SYMBOLIC LOGIC. - It is the study of formal logic by means of symbols, also called ideographic logic, mathematical logic, logistics.
Already in Aristotle there are elements of symbolic logic, for example, the use of letters to indicate variables, the linking of reasoning with calculation, the idea of a universal mathematics, etc. The logic of propositions was extensively treated by the Stoics and by the medieval Scholastics. Leibniz (v.) was a precursor of symbolic logic, but he was not followed. In 1830-50 the formalist attitude in mathematics prepared the beginning of symbolic logic. The first to treat it was G. Boole, who drew too heavily on algebra, applying its methods to logic. In America C. S. Peirce worked. He for logic, G. Cantor and R. Dedekind for mathematics, prepared the way for G. Frege, who attempted to apply the new symbolic logic to the investigation of the foundations of mathematics, wishing to derive the content of pure mathematics from logical principles alone. G. Frege's symbolism is obscure. Later G. Peano, with a clearer symbolism, constructed an axiomatic system for arithmetic; he had a numerous school (G. Loria, C. Burali-Forti, G. Vacca, etc.). B. Russell, modifying G. Peano's symbolism and using G. Frege's discoveries, set out to prove that all mathematics can be deduced from a few very simple principles; but having discovered contradictions in Frege's system, he invented the theory of types and together with A. N. Whitehead wrote the Principia mathematica (3 vols., Cambridge 1909-13; 1925-27), today the most complete work. The neo-positivist circle of Vienna (v. neopositivism), wishing to replace metaphysics with the logical study of scientific language, took an interest in symbolic logic especially with L. Wittgenstein, R. Carnap, H. Reichenbach, K. Gödel, etc. Very important is the school of symbolic logic of Warsaw: S. Lesniewski constructed a more exact system than that of B. Russell and initiated the new discipline called semantics; J. Lukasiewicz invented a simpler symbolism (today there are about ten symbolisms) and discovered many-valued logics; A. Tarski studied methodology and semantics in depth. The school of symbolic logic of America has C. I. Lewis, C. H. Langford, W. V. Quine, C. W. Morris, etc., to whom, after the victory of Nazism, were added Europeans who had taken refuge in America: R. Carnap, A. Tarski, etc.
For an idea of symbolic logic, here are some elements. In the sentence: "if it snows, then it is cold", "if... then..." is the predicate, "it snows... it is cold" the argument. This is replaced by variables "p", "q", ... which indicate propositions or terms. An expression with variables is called a function. Predicates, whose variables are propositions, are indicated by symbols; according to Lukasiewicz's notations: Np = not p (negation); Cpq = if p then q (material implication); Apq = p or q (alternation); Dpq = p excludes q (exclusive); Epq = p if and only if q (equivalence); Kpq = p and q (copulative). All predicates are written before their arguments; for example, "if p, then q; but p, therefore q" is written: "C K Cpq p q".
The most elementary part of symbolic logic is the logic of propositions, which studies the functions in which the variables are propositions. By replacing the variables with propositions, true and false expressions are obtained. If the values are two, truth and falsity, the logic is called bivalent; a trivalent logic is possible — truth, falsity, doubt; or many-valued. H. Reichenbach has developed a logic with infinitely many values for the calculus of probabilities, suited to modern theoretical physics. Besides the logic of propositions, there is the logic of terms: it comprises the logic of predicates, which considers terminal predicates; the logic of classes, which studies sets of individuals, called classes; the logic of relations, most important in modern symbolic logic, which studies logical relations.
Symbolic logic, by means of symbolism and the formalistic and assimilative method, tends to bring out all the elements of deductive thought that are often implicit, showing its inner structure, and thus to achieve a perfect logicality of deduction. All this is of great help in analyzing the meaning of questions and in solving them. It has been usefully applied in the analysis of the foundations of mathematics and can also serve in other sciences.