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.
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.
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.
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.
