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In Western thought, systematic logic is considered to have begun with Aristotle's collection of treatises, the Organon [tool]. Aristotle introduced the use of variables: While his contemporaries illustrated principles by the use of examples, Aristotle generalized, as in: All x are y; all y are z; therefore, all x are z. Aristotle posited three laws as basic to all valid thought: the law of identity, A is A; the law of contradiction, A cannot be both A and not A; and the law of the excluded middle, A must be either A or not A.
Aristotle believed that any logical argument could be reduced to a standard form, known as a syllogism. A syllogism is a sequence of three propositions: two premises and the conclusion. By varying the form of the proposition and the modifiers (such as all, no, and some), a few specific forms may be delimited. Although Aristotle was concerned with problems in modal logic and other minor branches, it is usually agreed that his major contribution in the field of logic was his elaboration of syllogistic logic; indeed, the Aristotelian statement of logic held sway in the Western world for 2,000 years. Nonetheless, various logicians did, during that time, take issue with parts of Aristotle's thought.
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