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7 Boole's Work

During Boole's time in Doncaster as assistant teacher he had a flash one afternoon as he was walking across a field, and the initial idea of a logic of human thought was born. Fourteen years later, early in the spring of 1847, his interest in logic was re-awakened with a quarrel raging between De Morgan and Sir William Hamilton, the Scottish Philosopher and Metaphysician, concerning the origin of the quantification of the predicate. Hamilton extended classical logic, which dealt with propositions of the form ``all A are B, no A are B, some A are B, some A are not B'', to a quantification of the second term, as in ``all A are all B, any A is not some B''. A similar approach was taken by De Morgan and consequently Hamilton accused De Morgan of plagiarism, though the idea in question was not new.

The controversy attracted much attention and, in particular, it inspired Boole's work on logic, as he states in the preface of his 1847 book:

In presenting this Work to public notice, I deem it not irrelevant to observe that speculations similar to those which it records have, at different periods, occupied my thoughts. In the spring of the present year, my attention was directed to the question then moved between Sir W. Hamilton and Professor De Morgan; and I was induced by the interest which it inspired, to resume the almost-forgotten thread of former inquiries. It appears to me that, although Logic might be viewed with reference to the idea of quantity, it had also another and deeper system of relations. If it was lawful to regard it from without, as connecting itself through the medium of Number with the intuitions of Space and Time, it was lawful to regard it from within, as based upon facts of another order which have their abode in the constitution of the Mind.²⁶

He wrote the book, which was published in 1847, on the same day as De Morgan's ``Formal Logic'', at a furious pace, and so it contained several flaws which led him--seven years later--to write the corrected, more settled book ``An Investigation into the Laws of Thought''. In the sequel we review the logic he developed in the ``Laws of Thought''.

Concepts are independent of their representation: ``Romans expressed by the word `civitas' what we designate by the word `state'. But both they and we might equally well have employed any other word to represent the same conception''.²⁷ Besides such literal symbols which stand for concepts, he introduces operations to relate and manipulate the concepts. The only relation Boole uses is equality. The similarity to Leibniz's idea of the ``calculus ratiocinator'' is striking, but Boole did not know Leibniz's work.

Proposition I.
All the operations of Language, as an instrument of reasoning, may be conducted by a system of signs composed of the following elements, viz:

1st. Literal symbols, as x, y, & c., representing things as subject of our conceptions.

2nd. Signs of operations, as +, -, × , standing for those operations of the mind by which the conceptions of things are combined or resolved so as to form new conceptions involving the same elements.

3rd. The sign of identity, =.

And these symbols of Logic are in their use subject to definite laws, partly agreeing with and partly differing from the laws of the corresponding symbols in the science of Algebra.²⁸

Boole's idea of a mind selecting items from classes is central in his arguments for the laws that hold in his algebra. For example, to get all ``good men'' the mind selects from the class ``men'' those who possess the further quality ``good''. Obviously, the order of the selection does not affect the result, so that Boole argues that xy = yx, the commutative law, holds.²⁹ In a similar way, he proves that the distributive law holds. Interestingly, he also uses the associative law, but never mentions it explicitly. As arithmetic provides the laws 0y = 0 and 1y = y he states that 0 is nothing and 1 is the universe. In his first book, Boole still assumes a fixed universe of all things in existence. Influenced by De Morgan's universe of discourse, he adopts this idea in the ``Laws of Thought''. Though the universe may possibly be empty, Boole excludes this trivial case³⁰ and implicitly assumes that the law 0 $\neq$ 1 holds. The idea of selecting from classes also leads him to a law that holds in arithmetic only for 0 and 1, namely x² = x. Together with an additive inverse and the rule that x + x = 0 implies x = 0, Boole's Algebra, as summarized in Figure 3, is completely specified including also the laws he used but which he did not mention explicitly.

**Figure 3:** Boole's Algebra
$\begin{figure} % latex2html id marker 385 \ignorespaces\begin{equation} \begin{... ... 1\\ \ x+x=0 \mbox{ implies }x=0\qquad\\ \end{array}\end{equation}\end{figure}$

As an illustration Boole uses the definition of wealth due to the economist N.W. Senoir: ``Wealth consists of things transferable, limited in supply, and either productive of pleasure or preventative of pain.'' With w representing wealth, t transferable things, s limited in supply, p productive of pleasure and r preventative of pain, Boole obtains the equation:

w = st(p + r(1 - p)) $\displaystyle \qquad$ or $\displaystyle \qquad$ w = st(p(1 - r) + r(1 - p))

depending on whether the ``or'' in the definition is read inclusively or exclusively.

In order to cope with statements such as ``all men are mortal'', or ``all men are some mortal beings'', Boole introduces a special symbol v,

a class indefinite in every respect but this, viz., that some of its members are mortal beings, and let x stand for `mortal beings', then will vx represent `some mortal beings'. Hence if y represents men, the equation sought will be y = vx.
... it is obvious that v is a symbol of the same kind as x, y, etc., and that it is subject to the general law v² = v, or v(1 - v) = 0.³¹

Later we will see the problems and misunderstandings that the symbol causes.

Boole solved equations guided by the three principles of division, development, and interpretation. Probably he transferred the idea from differential equations, where he enhanced a solution method presented by Duncan Farquharson Gregory (1813-1844) in 1840.³² As an example for the solution of equations, he presents the ``definition of `clean beasts' as laid down in the Jewish law, viz., `clean beasts are those which both divide the hoof and chew the cud'''³³, i.e.

x = yz,

where x represents the clean beasts, y beasts dividing the hoof and z beasts chewing the cud.

In order to obtain the relation in which `beasts chewing the cud' stand to `clean beasts' and `beasts dividing the hoof' we divide by y:

z = $\displaystyle {\frac{x}{y}}$

which is developed by a case analysis of x and y. Each summand is one of the four cases for bindings of x and y and consists of the fraction with the values replaced and multiplied by the case:

$\displaystyle \begin{array} {rcr@{\ \,+\ \,}r@{\ \,+\ \,}r@{\ \,+\ \,}r} z& =&... ...ineskip] & =& xy& \frac{1}{0}x(1-y)& 0(1-x)y& \frac{0}{0}(1-x)(1-y)\end{array}$

The equation can be read as

Beasts which chew the cud [z] consists of all clean beasts (which also divide the hoof) [xy] together with an indefinite remainder (some, none, or all) [indicated by ${\frac{0}{0}}$ ] of unclean beasts which do not divide the hoof [ (1 - x)(1 - y)].³⁴

Thus, the terms 0(1 - x)y and ${\frac{1}{0}}$ x(1 - y) are omitted. The former because it does not make a statement at all, the latter because it does not make a statement about z. By multiplication with 0, the equation z = ${\frac{1}{0}}$ x(1 - y) can be rewritten as 0z = x(1 - y), i.e. there are no clean beasts that do not divide the hoof.

No doubt, Boole's notation was ingenious and the work was developed in a formal style, which was a very important innovation, but the work also produced some unintuitive or even wrong results. In the next section we review De Morgan's criticism of Boole's work.

Footnotes

...Mind.²⁶: Boole 1847, Preface.
...conception''.²⁷: Boole 1854, p. 26.
...Algebra.²⁸: Boole 1854, p. 27.
...holds.²⁹: Boole 1854, p. 29.
...case³⁰: Hailperin 1986, p. 84, in case 0 = 1 the laws lose their intuitive meaning since all classes are empty.
...0.³¹: Boole 1854, p. 61.
...1840.³²: Hailperin 1986, p. 66.
...cud'''³³: Boole 1854, p. 86.
...y)].³⁴: Boole 1854, p. 87.

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