**Symbolic Logic.** — The study of formal logic by means of symbols, also called ideographic logic, mathematical logic, or logistic.
Elements of symbolic logic are already found in Aristotle, for example, the use of letters to denote variables, the analogy between reasoning and calculation, the idea of a universal mathematics, etc. The logic of propositions was extensively treated by the Stoics and medieval Scholastics. Leibniz (v.) was a precursor of symbolic logic, but he was not followed. Between 1830 and 1850, the formalist approach in mathematics prepared the way for the beginning of symbolic logic. The first to treat it was G. Boole, who drew too much from algebra, applying its methods to logic. In America, C. S. Peirce worked on logic, while G. Cantor and R. Dedekind prepared the ground in mathematics for G. Frege, who attempted to apply the new symbolic logic to the foundations of mathematics, aiming to derive the content of pure mathematics from purely logical principles. Frege’s symbolism is obscure. Later, G. Peano constructed an axiomatic system for arithmetic with clearer symbolism and gathered a large school (G. Loria, C. Buralli-Forti, G. Vacca, etc.). B. Russell, modifying Peano’s symbolism and utilizing Frege’s discoveries, sought to prove that all mathematics can be deduced from a few very simple principles; but upon discovering contradictions in Frege’s system, he invented the theory of types and, together with A. N. Whitehead, wrote *Principia mathematica* (3 vols., Cambridge 1909–13; 1925–27), the most comprehensive work today. The neo-positivist circle of Vienna (v. POSITIVISMO), seeking to replace metaphysics with the logical study of scientific language, took an interest in symbolic logic, particularly through L. Wittgenstein, R. Carnap, H. Reichenbach, K. Gödel, etc. The Warsaw school of symbolic logic was very important: S. Leśniewski constructed a system more exact than Russell’s and initiated the new discipline called semantics; J. Łukasiewicz invented a simpler symbolism (there are now about ten systems) and discovered many-valued logics; A. Tarski conducted deep studies in methodology and semantics. The American school of symbolic logic includes C. I. Lewis, C. H. Langford, W. V. Quine, C. W. Morris, etc., to whom, after the Nazi victory, European refugees such as R. Carnap and A. Tarski were added.
To give an idea of symbolic logic, here are some elements. In the sentence: “If it snows, then it is cold,” “if... then...” is the predicate, and “it snows... it is cold” is the argument. The argument is replaced by variables “p,” “q,” etc., which denote propositions or terms. An expression with variables is called a function. Predicates whose variables are propositions are denoted by symbols; according to Łukasiewicz’s notation: Np = not p (negation); Cpq = if p then q (material implication); Apq = p or q (alternative); Dpq = p excludes q (exclusive); Epq = p if and only if q (equivalence); Kpq = p and q (conjunctive). All predicates are written before their arguments; for example, “If p, then q; but p, therefore q” is written as “C K Cpq p q.”
The most elementary part of symbolic logic is propositional logic, which studies functions in which the variables are propositions. By substituting variables with propositions, one obtains expressions that are either true or false. If there are only two values, truth and falsity, the logic is called bivalent; a trivalent logic is also possible, with truth, falsity, and doubt; or even many-valued logics. H. Reichenbach developed an infinite-valued logic for the calculus of probabilities, suitable for modern theoretical physics. Beyond propositional logic, there is term logic, which includes predicate logic (considering terminal predicates), class logic (studying sets of individuals, called classes), and relation logic, which is very important in modern symbolic logic and studies logical relations.
Symbolic logic, through its symbolism and formalistic and axiomatic methods, aims to highlight all the elements of deductive thought often implied, revealing their inner structure and achieving perfect logical deduction. 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.
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Indispensable is the journal *The Journal of Symbolic Logic*, Brown University, Providence, R.I. (rich in bibliography since its first volume [1936]); E. Schröder, *Vorlesungen über die Algebra der Logik*, Leipzig 1890–1905; J. Jørgensen, *A Treatise of Formal Logic*, Copenhagen 1931, with a history of symbolic logic; C. I. Lewis and C. H. Langford, *Symbolic Logic*, New York and London 1932; R. Carnap, *The Logical Syntax of Language*, London 1937; I. M. Bochenski, *Nove lezioni di logica simbolica*, Rome 1938; id., *Précis de logique mathématique*, Bussum 1940; W. V. Quine, *Mathematical Logic*, New York 1940; id., *Elementary Logic*, Boston 1941; A. Tarski, *Introduction to Logic*, 2nd ed., New York 1946; H. Veatch, *Aristotelian and Mathematical Logic*, in *Thomist*, 13 (1950), pp. 50–56.