarding school.[1] Boole immediately became involved in the Lincoln Topographical Society, serving as a member of the committee, and presenting a paper entitled "On the origin, progress, and tendencies of polytheism, especially amongst the ancient Egyptians and Persians, and in modern India".[16]
From 1838 onwards, Boole was making contacts with sympathetic British academic mathematicians and reading more widely. He studiedalgebra in the form of symbolic methods, as far as these were understood at the time, and began to publish research papers.[1]
The house at 5 Grenville Place inCork, in which Boole lived between 1849 and 1855, and where he wroteThe Laws of Thought (picture taken during renovation)
Boole's status as a mathematician was recognised by his appointment in 1849 as the first professor of mathematics atQueen's College, Cork (nowUniversity College Cork (UCC)) in Ireland. He met his future wife,Mary Everest, there in 1850 while she was visiting her uncle John Ryall who was professor of Greek. They married in 1855.[20][21] He maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution.[22]
In 1861, Boole was involved in aJudgement in theCourt of Queen's Bench in Ireland against one John Hewitt Wheatley of Craig House, Sligo for the sum of £400, whereby Wheatley's estate and interest in lands of Maghan/Mahon, County Cork became vested in Boole.[23]
In March 1863, Boole leased Litchfield Cottage, Cork, the house in which he would live with his wife Mary until his death in December of the following year.[24] The premises was described in the deeds as "all that and those the dwelling house called Litchfield Cottage with the premises and appurtenances thereunto belonging and the Garden and Walled in field to the rere thereof". Boole's will bequeathed all his 'estate term and interest' in the lease of Litchfield Cottage unto his wife.[25] In August 1865, some 8 months after his death, Mary (by then living at 68 Harley Street, London) passed the house on to Francis Heard of Ballintemple, Cork, Esquire, a captain in her Majesty's 87th Regiment of South Cork.
Detail depicting his favourite Bible passage (content suggested by his widow), God's calling of the prophet Samuel (1 Samuel 3:1–10), a child dedicated to God by his parents[26]
Boole's first published paper was "Researches in the theory of analytical transformations, with a special application to the reduction of the general equation of the second order", printed in theCambridge Mathematical Journal in February 1840 (Volume 2, No. 8, pp. 64–73), and it led to his friendship withDuncan Farquharson Gregory, the editor of the journal.[20] His works are in about 50 articles and a few separate publications.[30][22]
In 1841, Boole published an influential paper in earlyinvariant theory.[14] He received a medal from theRoyal Society for his memoir of 1844, "On a General Method in Analysis".[20] It was a contribution to the theory oflinear differential equations, moving from the case of constant coefficients on which he had already published, to variable coefficients.[31] The innovation in operational methods is to admit that operations may notcommute.[32] In 1847, Boole publishedThe Mathematical Analysis of Logic, the first of his works on symbolic logic.[33]
Boole completed two systematic treatises on mathematical subjects during his lifetime. TheTreatise onDifferential Equations[34] appeared in 1859, and was followed, the next year, by aTreatise on theCalculus of Finite Differences,[35] a sequel to the former work.[20] Shortly after his death, Todhunter republished Boole's treatise with some of Boole's revisions, along with a supplement that was originally intended to be merged in the making of the second edition.
In 1857, Boole published the treatise "On the Comparison of Transcendent, with Certain Applications to the Theory of Definite Integrals",[36] in which he studied the sum ofresidues of arational function. Among other results, he proved what is now called Boole's identity:
for any real numbersak > 0,bk, andt > 0.[37] Generalisations of this identity play an important role in the theory of theHilbert transform.[37]
In 1847, Boole published the pamphletMathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system and wantedAn 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.[20] Contrary to widespread belief, Boole never intended to criticise or disagree with the main principles ofAristotle's logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability.[38] Boole's initial involvement in logic was prompted by a current debate onquantification, betweenSir William Hamilton who supported the theory of "quantification of the predicate", and Boole's supporterAugustus De Morgan who advanced a version ofDe Morgan duality, as it is now called. Boole's approach was ultimately much further reaching than either sides' in the controversy.[39] It founded what was first known as the "algebra of logic" tradition.[40]
Among his many innovations is his principle ofwholistic reference, which was later, and probably independently, adopted byGottlob Frege and by logicians who subscribe to standard first-order logic. A 2003 article[41] provides a systematic comparison and critical evaluation ofAristotelian logic andBoolean logic; it also reveals the centrality of holistic reference in Boole'sphilosophy of logic.
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 theuniverse of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.[42]
Boole conceived of "elective symbols" of his kind as analgebraic structure. But this general concept was not available to him: he did not have the segregation standard inabstract algebra of postulated (axiomatic) properties of operations, and deduced properties.[43] His work was a beginning to thealgebra of sets, again not a concept available to Boole as a familiar model. His pioneering efforts encountered specific difficulties, and the treatment of addition was an obvious difficulty in the early days.
Boole replaced the operation of multiplication by the word "and" and addition by the word "or". But in Boole's original system, + was apartial operation: in the language ofset theory it would correspond only to theunion of disjoint subsets. Later authors changed the interpretation, commonly reading it asexclusive or, or in set theory termssymmetric difference; this step means that addition is always defined.[40][44]
In fact, there is the other possibility generalizing Boole's original partial operation, that + should be read asnon-exclusive or.[43] Handling this ambiguity was an early problem of the theory, reflecting the modern use of bothBoolean rings and Boolean algebras (which are simply different aspects of one type of structure). Boole andJevons struggled over just this issue in 1863, in the form of the correct evaluation ofx +x. Jevons argued for the resultx, which is correct for + as disjunction. Boole kept the result as something undefined. He argued against the result 0, which is correct for exclusive or, because he saw the equationx +x = 0 as implyingx = 0, a false analogy with ordinary algebra.[14]
The second part of theLaws of Thought contained a corresponding attempt to discover ageneral method in probabilities. Here the goal was algorithmic: from the given probabilities of any system of events, to determine the consequent probability of any other event logically connected with those events.[45][20]
In late November 1864, Boole walked, in heavy rain, from his home at Lichfield Cottage inBallintemple[46] to the university, a distance of three miles, and lectured wearing his wet clothes.[47] He soon became ill, developing pneumonia. As his wife believed that remedies should resemble their cause, she wrapped him in wet blankets – the wet having brought on his illness.[47][48][49] Boole's condition worsened and on 8 December 1864,[50] he died of fever-inducedpleural effusion.
He was buried in theChurch of Ireland cemetery of St Michael's, Church Road,Blackrock (a suburb ofCork). There is a commemorative plaque inside the adjoining church.[51]
Boole is the namesake of the branch of algebra known asBoolean algebra, as well as the namesake of thelunar craterBoole. The keywordBool represents aBoolean data type in many programming languages, thoughPascal andJava, among others, both use the full nameBoolean.[52] The library, underground lecture theatre complex and the Boole Centre for Research in Informatics[53] atUniversity College Cork are named in his honour. A road calledBoole Heights in Bracknell, Berkshire is named after him.
In modern notation, thefree Boolean algebra on basic propositionsp andq arranged in aHasse diagram. The Boolean combinations make up 16 different propositions, and the lines show which are logically related.
In 1921, the economistJohn Maynard Keynes published a book on probability theory,A Treatise of Probability. Keynes believed that Boole had made a fundamental error in his definition of independence which vitiated much of his analysis.[55] In his bookThe Last Challenge Problem, David Miller provides a general method in accord with Boole's system and attempts to solve the problems recognised earlier by Keynes and others. Theodore Hailperin showed much earlier that Boole had used the correct mathematical definition of independence in his worked out problems.[56]
Boole's work and that of later logicians initially appeared to have no engineering uses.Claude Shannon attended a philosophy class at theUniversity of Michigan which introduced him to Boole's studies. Shannon recognised that Boole's work could form the basis of mechanisms and processes in the real world and that it was therefore highly relevant. In 1937 Shannon went on to write a master's thesis, at theMassachusetts Institute of Technology, in which he showed how Boolean algebra could optimise the design of systems of electromechanicalrelays then used in telephone routing switches. He also proved that circuits with relays could solve Boolean algebra problems. Employing the properties of electrical switches to process logic is the basic concept that underlies all modern electronicdigital computers.Victor Shestakov at Moscow State University (1907–1987) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematiciansSofya Yanovskaya, Gaaze-Rapoport,Roland Dobrushin, Lupanov, Medvedev and Uspensky. But the first publication of Shestakov's result took place only in 1941 (in Russian). Hence, Boolean algebra became the foundation of practicaldigital circuit design; and Boole, via Shannon and Shestakov, provided the theoretical grounding for theInformation Age.[57]
"Boole's legacy surrounds us everywhere, in the computers, information storage and retrieval, electronic circuits and controls that support life, learning and communications in the 21st century. His pivotal advances in mathematics, logic and probability provided the essential groundwork for modern mathematics, microelectronic engineering and computer science."
The year 2015 saw the 200th anniversary of Boole's birth. To mark the bicentenary year,University College Cork joined admirers of Boole around the world to celebrate his life and legacy.
UCC's George Boole 200[58] project, featured events, student outreach activities and academic conferences on Boole's legacy in the digital age, including a new edition ofDesmond MacHale's 1985 biography The Life and Work of George Boole: A Prelude to the Digital Age,[59] 2014.
The search engineGoogle marked the 200th anniversary of his birth on 2 November 2015 with an algebraic reimaging of itsGoogle Doodle.[4]
5, Grenville Place in 2017 following restoration by UCCBronze statue of Boole located atLincoln Central Train Station. The design, by sculptorAntony Dufort, was funded in part by the Heslam Trust.
In September 2022, a statue of George Boole in his role as a teacher was unveiled atLincoln Central Train Station, in Boole's home town ofLincoln.
Boole's views were given in four published addresses:The Genius of Sir Isaac Newton;The Right Use of Leisure;The Claims of Science; andThe Social Aspect of Intellectual Culture.[20] The first of these was from 1835 whenCharles Anderson-Pelham, 1st Earl of Yarborough gave a bust of Newton to the Mechanics' Institute in Lincoln.[60] The second justified and celebrated in 1847 the outcome of the successful campaign for early closing in Lincoln, headed by Alexander Leslie-Melville, ofBranston Hall.[61]The Claims of Science was given in 1851 at Queen's College, Cork.[62]The Social Aspect of Intellectual Culture was also given in Cork, in 1855 to the Cuvierian Society.[63]
Though his biographer Des MacHale describes Boole as an "agnostic deist",[64][65] Boole read a wide variety of Christian theology. Combining his interests in mathematics and theology, he compared theChristian trinity of Father, Son, and Holy Ghost with the three dimensions of space, and was attracted to the Hebrew conception of God as an absolute unity. Boole considered converting toJudaism but in the end was said to have chosenUnitarianism.[reference?] Boole came to speak against what he saw as "prideful" scepticism, and instead favoured the belief in a "Supreme Intelligent Cause".[66] He also declared "I firmly believe, for the accomplishment of a purpose of theDivine Mind."[67][68] In addition, he stated "To infer the existence of an intelligent cause from the teeming evidence of surroundingdesign, to rise to the conception of a moral Governor of the World, from the study of the constitution and the moral provisions of our own nature;--these, though but the feeble steps of an understanding limited in its faculties and its materials of knowledge, are of more avail than the ambitious attempt to arrive at a certainty unattainable on the ground of natural religion. And as these were the most ancient, so are they still the most solid foundations, Revelation being set apart, of the belief that the course of this world is not abandoned to chance and inexorable fate."[69][70]
Two influences on Boole were later claimed by his wife,Mary Everest Boole: a universal mysticism tempered byJewish thought, andIndian logic.[71] Mary Boole stated that an adolescent mystical experience provided for his life's work:
My husband told me that when he was a lad of seventeen a thought struck him suddenly, which became the foundation of all his future discoveries. It was a flash of psychological insight into the conditions under which a mind most readily accumulates knowledge ... For a few years he supposed himself to be convinced of the truth of "the Bible" as a whole, and even intended to take orders as a clergyman of the English Church. But by the help of a learned Jew in Lincoln he found out the true nature of the discovery which had dawned on him. This was that man's mind works by means of some mechanism which "functions normally towardsMonism."[72]
In Ch. 13 ofLaws of Thought Boole used examples of propositions fromBaruch Spinoza andSamuel Clarke. The work contains some remarks on the relationship of logic to religion, but they are slight and cryptic.[73] Boole was apparently disconcerted at the book's reception just as a mathematical toolset:
George afterwards learned, to his great joy, that the same conception of the basis of Logic was held byLeibniz, the contemporary of Newton. De Morgan, of course, understood the formula in its true sense; he was Boole's collaborator all along. Herbert Spencer, Jowett, andRobert Leslie Ellis understood, I feel sure; and a few others, but nearly all the logicians and mathematicians ignored [953] the statement that the book was meant to throw light on the nature of the human mind; and treated the formula entirely as a wonderful new method of reducing to logical order masses of evidence about external fact.[72]
Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830–65. What share had it in generating theVector Analysis and the mathematics by which investigations in physical science are now conducted?[72]
Boole maintained that:
No general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.[75]
In 1855, Boole marriedMary Everest (niece ofGeorge Everest), who later wrote several educational works on her husband's principles.
The Booles had five daughters:
Mary Ellen (1856–1908)[76] who married the mathematician and authorCharles Howard Hinton and had four children. After the sudden death of her husband in April 1907, Mary Ellen committed suicide inWashington, D.C., in May 1908.[77]
George Hinton (1882–1943), mining engineer and botanist
Her son Leonard Stott, a medical doctor and tuberculosis pioneer, invented a portableX-ray machine, apneumothorax apparatus, and system of navigation based on spherical coordinates.[80]
Lucy Everest (1862–1904), who was the first female professor of chemistry in England.
^Rhees, Rush. (1954) "George Boole as Student and Teacher. By Some of His Friends and Pupils",Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. Vol. 57. Royal Irish Academy
^A Selection of Papers relative to the County of Lincoln, read before the Lincolnshire Topographical Society, 1841–1842. Printed by W. and B. Brooke, High-Street, Lincoln, 1843.
^Ronald Calinger,Vita mathematica: historical research and integration with teaching (1996), p. 292;Google BooksArchived 27 April 2016 at theWayback Machine.
^Registry of Deeds, Dublin. Memorial: 1863-007-257 (extract). A Memorial of an Indenture of Re-Conveyance bearing date the seventh day of February one thousand eight hundred and sixty three made between William Hutchinson Massey of Mountmassey in the County of Cork, Esquire, of the first part, George Boole of Blackrock in the County of Cork, Esquire, Professor of Mathematics of the second part, Wilhelmina Smithwick of Dunmanway in said County of Cork, Spinster, of the third part and John Hewitt Wheatley of Craig House in the County of Sligo, Esquire, of the fourth part reciting a certain Indenture of Mortgage bearing date the fifth day of January one thousand eight hundred and sixty one whereby the said John Hewitt Wheatley in consideration of the sum.... And reciting that George Boole by the name and description of George Boole of Blackrock in the County of Corke, Professor of Mathematics did in or as of Trinity Term one thousand eight hundred and sixty one obtain a Judgement in the Court of Queen's Bench in Ireland against said John Hewitt Wheatley for the sum of four hundred pounds debt besides costs And reciting that said Judgement was duly registered on the ninth of November one thousand eight hundred and sixty one whereby pursuant to the statute in such Case made and provided the Estate and interest of the said John Hewitt Wheatley in said lands and premises became vested in one George Boole but subject to Redemption...
^Registry of Deeds, Dublin. Memorial: 1863-011-164 (extract). Registered: 30/03/1863. Memorial of an Intended Deed ... made between Edwards Casey then of Waterloo Place in the City of Cork, Esquire ... and George Boole then of Blackrock in the County of Cork, Esquire, L.L.D. then Professor of Mathematics in the Queens College at Cork ... After reciting that by Indenture of Lease bearing date the Twenty seventh day of March one thousand eight hundred and fifty six, John Litchfield then of Ballymaloo in the County of Cork, Esquire, did for the considerations therein mentioned demise and set unto the said William Jackson Cummins All That and Those the dwelling house with the premises and the Garden and walled in field to the rere thereof hereinafter particularly mentioned and described To Hold the said demised premises ... To Hold the said dwelling house and premises with the appurtenances unto the said George Boole, his Executors, Administrators and Assigns, from thenceforth for the then residue of the said term of one hundred years then to come and unexpired vested in him the said Edwards Casey...
^Registry of Deeds, Dublin. Memorial: 1865-030-121 (extract). Registered: 20/10/1865. Memorial of a certain Deed of Assignment bearing date the Twenty first day of August one thousand eight hundred and sixty five and made between Mary Boole of 68 Harley Street, London, Widow and Executrix of the Last Will and Testament of George Boole late of Litchfield Cottage Blackrock in the County of Cork Esquire L.L.D. deceased of the one part and Francis Heard of Ballintemple in the County of Cork, Esquire, Captain in her Majesty's eighty seventh Regiment of South Cork, Militia of the other part Whereby after reciting that by Indenture of Lease bearing date the twenty seventh day of March one thousand eight hundred and fifty six made between John Litchfield of Ballymaloo in the County of Cork, Esquire, of the one part and William Jackson Cummins of the City of Cork, Doctor in Medicine of the other part, the said John Litchfield demised unto the said William Jackson Cummins All that and those the dwelling house called Litchfield Cottage with the premises and appurtenances thereunto ... also reciting that the said George Boole ... having before his death duly made and published his last Will and Testament in writing and thereby bequeathed all his Estate term and interest in said hereinbefore recited Indenture of Lease and premises thereby demised unto the said Mary Boole party of said deed of which this is the Memorial and said Will was afterwards duly proved by the said Mary Boole in the Court of Probate District of Cork... witnesses as to the execution of said Deed and this Memorial by the said Mary Boole are witnessed by John Knights, Porter at Queens College, Harley Street, London and Jane White, Housekeeper 68 Harley Street, London.
^A list of Boole's memoirs and papers is in theCatalogue of Scientific Memoirs published by theRoyal Society, and in the supplementary volume on differential equations, edited byIsaac Todhunter. To theCambridge Mathematical Journal and its successor, theCambridge and Dublin Mathematical Journal, Boole contributed 22 articles in all. In the third and fourth series of thePhilosophical Magazine are found 16 papers. The Royal Society printed six memoirs in thePhilosophical Transactions, and a few other memoirs are to be found in theTransactions of theRoyal Society of Edinburgh and of theRoyal Irish Academy, in theBulletin de l'Académie de St-Pétersbourg for 1862 (under the name G. Boldt, vol. iv. pp. 198–215), and inCrelle's Journal. Also included is a paper on the mathematical basis of logic, published inThe Mechanics' Magazine in 1848.
^abCima, Joseph A.; Matheson, Alec; Ross, William T. (2005). "The Cauchy transform".Quad domains and their applications. Oper. Theory Adv. Appl. Vol. 156. Basel: Birkhäuser. pp. 79–111.MR2129737.
^John Corcoran, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288.
^abWitold Marciszewski (editor),Dictionary of Logic as Applied in the Study of Language (1981), pp. 194–195.
^Corcoran, John (2003). "Aristotle's Prior Analytics and Boole's Laws of Thought".History and Philosophy of Logic,24: 261–288. Reviewed by Risto Vilkko.Bulletin of Symbolic Logic,11(2005) 89–91. Also by Marcel Guillaume,Mathematical Reviews 2033867 (2004m:03006).
^George Boole. 1854/2003.The Laws of Thought, facsimile of 1854 edition, with an introduction byJohn Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review.24 (2004) 167–169.
^Burris, Stanley (2 September 2018). Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.Archived from the original on 2 September 2019. Retrieved2 September 2019 – via Stanford Encyclopedia of Philosophy.
^"George Boole".Encyclopædia Britannica. 30 January 2017.Archived from the original on 7 December 2017. Retrieved7 December 2017.
^Chapter XVI, p. 167, section 6 ofA treatise on probability, volume 4: "The central error in his system of probability arises out of his giving two inconsistent definitions of 'independence' (2) He first wins the reader's acquiescence by giving a perfectly correct definition: "Two events are said to be independent when the probability of either of them is unaffected by ourexpectation of the occurrence or failure of the other." (3) But a moment later he interprets the term in quite a different sense; for, according to Boole's second definition, we must regard the events as independent unless we are told either that theymust concur or that theycannot concur. That is to say, they are independent unless we know for certain that there is, in fact, an invariable connection between them. "The simple events,x,y,z, will be said to beconditioned when they are not free to occur in every possible combination; in other words, when some compound event depending upon them is precluded from occurring. ... Simple unconditioned events are by definition independent." (1) In fact as long asxz ispossible,x andz are independent. This is plainly inconsistent with Boole's first definition, with which he makes no attempt to reconcile it. The consequences of his employing the term independence in a double sense are far-reaching. For he uses a method of reduction which is only valid when the arguments to which it is applied are independent in the first sense and assumes that it is valid if they are independent in the second sense. While his theorems are true if all propositions or events involved are independent in the first sense, they are not true, as he supposes them to be, if the events are independent only in the second sense."
^"That dissertation has since been hailed as one of the most significant master's theses of the 20th century. To all intents and purposes, its use of binary code and Boolean algebra paved the way for the digital circuitry that is crucial to the operation of modern computers and telecommunications equipment."Emerson, Andrew (8 March 2001)."Claude Shannon".The Guardian.Archived from the original on 10 April 2019. Retrieved14 December 2016.
^James Gasser,A Boole Anthology: recent and classical studies in the logic of George Boole (2000), p. 5;Google BooksArchived 10 May 2016 at theWayback Machine.
^"A tale of two amateurs".Semiotica, Volume 105. Mouton. 1995. p. 56.MacHale's biography calls George Boole 'an agnostic deist'. Both Booles' classification of 'religious philosophies' as monistic, dualistic, and trinitarian left little doubt about their preference for 'the unity religion', whether Judaic or Unitarian.
^Semiotica, Volume 105. Mouton. 1996. p. 17.MacHale does not repress this or other evidence of the Boole's nineteenth-century beliefs and practices in the paranormal and in religious mysticism. He even concedes that George Boole's many distinguished contributions to logic and mathematics may have been motivated by his distinctive religious beliefs as an "agnostic deist" and by an unusual personal sensitivity to the sufferings of other people.
^Boole, George. Studies in Logic and Probability. 2002. Courier Dover Publications. pp. 201-202
^Boole, George. Studies in Logic and Probability. 2002. Courier Dover Publications. p. 451
^Some-Side of a Scientific Mind (2013). pp. 112–193. The University Magazine, 1878. London: Forgotten Books. (Original work published 1878)
^Concluding remarks of his treatise of "Clarke and Spinoza", as found in Boole, George (2007). An Investigation of the Laws of Thought. Cosimo, Inc. Chap . XIII. p. 217-218. (Original work published 1854)
^Boole, George (1851). The claims of science, especially as founded in its relations to human nature; a lecture, Volume 15. p. 24
^abcBoole, Mary EverestIndian Thought and Western Science in the Nineteenth Century, Boole, Mary EverestCollected Works eds. E. M. Cobham and E. S. Dummer, London, Daniel 1931 pp. 947–967
^'My Right To Die', Woman Kills Self inThe Washington Times v. 28 May 1908 (PDFArchived 5 June 2012 at theWayback Machine);Mrs. Mary Hinton A Suicide inThe New York Times v. 29 May 1908 (PDFArchived 25 February 2021 at theWayback Machine).
The Calculus of Logic by George Boole; a transcription of an article which originally appeared inCambridge and Dublin Mathematical Journal, Vol. III (1848), pp. 183–198.
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