Editing History of logic (section)

===Algebraic period===
[[File:George Boole color.jpg|thumb|140px|George Boole]]
Modern logic begins with what is known as the "algebraic school", originating with Boole and including [[Charles Sanders Peirce|Peirce]], [[William Stanley Jevons|Jevons]], [[Ernst Schröder (mathematician)|Schröder]], and [[John Venn|Venn]].<ref>See e.g. Bochenski p. 296 and ''passim''</ref> Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work ''Mathematical Analysis of Logic'' which appeared in 1847, although [[Augustus De Morgan|De Morgan]] (1847) is its immediate precursor.<ref>Before publishing, he wrote to [[Augustus De Morgan|De Morgan]], who was just finishing his work ''Formal Logic''.  De Morgan suggested they should publish first, and thus the two books appeared at the same time, possibly even reaching the bookshops on the same day. cf. Kneale p. 404</ref> The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in [[Lincoln, Lincolnshire]].<ref>Kneale p. 404</ref> For example, let x and y stand for classes, let the symbol ''='' signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these ''elective symbols'', i.e. symbols which select certain objects for consideration.<ref name="Kneale p. 407">Kneale p. 407</ref> An expression in which elective symbols are used is called an ''elective function'', and an equation of which the members are elective functions, is an ''elective equation''.<ref>Boole (1847) p. 16</ref> The theory of elective functions and their "development" is essentially the modern idea of [[truth-function]]s and their expression in [[disjunctive normal form]].<ref name="Kneale p. 407"/>

Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between "primary propositions" which are the subject of syllogistic theory, and "secondary propositions", which are the subject of propositional logic, and showed how under different "interpretations" the same algebraic system could represent both. An example of a primary proposition is "All inhabitants are either Europeans or Asiatics." An example of a secondary proposition is "Either all inhabitants are Europeans or they are all Asiatics."<ref>Boole 1847 pp. 58–59</ref> These are easily distinguished in modern predicate logic, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system.<ref>Beaney p. 11</ref>

In his ''Symbolic Logic'' (1881), [[John Venn]] used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a "logical machine" which he showed to the [[Royal Society]] the following year.<ref name="Kneale p. 407"/> In 1885 [[Allan Marquand]] proposed an electrical version of the machine that is still extant ([https://web.archive.org/web/20080908073359/http://finelib.princeton.edu/instruction/wri172_demonstration.php picture at the Firestone Library]).
[[File:Charles Sanders Peirce.jpg|left|thumb|160px|Charles Sanders Peirce]]
The defects in Boole's system (such as the use of the letter ''v'' for existential propositions) were all remedied by his followers. Jevons published ''Pure Logic, or the Logic of Quality apart from Quantity'' in 1864, where he suggested a symbol to signify [[exclusive or]], which allowed Boole's system to be greatly simplified.<ref>Kneale p. 422</ref> This was usefully exploited by Schröder when he set out theorems in parallel columns in his ''Vorlesungen'' (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "[[Logical NOR|neither ... nor ...]]" and equally well "[[Sheffer stroke|not both ... and ...]]",<ref>Peirce, "A Booli<!-- sic! -->an Algebra with One Constant", 1880 MS, ''Collected Papers'' v. 4, paragraphs 12–20, reprinted ''Writings'' v. 4, pp. 218–221. Google [https://archive.org/details/writingsofcharle0002peir <!-- quote=378 Winter. --> Preview].</ref> however, like many of Peirce's innovations, this remained unknown or unnoticed until [[Henry M. Sheffer|Sheffer]] rediscovered it in 1913.<ref>''Trans. Amer. Math. Soc., xiv (1913)'', pp. 481–488. This is now known as the [[Sheffer stroke]]</ref> Boole's early work also lacks the idea of the [[logical sum]] which originates in Peirce (1867), [[Ernst Schröder (mathematician)|Schröder]] (1877) and Jevons (1890),<ref>Bochenski 296</ref> and the concept of [[Inclusion (logic)|inclusion]], first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).
[[File:Boolean multiples of 2 3 5.svg|alt=Coloured diagram of 4 interlocking sets|right|thumb|250px|Boolean multiples]]
The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schröder's monumental ''Vorlesungen über die Algebra der Logik'' ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce.<ref>See CP III</ref>

Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic [[John Corcoran (logician)|John Corcoran]] in an accessible introduction to ''[[The Laws of Thought|Laws of Thought]].''<ref>[[George Boole]]. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by J. Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review. 24 (2004) 167–169.</ref> Corcoran also wrote a point-by-point comparison of ''Prior Analytics'' and ''Laws of Thought''.<ref>JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288.</ref> According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat—from assessing validity to solving equations—and 3) expanding the range of applications it could handle—e.g. from propositions having only two terms to those having arbitrarily many.

More specifically, Boole agreed with what [[Aristotle]] said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. 
First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations—by itself a revolutionary idea. 
Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. 
Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle".