Mathematical Logic (Symbolic Logic)

Symbolic Logic - or mathematical logic, formalized system of deductive logic, employing abstract symbols for the various aspects of natural language. Symbolic logic draws on the concepts and techniques of mathematics, notably set theory, and in turn has contributed to the development of the foundations of mathematics. Symbolic logic dates from the work of Augustus De Morgan and George Boole in the mid-19th cent. and was further developed by W. S. Jevons, C. S. Peirce, Ernst Schröder, Gottlob Frege, Giuseppe Peano, Bertrand Russell, A. N. Whitehead, David Hilbert, and others.

Truth-functional Analysis

The first part of symbolic logic is known as truth-functional analysis, the propositional calculus, or the sentential calculus; it deals with statements that can be assigned truth values (true or false). Combinations of these statements are called truth functions, and their truth values can be determined from the truth values of their components.

The basic connectives in truth-functional analysis are usually negation, conjunction, and alternation. The negation of a statement is false if the original statement is true and true if the original statement is false; negation corresponds to "it is not the case that," or simply "not" in ordinary language. The conjunction of two statements is true only if both are true; it is false in all other instances. Conjunction corresponds to "and" in ordinary language. The alternation, or disjunction, of two statements is false only if both are false and is true in all other instances; alternation corresponds to the nonexclusive sense of "or" in ordinary language (Lat. vel), as opposed to the exclusive "either…or…but not both" (Lat. aut).

Other connectives commonly used in truth-functional analysis are the conditional and the biconditional. The conditional, or implication, corresponds to "if…then" or "implies" in ordinary language, but only in a weak sense. The conditional is false only if the antecedent is true and the consequent is false; it is true in all other instances. This kind of implication, in which the connection between the antecedent and the consequent is merely formal, is known as material implication. The biconditional, or double implication, is the equivalence relation and is true only if the two statements have the same truth value, either true or false. In any truth function one may substitute an equivalent expression for all or any part of the function. The validity of arguments may be analyzed by assigning all possible combinations of truth values to the component statements; such an array of truth values is called a truth table.

The Predicate Calculus

There are many valid argument forms, however, that cannot be analyzed by truth-functional methods, e.g., the classic syllogism: "All men are mortal. Socrates is a man. Therefore Socrates is mortal." The syllogism and many other more complicated arguments are the subject of the predicate calculus, or quantification theory, which is based on the calculus of classes. The predicate calculus of monadic (one-variable) predicates, also called uniform quantification theory, has been shown to be complete and has a decision procedure, analogous to truth tables for truth-functional analysis, whereby the validity or invalidity of any statement can be determined. The general predicate calculus, or quantification theory, was also shown to be complete by Kurt Gödel, but Alonso Church subsequently proved (1936) that it has no possible decision procedure.

Analysis of the Foundations of Mathematics

Symbolic logic has been extended to a description and analysis of the foundations of mathematics, particularly number theory. Gödel also made (1931) the surprising discovery that number theory cannot be complete, i.e., that no matter what axioms are chosen as a basis for number theory, there will always be some true statements that cannot be deducted from them, although they can be proved within the larger context of symbolic logic. Since many branches of mathematics are ultimately based on number theory, this result has been interpreted by some as affirming that mathematics is an open, creative discipline whose possibilities cannot be delineated. The work of Gödel, Church, and others has led to the development of proof theory, or metamathematics, which deals with the nature of mathematics itself.

Bibliography

See D. Hilbert and W. Ackermann, Principles of Mathematical Logic (tr. of 2d ed. 1950); W. V. Quine, Mathematical Logic (1968) and Methods of Logic (3d ed. 1972).

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