Editing George Boole (section)

==== Treatment of addition in logic ====
Boole conceived of "elective symbols" of his kind as an [[algebraic structure]]. But this general concept was not available to him: he did not have the segregation standard in [[abstract algebra]] of postulated (axiomatic) properties of operations, and deduced properties.<ref name=KY>[[Andrei Nikolaevich Kolmogorov]], [[Adolf Pavlovich Yushkevich]], ''Mathematics of the 19th century: mathematical logic, algebra, number theory, probability theory'' (2001), pp. 15 (note 15)–16; [https://books.google.com/books?id=X3u5hJCkobYC&pg=PA15 Google Books] {{Webarchive|url=https://web.archive.org/web/20160517144440/https://books.google.com/books?id=X3u5hJCkobYC&pg=PA15 |date=17 May 2016 }}.</ref> His work was a beginning to the [[algebra of sets]], again not a concept available to Boole as a familiar model. His pioneering efforts encountered specific difficulties, and the treatment of addition was an obvious difficulty in the early days.

Boole replaced the operation of multiplication by the word "and" and addition by the word "or". But in Boole's original system, + was a [[partial operation]]: in the language of [[set theory]] it would correspond only to the [[Union (set theory)|union]] of disjoint subsets. Later authors changed the interpretation, commonly reading it as [[exclusive or]], or in set theory terms [[symmetric difference]]; this step means that addition is always defined.<ref name=Marc /><ref>{{cite SEP |url-id=algebra-logic-tradition |title=The Algebra of Logic Tradition |last=Burris |first=Stanley}}</ref>

In fact, there is the other possibility generalizing Boole's original partial operation, that + should be read as [[disjunction|non-exclusive or]].<ref name=KY /> Handling this ambiguity was an early problem of the theory, reflecting the modern use of both [[Boolean ring]]s and Boolean algebras (which are simply different aspects of one type of structure). Boole and [[William Stanley Jevons|Jevons]] struggled over just this issue in 1863, in the form of the correct evaluation of ''x'' + ''x''. Jevons argued for the result ''x'', which is correct for + as disjunction. Boole kept the result as something undefined. He argued against the result 0, which is correct for exclusive or, because he saw the equation ''x'' + ''x'' = 0 as implying ''x'' = 0, a false analogy with ordinary algebra.<ref name=SED />