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In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of model theory appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these ) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic .
Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator .
"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion."
The conversion refers to the converse relation that always exists, contrary to function theory. A given relation may be represented by a logical matrix; then the converse relation is represented by the transpose matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained by matrix multiplication using Boolean arithmetic.
But a univalent relation is only a partial function, while a univalent total relation is a function. The formula for totality is Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation.G. Schmidt & T. Ströhlein (1993) Relations and Graphs Discrete Mathematics for Computer Scientists, page 54, EATCS Monographs on Theoretical Computer Science, Springer Verlag, G. Schmidt (2011) Relational Mathematics, Encyclopedia of Mathematics and its Applications, vol. 132, pages 49 and 57, Cambridge University Press
The facility of complementary relations inspired Augustus De Morgan and Ernst Schröder to introduce equivalences using for the complement of relation . These equivalences provide alternative formulas for univalent relations (), and total relations (). Therefore, mappings satisfy the formula Schmidt uses this principle as "slipping below negation from the left".G. Schmidt & M. Winter(2018) Relational Topology, page 8, Lecture Notes in Mathematics vol. 2208, Springer Verlag, For a mapping
In algebraic logic:
In the table below, the left column contains one or more logical system or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or thereof. modal logic and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators."
Algebraic formalisms going beyond first-order logic in at least some respects include:
| Classical sentential logic | Boolean algebra |
| Intuitionistic propositional logic | Heyting algebra |
| Łukasiewicz logic | MV-algebra |
| Modal logic K | Modal algebra |
| Lewis's modal logic | Interior algebra |
| Lewis's S5, monadic predicate logic | Monadic Boolean algebra |
| First-order logic | Complete Boolean algebra, polyadic algebra, predicate functor logic |
| First-order logic with equality | Cylindric algebra |
| Set theory | Combinatory logic, relation algebra |
Modern mathematical logic began in 1847, with two pamphlets whose respective authors were George BooleGeorge Boole, The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning (London, England: Macmillan, Barclay, & Macmillan, 1847). and Augustus De Morgan.Augustus De Morgan (1847), Formal Logic, London: Taylor & Walton, link from Hathi Trust In 1870 Charles Sanders Peirce published the first of several works on the logic of relatives. Alexander Macfarlane published his Principles of the Algebra of LogicAlexander Macfarlane (1879), Principles of the Algebra of Logic, via Internet Archive in 1879, and in 1883, Christine Ladd, a student of Peirce at Johns Hopkins University, published "On the Algebra of Logic".Christine Ladd (1883), On the Algebra of Logic via Google Books Logic turned more algebraic when were combined with composition of relations. For sets A and B, a binary relation over A and B is represented as a member of the power set of A× B with properties described by Boolean algebra. The "calculus of relations" is arguably the culmination of Leibniz's approach to logic. At the Hochschule Karlsruhe the calculus of relations was described by Ernst Schröder.Ernst Schröder, (1895), Algebra der Logik (Exakte Logik) Dritter Band, Algebra und Logik der Relative, Leibzig: B. G. Teubner via Internet Archive 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.B. Russell (1903) The Principles of Mathematics The "Boole–Schröder algebra of logic" was developed at University of California, Berkeley in a textbook by Clarence Lewis in 1918.Clarence Lewis (1918) A Survey of Symbolic Logic, University of California Press, second edition 1932, Dover edition 1960 He treated the logic of relations as derived from the propositional functions of two or more variables.
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".
According to Helena Rasiowa, "The years 1920-40 saw, in particular in the Polish school of logic, researches on non-classical propositional calculi conducted by what is termed the logical matrix method. Since logical matrices are certain abstract algebras, this led to the use of an algebraic method in logic."Helena Rasiowa (1974), "Post Algebras as Semantic Foundations of m-valued Logics", pages 92–142 in Studies in Algebraic Logic, edited by Aubert Daigneault, Mathematical Association of America
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 also extended ordering to the heterogeneous context by his note that a staircase logical matrix has a complement that is also a staircase, and that the theorem of N. M. Ferrers follows from interpretation of the transpose of a staircase. Riguet generated rectangular relations by taking the outer product of logical vectors; these contribute to the non-enlargeable rectangles of formal concept analysis.
Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the Parkinson and Loemker translations. Our present understanding of Leibniz as a logician stems mainly from the work of Wolfgang Lenzen, summarized in . To see how present-day work in logic and metaphysics can draw inspiration from, and shed light on, Leibniz's thought, see .
Historical perspective
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|set theory]] model theory as a major branch of contemporary mathematical logic, also:
See also
Sources
Further reading
(2026). 9780198531920, Oxford University Press. ISBN 9780198531920 Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra; the book covers these prerequisites at length. This book however has been criticized for poor and sometimes incorrect presentation of AAL results. Review by Janusz Czelakowski
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