Inmathematical logic,algebraic logic is the reasoning obtained by manipulating equations withfree variables.

What is now usually called classical algebraic logic focuses on the identification and algebraic description ofmodels appropriate for the study of various logics (in the form of classes of algebras that constitute thealgebraic semantics for thesedeductive systems) and connected problems likerepresentation and duality. Well known results like therepresentation theorem for Boolean algebras andStone duality fall under the umbrella of classical algebraic logic (Czelakowski 2003).

Works in the more recentabstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using theLeibniz operator (Czelakowski 2003).

A homogeneousbinary relation is found in thepower set ofX ×X for some setX, while aheterogeneous relation is found in the power set ofX ×Y, whereX ≠Y. Whether a given relation holds for two individuals is onebit of information, so relations are studied with Boolean arithmetic. Elements of the power set are partially ordered byinclusion, and lattice of these sets becomes an algebra throughrelative multiplication orcomposition of relations.

"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion."^[1]

Theconversion refers to theconverse relation that always exists, contrary to function theory. A given relation may be represented by alogical matrix; then the converse relation is represented by thetranspose matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained bymatrix multiplication using Boolean arithmetic.

An example of calculus of relations arises inerotetics, the theory of questions. In the universe of utterances there arestatementsS andquestionsQ. There are two relationsπ and α fromQ toS:q αa holds whena is a direct answer to questionq. The other relation,qπp holds whenp is apresupposition of questionq. The converse relationπ^T runs fromS toQ so that the compositionπ^Tα is a homogeneous relation onS.^[2] The art of putting the right question to elicit a sufficient answer is recognized inSocratic method dialogue.

The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relationR that satisfies the formula${\displaystyle R^{T}R\subseteq I,}$ whereI is the identity relation on the range ofR. The injective property corresponds to univalence of${\displaystyle R^T}$, or the formula${\displaystyle R{R}^T\subseteq I,}$ where this timeI is the identity on the domain ofR.

But a univalent relation is only apartial function, while a univalenttotal relation is afunction. The formula for totality is${\displaystyle I\subseteq R{R}^T.}$Charles Loewner andGunther Schmidt use the termmapping for a total, univalent relation.^[3][4]

The facility ofcomplementary relations inspiredAugustus De Morgan andErnst Schröder to introduceequivalences using${\displaystyle \overline{R}}$ for the complement of relationR. These equivalences provide alternative formulas for univalent relations (${\displaystyle R\overline{I}\subseteq \overline{R}}$), and total relations (${\displaystyle \overline{R}\subseteq R\overline{I}}$). Therefore, mappings satisfy the formula${\displaystyle \overline{R}=R\overline{I}.}$ Schmidt uses this principle as "slipping below negation from the left".^[5] For a mappingf, ${\displaystyle f\overline{A}=\overline{fA}.}$

Therelation algebra structure, based in set theory, was transcended by Tarski with axioms describing it. Then he asked if every algebra satisfying the axioms could be represented by a set relation. The negative answer^[6] opened the frontier ofabstract algebraic logic.^[7][8][9]

Algebraic logic treatsalgebraic structures, oftenbounded lattices, as models (interpretations) of certainlogics, making logic a branch oforder theory.

In algebraic logic:

• Variables are tacitlyuniversally quantified over someuniverse of discourse. There are noexistentially quantified variables oropen formulas;
• Terms are built up from variables using primitive and definedoperations. There are noconnectives;
• Formulas, built from terms in the usual way, can be equated if they arelogically equivalent. To express atautology, equate a formula with atruth value;
• The rules of proof are the substitution of equals for equals^{[clarification needed]}, and uniform replacement.Modus ponens remains valid, but is seldom employed.

In the table below, the left column contains one or morelogical 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 eitherBoolean algebras orproper extensions thereof.Modal and othernonclassical logics are typically modeled by what are called "Boolean algebras with operators."

Algebraic formalisms going beyondfirst-order logic in at least some respects include:

• Combinatory logic, having the expressive power ofset theory;
• Relation algebra, arguably the paradigmatic algebraic logic, can expressPeano arithmetic and mostaxiomatic set theories, including the canonicalZFC.

Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memorandaLeibniz wrote in the 1680s, some of which were published in the 19th century and translated into English byClarence Lewis in 1918.^{[10]: 291–305} But nearly all of Leibniz's known work on algebraic logic was published only in 1903 afterLouis Couturat discovered it in Leibniz'sNachlass.Parkinson (1966) andLoemker (1969) translated selections from Couturat's volume into English.

Modern mathematical logic began in 1847, with two pamphlets whose respective authors wereGeorge Boole^[11] andAugustus De Morgan.^[12] In 1870Charles Sanders Peirce published the first of several works on thelogic of relatives.Alexander Macfarlane published hisPrinciples of the Algebra of Logic^[13] in 1879, and in 1883,Christine Ladd, a student of Peirce atJohns Hopkins University, published "On the Algebra of Logic".^[14] Logic turned more algebraic whenbinary relations were combined withcomposition of relations. For setsA andB, arelation overA andB is represented as a member of thepower set ofA×B with properties described byBoolean algebra. The "calculus of relations"^[9] is arguably the culmination of Leibniz's approach to logic. At theHochschule Karlsruhe the calculus of relations was described byErnst Schröder.^[15] In particular he formulatedSchröder rules, though De Morgan had anticipated them with his Theorem K.

In 1903Bertrand Russell developed the calculus of relations andlogicism as his version of pure mathematics based on the operations of the calculus asprimitive notions.^[16] The "Boole–Schröder algebra of logic" was developed atUniversity of California, Berkeley in atextbook byClarence Lewis in 1918.^[10] He treated the logic of relations as derived from thepropositional functions of two or more variables.

Hugh MacColl,Gottlob Frege,Giuseppe Peano, andA. N. Whitehead all shared Leibniz's dream of combiningsymbolic logic,mathematics, andphilosophy.

Some writings byLeopold Löwenheim andThoralf Skolem on algebraic logic appeared after the 1910–13 publication ofPrincipia Mathematica, and Tarski revived interest in relations with his 1941 essay "On the Calculus of Relations".^[9]

According toHelena 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 thelogical matrix method. Since logical matrices are certain abstract algebras, this led to the use of an algebraic method in logic."^[17]

Brady (2000) discusses the rich historical connections between algebraic logic andmodel theory. The founders of model theory, Ernst Schröder and Leopold Loewenheim, were logicians in the algebraic tradition.Alfred Tarski, the founder ofset theoretic model theory as a major branch of contemporary mathematical logic, also:

• Initiated abstract algebraic logic withrelation algebras^[9]
• Inventedcylindric algebra
• Co-discoveredLindenbaum–Tarski algebra.

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 adifunctional 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 ofN. M. Ferrers follows from interpretation of thetranspose of a staircase. Riguet generatedrectangular relations by taking theouter product of logical vectors; these contribute to thenon-enlargeable rectangles offormal 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 inLenzen (2004). To see how present-day work in logic andmetaphysics can draw inspiration from, and shed light on, Leibniz's thought, seeZalta (2000).

• Boolean algebra
• Codd's theorem
• Computer algebra
• Universal algebra

• Brady, Geraldine (2000).From Peirce to Skolem: A Neglected Chapter in the History of Logic. Studies in the History and Philosophy of Mathematics. Amsterdam, Netherlands: North-Holland/Elsevier Science BV.ISBN 9780080532028.
• Czelakowski, Janusz (2003). "Review: Algebraic Methods in Philosophical Logic by J. Michael Dunn and Gary M. Hardegree".The Bulletin of Symbolic Logic.9. Association for Symbolic Logic, Cambridge University Press.ISSN 1079-8986.JSTOR 3094793.
• Lenzen, Wolfgang, 2004, "Leibniz’s Logic" in Gabbay, D., and Woods, J., eds.,Handbook of the History of Logic, Vol. 3: The Rise of Modern Logic from Leibniz to Frege. North-Holland: 1-84.
• Loemker, Leroy (1969) [First edition 1956],Leibniz: Philosophical Papers and Letters (2nd ed.), Reidel.
• Parkinson, G.H.R (1966).Leibniz: Logical Papers. Oxford University Press.
• Zalta, E. N., 2000, "A (Leibnizian) Theory of Concepts,"Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137-183.

• J. Michael Dunn; Gary M. Hardegree (2001).Algebraic Methods in Philosophical Logic. Oxford University Press.ISBN 978-0-19-853192-0. Good introduction for readers with prior exposure tonon-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
• Hajnal Andréka, István Németi and Ildikó Sain (2001). "Algebraic logic". In Dov M. Gabbay, Franz Guenthner (ed.).Handbook of Philosophical Logic, vol 2 (2nd ed.). Springer.ISBN 978-0-7923-7126-7.Draft.
• Ramon Jansana (2011), "Propositional Consequence Relations and Algebraic Logic". Stanford Encyclopedia of Philosophy. Mainly about abstract algebraic logic.
• Stanley Burris (2015), "The Algebra of Logic Tradition". Stanford Encyclopedia of Philosophy.
• Willard Quine, 1976, "Algebraic Logic and Predicate Functors" pages 283 to 307 inThe Ways of Paradox,Harvard University Press.

Historical perspective

• Ivor Grattan-Guinness, 2000.The Search for Mathematical Roots. Princeton University Press.
• Irving Anellis & N. Houser (1991) "Nineteenth Century Roots of Algebraic Logic and Universal Algebra", pages 1–36 inAlgebraic Logic, Colloquia Mathematica Societatis János Bolyai # 54,János Bolyai Mathematical Society &ElsevierISBN 0444885439

• Algebraic logic atPhilPapers

• Algebraic logic
• History of logic

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