Finally, in , he obtained a professorship in a new university opening in Cork, Ireland. In the years he was a professor in Cork — he would occasionally inquire about the possibility of a position back in England. In addition Boole published 24 more papers on traditional mathematics during this period, while only one paper was written on logic, that being in He was awarded an honorary LL.
During the last 10 years of his career, from to , Boole published 17 papers on mathematics and two mathematics books, one on differential equations and one on difference equations. Both books were highly regarded, and used for instruction at Cambridge.
Also during this time significant honors came in:. Unfortunately his keen sense of duty led to his walking through a rainstorm in late , and then lecturing in wet clothes. Not long afterwards, on December 8, in Ballintemple, County Cork, Ireland, he died of pneumonia, at the age of Another paper on mathematics and a revised book on differential equations, giving considerable attention to singular solutions, were published post mortem.
The 19th century opened in England with mathematics in the doldrums. One of the obstacles to overcome in updating English mathematics was the fact that the great developments of algebra and analysis had been built on dubious foundations, and there were English mathematicians who were quite vocal about these shortcomings.
In ordinary algebra, it was the use of negative numbers and imaginary numbers that caused concern. The first major attempt among the English to clear up the foundation problems of algebra was the Treatise on Algebra , , by George Peacock — He divided the subject into two parts, the first part being arithmetical algebra , the algebra of the positive numbers which did not permit operations like subtraction in cases where the answer would not be a positive number.
The second part was symbolical algebra , which was governed not by a specific interpretation, as was the case for arithmetical algebra, but solely by laws. In symbolical algebra there were no restrictions on using subtraction, etc. Peacock believed that in order for symbolical algebra to be a useful subject its laws had to be closely related to those of arithmetical algebra. In this connection he introduced his principle of the permanence of equivalent forms , a principle connecting results in arithmetical algebra to those in symbolical algebra.
This principle has two parts:. This application of algebra captured the interest of Gregory who published a number of papers on the method of the separation of symbols , that is, the separation into operators and objects, in the CMJ. Unfortunately these laws fell far short of what is required to justify even some of the most elementary results in algebra, like those involving subtraction.
In On the foundation of algebra, , the first of four papers on this topic by De Morgan that appeared in the Transactions of the Cambridge Philosophical Society , one finds a tribute to the separation of symbols in algebra, and the claim that modern algebraists usually regard the symbols as denoting operators e.
The footnote. In the second foundations paper in De Morgan proposed what he considered to be a complete set of eight rules for working with symbolical algebra. Sheffer and Clarence I. Lewis and in due course by other Harvard professors and the world. It essentially referred to the modern version of the algebra of logic introduced in by William Stanley Jevons — , a version that Boole had rejected in their correspondence—see Section 5. In MAL , and more so in LT , Boole was interested in the insights that his algebra of logic gave to the inner workings of the mind.
This pursuit has met with little favor, and is not discussed in this article. In New Light on George Boole by MacHale and Cohen one finds, published for the first time, an edited version of the biography by MaryAnn Boole — of her famous brother, and on p.
In early he was stimulated to renew his investigations into logic by a trivial but very public dispute between De Morgan and the Scottish philosopher Sir William Hamilton — —not to be confused with his contemporary the Irish mathematician Sir William Rowan Hamilton — This dispute revolved around who deserved credit for the idea of quantifying the predicate e.
Within a few months Boole had written his 82 page monograph, Mathematical Analysis of Logic , first presenting an algebraic approach to Aristotelian logic, then looking briefly at the general theory. We are not told what the true method was that flashed upon Boole. One possibility is the discovery of the Expansion Theorem and the properties of constituents.
For his extended version of Aristotelian logic he stated MAL , p. These transformation rules did not appear in LT. It is somewhat curious that when it came to analyzing categorical syllogisms, it was only in the conclusion that he permitted his generalized categorical propositions to appear.
Among the vast possibilities for hypothetical syllogisms, the ones that he discussed were standard, with one new example added. The elective symbols denoted elective operators—for example the elective operator red when applied to a class would elect select the red items in the class. As pointed out by Theodore Hailperin — , p. This was the first mention of addition in MAL. He added MAL , p. After stating the above distributive and commutative laws, Boole believed he was entitled to fully employ the ordinary algebra of his time, saying MAL , p.
In the mid s the word algebra meant, for most mathematicians, simply the algebra of numbers. Similar results hold for polynomials in any number of variables MAL , pp. This is the first appearance of subtraction in MAL. Then in the next several pages he adds supplementary expressions; of these the main ones will be called the secondary expressions.
This was the first appearance of 0 in MAL. It was not introduced as the symbol for the empty class—indeed the empty class does not appear in MAL. In LT , what we call the empty class was introduced and denoted by 0. Syllogistic reasoning is just an exercise in elimination , namely the middle term is eliminated from the premises to give the conclusion.
One finds, using the improved reduction and elimination theorems of LT , that the best possible result of elimination is. Applying weak elimination to the primary equational expressions was not sufficient to derive all of the valid syllogisms.
Boole introduced the alternative equational expressions see MAL , p. Toward the end of the chapter on categorical syllogisms there is a long footnote MAL , pp. The footnote loses much of its force because the results it presents depend heavily on the weak elimination theorem being best possible, which is not the case. Regarding the secondary expressions, in the Postscript to MAL he says:. His justification of this claim would appear in LT. Indeed Boole used only the secondary expressions of MAL to express propositions as equations in LT , 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 in LT to analyze particular propositions.
Boole analyzed the seven hypothetical syllogisms that were standard in Aristotelian logic, from the Constructive and Destructive Conditionals to the Complex Destructive Dilemma. Evidently his notion of a case was an assignment of truth values to the propositional variables. Boole said the universe of a categorical proposition has two cases, true and false. To find an equational expression for a hypothetical proposition Boole resorted to a near relative of truth tables MAL , p.
His algebraic method of analyzing hypothetical syllogisms was to transform each of the hypothetical premises into an elective equation, and then apply his algebra of logic which was developed for categorical propositions.
Boole only considered rather simple hypothetical propositions on the grounds these were the only ones encountered in common usage see LT , p. His algebraic approach to propositional logic is easily extended to all propositional formulas as follows. Beginning with the chapter Properties 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.
He used the power series expansion of an elective function in his proof of the one-variable case of the Expansion Theorem MAL , p. The operation of division with polynomial functions was introduced in MAL but never successfully developed in his algebra of logic—there are no equational laws for how to deal with division. It was abandoned in LT except for being frequently used as a mnemonic device when solving a polynomial equation. This result generalizes to functions of several variables.
It will not be stated as such in LT , but will be absorbed in the much more general if somewhat opaquely stated result that will be called the Rule of 0 and 1. Furthermore this led MAL , p. The following table gives the constituents and modulii of the expansions:. The Solution Theorem described how to solve an elective equation for one of its symbols in terms of the others, often introducing constraint equations on the independent variables.
In his algebra of logic he could always solve an elective equation for any one of its elective symbols.
This theorem will be discussed in more detail in Step 7 of Section 6. This example used a well known technique for handling side conditions in analysis called Lagrange Multipliers—this method which reduced the three equations in the example to a single equation in five unknowns reappeared in LT p. It was superseded by the sum of squares reduction LT , p.
The Elimination Theorem that he borrowed from common algebra turned out to be weaker than what his algebra offered, and his method of reducing equations to a single equation was clumsier than the main one used in LT , but the Expansion Theorem and Solution Theorem were the same. Power series were not completely abandoned in LT —they appeared, but only in a footnote LT , p. At the end of Chapter I Boole mentioned the theoretical possibility of using probability theory, enhanced by his algebra of logic, to uncover fundamental laws governing society by analyzing large quantities of social data by large numbers of human computers.
Addition was introduced as aggregation when the classes were disjoint. The associative laws for addition and multiplication were conspicuously absent. But it is not the case that every pair of classes is disjoint. It was not until p. The dispersion of relevant facts about a topic, such as definitions of the 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.
One might expect that Boole was building toward claiming an axiomatic foundation for his algebra of logic that, as he had erroneously claimed in MAL , justified using all the processes of common algebra.
Indeed he did discuss the rules of inference, that adding or subtracting equals from equals gives equals, and multiplying equals by equals gives equals.
But then the development of an axiomatic approach came to an abrupt halt. There was no discussion as to whether the stated axioms which he called laws and rules of inference which he called axioms were sufficient for his algebra of logic. They were not. Instead he simply and briefly, with remarkably little fanfare, presented a radically new foundation for his algebra of logic LT pp.
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 assertion that this was to be a two-element algebra. Burris and Sankappanavar 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.
Further comments on this Rule are below in Section 5. Nonetheless it turns out one can prove that they do apply to his algebra of logic. In succeeding chapters he gave 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 an equation.
Boole turned to the topic of the interpretability of a logical function in Chapter VI Section He had already stated in MAL that every equation is interpretable by showing an equation was equivalent to a collection of constituent equations. However algebraic terms need not be interpretable, e. Some terms are partially interpretable, and equivalent terms can have distinct domains of interpretability. In Chapter XIII Boole selected arguments of Clarke and Spinoza, on the nature of an eternal being, to put under the magnifying glass of his algebra of logic, starting with the comment LT , p.
In the final chapter on logic, Chapter XV, Boole presented his analysis of the conversions and syllogisms of Aristotelian logic. He now considered this ancient logic to be a weak, fragmented attempt at a logical system. This is also the chapter where he stated incompletely the rules for working with some.
In Chapter XV Boole gave the reader a brief summary of traditional Aristotelian categorical logic, and analyzed some simple examples using ad hoc techniques with his algebra of logic. Then he launched into proving a comprehensive result by applying his General Method to the pair of equations:.
This was the case of like middle terms. The premises of many categorical syllogisms can be expressed in this form. His summary of the interpretation of this rather complicated algebraic analysis was simply that in the case of like middle terms with at least one middle term universal, equate the extremes. Then he noted that the remaining categorical syllogisms are such that their premises can be put in the form:. This is the case of unlike middle terms.
This led to another triple of large equations, again with details of the derivation omitted, but briefly summarized by Boole in two recipes. First, in the case of unlike middle terms with at least one universal extreme, change the quantity and quality of that extreme and equate it to the other extreme.
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.
Each of these two propositions is just the conversion by negation of the other. If my view be right, his system will come to be regarded as a most remarkable combination of truth and error. This would force every class symbol to denote the empty class.
It seems quite possible that Boole found the simplest way to construct a model—whose domain was classes contained in the universe of discourse—for an algebra of logic that allowed one to use all the equations and equational arguments that were valid for numbers. For example, one does not find a clear statement of what Boole meant by equivalent or interpretable.
For those familiar with partial algebras the latter word can easily be taken to mean is defined —the domain of definition of an algebraic term has a recursive definition, just as algebraic term has a recursive definition. In LT Boole gave detailed instructions on how to use his algebra to obtain valid propositional conclusions from propositional premises about classes when the propositions were universal.
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.
It is the story of a remarkable man, beautifully told. George Boole was born in Lincoln, England, the son of a struggling shoemaker. Boole was forced to leave school at the age of sixteen and never attended a university. He taught himself languages, natural philosophy and mathematics. He began to produce original mathematical research and, in , he was awarded the first gold medal for mathematics by the Royal Society. He also made important contributions to areas of mathematics such as invariant theory of which he was the founder , differential and difference equations and probability.
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