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Logic is the systematic study of valid inference. A distinction is drawn between logical validity and truth. Validity merely refers to formal properties of the process of inference. Thus, a conclusion whose value is true may be drawn from an invalid argument, and one whose value is false, from a valid sequence. For example, the argument All professors are brilliant; Smith is a professor, therefore, Smith is brilliant  is a valid inference, but the argument  All professors are brilliant; Smith is brilliant; therefore, Smith is a professor  is an invalid inference, even if Smith is a professor. Aristotelian Logic 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. Post-Aristotelian Logic One of Aristotle's tacit assumptions was that there is a correspondence linking the structures of reality, the mind, and language (and hence logic). This position came to be known in the Middle Ages as realism. The opposing school of thought, nominalism, is exemplified by  William of Occam , a medieval logician, who maintained that the structure of language and logic corresponds only to the structure of the mind, not to that of reality. Since knowledge is a study of generalizations, while nature occurs in myriad single instances, the distinction between the world and our conception of it is stressed by the nominalists. Inductive Reasoning In the 19th cent. John Stuart Mill  noticed the same dichotomy between man's generalizations and nature's instances, but moved toward a different conclusion. Mill held that the scientist or experimenter is not interested in moving from the general to the specific case, which characterizes deductive logic, but is concerned with inductive reasoning, moving from the specific to the general (see induction ). For example, the statement The sun will rise tomorrow  is not the result of a particular deductive process, but is based on a psychological calculation of general probability based on many specific past experiences. Mill's chief contribution to logic rests on his efforts to formulate rules of inductive logic. Although since the criticisms of David Hume there has been disagreement about the validity of induction, modern logicians have argued that inductive logic does not need justification any more than deductive logic does. The real problem is to establish rules of induction, just as Aristotle established rules of deduction. Mathematics and Logic With the development of symbolic logic  by George  Boole  and Augustus  De Morgan  in the 19th cent., logic has been studied in more purely mathematical terms, and mathematical symbols have replaced ordinary language. Reference to external interpretations of the symbols (formulated in ordinary language) was also rejected by the formalist movement of the early 20th cent. Bertrand Russell  and Alfred North  Whitehead, in  Principia Mathematica  (3 vol., 1910-13), attempted to develop logical theory as the basis for mathematics. Pure formal logic attempts to prove that a logical system is dependent only on the perceptual recognition and valid manipulation of symbols and requires no interpretive reference to content. Intuitionism, rejecting such formalism, holds that words and formulas have significance only as a reflection of activity in the mind. Thus a theorem has meaning only if it represents a mental construction of a mathematical or logical entity. Kurt Gödel, in the 1930s, brought forth his   "incompleteness theorem,"   which demonstrates that an infinitude of propositions that are underivable from the axioms of a system nevertheless have the value of true within the system. Neither these Gödel Propositions, as they are called, nor their negations are provable. One implication for the modern logician is that Aristotle's law of the excluded middle ( either A or not A ) is neither so simple nor so self-evident as it once seemed.