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logic, 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

## Post-Aristotelian Logic

## Inductive Reasoning

## Mathematics and 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.

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.

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.

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.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

Non sequitur (Latin for "it does not follow."), in formal logic, is an argument where its conclusion does not follow from its premises. In a non sequitur, the conclusion can be either true or false, but the argument is a fallacy because the conclusion does not follow from the premise. All formal fallacies are special cases of non sequitur. The term has special applicability in law, having a formal legal definition. Many types of known non sequitur argument forms have been classified into many different types of logical fallacies.

The term is often used in everyday speech and reasoning to describe a statement in which premise and conclusion are totally unrelated but which is used as if they were. An example might be: "If I buy this cell phone, all people will love me." However, there is no actual relation between buying a cell phone and the love of all people. This kind of reasoning is often used in Advertising to trigger an emotional purchase.

Other examples include:

- * "If you buy this car your family will be safer." (While some cars are safer than others, they will most likely decrease instead of increase your family's overall safety.)

- * "If you do not buy this type of pet food you are neglecting your dog." (Premise and conclusion are once again unrelated, this is also an example of an appeal to emotion.)

- * "Our product is so good, it was even given away in celebrity gift bags." (True, perhaps, but not relevant to the quality of the product)

The fallacy of the undistributed middle is a logical fallacy that is committed when the middle term in a categorical syllogism isn't distributed. It is thus a syllogistic fallacy. More specifically it is also a form of non sequitur.

The fallacy of the undistributed middle takes the following form:

- All Zs are Bs
- Y is a B
- Therefore, Y is a Z

It may or may not be the case that "all Zs are Bs," but in either case it is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that "all Bs are Zs," which is ignored in the argument.

Note that if the terms were swapped around in either the conclusion or the first co-premise or if the first premise was rewritten to "All Zs can only be Bs" then it would no longer be a fallacy although it could still be unsound. This also goes for the following two logical fallacies which are similar in nature to the fallacy of the undistributed middle and also non sequiturs.

An example can be given as follows:

- All men are human
- Ann is a human
- Therefore, Ann is a man

Any argument that takes the following form is a non sequitur

- If A is true, then B is true.
- B is stated to be true.
- Therefore, A must be true.

Even if the premises and conclusion are all true, the conclusion is not a necessary consequence of the premises. This sort of non sequitur is also called affirming the consequent.

An example of affirming the consequent would be:

- If I am a human (A) then I am a mammal. (B)
- I am a mammal. (B)
- Therefore, I am a human. (A)

"I" could be another type of mammal without being a human. While the conclusion may be true, it does not follow from the premises. This argument is still a fallacy even if the conclusion is true. It is a non sequitur.

Another common non sequitur is this:

- If A is true, then B is true.
- A is stated to be false.
- Therefore B must be false.

While the conclusion can indeed be false, this cannot be linked to the premise since the statement is a non sequitur. This is called denying the antecedent.

An example of denying the antecedent would be:

- If I am in Tokyo, I am in Japan.
- I am not in Tokyo.
- Therefore, I am not in Japan.

Whether or not the speaker is in Japan cannot be derived from the premise. He could either be outside Japan or anywhere in Japan except Tokyo.

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Last updated on Wednesday October 01, 2008 at 18:36:21 PDT (GMT -0700)

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