symbolic logic


Also found in: Dictionary, Thesaurus, Acronyms, Wikipedia.
Related to symbolic logic: propositional 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 setset,
in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g.
..... Click the link for more information.
 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 syllogismsyllogism,
a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years.
..... Click the link for more information.
: "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).

Symbolic Logic

 

a synonym for “mathematical logic.” In the words of P. S. Poretskii, mathematical logic is “logic in subject and mathematics in methodology.” According to A. Church, it is logic studied through the construction of formalized languages.

The term “symbolic logic” draws attention to the special nature of the basic elements of the formalized languages used in the mathematical method of studying logic. These elements are not the words employed in everyday discourse, even if such words are used in some special meanings. Instead, the elements are symbols that are selected (or constructed from previously selected symbols) and interpreted in a definite way specific to the given logical situation. In general, this way has no relation to any “traditional” usage, understanding, or functions of the symbols in other contexts.

symbolic logic

[sim′bäl·ik ′läj·ik]
(mathematics)
The formal study of symbolism and its use in the foundations of mathematical logic.

symbolic logic

(logic)
The discipline that treats formal logic by means of a formalised artificial language or symbolic calculus, whose purpose is to avoid the ambiguities and logical inadequacies of natural language.
References in periodicals archive ?
Skinner had been interested in the history and philosophy of science from his graduate student days, as reflected, for example, in the work of Mach and Poincare, but he never subscribed to the formal symbolic logic of logical empiricism.
Utilizing the new symbolic logic systematically synthesized by Frege and his definition of the natural numbers in term s of classes of classes, the first three volumes set out to axiomatize logic and to show how arithmetic could be deduced from the proposed system.
Delany's Analytics of Attention." I found it difficult to concentrate my own attention on the technical details of his argument, but anyone possessing a passing acquaintance with symbolic logic should have no trouble with it.
This new system of logic described the relationship of symbols to each other, or symbolic logic. The importance of the work by Whitehead and Russell lay in the fact that it did not reject the centuries of work by philosophers since Aristotle, but refined it through mathematics to a degree of precision never before seen.
Despite this fact, Britain provided us with a great philosopher, namely Bertrand Russell, who specialized in the purest forms of science and scholarship in areas such as symbolic logic, philosophy of mathematics, and the theory of knowledge.
goal of containing the spread of communism manifests itself in a kind of symbolic logic that maps global politics onto American gender politics of the 1950s, the nascent moment of the postmodern.
On the contrary, Robinson delighted in showing that symbolic logic could be used within mathematics to positive, creative effect (p.
It is "surprising" how many people take a "belletristic view," Jameson comments, making "the assumption, which they would never make in the area of nuclear physics, linguistics, symbolic logic, or urbanism, that such [cultural] problems can still be laid out with all the leisurely elegance of a coffee-table magazine."
For the knowledge that such a logic provides, the knowledge of positioning in the social order that makes this not simply "a logic" but the template of symbolic logic as such, predicates its assurance of "meaning" on the recurrent administration of countless invisible, because naturalized, acts of violent negation.
In 1847 he published The Mathematical Analysis of Logic, thus founding what might be called Boolean algebra or symbolic logic. This was later to be of much use in studying the rigorous foundations of mathematics and, eventually, in programming computers.
Paulos, who teaches freshmen as well as graduate stu- dents, is a recognized expert in symbolic logic, computer languages, and artificial intelligence.

Full browser ?