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Before 1854 when Boole published An Investigation of The Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities (henceforth: Laws of Thought), the subject of ?logic? in the western world was entirely restricted to the study of the Aristotelian syllogism, which was as old as 384-322 BC. In this work, Boole introduces a sort of algebra of propositions with probability. Boole?s logic forms the basis for present day Boolean Algebra, which in turn lies at the base of computer science. Because Boole is essentially inventing a number of new concepts, the discussions concerning his ideas of logic are both accessible to the non-specialist and fascinating for the historian or philosopher of mathematics and logic.
George Boole (1815 ? 1864) was born in Lincolnshire, England. He was born of humble origin as the first son of a shoemaker, but posthumously, he rose to the heights of the moon, by having a moon crater named after him in 1967. For financial reasons, he did not receive the best education, but neverthelesshe learned Latin from a local bookseller and taught himself French, German, and Greek. At the age of nineteen, Boole had to support his parents and siblings. He started teaching at a local school, and later set up and ran a boarding school with the help of his family.
Boole only later turned his attention to mathematics. He was encouraged in his mathematical pursuits by Duncan Gregory at Cambridge who recognized his deep understanding and imaginative approach to mathematics. Boole?s early contributions to mathematics were to calculus and analysis. Boole was more widely acclaimed for his contributions to mathematics in 1844, when he received the Royal Medal from the Royal Society for his paper On a General Method of Analysis. In 1849, Boole was appointed chair of mathematics at Queens College, Cork. In 1854, while teaching at Cork, Boole published his most important work The Laws of Thought:
I am now about to set seriously to work upon preparing for the press an account of my theory of Logic and Probabilities which in its present state I look upon as the most valuable if not the only valuable contribution that I have made or am likely to make to Science?.
Writing to his printers, De Morgan writes that Boole ?is meditating typographically on his mathematical logic, which is a very original thing, and, for power of thought, worthy to be printed?.? Boole had an important correspondence with De Morgan, and his logical innovations have inspired many logicians and mathematicians since, including Charles Saunders Pierce and J. Venn, inventor to the famous Venn diagrams.
Boole?s most famous works are ?The Mathematical Analysis of Logic,? published in 1847 and The Laws of Thought. It is from these two works that we learn some of Boole?s most important and influential ideas. One of Boole?s great and original ideas was to claim logic as an area of mathematics, where, previously, logic had belonged exclusively to philosophy. This freed up the conception of what logic is and allowed Boole to then introduce a new structure to logical reasoning. In Laws of Thought, Boole braids together three ideas: that logic should account for, and capture, the syllogistic reasoning while reaching beyond it, that there are law-like constraints on our reasoning, and that reasoning about probabilities is a logical primitive.
Boole invented, what is now called ?Boolean algebra.? He saw his logic as a formal representation of laws of reasoning, in the sense of marshalling our reasoning. One of Boole?s greatest innovations was to think of algebra as not only pertaining to number, but to other things such as terms or propositions. In his logical system, Boole introduces a notion of class. From this, Boole very naturally develops the notions of sub-class, the intersection of two classes, the union of two classes and the complement of a class. Today, these are the familiar Boolean connectives: IF THEN, AND, OR, NOT. Boole also introduces notions of quantification and probability. Semantics are captured by the numbers ?zero? and ?one? to signify ?nothing? and ?the universe,? respectively. This elegantly simple concept of semantics has significantly contributed to the shape of modern mathematics and computer science, where we re-interpret ?zero? and ?one? to mean ?off / on? or ?false / true.?
It is thanks to Boole?s Laws of Thought, that the topic of ?logic? was released from the constraints of the syllogism by applying concepts of proof from algebra to terms and propositions. This allowed logical proofs to be extended to include more than two premises. This is important because here we have the first glimpse of a notion of logical proof in which each step does not have to be obviously related to the conclusion. Logical proof is newly understood as a procedure that can take several steps, each one of which follows from pre-accepted principles of deduction. These principles of proof are taken from algebra and probability theory, which Boole thought epitomized standards of good human reasoning.
Charles Sanders Pierce (1839?1914) was impressed by Boole?s contribution to mathematics and worked on the electrical application of the Boolean logic. This became one of the pre-cursors of electrical computing machines. Pierce also taught one of the first courses in Boolean algebra. Pierce developed Boole?s logic further by making explicit the notion of truth-value assignment to propositions. He added the idea that a necessary truth is one that is true under any truth-value assignment. With this we have the stage set for the truth-table definitions of the logical connectives. These were developed independently by Post and Wittgenstein in the early 1920s.
Despite his monumental achievements, Boole?s Laws of Thought is too-little read. One reason for this is that Boole?s logic was greatly surpassed in 1879 with the publication of Frege?s Begriffsschrift. In this, Frege develops a logic of predicates, functions, and relations, with quantifiers quantifying over first and second-order variables. Frege?s system is clearer and much more supple and sophisticated than Boole?s. Indeed, apart from the notation, the logic we learn today, in the form of predicate logic or first-order logic, is descended from Frege. Lying in Frege?s shadow, Boole?s Laws of Thought has been too often dismissed as being, at best, of historical interest.
A second reason Boole is overlooked is that he was heavily criticized by Frege and Russell for being psychologistic, where this is understood in the standard sense of logic being essentially a mental construct, and thus, culpable of being subjective. This criticism is partly based on a confusion of the term. Bornet speculates that, despite his criticism of the work, Russell never read Boole?s Laws of Thought! But he did read the title, and it is based on this that Russell dismisses the work as psychologistic, thinking that it dealt with a description of how we in fact reason rather than with logic, which tells us how we ought to reason. Russell?s dismissal was enough to dissuade many potential readers.
Far from being psychologistic, in the ?mental construct? sense, one can detect hints of logicism and formalism in The Laws of Thought; where logicism is understood as the reduction of arithmetic to logic, and formalism is the idea that mathematics is essentially symbol manipulation. Formalism serves computer science well. By allowing long chains of reasoning in his logic, and allowing a mechanistic element in the reasoning, Boole anticipated the computer?s ability to carry out very long proofs.
Boole?s Laws of Thought is worthy of our careful re-examination. The book is of particular interest to any serious scholar of the philosophy of, or the history of, logic; anyone interested in the early history of the computer and notions of automated computation; anyone interested in the history of probability theory; and anyone interested in nineteenth-century mathematics. The wide variety of topics discussed makes Boole?s Laws of Thought well worth the read.
More specifically, philosophers of logic will be particularly interested in the first, thirteenth, fifteenth, and final chapter to discover what Boole thought was the importance of logic. In these chapters we see Boole?s formal system as a formal representation of the structure of reasoning. He takes logic to be a part of the philosophy of mind, which in turn is a subspecies of metaphysics. He is also very keen to show the allegiance of his logic to traditional Aristotelian syllogistic reasoning. In chapter thirteen he shows how one can recapture the Aristotelian logic in his formal system. At the time, this was deemed essential to uphold the claim that what he was discussing was logic, as opposed to mathematics. In chapter fifteen, Boole goes well beyond the syllogism and uses his formal system to analyze long arguments by Clarke and Spinoza, both noted for their careful philosophical arguments. To his credit, Boole is aware of the limitations of formalizing arguments originally written in a natural language. Not only are there problems in loyalty of representation of the basic propositions, but also of the connections made between them. Put another way, Boole is aware that, showing that one proposition is not derivable from another in the formal system under a particular translation, using only logical rules of inference, does not imply that there is no legitimate metaphysical connection between the propositions. Nevertheless, when philosophical arguments are made as meticulously as they are in the texts of Clarke and Spinoza, these lend themselves naturally to further elucidation through logical analysis. Such an analysis would have been almost impossible if restricted to Aristotelian syllogistic reasoning.
For those readers interested in the logical system minus probability, either for historical or conceptual reasons, you will be interested in chapters two through twelve. In these we have a charming and candid discussion of some basic logical concepts, some of which are seldom so described. Boole?s notion of quantity is quirky by today?s standards. Boole has a general operator, symbolized by the Greek letter ?n.? This is placed to the left of a term or proposition variable or constant. The letter ?n? can be interpreted as ?all? or ?not all,? i.e., ?some.? This has to be specified independently. For, what interests Boole is the algebra of the operator, and this remains the same under either interpretation.
Boole discusses probability theory in chapters sixteen to twenty-one. He not only sums up what has already been established in probability theory by mathematicians such as Laplace and Poisson. He brings his own innovative contribution by combining probability theory with propositional logic. In doing this, Boole is able to discuss the probability of the truth of a proposition, as opposed to the more usual ?probability of an event.? The merging of these two notions of proposition and probability is what allows Boole to combine various events, with attending probabilities, into one (compound) event with one probability. Formerly, these each had to be treated in isolation. Chapters nineteen to twenty-one give a more philosophical and general discussion of probability theory in the contexts of statistics, causation, and judgment. Boole is very careful to distinguish subjective probability from objective probability, although he does not use this convenient modern vocabulary.
As we can see, Boole?s Laws of Thought is a monumental contribution to human thinking. Not only does Boole venture into several areas which we would normally keep separate today, but, for him, these areas merge and form a coherent whole, each part informing another, and all this through deep philosophical reflection. Moreover, the deep philosophical reflection is not a metaphysical treatise made in isolation of any ?hard data.? Boole?s technical contribution is both original and substantial.
Michele Friend is Assistant Professor at The George Washington University. She teaches courses in formal logic, critical thinking, and the philosophy of mathematics. Her research is in the philosophy of mathematics and logic.
Posted June 10, 2011
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