LOGICA SIMBOLICA. – This is the study of formal logic by means of symbols, also called ideographic logic, mathematical logic, or logistic.
Already in Aristotle there are elements of s. l., for example, the use of letters to indicate variables, the association of reasoning with calculation, the idea of a universal mathematics, and so forth. The logic of propositions was extensively treated by the Stoics and by the medieval Scholastics. Leibniz (v.) was a precursor of s. l., but he was not followed. In 1830–50, the formalist attitude in mathematics prepared the beginning of s. l. The first to treat it was G. Boole, who was too strongly inspired by algebra, applying its methods to logic. C. S. Peirce worked in America. The latter in logic, and G. Cantor and R. Dedekind in mathematics, prepared the way for G. Frege, who attempted to apply the new s. l. 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 making use of 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), the most complete work to date. The neo-positivist circle of Vienna (v. neopositivismo), wishing to replace metaphysics with the logical study of scientific language, became interested in s. l., especially through L. Wittgenstein, R. Carnap, H. Reichenbach, K. Gödel, and others. The school of s. l. in Warsaw was very important: S. Lesniewski constructed a system more exact than that of B. Russell and initiated the new discipline called semantics; J. Lukasiewicz invented a simpler symbolism (there are now about ten symbolisms) and discovered many-valued logics; A. Tarski conducted profound studies in methodology and semantics. The American school of s. l. includes C. I. Lewis, C. H. Langford, W. V. Quine, C. W. Morris, and others, to whom, after the victory of Nazism, were added Europeans who had taken refuge in America: R. Carnap, A. Tarski, and others.
For an idea of s. l., 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» the argument. The latter is replaced by variables «p», «q», ... which indicate propositions or terms. An expression containing variables is called a function. Predicates whose variables are propositions are indicated by symbols; according to Lukasiewicz’s notation: Np = not p (negation); Cpq = if p then q (material implication); Apq = p or q (alternative); Dpq = p excludes q (exclusive alternative); Epq = p if and only if q (equivalence); Kpq = p and q (conjunction). 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 s. l. is the logic of propositions, which studies functions in which the variables are propositions. By replacing the variables with propositions, true and false expressions are obtained. If there are two values, truth and falsity, the logic is called bivalent; a trivalent logic is also possible—truth, falsity, doubt—or a many-valued logic. H. Reichenbach developed a logic with infinitely many values for the calculus of probabilities, suited to modern theoretical physics. In addition to the logic of propositions, there is the logic of terms: this includes the logic of predicates, which considers terminal predicates; the logic of classes, which studies sets of individuals, called classes; and the logic of relations, which is of the greatest importance in modern s. l. and studies logical relations.
S. l., by means of symbolism and the formalistic and assimilative method, tends to bring to light all the elements of deductive thought that are often implicit, showing their inner structure, and thus to attain perfect logicality in deduction. All this is of great assistance in analyzing the meaning of questions and in solving them. It has been usefully applied to the analysis of the foundations of mathematics and can also serve in other sciences.
BIBLI: The journal The journal of symbolic logic, Browne University, Providence R. I. (rich in bibliography from its first volume [1936] onward), is indispensable; E. Schröder, Vorlesungen über die Algebra der Logik, Lipsia 1890–1905; J. Jörgensen, A treatise of formal logic, Copenhagen 1931, with a history of s. l.; C. I. Lewis and C. H. Langford, Symbolic logic, Nuova York and Londra 1932; R. Carnap, The logical syntax of language, Londra 1937; I. M. Bochenicki, Nove lezioni di l. s., Roma 1938; id., Précis de logique mathématique, Bossum 1949; W. V. Quine, Mathematical logic, Nuova York 1940; id., Elementary logic, Boston 1941; A. Tarski, Introduction to logic, 2nd ed., Nuova York 1946; H. Veatch, Aristotelian and mathematical logic, in Thomist, 13 (1950), pp. 20–96.