The Laws of Thought
Encyclopedia
The Laws of Thought, more precisely, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, is a very influential 19th century book on logic
by George Boole
, the second of his two monographs on algebraic logic
. It was published in 1854.
George Boole was Professor
of Mathematics
of then Queen's College, Cork in Ireland
(now University College Cork
, where the Boole Centre for Research in Informatics is named in his honour)
Boole's work began the discipline of algebraic logic
, and is often, but mistakenly, credited as being the source of what we know today as Boolean algebra. In fact, however, Boole's algebra differs from modern Boolean algebra, in that in Boole's algebra A+B cannot be interpreted by set union, due to the permissibility of uninterpretable terms in Boole's calculus, and so algebras on Boole's account cannot be interpreted by sets under the operations of union, intersection and complement, as is the case with modern Boolean algebra. The task of developing the modern account of Boolean algebra fell to Boole's successors in the tradition of algebraic logic (Jevons 1869, Peirce 1880, Jevons 1890, Schröder 1890, Huntingdon 1904).
The coherences of the whole enterprise is justified by Boole in what Stanley Burris has later called the "rule of 0s and 1s", which justifies the claim that uninterpretable terms cannot be the ultimate result of equational manipulations from meaningful starting formulae (Burris 2000). Boole provided no proof of this rule, but the coherence of his system was proved by Theodore Hailperin, who provided an interpretation based on a fairly simple construction of rings
from the integers to provide an interpretation of Boole's theory (Hailperin 1976).
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
by George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
, the second of his two monographs on algebraic logic
Algebraic logic
In mathematical logic, algebraic logic is the study of logic presented in an algebraic style.What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics and connected problems...
. It was published in 1854.
George Boole was Professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
of Mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
of then Queen's College, Cork in Ireland
Ireland
Ireland is an island to the northwest of continental Europe. It is the third-largest island in Europe and the twentieth-largest island on Earth...
(now University College Cork
University College Cork - National University of Ireland, Cork
University College Cork is a constituent university of the National University of Ireland. The university is located in Cork....
, where the Boole Centre for Research in Informatics is named in his honour)
Boole's work began the discipline of algebraic logic
Algebraic logic
In mathematical logic, algebraic logic is the study of logic presented in an algebraic style.What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics and connected problems...
, and is often, but mistakenly, credited as being the source of what we know today as Boolean algebra. In fact, however, Boole's algebra differs from modern Boolean algebra, in that in Boole's algebra A+B cannot be interpreted by set union, due to the permissibility of uninterpretable terms in Boole's calculus, and so algebras on Boole's account cannot be interpreted by sets under the operations of union, intersection and complement, as is the case with modern Boolean algebra. The task of developing the modern account of Boolean algebra fell to Boole's successors in the tradition of algebraic logic (Jevons 1869, Peirce 1880, Jevons 1890, Schröder 1890, Huntingdon 1904).
Uninterpretable terms
In Boole's account of his algebra, terms are reasoned about equationally, without a systematic interpretation being assigned to them. In places, Boole talks of terms being interpreted by sets, but he also recognises terms that cannot always be so interpreted, such as the term 2AB, which arises in equational manipulations. Such terms he classes uninterpretable terms; although elsewhere he has some instances of such terms being interpreted by integers.The coherences of the whole enterprise is justified by Boole in what Stanley Burris has later called the "rule of 0s and 1s", which justifies the claim that uninterpretable terms cannot be the ultimate result of equational manipulations from meaningful starting formulae (Burris 2000). Boole provided no proof of this rule, but the coherence of his system was proved by Theodore Hailperin, who provided an interpretation based on a fairly simple construction of rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
from the integers to provide an interpretation of Boole's theory (Hailperin 1976).