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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.

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Licensed from Columbia University Press

Licensed from Columbia University Press

Formal systems incorporating modalities such as necessity, possibility, impossibility, contingency, strict implication, and certain other closely related concepts. The most straightforward way of constructing a modal logic is to add to some standard nonmodal logical system a new primitive operator intended to represent one of the modalities, to define other modal operators in terms of it, and to add axioms and/or transformation rules involving those modal operators. For example, one may add the symbol L, which means “It is necessary that,” to classical propositional calculus; thus, Lp is read as “It is necessary that p.” The possibility operator M (“It is possible that”) may be defined in terms of L as Mp = ¬L¬p (where ¬ means “not”). In addition to the axioms and rules of inference of classical propositional logic, such a system might have two axioms and one rule of inference of its own. Some characteristic axioms of modal logic are: (A1) Lp ⊃ p and (A2) L(p ⊃ q) ⊃ (Lp ⊃ Lq). The new rule of inference in this system is the Rule of Necessitation: If p is a theorem of the system, then so is Lp. Stronger systems of modal logic can be obtained by adding additional axioms. Some add the axiom Lp ⊃ LLp; others add the axiom Mp ⊃ LMp.

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Philosophical study of the nature and scope of logic. Examples of questions raised in the philosophy of logic are: “In virtue of what features of reality are the laws of logic true?”; “How do we know the truths of logic?”; and “Could the laws of logic ever be falsified by experience?” The subject matter of logic has been variously characterized as the laws of thought, “the rules of right reasoning,” “the principles of valid argumentation,” “the use of certain words called logical constants,” and “truths based solely on the meanings of the terms they contain.”

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Study of inference and argument. Inferences are rule-governed steps from one or more propositions, known as premises, to another proposition, called the conclusion. A deductive inference is one that is intended to be valid, where a valid inference is one in which the conclusion must be true if the premises are true (*see* deduction; validity). All other inferences are called inductive (*see* induction). In a narrow sense, logic is the study of deductive inferences. In a still narrower sense, it is the study of inferences that depend on concepts that are expressed by the “logical constants,” including: (1) propositional connectives such as “not,” (symbolized as ¬), “and” (symbolized as ∧), “or” (symbolized as ∨), and “if-then” (symbolized as ⊃), (2) the existential and universal quantifiers, “(∃x)” and “(∀x),” often rendered in English as “There is an *x* such that *elipsis*” and “For any (all) *x*, *elipsis*,” respectively, (3) the concept of identity (expressed by “=”), and (4) some notion of predication. The study of the logical constants in (1) alone is known as the propositional calculus; the study of (1) through (4) is called first-order predicate calculus with identity. The logical form of a proposition is the entity obtained by replacing all nonlogical concepts in the proposition by variables. The study of the relations between such uninterpreted formulas is called formal logic. *Seealso* deontic logic; modal logic.

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Logic based on the concept of fuzzy sets, in which membership is expressed in varying probabilities or degrees of truth—that is, as a continuum of values ranging from 0 (does not occur) to 1 (definitely occurs). As additional data are gathered, many fuzzy-logic systems are able to adjust the probability values assigned to different parameters. Because some such systems appear able to learn from their mistakes, they are often considered a crude form of artificial intelligence. The term and concept date from a 1965 paper by Lotfi A. Zadeh (born 1921). Fuzzy-logic systems achieved commercial application in the early 1990s. Advanced clothes-washing machines, for example, use fuzzy-logic systems to detect and adapt to patterns of water movement during a wash cycle, increasing efficiency and reducing water consumption. Other products using fuzzy logic include camcorders, microwave ovens, and dishwashers. Other applications include expert systems, self-regulating industrial controls, and computerized speech- and handwriting-recognition programs.

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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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