symbolic logic

Also found in: Dictionary, Thesaurus, Acronyms, Wikipedia.
Related to symbolic logic: propositional logic

symbolic logic


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.


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]
The formal study of symbolism and its use in the foundations of mathematical logic.

symbolic 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 ?
This is perhaps the first utilization of the new symbolic logic to axiomatize a subject matter beyond mathematics, but more importantly than that, it is a good clue as to the way in which Whitehead conceives of a conceptual framework,.
His discussion of The Ten Commandments (1956), which appeared earlier in PMLA, reveals the extent to which this film participates in perhaps the central authorizing of America's cold war sense of national identity: in a symbolic logic wherein nuclear power stands for God's power and favor, America's initial period of exclusive possession of nuclear weapons renders Americans analogous to the Israelites as the chosen people.
Paulos, who teaches freshmen as well as graduate students, is a recognized expert in symbolic logic, computer languages, and artificial intelligence.
Euclid and His Modern Rivals (1879), Curiosa Mathematica (1888), and Symbolic Logic (1896).
Concepts explored in this section include 'food is not medicine', the loss of folk remedy consciousness, the role of anatomy in the plausibility of functional products, and the role of symbolic logic.
Topics in the second section include Giuseppe Peano and symbolic logic, E.
Having in the past written both recreational puzzle books and more technical writings in the field of symbolic logic, Smullyan here presents a hybrid intended to serve as a text for a one- or two-semester introductory course in logic.
Hanson's article, 'Formalization in Philosophy' [Bulletin of Symbolic Logic 6, 2000], contains further information on this topic.
Nurtured on Frege's anti-psychologism, with its radical distinction between pure, static concepts and the contingencies of the mind's ideas or representations, modern symbolic logic works only with forms, assuming that concepts can be plugged into the various slots without distortion and without residue.
These results raise doubts about the validity and efficacy of using natural-language examples to promote and test the understanding of symbolic logic.
A first or second undergraduate course in symbolic logic is probably sufficient for most of the book, all but the sections and exercises on metatheory, which are marked as optional.
Just as arithmetic became algebra with the introduction of symbols to stand for variables (unknown quantities), so classical logic became symbolic logic with the introduction of the propositional function and of symbols to stand for propositions.

Full browser ?