"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion.
The converse relation πT runs from S to Q so that the composition πTα is a homogeneous relation on S.[2] The art of putting the right question to elicit a sufficient answer is recognized in Socratic method dialogue.
Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation.
[3][4] The facility of complementary relations inspired Augustus De Morgan and Ernst Schröder to introduce equivalences using
The relation algebra structure, based in set theory, was transcended by Tarski with axioms describing it.
The negative answer[6] opened the frontier of abstract algebraic logic.
Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators."
Parkinson (1966) and Loemker (1969) translated selections from Couturat's volume into English.
Modern mathematical logic began in 1847, with two pamphlets whose respective authors were George Boole[11] and Augustus De Morgan.
[12] In 1870 Charles Sanders Peirce published the first of several works on the logic of relatives.
[15] In particular he formulated Schröder rules, though De Morgan had anticipated them with his Theorem K. In 1903 Bertrand Russell developed the calculus of relations and logicism as his version of pure mathematics based on the operations of the calculus as primitive notions.
[16] The "Boole–Schröder algebra of logic" was developed at University of California, Berkeley in a textbook by Clarence Lewis in 1918.
Hugh MacColl, Gottlob Frege, Giuseppe Peano, and A. N. Whitehead all shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy.
Some writings by Leopold Löwenheim and Thoralf Skolem on algebraic logic appeared after the 1910–13 publication of Principia Mathematica, and Tarski revived interest in relations with his 1941 essay "On the Calculus of Relations".
"[17] Brady (2000) discusses the rich historical connections between algebraic logic and model theory.
The founders of model theory, Ernst Schröder and Leopold Loewenheim, were logicians in the algebraic tradition.
Alfred Tarski, the founder of set theoretic model theory as a major branch of contemporary mathematical logic, also: In the practice of the calculus of relations, Jacques Riguet used the algebraic logic to advance useful concepts: he extended the concept of an equivalence relation (on a set) to the heterogeneous case with the notion of a difunctional relation.
Riguet generated rectangular relations by taking the outer product of logical vectors; these contribute to the non-enlargeable rectangles of formal concept analysis.
To see how present-day work in logic and metaphysics can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000).