Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as (...) a system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind. (shrink)

Appropriate for upper-level undergraduates and graduate students, this volume includes a variety of Boole's writings on logical subjects, along with papers on related questions of probability. His earlier work, The Mathematical Analysis of Logic, appears here, together with an account of the notes Boole made on his own interleaved copy. In addition, the appendices contain relevant papers by contemporaries with whom the author engaged in discussion, making it possible to trace interesting developments in Boolean reasoning-particularly in regard to (...) his extended treatment of the relation between formal logic and the theory of probabilities. 1952 ed. (shrink)

This paper studies the possible influences on Boole's logic of the writings related to the controversy over the quantification of the predicate between the philosopher William Hamilton and the mathematician Augustus De Morgan. As Boole himself testified in the introduction to his book The mathematical analysis of logic , this controversy was the external agent that stimulated him into writing up his earlier thoughts about a new conception of logic. But in addition to the external role (...) that was played by the controversy, it seems that other kinds of influences on Boole of Hamilton's and De Morgan's ideas about logic, which were made explicit in writings related to the controversy, can also be traced. (shrink)

This groundbreaking work on logic by the brilliant 19th-century English mathematician George Boole remains influential to this day. Boole's major contribution was to demonstrate conclusively that the symbolic expressions of algebra could be adapted to convey the fundamental principles and operations of logic, which hitherto had been expressed only in words. Boole was thus the founder of today's science of symbolic logic. Summing up his innovative approach, Boole stated, "We ought no longer to associate Logic (...) and Metaphysics, but Logic and Mathematics." As the great English logician Augustus De Morgan later put it, in praise of Boole, his genius consisted in showing that "the symbolic processes of algebra, invented as tools of numerical calculation, should be competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic." The Laws of Thought lays out this new system in detail and also explores a "calculus of probability." The story of Boole's life is as impressive as his work. Besides rudimentary lessons from his father and a few years at local schools, Boole was largely self-taught. Revealing his aptitude for many subjects at an early age, he began his career already at age 16 as a teacher at a village school. In his leisure time he tackled the daunting works of Newton, Laplace, and Lagrange on physics and mathematics. By the age of twenty-four he was submitting original papers to the Cambridge Mathematical Journal and at age twenty-nine he won a medal from the Royal Society for his contributions to mathematical analysis. He continued to so impress his contemporaries that five years later he was appointed professor of mathematics at Queens College, Cork in Ireland, even though he had no university degree. At his untimely death of forty-nine, Boole could never have guessed that his new symbolic logic would become essential in the next century for telephone switching and the design of computers. For this practical reason, as well as the sheer intellectual importance of his accomplishment, The Laws of Thought merits our attention today. (shrink)

In the Laws of Thought, Boole establishes a theory of secondary propositions based upon the notion of time. This temporal interpretation of secondary propositions has historically been met with wide disapproval and is usually dismissed in the modern literature as a philosophical non-starter. What was Boole thinking? This paper attempts to give an answer to this question. Specifically, it provides an account according to which Boole’s temporal interpretation follows from his psychologistic conception of logic, in addition to certain background (...) assumptions regarding the psychological necessity of the concept of time. Once Boole’s psychological premises are laid bare, it becomes clearer how he might have viewed the temporal interpretation to be an essential feature of his theory of secondary propositions. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. (...) These practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

George Boole published his account on hypotheticals in his pamphlet The Mathematical Analysis of Logic in 1847. Hypothetical deductions were not as developed as categorical ones by Boole’s time. It was still common practice to reduce hypotheticals to categoricals. Boole innovated by proposing an algebraic method to derive (equations expressing) the conclusions of hypotheticals. He had developed a calculus of classes for categoricals in his first pamphlet chapters and seemingly intended extending it to hypotheticals. Nonetheless, propositions can be only (...) true or false. In contradistinction with his approach for categorical propositions, here it was not possible to consider things or classes of things but only the cases and conjunctures of circumstances in which the propositions are true. Some criticisms have been raised concerning Boole’s view of logic as the solution of algebraic equations. In order to circumvent such criticisms, we construct a deductive system based on Boole's algebraic treatment and apply it to deductions of those hypotheticals analyzed by Boole. We found that Barbara translated in our notation is useful to eliminate symbols in such deductions. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to hypotheticals in The Mathematical Analysis of Logic. (shrink)

In the mid-nineteenth century George Boole formulated his ‘conditions of possible experience’. These are equations and ineqaulities that the relative frequencies of events must satisfy. Some of Boole's conditions have been rediscovered in more recent years by physicists, including Bell inequalities, Clauser Horne inequalities, and many others. In this paper, the nature of Boole's conditions and their relation to propositional logic is explained, and the puzzle associated with their violation by quantum frequencies is investigated in relation to (...) a variety of approaches to the interpretation of quantum mechanics. * While preparing this paper for publication I have learnt of the untimely death of Professor J. S. Bell, and I wish to dedicate the paper to his memory. This research was undertaken while I spent a sabbatical leave at Wolfson College, and the History and Philosophy of Science Department at the University of Cambridge. I would like to thank Michael Redhead and Jeremy Butterfield for their hospitality and for helpful discussions. A first draft of this paper has been distributed among the participants of the conference 'Einstein in Context' which was held in Israel, in April 1990.1 have benefited from the comments of many colleagues. I would like to thank in particular Arthur Fine who enlightened me on the prism models, David Albert. Maya Bar-Hillel. Yemima Ben-Menachem, Mara Beller. Simon Saunders, and Mark Steiner. This research is partially supported by the Edelstein Center for the History and Philosophy of Science at the Hebrew University. (shrink)

The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call proto-Boolean, and a "general method in Logic" making use of that algebra. Our object is to elucidate these two components of Boole's system, to prove his principal results, and to draw some conclusions not explicit in LT. We also discuss some examples of incoherence in LT; these mask the genius of Boole's design and account for much of the puzzled and (...) disparaging commentary LT has received. Our evaluation of Boole's logical system does not differ substantially from that advanced in Hailperin's exhaustive study, Boole's Logic and Probability. Unlike the latter work, however, we make direct use of the polynomials native to LT rather than appealing to formalisms such as multisets and rings. (shrink)

George Boole collected ideas for the improvement of his Mathematical analysis of logic(1847) on interleaved copies of that work. Some of the notes on the interleaves are merely minor changes in explanation. Others amount to considerable extension of method in his mathematical approach to logic. In particular, he developed his technique in solving simultaneous elective equations and handling hypotheticals and elective functions. These notes and extensions provided a source for his later book Laws of thought(1854).

The approach used by Boole in Mathematical analysis of logic to develop propositional logic was based on the idea of ?cases? or ?conjunctures of circumstances?. But this was dropped in Laws of thought in favor of one which Boole considered to be more satisfactory, that of using the notion of ?time for which a proposition is true?. We show that, when suitable clarifications and corrections are made, the earlier approach? which accords with modern logic in eschewing the (...) extraneous notion of time?is indeed a viable one. (shrink)

We show how one can give a clear formal account of Boole’s notorious “indefinite" (or “auxiliary”) symbols by treating them as variables that range over functions from classes to classes rather than just over classes while, at the same time, following Hailperin’s proposal of binding them existentially.

This paper deals with the influence exerted by Boole's own work on differential equations on his creation of algebraic logic. The main traits of Boole's methodology of logic, and the particular algorithms which he used in his 1847 The mathematical analysis of logic, are first pointed out. An examination of the mathematical papers which Boole wrote before the publication of the mentioned logical treatise shows that both the methodology leading to the production of his (...) class='Hi'>logic and the algorithms used in its development were repeatedly used by him in his earlier work in analysis. (shrink)

This paper discusses George Boole’s two distinct approaches to the explanatory relationship between logical and psychological theory. It is argued that, whereas in his first book he attributes a substantive role to psychology in the foundation of logical theory, in his second work he abandons that position in favour of a linguistically conceived foundation. The early Boole espoused a type of psychologism and later came to adopt a type of anti-psychologism. To appreciate this invites a far-reaching reassessment of his philosophy (...) of logic. (shrink)

This paper augments Hailperin's substantial efforts to place Boole's algebra of logic on a solid footing. Namely Horn sentences are used to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic—we call this principle The Rule of 0 and 1.

Prior Analytics by the Greek philosopher Aristotle and Laws of Thought by the English mathematician George Boole are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle's system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss many other historically and philosophically important (...) aspects of Boole's book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole's contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology.… using mathematical methods … has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to … when Boole's masterpiece was published.Paul Rosenbloom 1950. (shrink)

George Boole (1815-1864) is well known to mathematicians for his research and textbooks on the calculus, but his name has spread world-wide for his innovations in symbolic logic and the development and applications made since his day. The utility of "Boolean algebra" in computing has greatly increased curiosity in the nature and extent of his achievements. His work is most accessible in his two books on logic, "A mathematical analysis of logic" (1947) and "An investigation of the (...) laws of thought" (1954). But at various times he wrote manuscript essays, especially after the publication of the second book; several were intended for a non-technical work, "The Philosophy of logic", which he was not able to complete. This volume contains an edited selection which not only relates them to Boole's publications and the historical context of his time, but also describes their strange history of family, followers and scholars have treid to confect an edition. The book will appeal to logicians, mathematicians and philosophers, and those interested in the histories of the corresponding subjects; and also students of the early Victorian Britain in which they were written. (shrink)

2006. George Boole. Encyclopedia of Philosophy. 2nd edition. Detroit: Macmillan Reference USA. -/- George Boole (1815-1864), whose name lives among modern computer-related sciences in Boolean Algebra, Boolean Logic, Boolean Operations, and the like, is one of the most celebrated logicians of all time. Ironically, his actual writings often go unread and his actual contributions to logic are virtually unknown—despite the fact that he was one of the clearest writers in the field. Working with various students including Susan Wood (...) and Sriram Nambiar, I have written several publications trying to set the record straight—but so far to little avail. This encyclopedia entry is one more attempt to set the record straight in a way that can be appreciated by non-experts.Also see https://www.academia.edu/10161999/Booles_criteria_of_validity_and_invalidity . (shrink)

Charles Peirce is often credited for being among the first, perhaps even the first, to develop a scientific metaphysics of indeterminism. After rejecting the received view that Peirce developed his views from Darwin and Maxwell, I argue that Peirce's view results from his synthesis of Immanuel Kant's critical philosophy and George Boole's contributions to formal logic. Specifically, I claim that Kant's conception of the laws of logic as the basis for his architectonic, when combined with Boole's (...) view of probability, yields Peirce's metaphysics of probabilistic laws. Indeterminism provides, therefore, an excellent illustration of how Peirce attempted to use logic to clarify metaphysical problems.Since everyone must have conceptions of things in general, it is most important that they should be carefully constructed. I shall enter into no criticism of the different methods of metaphysical research, but shall merely say that in the opinions of several great thinkers, the only successful mode yet lighted upon is that of adopting our logic as our metaphysics. (W1: 490, 1866)2. (shrink)

In §1 I examine Boole’s “principle of wholistic reference” in relation to Frege’s postulation of truth-values as referents for sentences. I also consider in this connection Frege’s interpretation of quantification and his view that functions and concepts must be defined for all objects. I then present my own contrasting views on the reference of sentences. In §2 I discuss Boole’s introduction of the notion of universe of discourse and consider whether one of the issues implicit in John’s paper is a (...) confrontation between absolute interpretations of logic and relativistic interpretations of logic. I conclude with a brief examination of Tarski’s views on this issue. (shrink)

Mary Everest, Boole's wife, claimed after the death of her husband that his logic had a psychological, pedagogical, and religious origin and aim rather than the mathematico-logical ones assigned to it by critics and scientists. It is the purpose of this paper to examine the validity of such a claim. The first section consists of an exposition of the claim without discussing its truthfulness; the discussion is left for the sections 2?4, in which some arguments provided by the (...) examination of the inner consistency of Mary Everest's writings, Boole's own writings, and other sources, lead to the conclusion that there are sound reasons to accept Mary Everest's viewpoint. (shrink)

Although the existence of correspondence between George Boole (1815?1864) and William Stanley Jevons (1835?1882) has been known for a long time and part was even published in 1913, it has never been fully noted; in particular, it is not in the recent edition of Jevons's letters and papers. The texts are transcribed here, with indication of their significance. Jevons proposed certain quite radical changes to Boole's system, which Boole did not accept; nevertheless, they were to become well established.

We aim at clarifying to what extent the work of the English mathematician George Boole on the algebra of logic is taken into consideration and discussed in the work of early Husserl, focusing in particular on Husserl’s lecture “Über die neueren Forschungen zur deduktiven Logik” of 1895, in which an entire section is devoted to Boole. We confront Husserl’s representation of the problem-solving processes with the analysis of “symbolic reasoning” proposed by George Boole in the Laws of Thought and (...) try to show how and why Husserl, while praising Boole’s calculus, strongly criticizes his attempt at a philosophical clarification and justification of it. (shrink)

By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, (...) is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems. (shrink)

SummaryThe germs of future development, contained in Aristotle's logical works, are indicated, and their influence on the later evolution of logic is explained.The history of symbolic logic since Boole's Mathematical analysis and De Morgan's Formal logic, both of which were published in 1847, is divided into four approximately subsequent phases, viz.:1. algebra of logic; this phase is characterized by Boole's work;2. logical foundation of mathematics; this phase is characterized by Frege's, Peano's and Russell's work, (...) by the discovery of the antonomies of logic and set theory, by the Couturat‐Poincaré debate and by the rise of Brouwer's intuitionism;3. intuitive metamathematics, concentrating in Hilbert's attempts towards a consistency proof for classical analysis; the discovery of Gödel's theorem necessitated a change of policy;4. axiomatic methodology, deriving from Hilbert's metamathematics and culminating so far in Tarski's semantics. — E. W. B. (shrink)

In the 1860s and 1870s the logic of Boole and the calculating machines of Babbage were key resources in W. S. Jevons’s attempt to construct a mechanical model of the mind, and both therefore played an important role in Jevons’s attempted revolution in economic theory. In this same period both Boole and Babbage were studied within the Cambridge Moral Sciences Tripos, but the Cambridge reading of Boole and Babbage was much more circumspect. Implicitly following the division of the moral (...) sciences into material and ‘real’ as established by the Rev. Grote, John Venn treated Boole’s logic as a purely formal science, while Alfred Marshall based his psychological model of the mechanical part of the human mind upon Babbage’s two-level machine. From the different perspectives of logic and psychology, Venn and Marshall did not simply incorporate their readings of Boole and Babbage, but also attempted to establish the limits to any mechanical explanation of the mind. This comparison of the attitudes to mental science of Jevons and Marshall provides a foundation from which the differing conceptions of economic theory of the two men can be established.Keywords: Alfred Marshall; Charles Babbage; George Boole; John Venn; Logic; Psychology. (shrink)

This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole’s theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will (...) appear as commuting projection operators and the truth values, which take the binary values \, are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches. (shrink)

“There’s Plenty of Room at the Bottom”, said the title of Richard Feynman’s 1959 seminal conference at the California Institute of Technology. Fifty years on, nanotechnologies have led computer scientists to pay close attention to the links between physical reality and information processing. Not all the physical requirements of optimal computation are captured by traditional models—one still largely missing is reversibility. The dynamic laws of physics are reversible at microphysical level, distinct initial states of a system leading to distinct final (...) states. On the other hand, as von Neumann already conjectured, irreversible information processing is expensive: to erase a single bit of information costs ~3 × 10−21 joules at room temperature. Information entropy is a thermodynamic cost, to be paid in non-computational energy dissipation. This paper addresses the problem drawing on Edward Fredkin’s Finite Nature hypothesis: the ultimate nature of the universe is discrete and finite, satisfying the axioms of classical, atomistic mereology. The chosen model is a cellular automaton with reversible dynamics, capable of retaining memory of the information present at the beginning of the universe. Such a CA can implement the Boolean logical operations and the other building bricks of computation: it can develop and host all-purpose computers. The model is a candidate for the realization of computational systems, capable of exploiting the resources of the physical world in an efficient way, for they can host logical circuits with negligible internal energy dissipation. (shrink)

In the ‘analytic tradition’, Hans Sluga wrote thirty years ago in his book Gottlob Frege, there has been a ‘lack of interest in historical questions — even in the question of its own roots. Anti-historicism has been the baggage of the tradition since Frege’ (Sluga, 1980, p. 2). The state of the discussion of Frege among analytic philosophers, Sluga claimed, illustrated well this indifference. Despite the numbers of pages devoted to Frege, there was still, Sluga claimed, little understanding of the (...) sources of Frege’s ideas, his intellectual debts, and the historical circumstances of his thought. Sluga singled out Michael Dummett’s Frege: Philosophy of Language ‘as a paradigm for the failure of analytic philosophers to come to grips with the actual, historical Frege’. In that book, Dummett famously wrote that the logical system that Frege put forward in his 1879 Begriffsschrift ‘is astonishing because it has no predecessors: it appears to have been born from Frege’s brain unfertilized by external influences’ (Dummett, 1981a, p. xxxv) — and Dummett devoted almost none of its 700 pages to the relationship between Frege and his contemporaries. (shrink)

In this note we reproduce the book reviews that De Morgan wrote on Boole's and Jevons's first logical works. The most notable property of these documents is the mere fact of their existence and the absence of any reference to them in the specialized literature.