Logic

I

Introduction

Logic (Greek logos, “word,” “speech,” “reason”), science dealing with the principles of valid reasoning and argument. The study of logic is the effort to determine the conditions under which one is justified in passing from given statements, called premises, to a conclusion that is claimed to follow from them. Logical validity is a relationship between the premises and the conclusion such that if the premises are true then the conclusion is true.

The validity of an argument should be distinguished from the truth of the conclusion. If one or more of the premises is false, the conclusion of a valid argument may be false. For example, “All mammals are four-footed animals; all people are mammals; therefore, all people are four-footed animals” is a valid argument with a false conclusion. On the other hand, an invalid argument may by chance have a true conclusion. “Some animals are two-footed; all people are animals; therefore, all people are two-footed” happens to have a true conclusion, but the argument is not valid. Logical validity depends on the form of the argument, not on its content. If the argument were valid, some other term could be substituted for all occurrences of any one of those used and validity would not be affected. By substituting “four-footed” for “two-footed,” it can be seen that the premises could both be true and the conclusion false. Thus the argument is invalid, even though it has a true conclusion.

II

Aristotelian Logic

What is now known as classical or traditional logic was first formulated by Aristotle, who developed rules for correct syllogistic reasoning. A syllogism is an argument made up of statements in one of four forms: “All A's are B's” (universal affirmative), “No A's are B's” (universal negative), “Some A's are B's” (particular affirmative), or “Some A's are not B's” (particular negative). The letters stand for common nouns, such as “dog,””four-footed animal,””living thing,” which are called the terms of the syllogism. A well-formed syllogism consists of two premises and a conclusion, each premise having one term in common with the conclusion and one in common with the other premise. In classical logic, rules are formulated by which all well-formed syllogisms are identified as valid or invalid forms of argument.

III

Modern Logic

In the middle of the 19th century, the British mathematicians George Boole and Augustus De Morgan opened a new field of logic, now known as symbolic or modern logic, which was further developed by the German mathematician Gottlob Frege and especially by the British mathematicians Bertrand Russell and Alfred North Whitehead in Principia Mathematica (3 volumes, 1910-13). The logical system of Russell and Whitehead covers a far greater range of possible arguments than those that can be cast into syllogistic form. It introduces symbols for complete sentences and for the conjunctions that connect them, such as “or,””and,” and “If . . . then. . . .” It has different symbols for the logical subject and the logical predicate of a sentence; and it has symbols for classes, for members of classes, and for the relationships of class membership and class inclusion. It also differs from classical logic in its assumptions as to the existence of the things referred to in its universal statements. The statement “All A's are B's” is rendered in modern logic to mean, “If anything is an A, then it is a B,” which, unlike classical logic, does not assume that any A's exist.

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Both classical logic and modern logic are systems of deductive logic. In a sense, the premises of a valid argument contain the conclusion, and the truth of the conclusion follows from the truth of the premises with certainty. Efforts have also been made to develop systems of inductive logic, such that the premises are evidence for the conclusion, but the truth of the conclusion follows from the truth of the evidence only with a certain probability. The most notable contribution to inductive logic is that of the British philosopher John Stuart Mill, who in his System of Logic (1843) formulated the methods of proof that he believed to characterize empirical science. This inquiry has developed in the 20th century into the field known as philosophy of science. Closely related is the branch of mathematics known as probability theory.

Both classical and modern logic in their usual forms assume that any well-formed sentence is either true or false. In recent years efforts have been made to develop systems of so-called many-valued logic, such that an assertion may have some value other than true or false. In some this is merely a third neutral value; in others it is a probability value expressed as a fraction ranging between 0 and 1 or between -1 and +1. Another development in recent years has been the effort to develop systems of modal logic, to represent the logical relations between assertions of possibility and impossibility, necessity and contingency. Still another development is deontic logic, the investigation of the logical relations between commands or between statements of obligation.

IV

Related Aspects

Closely related to logic is semantics, or the philosophy of language, which concerns the meaning of words and sentences; epistemology, or the theory of knowledge, which concerns the conditions under which assertions are true; and the psychology of reasoning, which concerns the mental processes involved in reasoning. Some treatises on logic include these subjects, but usually attention is restricted to the logical relations between statements.

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