Logic
From Encyclopædia
Logic is the systematic study of reasoning that provides standards by which valid reasoning can be recognized. It clarifies the reasoning process and provides a means for analyzing the consistency of basic concepts. Logic has played an important role in the history of
philosophy.TRADITIONAL LOGICThe history of Western logic can be traced to ancient Greece. Logic had developed independently in China and India but apparently had little influence in the West.
Aristotle is usually considered the first major Western logician, although earlier contributions to logic were made by
Plato,
Socrates, ZENO OF ELEA, and others.
Aristotle's logical system treated the categorical
syllogism and the laws of logic. His many books on logic became the basis for its study up to the 19th century. The next school of logicians, the Megarians, flourished in the 4th century BC. Rather than study categorical inferences, they sought to define the conditions under which a conditional, "if A, then B," is true. The Stoics, especially one of their leaders, Chrysippus, took over and developed the logical ideas of the Megarians (see STOICISM).
Aristotle had set forth a logic of terms; Chrysippus worked out a logic of propositions. Stoic thinkers after Chrysippus apparently did not contribute further to the development of logic. Other late Greek and Roman logicians mainly codified the work of their predecessors, although Galen (AD 129-199) added some theories about special kinds of
syllogisms.The logic of the Greeks endured in the Middle East after the fall of Rome and the conquest of western
Europe by the barbarian tribes. In the
middle ages logic was again brought to the
attention of the Western
world by logicians of the Islamic empire, mainly those working in
Baghdad and southern Spain. Many of
Aristotle's writings as well as commentaries preserved by heretical Christian sects were available to these logicians. Al-Farabi (870-950), one of the greatest Muslim logicians, wrote commentaries on most of
Aristotle's logical works. Other logicians translated other Greek works on logic into Arabic. (From Arabic they were usually translated by Jewish scholars into Hebrew, and then from Hebrew into Latin; in this way these texts became available to scholars in Christian
Europe.) The great Muslim philosopher AVICENNA made logic independent of the
teachings of
Aristotle and the Stoics. By the 14th century, intellectuals were reading handbooks on logic by various Muslim scholars, instead of the classics.Meanwhile, the logical materials that were being studied in the Islamic empire became known in Christian
Europe. The Christians had previously had only a few of
Aristotle's works and a few commentaries. The first significant logician of the Christian
middle ages, Peter ABELARD, wrote before most of the Aristotelian materials became available, but developed detailed and critical evaluations of the material that had been preserved in the West. The rest of
Aristotle's logical writings became available by about 1200, resulting in the emergence of the logica moderna, a "new logic" taught mainly in the arts faculties of the universities rather than in the theological schools. Perhaps the greatest of the modern logicians was
William OF OCCAM, who wrote the Summa Logicae (1326?). Other logicians restructured the domain of logic so that the Aristotelian heritage was incorporated into a broader logic.As the
Renaissance began, so did an attack on medieval (Scholastic) thought and on the Aristotelian theories that provided its basis. The major
Renaissance opponent of Aristotelian logic was the 16th-century French Protestant thinker Petrus RAMUS, who is supposed to have debated "that every proposition in
Aristotle is false." Ramus attacked nearly all of
Aristotle's logical doctrines and proposed instead that there be a logic of invention or discovery and a logic of judgment.With the rejection of
Aristotle's metaphysics by the great 17th-century thinkers Francis Bacon, Rene Descartes, Baruch Spinoza, and John Locke, a search began for a new logic that would fit with a new picture of reality. Gottfried Wilhelm von Leibniz made major contributions to this new logic, although most of his logical work did not become generally available until the end of the 19th century. Leibniz tried to work out a universal logical
language, and he also developed a logical
calculus. He was apparently not ready to reject Aristotelian logic, but his examinations of other possibilities when discovered at the end of the 19th century nevertheless aided in the development of modern logic.Richard H. PopkinMATHEMATICAL LOGICLogic is the theory of argument and reasoning. Mathematical logic is that branch of logic which uses exacting formal methods to achieve precision and objectivity in explaining what it is to be logical in argument and reasoning. It concentrates on the explanation and development of proof and on the nature of formal systems used in constructing proofs. Although its greatest successes have been in application to
mathematics and computers, its generality makes it potentially applicable to virtually any field. Different branches of mathematical logic treat different types of questions. Alethic modal logics have been applied to such diverse questions as the nature of God and the structure of scientific laws; deontic logics, to normative systems of legal and moral behavior.Mathematical logic developed from the desire to provide systematic foundations for the practice of
mathematics, explaining the nature of
numbers and the laws of arithmetic and replacing
intuition with rigorous proof. Noteworthy among its founders is the Italian mathematician Giuseppe Peano (1858-1932), but the German mathematical philosopher Gottlob Frege (1848-1925) is considered the father of mathematical logic. Their early work was advanced by Bertrand Russell and Alfred North Whitehead in
Principia Mathematica (3 vols., 1910-13), and by many others, including Alonzo Church, Kurt Godel, David Hilbert, Emil Post, and Alfred Tarski. Among important precursors of modern logic were
Aristotle, George Boole, Augustus De Morgan, Gottfried Wilhelm von Leibniz, Charles Sanders Peirce, and Ernst Schroder.Valid argument is basic to mathematical logic. An argument is composed of a conclusion, which is argued for, and premises, which are the reasons for the conclusion. In a valid argument the conclusion follows from the premises: the argument is of such a form that no argument having that form can have true premises and a false conclusion. Thus, the following argument is valid, and its corresponding form exemplifies a logical way of thinking.Argument I Form INo one putting
profits first is putting human No F is G rights firstThis person is putting
profits first This H is FTherefore: Therefore:This person is not putting
human rights first This H is not G---------------------------------------------------------------In contrast, the following argument and corresponding form are invalid, as is shown by the counterexample.Argument II Form II Counter-exampleAll communists are All F are G All tigers are cats dissentersAll communists are All F are H All tigers are striped subversivesTherefore: Therefore: Therefore:All dissenters are All G are H All cats are striped subversive------------------------------------------------------?--------Counterexample II shows Form II to constitute an illogical way of thinking, in which truth can
lead to falsehood. Arguments I and II are deductive: certainty of the premises would be intended to make the conclusion certain (see DEDUCTION). In contrast, inductive arguments seek
probability rather than certainty (see
induction). Some theorists posit other kinds of arguments: Peirce, the abductive;
Aristotle, the practical. The nature and significance of nondeductive arguments remains to be fully investigated.Alternative deductive logics are produced by changes in the conception of validity. The best-known such school is intuitionist logic, founded by the Dutch mathematician L. E. J. Brouwer (1881). Many-valued logics, which reject the assumption, questioned since the time of
Aristotle, that there are only two values--truth and falsehood--have been rigorously developed, beginning with the work of the Polish logician Jan Lukasiewicz in the 1920s.Mathematical logic achieves precision, clarity, and manipulability through the use of artificial
languages (see
languages, ARTIFICIAL), symbol systems deliberately constructed for use in logic. Sentences of such a system may represent sentences of natural
languages like English, directly representing logical words such as no or all and suppressing structures considered irrelevant.
