Editing History of logic (section)

===Embryonic period===
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|upright|Leibniz]]
The idea that inference could be represented by a purely mechanical process is found as early as [[Ramon Llull|Raymond Llull]], who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the [[Oxford Calculators]]<ref>Edith Sylla (1999), "Oxford Calculators", in ''The Cambridge Dictionary of Philosophy'', Cambridge, Cambridgeshire:  Cambridge.</ref> led to a method of using letters instead of writing out logical calculations (''calculationes'') in words, a method used, for instance, in the ''Logica magna'' by [[Paul of Venice]]. Three hundred years after Llull, the English philosopher and logician [[Thomas Hobbes]] suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction.<ref>El. philos. sect. I de corp 1.1.2.</ref> The same idea is found in the work of [[Gottfried Wilhelm Leibniz|Leibniz]], who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages are subject to "countless ambiguities" and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words;<ref>Bochenski p. 274</ref> hence, he proposed to identify an [[alphabet of human thought]] comprising fundamental concepts which could be composed to express complex ideas,<ref>Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., ''The Cambridge Companion to Leibniz''. Cambridge Univ. Press.</ref> and create a ''[[calculus ratiocinator]]'' that would make all arguments "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate."<ref>Wiener, Philip, 1951. ''Leibniz: Selections''. Scribner.</ref>

[[Joseph Diaz Gergonne|Gergonne]] (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning of the symbols involved.<ref>''Essai de dialectique rationelle'', 211n, quoted in Bochenski p. 277.</ref> [[Bernard Bolzano|Bolzano]] anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables:<ref>{{cite book |author-last=Bolzano |author-first=Bernard |url=https://books.google.com/books?id=oA1NDDirneQC&q=%22deducible%20from%20propositions%22&pg=PA209 |title=The Theory of Science: Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik |date=1972 |publisher=[[University of California Press]] |isbn=978-0-52001787-0 |editor-last=George |editor-first=Rolf |page=209 |translator-last=Rolf |translator-first=George}}</ref><blockquote>Hence I say that propositions <math>M</math>, <math>N</math>, <math>O</math>,... are ''deducible'' from propositions <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,... with respect to variable parts <math>i</math>, <math>j</math>,..., if every class of ideas whose substitution for <math>i</math>, <math>j</math>,... makes all of <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,... true, also makes all of <math>M</math>, <math>N</math>, <math>O</math>,... true. Occasionally, since it is customary, I shall say that propositions <math>M</math>, <math>N</math>, <math>O</math>,... ''follow'', or can be ''inferred'' or ''derived'', from <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,.... Propositions <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,... I shall call the ''premises'', <math>M</math>, <math>N</math>, <math>O</math>,... the ''conclusions.''</blockquote>This is now known as [[semantic validity]].