George Boole (1815–1864) was an English mathematician and afounder of the algebraic tradition in logic. He worked as aschoolmaster in England and from 1849 until his death as professor ofmathematics at Queen’s University, Cork, Ireland. Herevolutionized logic by applying methods from the then-emerging fieldof symbolic algebra to logic. Where traditional (Aristotelian) logicrelied on cataloging the valid syllogisms of various simple forms,Boole’s method provided general algorithms in an algebraiclanguage which applied to an infinite variety of arguments ofarbitrary complexity. These results appeared in two major works,The Mathematical Analysis of Logic (1847) andThe Laws ofThought (1854).

• 1. Life and Work
• 2. The Context and Background of Boole’s Work In Logic
• 3. The Mathematical Analysis of Logic (1847)
  • 3.1 Boole’s Version Of Aristotelian Logic
  • 3.2 Class Symbols and Elective Symbols
  • 3.3 Operations and Laws for Elective Symbols
  • 3.4 Common Algebra
  • 3.5 Impact of the Index Law
  • 3.6 Equational Expressions of Categorical Propositions
  • 3.7 Hypothetical Syllogisms
  • 3.8 General Theorems of Boole’s Algebra inMAL
• 4. The Laws of Thought (1854)
• 5. Later Developments
  • 5.1 Objections to Boole’s Algebra of Logic
  • 5.2 Modern Reconstruction of Boole’s System
• 6. Boole’s Methods
  • 6.1 The Three Methods of Argument Analysis Used by Boole inLT
  • 6.2. Boole’s General Method for Primary Propositions
  • 6.3. Boole’s General Method for Secondary Propositions
• Bibliography
  • Primary Literature
  • Secondary Literature
• 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 wasevidently more of a good companion than a good breadwinner. His fatherwas a shoemaker whose real passion was being a devoted dilettante inthe realm of science and technology, one who enjoyed participating inthe Lincoln Mechanics’ Institution; this was essentially a communitysocial club promoting reading, discussions, and lectures regardingscience. It was founded in 1833, and in 1834 Boole’s father became thecurator of its library. This love of learning was clearly inherited byBoole. Without the benefit of an elite schooling, but with asupportive family and access to excellent books, in particular fromSir Edward Bromhead, FRS, who lived only a few miles from Lincoln,Boole was able to essentially teach himself foreign languages andadvanced mathematics.

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

There is a widespread belief that Boole was primarily alogician—in reality he became a recognized mathematician wellbefore he had penned a single word about logic, all the while runninghis private school to care for his parents and siblings. Boole’sability to read French, German and Italian put him in a good positionto start serious mathematical studies when, at the age of 16, he readLacroix’sCalcul Différentiel, a gift from his friendReverend G.S. Dickson of Lincoln. Seven years later, in 1838, he wouldwrite his first mathematical paper (although not the first to bepublished), “On certain theorems in the calculus ofvariations,” focusing on improving results he had read inLagrange’sMéchanique Analytique.

In early 1839 Boole travelled to Cambridge to meet with the youngmathematician Duncan F. Gregory (1813–1844), the editorof theCambridge Mathematical Journal(CMJ)—Gregory had co-founded this journal in 1837 andedited it until his health failed in 1843 (he died in early 1844, atthe age of 30). Gregory, though only 2 years beyond his degree in1839, 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 in1841.

Boole’s mathematical publications span the 24 years from 1841 to 1864,the year he died from pneumonia. Breaking these 24 years into threesegments, the first 6 years (1841–1846), the second 8 years (1847–1854), and the last 10 years (1855–1864), we find that his published 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 theCMJ and its 1846 successor,TheCambridge and Dublin Mathematical Journal. He wrote on standardmathematical topics, mainly differential equations, integration and thecalculus of variations. Boole enjoyed early success in using the newsymbolical method in analysis, a method which took a differentialequation, say:

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

leading to an expression of the differential equation as:

Now symbolical algebra came into play by simply treating the operator\(D^2 - D - 2\) as though it were an ordinary polynomial inalgebra. Boole’s 1841 paperOn the integration of lineardifferential equations with constant coefficients gave a niceimprovement to Gregory’s method for solving such differentialequations, an improvement based on a standard tool in algebra, partial fractions, which he applied to the reciprocal of differential operators like the above.

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

In 1843 the schoolmaster Boole finished a lengthy paper ondifferential equations, combining an exponential substitution andvariation of parameters with the separation of symbols method. Thepaper was too long for theCMJ—Gregory, and later DeMorgan, encouraged him to submit it to the Royal Society. The firstreferee rejected Boole’s paper, but the second recommended it for theGold Medal for the best mathematical paper written in the years1841–1844, and this recommendation was accepted. In 1844 theRoyal 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 BritishAssociation 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 tofind a way to become affiliated with an institution of higher learning.He considered attending Cambridge University to obtain a degree, butwas counselled that fulfilling the various requirements would likelyseriously interfere with his research program, not to mention theproblems of obtaining financing. Finally, in 1849, he obtained aprofessorship in a new university opening in Cork, Ireland. In theyears he was a professor in Cork (1849–1864) he wouldoccasionally inquire about the possibility of a position back inEngland.

The 8 year stretch from 1847 to 1854 starts and ends with Boole’stwo books on mathematical logic. In addition Boole published 24 morepapers on traditional mathematics during this period, while only onepaper was written on logic, that being in 1848. He was awarded anhonorary LL.D. degree by the University of Dublin in 1851, and this wasthe title that he used beside his name in his 1854 book on logic.

Boole’s 1847 book,Mathematical Analysis of Logic, will bereferred to asMAL; the 1854 book,Laws of Thought,asLT.

During the last 10 years of his career, from 1855 to 1864, Boolepublished 17 papers on mathematics and two mathematics books, one ondifferential equations and one on difference equations. Both books werehighly regarded, and used for instruction at Cambridge. Alsoduring 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 arainstorm in late 1864, and then lecturing in wet clothes. Not longafterwards, on December 8, 1864 in Ballintemple, County Cork, Ireland,he died of pneumonia, at the age of 49. Another paper on mathematicsand a revised book on differential equations, giving considerableattention to singular solutions, were published post mortem.

The reader interested in excellent accounts ofBoole’s personal life is referred to Desmond MacHale’sGeorgeBoole, His Life and Work, 1985/2014,and the more recent bookNew Light on George Boole, 2018, by Desmond MacHale and Yvonne Cohen,sources to which this article is greatlyindebted.

• 1815 — Birth in Lincoln, England
• 1830 — His translation of a Greek poem printed in a local paper
• 1831 — Reads Lacroix’sCalcul 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 theCambridge Mathematical Journal (CMJ)
• 1841 — First four mathematical publications (all in theCMJ)
• 1842 — Initiates correspondence with Augustus De Morgan— they become lifelong friends
• 1844 — Correspondence with Cayley starts (initiated byCayley) — they become lifelong friends
• 1844 — Gold Medal from the Royal Society for a paper ondifferential equations
• 1845 — Gives talk at the Annual Meeting of the BritishAssociation for the Advancement of Science, and meets William Thomson(later Lord Kelvin) — they become lifelong friends
• 1847 — PublishesMathematical Analysis of Logic
• 1848 — Publishes his only paper on the algebra of logic
• Professor of Mathematics
• 1849 — Accepts position as (the first) Professor of Mathematics at the new Queen’s University in Cork, Ireland
• 1851 — Honorary Degree, LL.D., from Trinity College, Dublin
• 1854 — PublishesLaws 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 — PublishesDifferential Equations; used as atextbook at Cambridge
• 1860 — Birth of Alicia Boole, who will coin the word“polytope”
• 1860 — PublishesDifference Equations ; used as atextbook at Cambridge
• 1862 — Birth of Lucy Everest Boole
• 1864 — Birth of daughter Ethel Lilian Boole, who would writeThe Gadfly, an extraordinarily popular book in Russia afterthe 1917 revolution
• 1864 — Death from pneumonia, Cork, Ireland

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

To understand how Boole developed hisalgebra of logic, it is useful to review the broadoutlines of the work on the foundations of algebra that had beenundertaken by mathematicians affiliated with Cambridge University inthe 1800s prior to the beginning of Boole’s mathematical publishingcareer. An excellent reference for further reading connected to thissection is the annotated sourcebookFrom Kant to Hilbert,1996, by William Ewald, which contains a complete copy of Boole’sMathematical Analysis of Logic.

The 19th century opened in England with mathematics in the doldrums.The English mathematicians had feuded with the continentalmathematicians over the issues of priority in the development of thecalculus, resulting in the English following Newton’s notation, andthose on the continent following that of Leibniz. One of the obstaclesto overcome in updating English mathematics was the fact that the greatdevelopments of algebra and analysis had been built on dubiousfoundations, and there were English mathematicians who were quite vocalabout these shortcomings. In ordinary algebra, it was the use ofnegative numbers and imaginary numbers that caused concern.

The first major attempt among the English to clear up the foundationproblems of algebra was theTreatise on Algebra, 1830, byGeorge Peacock (1791–1858). A second edition appeared as twovolumes, 1842/1845. He divided the subject into two parts, the firstpart beingarithmetical algebra, the algebra of the positivenumbers (which did not permit operations like subtraction in caseswhere the answer would not be a positive number). The second part wassymbolical algebra, which was governed not by a specificinterpretation, as was the case for arithmetical algebra, but solelyby laws. In symbolical algebra there were no restrictions on usingsubtraction, etc.

Peacock believed that in order for symbolical algebra to be a usefulsubject its laws had to be closely related to those of arithmeticalalgebra. In this connection he introduced hisprinciple of thepermanence of equivalent forms, a principle connecting results inarithmetical algebra to those in symbolical algebra. This principle hastwo parts:

1. General results in arithmetical algebra belong to the lawsof symbolical algebra.
2. Whenever an interpretation of a result of symbolical algebramade sense in the setting of arithmetical algebra, the result wouldgive a correct result in arithmetic.

A fascinating use of algebra was introduced in 1814 byFrançois-Joseph Servois (1776–1847) when he tackleddifferential equations by separating the differential operator from the subject, as described in an example givenabove. This application of algebra captured the interest of Gregorywho published a number of papers on the method of theseparationof symbols, that is, the separation into operators and objects,in theCMJ. He also wrote on the foundation of algebra, andit was Gregory’s foundation that Boole embraced almost verbatim prior to writingLT.Gregory had abandoned Peacock’s principle of the permanence ofequivalent forms in favor of three simple laws, one of which Booleregarded as merely a notation convention. Unfortunately these lawsfell far short of what is required to justify even some of the mostelementary results in algebra, like those involving subtraction.

InOn the foundation of algebra, 1839, the first of four papers on this topic by De Morgan that appeared in theTransactions of the CambridgePhilosophical Society, one finds a tribute to the separation ofsymbols in algebra, and the claim that modern algebraists usuallyregard the symbols as denoting operators (e.g., the derivativeoperation) instead of objects like numbers. The footnote

Professor Peacock is the first, I believe, who distinctlyset forth the difference between what I have called the technical andthe logical branches of algebra

credits Peacock with being the first to separate what are now called the syntactic [technical] and the semantic [logical] aspects of algebra. In the secondfoundations paper (in 1841) De Morgan proposed what he considered to bea complete set of eight rules for working with symbolical algebra.

Regarding the origin of the nameBoolean algebra, CharlesSanders Peirce (1839–1914) introduced, among several otherphrases, the nameBoolian algebra for the algebra thatresulted from dispensing with the arithmetical scaffolding ofBoole’s equational algebra of logic. With the spellingBoolean algebra this was embraced by his close friend theHarvard philosopher Josiah Royce (1855–1916) around 1900, andthen by Royce’s students (including Norbert Wiener, HenryM. Sheffer and Clarence I. Lewis) and in due course by other Harvardprofessors and the world. It essentially referred to the modernversion of the algebra of logic introduced in 1864 by William StanleyJevons (1835–1882), a version that Boole had rejected in theircorrespondence—see Section 5.1. For this reason the wordBoolean will not be used in this article to describe thealgebra of logic that Boole actually created; instead the nameBoole’s algebra will be used. (See the 2015 article“George Boole and Boolean Algebra” by Burris.)

InMAL, and more so inLT, Boole was interested in the insightsthat his algebra of logic gave to the inner workings of the mind. This pursuithas met with little favor, and is not discussed in this article.

3. The Mathematical Analysis of Logic (1847)

InNew Light on George Boole (2018) by MacHale and Cohen onefinds, published for the first time, (an edited version of) thebiography by MaryAnn Boole (1818–1887) of her famous brother,and on p. 41 there is the following passage:

He told me that from boyhood he had had the conviction that Logiccould be reduced to a mathematical science, and that he had often madehimself ill on the attempt to prove it, but it was not until 1847 thatthe true method flashed upon him.

Boole’s final path to logic fame occurred in a curious way. Inearly 1847 he was stimulated to renew his investigations into logic bya trivial but very public dispute between De Morgan and the Scottishphilosopher Sir William Hamilton (1788–1856)—not to beconfused with his contemporary the Irish mathematician Sir WilliamRowan Hamilton (1805–1865). This dispute revolved around whodeserved credit for the idea of quantifying the predicate (e.g.,All \(A\) is all \(B\),All \(A\) is some \(B\),etc.). MaryAnn wrote that when the true method flashed upon him in1847, “he was literally like a man dazzled with excess oflight”. Within a few months Boole had written his 82 pagemonograph,Mathematical Analysis of Logic, first presentingan algebraic approach to Aristotelian logic, then looking briefly atthe general theory. (Some say that this monograph and DeMorgan’s bookFormal Logic appeared on the same day inNovember 1847.)

We are not told what the true method was that flashed upon Boole. Onepossibility is the discovery of the Expansion Theorem and theproperties of constituents.

3.1 Boole’s Version Of Aristotelian Logic

In pages 15–59, a little more than half of the 82 pages inMAL, Boole focused on a slight generalization of Aristotelianlogic, namely augmenting its four types of categorical propositions bypermitting the subject and/or predicate to be of the formnot-\(X\). In the chapter on conversions, such as Conversion byLimitation—All \(X\) is \(Y\), therefore Some \(Y\) is\(X\)—Boole found the Aristotelian classification defective inthat it did not treat contraries, such as not-\(X\), on the samefooting as the named classes \(X, Y, Z\), etc. For example, he convertedNo \(X\) is \(Y\) intoAll\(Y\) is not-\(X\), andAll X is Y intoAll not-Y is not-X.

For his extended version of Aristotelian logic he stated (MAL, p. 30) a setof three transformation rules which he claimed allowed one to construct all validtwo-line categorical arguments. These transformation rules did not appear inLT. It is somewhatcurious that when it came to analyzing categorical syllogisms, it was onlyin the conclusion that he permitted his generalized categorical propositionsto appear.Among the vast possibilities for hypothetical syllogisms, the ones that he discussed were standard, with one new example added.

3.2 Class Symbols and Elective Symbols

TheIntroduction chapter ofMAL starts with Boole reviewing thesymbolical method:

the validity of the processes of analysis does not depend uponthe interpretation of the symbols which are employed, but solely uponthe laws of their combination.

The second chapter,FirstPrinciples, lets the symbol 1 represent the Universe “comprehending every conceivable class of objects, whetheractually existing or not.” Capital letters \(X, Y,Z,\ldots\) denoted classes. Then, no doubt heavilyinfluenced by his very successful work using algebraic techniques ondifferential operators, and consistent with De Morgan’s 1839 assertionthat algebraists preferred interpreting symbols as operators, Booleintroduced the elective symbol \(x\) corresponding to the class\(X\), the elective symbol \(y\) corresponding to\(Y\), etc. Theelective symbols denoted electiveoperators—for example the elective operatorredwhen applied to a class would elect (select) the red items in theclass.

Then Boole said “When no subject is expressed, we shall suppose 1 (the Universe) to be the subject understood”. He goes on to explain that \(x(1)\) is just \(X\). Evidently this means, asidefrom defining the elective symbol \(x\), when presented with the term \(x\) withouta subject one is actually dealing with \(X\).

3.3 Operations and Laws for Elective Symbols

The first operation that Boole introduced wasmultiplication\(xy\). The standard juxtaposition notation \(xy\) for multiplication also had a standard meaning for operators (for example, differential operators), namely one applied \(y\) to an object and then \(x\) is applied to theresult. As pointed out by Theodore Hailperin (1916–2014) (1981, p. 176; and 1986, pp. 67,68), this established notationconvention handed Boole his interpretation of the multiplication \(xy\) ofelective symbols as thecomposition of the two operators. Thus when encounteringthe expression \(xy\) without a subject one was dealing with the class\(x(y(1))\), “the result being the class whose members are both Xs and Ys”. We call this class the intersectionof \(X\) and \(Y\). InLT Boole dropped the (unnecessary) use of elective symbols and simply let \(x\), \(y\) denote classes, with \(xy\) being their intersection.

The first law inMAL (p. 16) was thedistributive law

where Boole said that \(u+v\) corresponded to dividing a class into two parts, evidently meaning \(U\) and \(V\) are disjoint classes. This was the first mention of addition inMAL.

He added (MAL, p. 17) thecommutative law \(xy =yx\) and theindex law \(x^n = x\)—inLT thelatter would be replaced by thelaw of duality \(x^2 = x\)(called theidempotent law in 1870 by the Harvardmathematician Benjamin Peirce (1809–1880), in anothercontext).

After stating the above distributive and commutative laws, Boolebelieved he was entitled to fully employ the ordinary algebra of histime, saying (MAL, p. 18) that

all the processes of Common Algebra are applicable to the present system.

Boole went beyond the foundations of symbolical algebra thatGregory had used in 1840—he added De Morgan’s 1841 singlerule of inference, that equivalent operations performed uponequivalent subjects produce equivalent results.

3.4 Common Algebra

It is likely more difficult for the modern reader to come to gripswith the idea that Boole’s algebra is based on Common Algebra, the algebra of numbers, than would have been the case with Boole’s contemporaries—the modernreader has been exposed to modern Boolean algebra (and perhaps Booleanrings). In the mid 1800s the wordalgebra meant, for most mathematicians, simply the algebra of numbers.

Boole’s three laws for his algebra of logic are woefully inadequatefor what follows inMAL. The reader will, for the most part,be well served by assuming that Boole is doing ordinary polynomialalgebra augmented by the assumption that any power\(x^n\) of an elective symbol \(x\) can be replaced by \(x\). One can safely assume that any polynomial equationp = q that holds in Common Algebra is validin Boole’s algebra as is any equational argument

that holds in Common Algebra.

[A note of caution: the argument “\(x^2 = x \therefore x = 1\) or\(x = 0\)” is valid in Common Algebra, but it isnot anequational argument since the conclusion is a disjunction ofequations, not a single equation.]

Boole’s algebrawas mainly concerned with polynomials with integer coefficients, andwith their values when the variables were restricted to taking on onlythe values 0 and 1.Some of the key polynomials in Boole’s work, along with their valueson \(\{0,1\}\), are presented in the following table:

Note that all of the polynomials \(p\)(x,y) in the abovetable, except for addition and subtraction, take values in \(\{0,1\}\) whenthe variables take values in \(\{0,1\}\). Such polynomials are calledswitching functions in computer science and electricalengineering, and as functions on \(\{0,1\}\) they are idempotent, that is,p\(^2 =\) p. The switching functions are exactlythe idempotent polynomials in Boole’s algebra.

3.5 Impact of the Index Law

In Boole’s algebra, any polynomial \(p(x)\)in one variable can be reduced to a linear polynomial \(ax + b\)since one has

Likewise any polynomial \(p(x, y)\) can be expressed as \(axy + bx +cy + d\). Etc.

However Boole was much more interested in the fact that \(ax + b\)can be written as a linear combination of \(x\) and \(1-x\), namely

This gives hisExpansion Theorem in one variable:

The Expansion Theorem for polynomials in two variables is

For example,

The expressions \(xy, \ldots, (1 - x)(1 - y)\), were calledtheconstituents of \(p(x,y)\)—it would be better tocall them the constituents of the variables \(x, y\)—and thecoefficients \(p(1,1), \ldots, p(0,0)\) were themodulii of\(p(x,y)\).

Similar results hold for polynomials in any number of variables(MAL, pp. 62–64), and there are threeimportant facts about the constituents for a given list ofvariables:

1. each constituent is idempotent,
2. the product of two distinct constituents is 0,
3. the sum of all the constituents is 1.

3.6 Equational Expressions of Categorical Propositions

In the chapterOf Expression and Interpretation, Boolesaid “the class not-X will be determined by the symbol \(1-x\)”. This is the first appearance ofsubtractioninMAL. Boole’s initial equational expressions of theAristotelian categorical propositions (MAL, pp. 21,22) will be called hisprimary expressions. Then in the next several pages he addssupplementary expressions; of these the main ones will be called thesecondary expressions.

The first primary expression given was forAll \(X\) is\(Y\), an equation which he then converted into\(x(1-y) = 0\). This was the first appearance of0 inMAL. It was not introduced as the symbol for the emptyclass—indeed the empty class does not appear inMAL.Evidently “\(= 0\)” performed the role of a predicate inMAL, with an equation \(E = 0\) asserting that the classdenoted by \(E\) simply did not exist. (InLT, what we call theempty class was introduced and denoted by 0.)

Syllogistic reasoning is just an exercise inelimination,namely the middle term is eliminated from the premises to give theconclusion. Elimination is a standard topic in the theory ofequations, and Boole borrowed a simple elimination result regardingtwo equations to use in his algebra of logic—if the premises ofa syllogism involved the classes \(X, Y\), and\(Z\), and one wanted to eliminate the middle term \(Y\),then Boole put the equations for the two premises in the form

where \(y\) does not appear in the coefficientsa,b,c,d.The result of eliminating \(y\) in ordinary algebra gives theequation

and this is what Boole used inMAL. Unfortunately this is aweak elimination result for Boole’s algebra. One finds, using theimproved reduction and elimination theorems ofLT, that thebest possible result of elimination is

Applying weak elimination to the primary equational expressions was not sufficient to derive all of the valid syllogisms. For example, in the cases where the premises had primary expressions \(ay = 0\) and \(cy = 0\),this elimination gave \(0 = 0\), even when there was a non-trivial conclusion. Boole introduced the alternative equationalexpressions (seeMAL, p. 32) of categorical propositions tobe able to derive all of the valid syllogisms.

Toward the end of the chapter on categorical syllogisms there is along footnote (MAL, pp. 42–45) claiming(MAL, pp. 42, 43) that secondary expressions alone aresufficient for the analysis of [his generalization of] Aristoteliancategorical logic. The footnote loses much of its force because theresults it presents depend heavily on the weak elimination theorembeing best possible, which is not the case. Regarding the secondaryexpressions, in the Postscript toMAL he says:

The system of equations there given for the expression of Propositionsin Syllogism is always preferable to the one beforeemployed—first, in generality—secondly, in facility ofinterpretation.

His justification of this claim would appear inLT. Indeed Boole used only the secondary expressions ofMAL to express propositions as equations inLT, but there the reader will no longer find a leisurely and detailed treatment of Aristotelian logic—the discussion of this subject is delayed until the last chapter on logic, namely Chapter XV (the only one inLT to analyze particular propositions). In this chapter the application of hisalgebra of logic to Aristotelian logic is presented in such a compressed form (by omitting all details of the reduction, elimination and solution steps), concluding with such long equations, that the reader is not likely to want to check that Boole’s analysis is correct.

3.7 Hypothetical Syllogisms

On p. 48 ofMAL Boole said:

A hypothetical Proposition is defined to betwo or morecategoricals united by a copula (or conjunction), and thedifferent kinds of hypothetical Propositions are named from theirrespective conjunctions, viz. conditional (if), disjunctive (either,or), &c.

Boole analyzed the sevenhypothetical syllogisms that werestandard in Aristotelian logic, from the Constructive and DestructiveConditionals to the Complex Destructive Dilemma. Letting capitalletters \(X, Y, \ldots\) represent categorical propositions, thehypothetical propositions traditionally involved inhypothetical syllogisms were in one of the forms\(X\) istrue,\(X\) is false,If \(X\) is truethen \(Y\) is true,\(X\) is true or \(Y\) is true or …, as well as\(X\) is true and \(Y\) is true and … At the end of thechapter on hypothetical syllogisms he noted that it was easy to createnew ones, and one could enrich the collection by using mixedhypothetical propositions such asIf \(X\) is true, theneither \(Y\) is true, or \(Z\) is true.

Most important in this chapter was Boole’s claim that his algebra oflogic for categorical propositions was equally suited to the study ofhypothetical syllogisms. This was based on adopting the standardreduction of hypothetical propositions to propositions about classesby letting thehypothetical universe, also denoted by 1, “comprehend all conceivablecases and conjunctures of circumstances”.Evidently his notion of acase was an assignment of truth values to the propositional variables.For \(X\) a categorical proposition Boole let \(x\) denote the electiveoperator that selects the cases for which \(X\) is true.

Boole said theuniverse of a categorical proposition has twocases,true andfalse. To find an equationalexpression for a hypothetical proposition Boole resorted to a nearrelative of truth tables (MAL, p. 50). To each case, that is,assignment of truth values to \(X\) and \(Y\), he associated anelective expression as follows:

These elective expressions are, of course, theconstituentsof \(x\), \(y\).

Boole expressed a propositional formula \(\Phi(X,Y, \ldots)\) byan elective equation \(\phi(x,y, \ldots)\) = 1 by ascertaining all thedistinct cases (assignments of truth values) for which the formulaholds,and summing their corresponding elective expressions to obtain \(\phi\).

For example, the elective expression for\(X\) is true or \(Y\) is true, withor inclusive, is thus \(xy + x(1 - y) + (1 - x)y = 1\),which simplifies to \(x + y - xy = 1\).

Boole did not have the modern view that a propositional formula can beconsidered as a function on the truth values \(\{\rT, \rF\}\), taking values in \(\{\rT,\rF\}\). The function viewpoint gives us an algorithm to determine whichconstituents are to be summed to give the desired elective expression,namely those constituents associated with the cases for which thepropositional formula has the value \(\rT\).

By not viewing propositional formulas as functions on \(\{\)T, F\(\}\) Boolemissed out on being the inventor of truth tables. His algebraicmethod of analyzing hypothetical syllogisms was to transform each ofthe hypothetical premises into an elective equation, and then applyhis algebra of logic (which was developed for categoricalpropositions). For example, the premises\(X\) is trueor \(Y\) is true, withor inclusive, and\(X\) is false are expressed by the equations\(x + y - xy = 1\) and\(x = 0\). From these it immediately follows that\(y = 1\), giving the conclusion\(Y\) is true.

Boole only considered rather simple hypothetical propositions on thegrounds these were the only ones encountered in common usage (seeLT, p. 172). His algebraic approach to propositional logicis easily extended to all propositional formulas as follows. For\(\Phi\) a propositional formula the associated elective function\(\Phi^*\) is defined recursively as follows:

• \(0^* = 0\); \(1^* = 1\); \(X^* = x\);
• \((\text{not-}\Phi)^* = 1 - \Phi^*\);
• \((\Phi \text{ and } \Psi)^* = \Phi^* \cdot \Psi^*\);
• \((\Phi \text{ or } \Psi)^* = \Phi^* + \Psi^* - \Phi^* \cdot\Psi^*\),  where “or” is inclusive;
• \((\Phi \text{ or } \Psi)^* = \Phi^* + \Psi^* - 2\Phi^* \cdot \Psi^*\), where “or” is exclusive;
• \((\Phi \text{ implies } \Psi)^* = 1 - \Phi^* + \Phi^* \cdot\Psi^*\);
• \((\Phi \text{ iff } \Psi)^* = \Phi^* \cdot \Psi^* + (1 - \Phi^*)\cdot(1 - \Psi^*)\).

Then one has:

1. \(\Phi\) is a tautology iff \(\Phi^* = 1\) is valid in Boole’s algebra.
2. \(\Phi_1\), ... , \(\Phi_k \therefore \Phi\) is valid inpropositional logic iff
   \(\Phi_{1}^* = 1, \ldots , \Phi_{k}^* = 1\therefore \Phi^* = 1\) is valid in Boole’s algebra.

This looks quite different from modern propositional logic where onetakes a few tautologies, such as \(X \rightarrow(Y \rightarrow X)\), as axioms, and inference rules such as modus ponens toform a deductive system.

This translation, from \(\Phi\) to \(\Phi^*\), viewed as mappingexpressions from modern Boolean algebra to polynomials, would bepresented in the 1933 paperCharacteristic functions and the algebra of logic by Hassler Whitney (1907–1989), withthe objective of showing that one does not need to learn the algebraof logic [modern Boolean algebra] to verify the equational laws andequational arguments of Boolean algebra—they can be translatedinto the ordinary algebra with which one is familiar. Howard Aiken(1900–1973), Director of the Harvard Computation Laboratory,would use such translations of logical functions into ordinary algebrain his 1951 bookSynthesis of Electronic Computing and ControlCircuits, specifically stating that he preferred Boole’snumerical function approach to that of Boolean algebra orpropositional logic.

3.8 General Theorems of Boole’s Algebra inMAL

Beginning with the chapterProperties of Elective Functions, Boole developed general theorems for working with elective functions and equations in his algebra of logic—the Expansion (or Development) Theorem (described above in Section 3.5) and the properties of constituents are discussed in this chapter. He used the power series expansion of an elective function in his proof of the one-variable case of the Expansion Theorem (MAL, p. 60), perhaps intending to apply it to rational elective functions.

The operation of division with polynomial functions was introduced inMAL but never successfully developed in his algebra of logic—there are no equational laws for how to deal with division. It was abandoned inLT except for being frequently used as a mnemonic device when solving a polynomial equation. From the Expansion Theorem and the properties of constituents he showed that themodulii of the sum/difference/product of two elective functions arethe sums/differences/products of the corresponding modulii of the twofunctions.

The Expansion Theorem is used (MAL, p. 61) to prove an important result, that p(x) and q(x) are equivalent in Boole’s algebra if and only if correspondingmodulii are the same, that is, \(p(1)=q(1)\) and \(p(0)=q(0)\). This result generalizes to functions of several variables. It will not be stated as such inLT, but will be absorbed in the much more general (if somewhat opaquely stated) result that will be called the Rule of 0 and 1.

Using the Expansion Theorem Boole showed (MAL, p. 64)that every elective equation \(p = 0\) is equivalent to the collectionof constituent equations \(r = 0\) where the modulus (coefficient) of\(r\) in the expansion of \(p\) is not zero, and thuseveryelective equation is interpretable. Furthermore this led(MAL, p. 65) to the fact that \(p = 0\) is equivalent to theequation \(q = 0\) where \(q\) is the sum of the constituents in theexpansion of \(p\) whose modulus is non-zero.

As examples, considerthe equations \(x + y = 0\) and \(x - y = 0\). The following tablegives the constituents and modulii of the expansions:

Thus \(x + y = 0\) is equivalent to the collection of constituentequations

as well as to the single equation

and \(x - y = 0\) is equivalent to the collection of constituentequations

as well as to the single equation

TheSolution Theorem described how to solve an elective equation for one of its symbols in terms of the others, often introducing constraintequations on the independent variables. In his algebra of logic he could always solve an elective equation for any one of its elective symbols. For example, the equation \(q(x)y = p(x)\) was solved by using formal division \(y = p(x)/q(x)\) and then using formal expansion to obtain \(y = ax + b(1-x)\)where \(a = p(1)/q(1)\) and \(b = p(0)/q(0)\), and then decoding thefractional coefficients. This theorem will be discussed in more detail in Step 7 of Section 6.2.

Boole’s final example (MAL, p. 78) was solving threeequations in three unknowns for one of the unknowns in terms of theother two. This example used a well known technique for handling sideconditions in analysis called Lagrange Multipliers—this method(which reduced the three equations in the example to a single equationin five unknowns) reappeared inLT (p. 117), but was onlyused once. It was superseded by the sum of squares reduction(LT, p. 121) which does not introduce new variables. Usingthe Reduction and Elimination Theorems inLT one discoversthat Boole’s constraint equations (3) (MAL, p. 80) forhis three equation example are much too weak—each of theproducts should be 0, and there are additional constraint equations.

MAL shows more clearly thanLT how closely Boole’salgebra of logic is based on Common Algebra plus the idempotent law. The Elimination Theorem that he borrowed from common algebraturned out to be weaker than what his algebra offered, and his methodof reducing equations to a single equation was clumsier than the mainone used inLT, but the Expansion Theorem and SolutionTheorem were the same. One sees thatMAL contained not onlythe basic outline forLT, but also some parts fullydeveloped. Power series were not completely abandoned inLT—theyappeared, but only in a footnote (LT, p. 72).

4. The Laws of Thought (1854)

The logic portion of Boole’s second logic book,An Investigation of The Laws of Thoughton which are founded the Mathematical Theories of Logic andProbabilities, published in 1854, would be devoted to trying to clarify andcorrect what was said inMAL, and providing moresubstantial applications, the main one being his considerable work inprobability theory. At the end of Chapter I Boole mentioned the theoreticalpossibility of using probability theory, enhanced by his algebra oflogic, to uncover fundamental laws governing society by analyzing largequantities of social data by large numbers of (human) computers.

Boole used lower case Latin letters at the end of the alphabet, like \(x,y,z\), to represent classes. Theuniverse was a class, denoted by 1; and there wasa class described as “Nothing”, denoted by 0,which we call theempty class. The operation ofmultiplication was defined to be what we call intersection, and this ledto his first law, \(xy = yx\), and then to theidempotent law \(x^2 = x\).Addition was introduced asaggregation when the classes were disjoint. He stated the commutativelaw for addition, \(x + y = y + x\), and the distributive law \(z(x +y) = zx + zy\). Then followed \(x - y = - y + x\) and \(z(x - y) = zx- zy\). The associative laws for addition and multiplication wereconspicuously absent.

The idempotent law \(x^2 = x\) was different from Boole’s laws for the common algebra—it only applied to theindividual class symbols, not in general to compound terms that onecould build from these symbols. For example, one does not have \((x + y)^2 = x + y\) in Boole’s system, otherwise by ordinaryalgebra with idempotent class symbols, this would imply \(2xy = 0\),and then \(xy = 0\), which would force \(x\) and \(y\) to representdisjoint classes. But it is not the case that every pair of classes isdisjoint.

It was this equational argument, that \((x + y)^2 = x + y\) implies \(xy = 0\), that led Boole to view addition \(x + y\) as a partial operation, only defined when \(xy = 0\), that is, when \(x\) and \(y\) are disjoint classes. The only place where he wrote down this argument was in his unpublishednotes—seeBoole: Selected Manuscripts …, 1997,edited by Ivor Grattan-Guiness and Gérard Bornet, pp. 91,92.A similar equational argument, that \((x - y)^2 = x - y\) implies \(y = xy\), led to \(x - y\) being only defined when \(y = xy\), that is, when \(y = x \cap y\), which is the same as \(y \subseteq x\).

It was not until p. 66 ofLT that Boole clearly informed the reader that addition, which had been introduced on p. 33, was a partial operation on classes:

The expression \(x+y\) seems indeed uninterpretable, unless it is assumed thatthe things represented by \(x\) and the things represented by \(y\) are entirelyseparate; that they embrace no individuals in common.

A similar statement about subtraction being a partial operation did not appearuntil p. 93:

The latter function presupposes, as a condition of its interpretation,that the class represented by \(y\) is wholly contained in the class represented by \(x\).

Here the “latter function” means\(x-y\). The dispersion of relevant facts about a topic, such as definitions ofthe fundamental operations of addition and subtraction, is not helpful to the reader.

Another example of the need for caution when working with partial algebras is how important it was that Boole chose the fundamental operation minus to be binary subtraction, and not the unary operation of additive inverse that is standard in ring theory. Note that the standard operations union, symmetricdifference, intersection and complement of the Boolean algebra of classes are definable in Boole’s partial-algebra by totally defined terms—he used these to be able to express propositions about classes by equations:

Subtraction was needed to find a totally defined term, namely \(1-x\), that expresses the complement of \(x\). The term \(1+(-x)\), like \(-x\), is only defined for \(x=0\) in Boole’s partial-algebra.

The same three equations define the Boolean algebra of classes in the standardBoolean ring of classes where addition is symmetric difference, except in this case subtraction is a derived operation, namely \(1-x\) is defined to be \(1 + (-x)\), the unary minus being a fundamental operation in the Boolean ring, indeed in any ring.

One might expect that Boole was building toward claiming an axiomaticfoundation for his algebra of logic that, as he had (erroneously)claimed inMAL,justified using all the processes of common algebra.Indeed he did discuss the rules of inference, that adding orsubtracting equals from equals gives equals, and multiplying equals byequals gives equals. But then the development of an axiomaticapproach came to an abrupt halt. There was no discussion as to whetherthe stated axioms (which he calledlaws) and rules of inference (which hecalledaxioms) were sufficient for his algebra oflogic. (They were not.) Instead he simply and briefly, withremarkably little fanfare, presented a radically new foundation forhis algebra of logic (LT pp. 37,38):

Let us conceive then of an algebra in which the symbols \(x, y, z\), &c. admit indifferently of the values 0 and 1, and of these values alone. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.

Note that this algebra restricted the values of the variables to 0 and 1,but placed no such restriction on the values of terms. There was no assertionthat this was to be a two-element algebra. Burris and Sankappanavar (2013) viewed the quote as saying that this algebra was just the ordinary algebra of numbers modified by restricting the variables to the values 0 and 1 to determine the validity of an argument. They called this Boole’sRule of 0 and 1, and said he used this Rule to justify three of his main theorems (Expansion, Reduction, Elimination). These main theorems along with the Solution Theorem yielded Boole’sGeneral Method for discovering the strongest possible consequences ofpropositional premises under certain desired constraints (such aseliminating some of the variables). Further comments on this Rule are below in Section 5.2.

In Chapter V he defended the use ofuninterpretables in hiswork; as part of his justification for the use of uninterpretablesteps in symbolic algebra he pointed to the well known use of\(\sqrt{-1}\) to obtain trigonometric identities. Unfortunately hisPrinciples of SymbolicalReasoning do not, in general, apply to partial algebras, that is,where some of the operations are only partially defined, such asaddition and subtraction in Boole’s algebra. Nonetheless it turns out one can prove that they do apply to his algebra of logic. In succeeding chapters hegave the Expansion Theorem, the new full-strength Elimination Theorem,an improved Reduction Theorem, and the Solution Theorem where formal division and formal expansion were used to solve anequation.

Boole turned to the topic of the interpretability of alogical function in Chapter VI Section 13. He had already stated inMAL thatevery equation isinterpretable (by showing an equation was equivalent to a collection ofconstituent equations). However algebraic terms need not be interpretable, e.g.,\(1+1\) is not interpretable. Some terms are partially interpretable,and equivalent terms can have distinct domains of interpretability. In Chapter VI Section 13, hecomes to the conclusion that the condition for a polynomial \(p\) to beequivalent to a (totally) interpretable function is that itsatisfy \(p^2 = p\), in which case it is equivalent to a sum ofdistinct constituents, namely those belonging to the non-vanishingmodulii of \(p\). A polynomial is idempotent if and only ifall of its modulii are idempotent, that is, they are in \(\{0, 1\}\),in which case the expansion of the polynomial is a sum of distinctconstituents (or it is 0).

Boole’s Chapter XIOf Secondary Propositions is parallel to the treatmentinMAL except that he changed from usingthe caseswhen \(X\) is true tothe times when \(X\) istrue. In Chapter XIII Boole selected argumentsof Clarke and Spinoza, on the nature of an eternal being, to put underthe magnifying glass of his algebra of logic, starting with thecomment (LT, p. 185):

2. The chief practical difficulty of this inquiry willconsist, not in the application of the method to the premises oncedetermined, but in ascertaining what the premisesare.

One conclusion was (LT, p. 216):

19. It is not possible, I think, to rise from theperusal of the arguments of Clarke and Spinoza without a deepconviction of the futility of all endeavours to establish, entirely apriori, the existence of an Infinite Being, His attributes, and Hisrelation to the universe.

In the final chapter on logic, Chapter XV, Boole presented hisanalysis of the conversions and syllogisms of Aristotelian logic. He now considered this ancient logic to be a weak, fragmented attempt at alogical system. This neglected chapter is quite interestingbecause it is theonly chapter where he analyzed particularpropositions, making essential use of additional letters like\(v\) to encodesome. This is also the chapter where he stated (incompletely) the rules for workingwithsome.

Boole noted in Chapter XV ofLT that when a premise about \(X\) and \(Y\)is expressed as an equation involving \(x, y\) and\(v\), the symbol \(v\) expressedsome, butonly in the context in which it appeared in the premise. For example,All \(X\) is \(Y\) has the expression \(x = vy\), which implies \(vx = vy\). This could be interpreted asSome \(X\) is \(Y\). A consequence of \(vx = vy\) is \(v(1-x) = v(1-y)\). However it was not permitted to readthis asSome not-\(X\) is not-\(Y\) since\(v\) did not appear with \(1-x\) or\(1-y\) in the premise.

In Chapter XV Boole gave the reader a brief summary of traditionalAristotelian categorical logic, and analyzed some simple examplesusing ad hoc techniques with his algebra of logic. Then he launchedinto proving a comprehensive result by applying his General Method tothe pair of equations:

This was the case oflike middle terms.He permitted some of the parameters \(v, v', w, w'\) to be replaced by 1,but not both \(v, v'\) and not both \(w, w'\) can be replaced by 1.One could also replace \(y\) by \(1-y\) in both equations, and independently replace \(x\) by \(1-x\) and \(z\) by \(1-z\).The premises of many categorical syllogisms can be expressed inthis form. His goal was to eliminate \(y\) and find expressions for\(x, 1-x\) and \(vx\) in terms of \(z, v, v', w, w'\).

Boole omitted writing out the reduction of the pair of equations to a singleequation, as well as the elimination of the middle term \(y\) from this equation and the details of applying the Solution Theorem to obtain thedesired expressions for \(x, 1-x\) and \(vx\), three equations involving large algebraic expressions.

His summary of the interpretation of this rather complicated algebraic analysis was simply thatin the case of like middle terms with at least onemiddle term universal, equate the extremes. For example the premisesAll \(y\) is \(x\) andSome \(z\) is \(y\) are expressed by the pair of equations

Thus the conclusion equation is \(vx = wz\), which has the interpretationSome \(x\) is \(z\).

Then he noted that the remaining categorical syllogisms are such that their premises can be put in the form:

This is the case ofunlike middle terms.This led to another triple of large equations, again with details of thederivation omitted, but briefly summarized by Boole in two recipes.

First,in the case of unlike middle terms with at least one universalextreme, change the quantity and quality of that extreme and equate it to the other extreme. For example the premisesAll \(x\) is not-\(y\) andSome \(z\) is \(y\) gives the pair of equations

Thus the conclusion equation is \(v(1-x) = wz\), which has the interpretationSome not-\(x\) is \(z\).

Secondly,in the case of unlike middle terms, both of which are universal,change the quantity and quality of one extreme and equate it to the other extreme. For example the premisesAll not-\(y\) is \(x\) andAll \(y\) is \(z\) gives the pair of equations

Thus one conclusion equation is \(1-x = wz\), which has the interpretationAll not-\(x\) is \(z\).The other is \(vx = 1-z\), which has the interpretationAll not-\(z\) is \(x\). Each of these two propositions is just theconversion by negation of the other.

Boole noted (LT p. 237) that:

The process of investigation by which they are deduced will probablyappear to be of needless complexity; and it is certain that they mighthave been obtained with greater facility, and without the aid of anysymbolical instrument whatever.

5. Later Developments

5.1 Objections to Boole’s Algebra of Logic

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

• the dependency on Common Algebra,
• the use of uninterpretable expressions in derivations,
• the treatment of particular propositions by equations, and
• the method of dealing with division.

For example, Boole’s use of \(v\) in theequational expression of propositions has been a long-standing bone ofcontention. Ernst Schröder (1841–1902) argued in Volume II of hisAlgebra der Logik (1891, p. 91) that the particular propositions about classes simply could not be expressed by equations in the algebra of logic.

We look at a different objection, namely at the Boole/Jevons disputeover adding \(x + x = x\) as a law.

[The following details are fromThe development of the theoriesof mathematical logic and the principles of mathematics, WilliamStanley Jevons, by Philip Jourdain, 1914.]

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

It is surely obvious, however, that \(x+x\) is equivalent only to\(x,\ldots\);

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

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

Boole replied:

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

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

Boole, clearly exasperated, replies:

To be explicit, I now, however, reply that it is not true that inLogic \(x + x = x\), though it is true that \(x + x = 0\) isequivalent to \(x = 0\). If I do not write more it is not from anyunwillingness to discuss the subject with you, but simply because ifwe differ on this fundamental point it is impossible that we shouldagree 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 thisexplanation your system probably is perfectly consistent with itself… But the question then becomes a wider one—does yoursystem correspond to the Logic of common thought?

Jevons’s new law, \(x + x = x\), resulted fromhis conviction that + should denote what we now callunion, where the membership of \(x + y\) is given by aninclusiveor. 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 notcomprehend the possibility of Jevons’s suggestion. Boole clearly hadthe semantic concept of union—he expressed the unionof \(x\) and \(y\) as \(x + (1-x)y\), a sum of two disjoint classes, andpointed out that the elements of this class are the ones that belongto either \(x\) or \(y\) or both. So how could he socompletely fail to see the possibility of taking union for hisfundamental operation + instead of his curious partial unionoperation?

The answer is simple: the law \(x + x = x\) would have destroyed hisability to useordinary algebra: from \(x + x = x\) onehas, by ordinary algebra, \(x = 0\). This would force every classsymbol to denote the empty class. Jevons’s proposed law \(x + x= x\) was simply not true if one was committed to constructing thealgebra of logic on top of the laws and inference rules of ordinaryalgebra. (Boolean rings satisfy all the laws of ordinary algebra, but not allof the inferences, for example, \(2x = 0\) implies \(x = 0\) doesnot hold in Boolean rings.) It seems quite possible that Boole foundthe simplest way to construct a model—whose domain was classes containedin the universe of discourse—for an algebra of logic thatallowed one to useall the equations and equational argumentsthat were valid for numbers.

5.2 Modern Reconstruction of Boole’s System

A popular misconception is that Boole’s algebra of logic is theBoolean algebra of classes with the usual operations of union,intersection and complement. This error was forcefully pointed out byHailperin in his 1981 paper “Boole’s algebra isn’tBoolean algebra,” a theme repeated in his path-breaking bookBoole’s Logic and Probability (1986). Nonetheless thegoal of the two algebras, Boole’s algebra and Boolean algebra,is the same, to provide an equational logic for the calculus ofclasses and for propositional logic. Thanks to Hailperin’swritings, for the first time there was clarity as to why Boole’salgebra gave correct results.

In his 1959 JSL review article Michael Dummett said:

anyone unacquainted with Boole’s works will receive anunpleasant surprise when he discovers how ill constructed his theoryactually was and how confused his explanations of it.

For example, one does not find a clear statement of what Boole meant byequivalent orinterpretable. For those familiar with partial algebras the latter word can easily be taken to meanis defined—the domain ofdefinition of an algebraic term has a recursive definition, just as algebraic term has a recursive definition. After verifying examples like thoseinLT that show Boole’s algebraic methods give correct results for propositions about classes, the challenge for thosewho want to make sense of Boole’s algebra of logic is to make enough of the foundations of Boole’s algebra sufficiently precise so as to be able to justify the algebraic procedures he used.

InLT Boole gave detailed instructions onhow to use his algebra to obtain valid propositional conclusions from propositional premises about classes when the propositions were universal. (He avoided particular propositions until Chapter XV, the last chapter ofLT on logic.) He showed how to express English language propositions as equations, the steps needed to obtain desired conclusion equations, and how they are to be interpreted as conclusion propositions to the premises. The algebraic steps can be lengthy and he gave some shortcuts in Chapter IX, but we now know that any method to carry out such deductions will confront a complexity of computation that grows rapidly with the number of propositional variables.

As stated in Dummett’s comment above, Boole was anything but clear as towhy his algebra worked as claimed, to give best possible conclusions to premises. This has inspired considerable commentary on what Boole meant to say, or should have said, and to what extent his justifications are valid.

Jevons (1864) gave sharp criticisms on the shortcomings of Boole’s algebra of logic and abandoned it to create the firstversion of modern Boolean algebra. (He did not have the unary complement operationthat is now standard, but instead used De Morgan’s convention that the complement of a class \(A\) is \(a\).)The title of Jevon’s 1864 bookstarted out with the wordsPure Logic, referring to the factthat his version of the algebra of logic had been cleansed fromconnections to the algebra of numbers. The same point would be made inthe introduction to Whitehead and Russell’sPrincipiaMathematica, that they had adopted the notation of Peano in partto free their work from such connections.

According to Hailperin(1986), the proof-theoretic side of Boole’s algebra is simply that of non-trivial commutative rings with unit and distinguished idempotent elements, but without non-zero additively or multiplicatively nilpotent elements. His favorite models were rings of signed multi-sets and he used them to explain why Boole’s theorems are correct for the algebra of logic ofuniversal propositions. (Hailperin’s analysis did not apply to particular propositions.)

Frank W. Brown’s paperGeorge Boole’s deductive system (2009) claims that one can avoid Hailperin’s signed multi-sets by working with the ring of polynomials Z[X] modulo a certain ideal.

Burris and Sankappanavar (2013) use the fact that Boole’s model, a partial-algebra, is isomorphic to the restriction of the operations of addition, multiplication and subtraction in thering \(Z^U\) to the idempotent elements of the ring. Here \(Z\) is the ring ofintegers, and \(U\) is the universe of discourse.From this one can deduce that any Horn sentence which holds in \(Z\) when the variables are restricted to 0 and 1 will hold in \(Z^U\) when the variables are restricted to idempotent elements, and thus will hold in Boole’s partial-algebra. This gives an expanded version of Boole’s Rule of 0 and 1, and since his main results (Expansion, Reduction, Elimination and Solution) can be expressed by such Horn sentences, one has a quickproof that they are indeed valid.

6. Boole’s Methods

While reading through this section, on the technical details ofBoole’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 eachstep of a derivation by Boole, which aspect of his methods is beingemployed.

6.1 The Three Methods of Argument Analysis Used by Boole inLT

Boole used three methods to analyze arguments inLT:

1. The first was the purely ad hoc algebraic manipulations thatwere used (in conjunction with a weak version of the EliminationTheorem) on the Aristotelian arguments inMAL.
2. Secondly, in section 15 of Chapter II ofLT, one findsthe method that, in this article, is called the Rule of 0 and 1.

The theorems ofLT combine to yield the master result,

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

When applying the ad hoc method, he used ordinary algebraalong with the idempotent law \(x^2 = x\) tomanipulate equations. There was no pre-established procedure tofollow—success with this method depended on intuitive skillsdeveloped through experience.

The second method, the Rule of 0 and 1, is very powerful, but itdepends on being given a collection of premise equations and aconclusion equation. It is a truth-table like method (but Boole neverdrew a table when applying the method) to determine if the argument iscorrect. He only used this method to establish the theorems thatjustified his General Method, even though it is an excellent tool forverifying simple arguments like syllogisms. Boole was mainlyinterested in finding the most general conclusion from given premises,modulo certain conditions, and aside from his general theorems, showedno interest in simply verifying logical arguments. The Rule of 0 and1 is a somewhat shadowy figure inLT—it has no name,and is reformulated in Section 6 of Chapter V as a procedure to use when carrying out a derivation and encountering uninterpretable terms.

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

The final version (fromLT) of his General Method foranalyzing arguments is, briefly stated, to:

1. express the premise propositions as equations,
2. apply a prescribed sequence of algebraic processes to theequations, processes which yield desired conclusion equations, andthen
3. interpret the equational conclusions as propositionalconclusions, yielding the desired consequences of the originalcollection of propositions.

With this method Boole had replaced the art of reasoning frompremise propositions to conclusion propositions by a routine mechanicalalgebraic procedure. On p. 240 ofLT he said that it was always theoretically possible to carry out elimination by piecing together syllogisms, but there was no method (i.e., algorithm) given for doing this.

InLT Boole divided propositions into two kinds, primaryand secondary. These correspond to, but are not exactly the same as,the Aristotelian division into categorical and hypotheticalpropositions. First we discuss his General Method applied to primarypropositions.

6.2. Boole’s General Method for Primary Propositions

Boole recognized three “great leading types” of primary propositions (LT, p. 64):

• 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. Theterms \(X\) and \(Y\) could be complex, for example, \(X\) couldbe “Either \(u\) and not-\(v\), or else \(w\)”. For Boole such terms were not very complicated, at most a disjunction of conjunctions of simple terms and their contraries, no doubt a reflectionof the fact that natural language terms are not very complex.

STEP 1: Propositional terms were expressed by algebraic terms as in the following; one can substitute more complex terms for \(x\), \(y\). Boole did not give a recursive definition, only some simple examples:

STEP 2: Having expressed the propositional terms as algebraic terms,one then expressed the propositions as equations using thefollowing; again one can substitute more complex terms for \(x\), \(y\), but not for \(v\):

InLT, prior to chapter XV, the one on Aristotelian logic, Boole’sexamples only used universal propositions. (One can speculate that hehad encountered difficulties with particular propositions and avoidedthem.) Those of the formAll x is y were first expressedas \(x = vy\), and then \(v\) was promptly eliminated, giving \(x =xy\). (Similarly if \(x\) was replaced by not-\(x\), etc.) Boole saidthe elimination of \(v\) was a convenient but unnecessary step. For the examples ofAll x is y inthe first fourteen chapters he could simply have used the expression\(x = xy\), skipping the use of the parameter \(v\).

To simplify thenotation he used thesame letter, say \(v\), forsome when there wereseveral universal premises, an incorrect step if one acceptsBoole’s claim that it is not necessary to eliminate the\(v\)’s immediately. Distinct universal propositions requiredifferent \(v\)’s in their translation; else one can run intothe following situation. Consider the two premisesAll \(x\) is\(z\) andAll \(y\) is \(z\). Using the same\(v\) for their equational expressions gives \(x = vz\) and \(y =vz\), leading to the equation \(x = y\), and then to the falseconclusion \(x\) equals \(y\). In chapter XV he was careful to usedistinct \(v\)’s for the expressions of distinct premises.

Boole used the four categorical propositions as his primary forms in1847, but in 1854 he eliminated the negative propositional forms,noting that one could changenot \(y\) tonot-\(y\). Thus in 1854 he would expressNo \(x\)is \(y\) byAll \(x\) is not-\(y\), with thetranslation \(x = v(1-y)\), and then eliminating \(v\) to obtain

which simplifies to \(xy = 0\).

STEP 3: After expressing the premises in algebraic form one has acollection of equations, say

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

with

An alternative way of forming the \(r_i\) to preserve the idempotent property,in case the \(p_i\) and \(q_i\) have this property, is given in Section 3 ofChapter X.

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

Reduce the system of equations

to a single equation \(r = 0\). Boole had three differentmethods for doing this—one of them was only forthe case that the \(r_i\) were idempotent. He had a strong preference forsumming the squares:

An alternative way of forming \(r\) to preserve the idempotent property,in case the \(r_i\) have this property, is also given in Section 3 ofChapter X.

Steps 1 through 4 are mandatory in Boole’s General Method. Afterexecuting 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 derivedfrom \(r = 0\) that involves some, but not all, of the classsymbols in \(r\). Then one wants to eliminate certain symbols. Suppose \(r\)involves the class symbols

Then one can write \(r\) as \(r(x_1, \ldots, x_j, y_1, \ldots ,y_k)\).

Boole’s procedure to eliminate the symbols \(x_1, \ldots ,x_j\)from

to obtain

was as follows:

1. form all possible expressions \(r(a_1, \ldots, a_j, y_1, \ldots,y_k)\) where \(a_1, \ldots, a_j\) are each either 0 or 1, then
2. multiply all of these expressions together to obtain \(s(y_1,\ldots, y_k)\).

For example, eliminating \(x_1, x_2\) from

gives

where

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

Given a term, say \(r(x_1, \ldots, x_j, y_1, \ldots, y_k)\), one canexpand the term with respect to a subset of the class symbols. Toexpand with respect to \(x_1, \ldots, x_j\) gives

where \(a_1 , \ldots ,a_j\) range over all sequences of 0s and 1s oflength \(j\), and where the \(C(a_i, x_i)\) are defined by:

The products

are theconstituents of \(x_1 , \ldots ,x_j\). There are \(2^j\)different constituents for \(j\) symbols—the regions of a Venndiagram give a popular way to visualize constituents. It will be convenient tosay an equation of the form

is aconstituent equation.

STEP 7: (DIVISION: SOLVING FOR A CLASS SYMBOL) [MAL (p. 73),LT (pp. 86–92) ]

Given an equation \(r = 0\), suppose one wants to solve this equationfor one of the class symbols, say \(x\), in terms of the other classsymbols, say they are \(y_1 , \ldots ,y_k\). To solve:

for \(x\), first let:

Then:

where \(s(y_1 ,\ldots ,y_k)\) is:

1. the sum of all constituents \(C(a_1, y_1) \cdots C(a_k, y_k)\) where \(a_1 , \ldots ,a_k\) range over all sequences of 0s and 1s for which:

   \[ N(a_1 , \ldots ,a_k) = D(a_1 , \ldots ,a_k) \ne 0,\]

plus

2. the sum of all the terms of the form \(v_{a_1 \ldots a_k} \cdot C(a_1, y_1) \cdots C(a_k, y_k)\) for which:

   \[N(a_1 , \ldots ,a_k) = D(a_1 , \ldots ,a_k) = 0.\]

The \(v_{a_1 \ldots a_k}\) are parameters, denoting arbitrary classes (the appearance of parameters is similar to what one sees in the solution of lineardifferential equations, a subject in which Boole was an expert).

To the equation (*) for \(x\) adjoin the constraint conditions (these are constituent equations that Boole called “independent relations”)

whenever

Note that one is to evaluate the terms:

using ordinary arithmetic. Thus solving an equation \(r = 0\) for aclass symbol \(x\) gives an equation

perhaps with constraint constituent equations. On p. 92 Boole noted that thesolution plus constraint equations could be written simply as

where A, B and C are each sums of distinct constituents.

This presentation gives exactly the same solution as that of Boole, but without the mysterious use of fractions like \(0/0\) and \(1/0\). Boole used formal divisionto express \(x\) as \(N\) divided by \(D\), and then a formal expansion where the coefficient of the constituent \(C(a_1, y_1) \cdots C(a_k, y_k)\) is the fraction

He had the following rules for how to decode the effect of the coefficients on their constituents: (1) for the coefficient\(m/m\) with \(m \neq 0\) the constituent is kept in the solution;(2) for \(0/m\) with \(m \neq 0\) the constituent is deleted;(3) the coefficient \(0/0\) is changed into an arbitrary parameter; and (4) any other coefficient indicated that the constituent was to be removed and set equal to 0. This use of formal division and formalexpansion is best regarded as a clever mnemonic device.

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

Any polynomial equation \(p(y_1 , \ldots ,y_k) = 0\) is equivalent to the collection of constituent equations

for which \(p(a_1 , \ldots ,a_k)\) is not 0. A constituent equationmerely asserts that a certain intersection of the original classes andtheir complements is empty. For example,

expresses the propositionAll \(y_1\) is \(y_2\) or\(y_3\), or equivalently,All \(y_1\) and not \(y_2\) is\(y_3\). It is routine to interpret constituent equations aspropositions.

6.3. Boole’s General Method for Secondary Propositions

Secondary propositions were Boole’s version of the propositions thatone encounters in the study of hypothetical syllogisms in Aristotelianlogic, statements likeIf \(X\) is true or \(Y\) is true then\(Z\) is true. The symbols \(X, Y, Z\), etc. referred to primary propositions. In keeping with theincomplete nature of the Aristotelian treatment of hypotheticalpropositions, Boole did not give a precise description of possibleforms for his secondary propositions.

The key (but not original) observation that Boole used was simply thatone can convert secondary propositions into primary propositions. InMAL he adopted the convention found in Whately (1826), thatgiven a propositional symbol \(X\), the symbol \(x\) willdenotethe cases in which \(X\) is true, whereas inLT Boole let \(x\) denotethe times for which\(X\) is true. With this the secondary propositionIf \(X\) is true or \(Y\) is true then \(Z\) is true is expressedbyAll \(x\) or \(y\) is \(z\). Theequation \(x = 1\) is the equational translation of\(X\) is true (in all cases, or for all times), and\(x = 0\) says\(X\) is false (inall cases, or for all times). The concepts ofall cases andall times depend on the choice of the universe ofdiscourse.

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

Boole worked mainly with Aristotelian propositions inMAL,using the traditional division into categoricals and hypotheticals. InLT thisdivision was replaced by the similar but more general primary versussecondary classification, where the subject and predicate were allowedto become complex names, and the number of propositions in an argumentbecame unrestricted. With this the parallels between the logic ofprimary propositions and that of secondary propositions became clear,with one notable difference, namely it seems that the secondarypropositions that Boole considered always translated into universalprimary propositions.

Bibliography

Primary Literature

• Boole, G., 1841, “On the Integration of LinearDifferential Equations with Constant Coefficients,”TheCambridge Mathematical Journal, 2: 114–119.
• –––, 1841, “Researches on the Theory of AnalyticalTransformations, with a special application to the Reduction of theGeneral Equation of the Second Order,”The CambridgeMathematical Journal, 2: 64–73.
• –––, 1841, “On Certain Theorems in theCalculus of Variations,”The Cambridge MathematicalJournal, 2: 97–102.
• –––, 1847,The Mathematical Analysis ofLogic, Being an Essay Towards a Calculus of Deductive Reasoning,Cambridge: Macmillan, Barclay, & Macmillan, and London: GeorgeBell; reprinted, Oxford: Blackwell, 1951.
• –––, 1848, “The Calculus of Logic,”The Cambridge and Dublin Mathematical Journal, 3:183–198.
• –––, 1854,An Investigation of The Laws ofThought on Which are Founded the Mathematical Theories of Logic andProbabilities, London: Walton and Maberly, Cambridge: Macmillanand Co.; reprinted, New York: Dover, 1958.
• –––, 1859,A Treatise on DifferentialEquations, Cambridge: Cambridge University Press, and London:Macmillan.
• –––, 1860,A Treatise on the Calculus ofFinite Differences, Cambridge: Cambridge University Press, andLondon: Macmillan.
• –––, 1997,Selected Manuscripts on Logic and itsPhilosophy (Science Networks Historical Studies: Volume 20),edited by Ivor Grattan-Guinness and Gérard Bornet. Basel,Boston, and Berlin: Birkhäuser Verlag.
• De Morgan, A., 1839, “On the Foundation of Algebra,”Transactions of the Cambridge Philosophical Society, VII,174–187.
• –––, 1841, “On the Foundation of Algebra,No. II,”Transactions of the Cambridge PhilosophicalSociety VII, 287–300.
• –––, 1847,Formal Logic: or, the Calculus ofInference, Necessary and Probable, Originally published in Londonby Taylor and Walton. Reprinted in London by The Open Court Company,1926.
• –––,On the Syllogism, and Other LogicalWritings, P. Heath (ed.), New Haven: Yale University Press,1966. (A posthumous collection of De Morgan’s papers onlogic.)
• Gregory, D.F., 1839, “Demonstrations in the DifferentialCalculus and the Calculus of Finite Differences,”TheCambridge Mathematical Journal, I: 212–222.
• –––, 1839, “I.–On the ElementaryPrinciples of the Application of Algebraical Symbols toGeometry,”The Cambridge Mathematical Journal, II(No. VII): 1–9.
• –––, 1840, “On the Real Nature ofSymbolical Algebra.”Transactions of the Royal Society ofEdinburgh, 14: 208–216. Also in [Gregory 1865, pp.1–13].
• –––, 1865,The Mathematical Writings ofDuncan Farquharson Gregory, M.A., W. Walton (ed.), Cambridge:Deighton, Bell.
• Jevons, W.S., 1864,Pure Logic, or the Logic of Quality apartfrom Quantity: with Remarks on Boole’s System and on the Relation ofLogic and Mathematics, London: Edward Stanford. Reprinted 1971 inPure 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 CalculDifférentiel et du Calcul Integral, Paris: ChezCourcier.
• Lagrange, J.L., 1788,Méchanique Analytique,Paris: Desaint.
• –––, 1797,Théorie des FonctionsAnalytique, Paris: Imprimerie de la Republique.
• Peacock, G., 1830,Treatise on Algebra, 2nd ed., 2 vols.,Cambridge: J.&J.J. Deighton, 1842/1845.
• –––, 1833, “Report on the Recent Progressand Present State of certain Branches of Analysis”,inReport of the Third Meeting of the British Association for theAdvancement of Science (held at Cambridge in 1833), London: JohnMurray, pp. 185–352.
• Schröder, E., 1890–1910,Algebra der Logik(Volumes I–III), Leipzig: B.G. Teubner; reprinted Chelsea1966.
• Whately, R., 1826,Elements of Logic, London: J. Mawman.

Secondary Literature

Cited Works

• Aiken, H.A., 1951,Synthesis of Electronic Computing andControl Circuits, Cambridge, MA: Harvard University Press.
• 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 Theoryof Boole’s Partial Algebras”,The Bulletin of SymbolicLogic, 19: 97–105.
• Burris, S.N., 2015, “George Boole and BooleanAlgebra”,European Mathematical Society Newsletter,98: 27–31.
• Dummett, M., 1959, “Review ofStudies in Logic andProbability, by George Boole”, Watts & Co., London, 1952,edited by R. Rhees,The Journal of Symbolic Logic, 24:203–209.
• Ewald, W. (ed.), 1996,From Kant to Hilbert. A Source Book inthe History of Mathematics, 2 volumes, Oxford: Oxford UniversityPress.
• Grattan-Guiness, I., 2001,The Search for MathematicalRoots, Princeton, NJ: Princeton University Press.
• 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 Booleanalgebra”,Mathematics Magazine, 54: 172–184.
• Jourdain, P.E.B., 1914, “The Development of the Theories ofMathematical Logic and the Principles of Mathematics. William StanleyJevons”,Quarterly Journal of Pure and AppliedMathematics, 44: 113–128.
• MacHale, D., 1985,George Boole, His Life and Work,Dublin: Boole Press; 2nd edition, 2014, Cork University Press.
• MacHale, D. and Cohen, Y., 2018,New Light on George Boole, Cork University Press.
• Schröder, E., 1890–1910,Algebra der Logik (VolumesI–III), Leipzig: B.G. Teubner; reprinted Chelsea 1966.
• Whitney, H., 1933, “Characteristic Functions and the Algebraof Logic”,Annals of Mathematics (Second Series), 34:405–414.

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 PublishingCo., Chicago & London, 1914; reprinted by Mineola: DoverPublications, 2006.
• Frege, G., 1880, “Boole’s Logical Calculus and theConcept-Script”, inGottlob Frege: Posthumous Writings,Basil Blackwell, Oxford, 1979. English translation ofNachgelassene Schriften, vol. 1, edited by H. Hermes,F. Kambartel, and F. Kaulbach, Felix Meiner, Hamburg, 1969.
• Kneale, W., and M. Kneale, 1962,The Development ofLogic, Oxford: The Clarendon Press.
• Lewis, C. I., 1918,A Survey of Symbolic Logic,University of California Press, Berkeley; reprinted New York: DoverPublications, 1960. (See Chap. II, “The Classic, orBoole-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 onTheMathematical Analysis of Logic”,History and Philosophyof Logic, 4: 27–39.
• Styazhkin, N. I., 1969,Concise History of Mathematical Logicfrom Leibniz to Peano, Cambridge, MA: The MIT Press.
• van Evra, J. W., 1977, “A Reassessment of George Boole’sTheory of Logic”,Notre Dame Journal of Formal Logic,18: 363–77.
• Venn, J., 1894,Symbolic Logic, 2nd edition (revised),London: Macmillan; reprinted Bronx: Chelsea Publishing Co., 1971.

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Other Internet Resources

• Versions of Boole's work annotated by Stanley Burris:
  • Algebra of Logic (1847)
  • “The Calculus of Logic” (1848)
  • An Investigation of The Laws of Thought (1854), first 15 chapters.
• Other entries at the MacTutor History of MathematicsArchive:
• Algebraic Logic Group, Alfred Reyni Institute ofMathematics, Hungarian Academy of Sciences
• George Boole 200, maintained at University College Cork.

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