Log cabin, Abraham Lincoln's boyhood home, Knob Creek, Kentucky, origi¬ nally built early 19th century.
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logarithm In mathematics, the power to which a base must be raised to yield a given number (e.g., the logarithm to the base 3 of 9, or log 3 9, is 2, because 3 2 = 9). A common logarithm is a logarithm to the base 10. Thus, the common logarithm of 100 (log 100) is 2, because 10 2 = 100. Logarithms to the base e, in which e = 2.71828..., called natural loga¬ rithms (In), are especially useful in calculus. Logarithms were invented to simplify cumbersome calculations, since exponents can be added or subtracted to multiply or divide their bases. These processes have been further simplified by the incorporation of logarithmic functions into digi¬ tal calculators and computers. See also John Napier.
loggia \'lo-je-9\ Hall, gallery, or porch open to the air on one or more sides. It evolved in the Mediterranean region as an open sitting room with protection from the sun. It is often a roofed, arcaded open gallery on an upper story overlooking a court, though it can also be a separate arcaded or colonnaded structure. In medieval and Renaissance Italy, it was often used in conjunction with a public square, as in Florence’s Loggia dei Lanzi (begun 1376).
logic 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 conclu¬ sion 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,” (sym¬ bolized as — i), “and” (symbolized as a), “or” (symbolized as v), and “if- then” (symbolized as z>), (2) the existential and universal quantifiers, “(3x)” and “(Vx),” often rendered in English as “There is an x such that ...” and “For any (all) x, 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 iden¬ tity. 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. See also deontic logic; modal logic.
logic, many-valued Formal system in which the well-formed formu¬ lae are interpreted as being able to take on values other than the two clas¬ sical values of truth or falsity. The number of values possible for well- formed formulae in systems of many-valued logic ranges from three to uncountably many.
logic, philosophy of 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 fal¬ sified 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 logi¬ cal constants,” and “truths based solely on the meanings of the terms they contain.”
logic design Basic organization of the circuitry of a digital computer. All digital computers are based on a two-valued logic system—1/0, on/off, yes/no (see binary code). Computers perform calculations using compo¬ nents called logic gates, which are made up of integrated circuits that receive an input signal, process it, and change it into an output signal. The components of the gates pass or block a clock pulse as it travels through them, and the output bits of the gates control other gates or output the result. There are three basic kinds of logic gates, called “and,” “or,” and “not.” By connecting logic gates together, a device can be constructed that can perform basic arithmetic functions.
logical positivism Early school of analytic philosophy, inspired by David Hume, the mathematical logic of Bertrand Russell and Alfred North Whitehead, and Ludwig Wittgenstein’s Tractatus (1921). The school, for¬ mally instituted at the University of Vienna in a seminar of Moritz Schlick (1882-1936) in 1922, continued there as the Vienna Circle until 1938. It proposed several revolutionary theses: (1) All meaningful discourse con¬ sists either of (a) the formal sentences of logic and mathematics or (b) the factual propositions of the special sciences; (2) Any assertion that claims to be factual has meaning only if it is possible to say how it might be verified; (3) Metaphysical assertions, including the pronouncements of
religion, belong to neither of the two classes of (1) and are therefore meaningless. Some logical positivists, notably A.J. Ayer, held that asser¬ tions in ethics (e.g., “It is wrong to steal”) do not function logically as statements of fact but only as expressions of the speaker’s feelings of approval or disapproval toward some action. See also Rudolf Carnap; emotivism; verifiability principle.
logicism Vla-ji-.si-zomN In the philosophy of mathematics, the thesis that all mathematical propositions are expressible as or derivable from the propositions of pure logic. Gottlob Frege attempted to establish the thesis in his Die Grundlagen der Arithmetik (1884) and other works; Bertrand Russell argued for logicism in The Principles of Mathematics (1903) and attempted a formal proof with Alfred North Whitehead in Principia Math- ematica (1910-13).
logistic system See formal system
logistics \lo-'jis-tiks, b-'jis-tiks\ In military science, all the activities of armed-force units in support of combat units, including transport, supply, communications, and medical aid. The term, first used by Henri Jomini, Alfred Thayer Mahan, and others, was adopted by the U.S. military in World War I and gained currency in other nations in World War II. Its importance grew in the 20th century with the increasing complexity of modern warfare. The ability to mobilize large populations has escalated military demands for supplies and provisions, and sophisticated technol¬ ogy has added to the cost and intricacy of weapons, communications sys¬ tems, and medical care, creating the need for a vast network of support systems. In World War II, for instance, only about three in 10 U.S. sol¬ diers served in a combat role.
Logone \lo-'gon\ River River, north-central Africa. The chief tributary of the Chari River of the Lake Chad basin in equatorial Africa, it drains northeastern Cameroon and Chad. It flows 240 mi (390 km) northwest to join the Chari at N'Djamena, Chad. It is seasonally navigable for small steamers below Bongor.
logos (Greek: “word,” “reason,” “plan”) In Greek philosophy and the¬ ology, the divine reason that orders the cosmos and gives it form and meaning. The concept is found in the writings of Heracleitus (6th century bc) and in Persian, Indian, and Egyptian philosophical and theological systems as well. It is particularly significant in Christian theology, where it is used to describe the role of Jesus as the principle of God active in the creation and ordering of the cosmos and in the revelation of the divine plan of salvation. This is most clearly stated in the Gospel of John the Apostle, which identifies Christ as the Word (Logos) made flesh.