Synopsis of George Boole, ‘L’analyse mathématique de la logique’
[‘The Mathematical Analysis of Logic’]
George Boole (1815-1864) was a British mathematician and philosopher, who made a decisive contribution to the development of modern (post-Aristotelian or post-syllogistic) logic. Boole conceives of logic as a ‘calculus of deductive reasoning’, one that applies not only to ‘external’ ideas (via ‘intuitions of space and time’) of quantity and number but also to a ‘deeper set of relations’, grounded in the inner ‘constitution of the mind’ (Boole, Mathematical Analysis of Logic, preface). Adapting a version of algebraic formalisation, Boole defines logical symbols x or y in terms of their conceptual extension, i.e. in terms of the class or group of objects they delimit. His approach anticipates some of the principles of Cantorian set theory, and allows for a clarification and generalisation of the rules of logical inference.
Boole presented an initial overview of his new approach to logic in a short book entitled the Mathematical Analysis of Logic (1847), which has not yet been translated into French. He provided a more substantial and more systematic exposition in his Investigation of the Laws of Thought (1854; a French version was published in 1992). Translated by Yves Michaud, the extract published in the tenth volume of the Cahiers pour l’Analyse is taken from the opening pages of the 1847 book.
In the brief preface to the Mathematical Analysis of Logic (which isn’t included in the Cahiers’ translation) Boole asks his readers, before they assess the merits of his project, to respect the condition that ‘no preconceived notion of the impossibility of its objects shall be permitted to interfere with that candour and impartiality which the investigation of Truth demands’ – a condition which might be said to apply to the general project of the Cahiers itself. The uninhibited work of analysis, Boole notes, must ‘dismiss all regard for precedent and authority, to interrogate the method itself for an expression of the just limits of its application’ (CpA 10.2:29).
Boole begins the Introduction to his book by emphasising the fact that the validity of processes of algebraic analysis ‘does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination’, which may then be applied or interpreted according to diverse domains (geometry, optics, number, etc.) (CpA 10.2:27). For Boole, ‘the mathematics we have to construct are the mathematics of the human intellect’ (29), so it is a mistake to conceive of mathematics as not just ‘actually’ but also ‘essentially’ the ‘science of magnitude’. A more adequate and more general definition of a ‘true calculus’ will conceive it as a method that – whatever the domain of its application – ‘rests upon the employment of symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation’ (28).
The key to such a method is our ability (which Boole here takes for granted) to relate, via our faculty of language, symbols to general notions or classes of objects. ‘That which renders Logic possible, is the existence in our minds of general notions, our ability to conceive of a class, and to designate its individual members by a common name’ (CpA 10.2:28). For an individual to belong to a class of objects is to share a ‘quality in common with other individuals’ (31).
Application of this ability to classify, or to treat ‘any conceivable collection of objects’ as a distinct group, involves ‘electing’ or isolating them from other objects. The process of such election can be repeated or qualified; Boole’s main concern here is to grasp the general principles that govern such processes, so as to ensure that our use of symbols is never a matter of ‘unreasoning reliance’ or ‘authority’ but rather a practice based on ‘perfect comprehension of that which renders their use lawful’ (31).
In the first chapter of his book (corresponding to section 1 of the Cahiers translation) Boole lists the ‘first principles’ which underlie the lawful use of symbols and classes. He proposes to use the symbol of unity, 1, to represent ‘the universe’, i.e. all conceivable objects. The symbols X, Y, Z etc. here represent individual i.e. typical members of classes (e.g. a member of the class of prime numbers, or mammals, or sheep...). The ‘elective symbol [symbole de choix]’ x (or y, z etc.), will then represent the operation that selects or isolates, from any given collection of classes or individuals, all the Xs which that collection contains. The product of two elective symbols combined, xy, will represent the selection, from our initial collection, of all those elements that can be classed as both X and Y. The class of ‘horned sheep’, for instance, selects from the collection of animals or of all possible objects (or any other collection) those which (a) have the quality of being a sheep, and (b) have horns. If x is here the symbol for the operation of selecting sheep and y of horned animals, then 1 – x would represent the collection of all objects in the universe that are not sheep, and 1 – x – y the collection of all things that are neither sheep nor have horns.
Boole then identifies three ‘laws of combination and succession’ which apply to the use of elective symbols, and which together provide the ‘sufficient basis of a calculus’ (CpA 10.2:33). First, the result of an act of classification or election is indifferent to the organisation or classification of the original collection. Once isolated, our class of ‘horned sheep’ will be independent of the initial collection to which they belong(ed). This guarantees the ‘distributive’ quality of elective symbols. Second, the order in which two or more successive operations of election are applied is irrelevant to the outcome: whether you first isolate horned animals, or sheep, the resulting collection will be the same (32). This ensures the ‘commutative’ quality of elective symbols. Boole calls his third law the ‘index law’ (which Michaud translates here as the ‘loi de fabulation’): ‘the result of a given act of election performed twice, or any number of times in succession, is the result of the same act performed once’; i.e. xx = x.
Boole further proposes that the systematic use of such elective symbols, in keeping with these three laws, would allow for all propositions to be conceived in the form of equations, whose logical validity could be determined by general laws of inference. He anticipates that, if his approach is successful, ‘every logical proposition, whether categorical or hypothetical, will be found to be capable of exact and rigorous expression, and not only will the laws of conversion and of syllogism be thence deducible, but the resolution of the most complex systems of propositions, the separation of any proposed element, and the expression of its value in terms of the remaining elements, with every subsidiary relation involved. Every process will represent deduction, every mathematical consequence will express a logical inference’ (CpA 10.2:29).
References to this text in other articles in the Cahiers pour l’Analyse:
- Badiou, Alain. ‘Marque et manque’, CpA 10.8:156.
- Boole, George. The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning. Cambridge: MacMillan, Barclay and MacMillan, 1847. Reprinted in Boole, Studies in Logic and Probability, ed. Rush Rhees. London: Watts & Co, 1952. Online at http://www.archive.org/details/mathematicalanal00booluoft
- Boole, George, An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities. London: Walton & Maberly, 1854. Reprinted in George Boole’s Collected Logical Works. Chicago: Open Court, 1916, vol. 2. Online at http://www.gutenberg.org/ebooks/15114.
Selected secondary literature:
- Gasser, James, ed. A Boole Anthology: Recent and Classical Studies in the Logic of George Boole. Dordrecht: Kluwer Academic, 2000.
- Grattan-Guinness, Ivor. The Search for Mathematical Roots 1870-1940. Princeton: Princeton University Press, 2000.
- Hailperin, Theodore. Boole’s Logic and Probability: A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory. Amsterdam: North Holland, 1986.
- MacHale, Des. George Boole: His Life and Work. Dublin: Boole Press, 1985.