- Algebraic logic
In

mathematical logic ,**algebraic logic**formalizes logic using the methods ofabstract algebra .**Logics as models of algebras**Algebraic logic treats

logic s as models (interpretations) of certainalgebraic structure s, specifically as models of bounded lattices and hence as a branch oforder theory .In algebraic logic:

* Variables are tacitly universally quantified over someuniverse of discourse . There are no existentially quantified variables or open formulas;

* Terms are built up from variables using primitive and defined operations. There are no connectives;

*Formula s, built from terms in the usual way, can be equated if they are logically equivalent. To express atautology , equate a formula with a truth value;

* The rules of proof are the substitution of equals for equals, and uniform replacement.Modus ponens remains valid, but is seldom employed.In the table below, the left column contains one or more logical or mathematical systems that are models of the

algebraic structure s shown on the right in the same row. These structures are either Boolean algebras orproper extension s thereof. Modal and other nonclassical logics are typically models of what are called "Boolean algebras with operators."Algebraic formalisms going beyond

first-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 most axiomatic set theories, including the canonicalZFC .**History**On the history of algebraic logic before

WWII , see Brady (2000) and Grattan-Guinness (2000) and their ample references. On the postwar history, see Maddux (1991) and Quine (1976)."Algebraic logic" has at least two meanings:

* The study of Boolean algebra, begun byGeorge Boole , and ofrelation algebra , begun byAugustus DeMorgan , extended byCharles Peirce , and taking definitive form in the work ofErnst Schröder ;

*Abstract algebraic logic , a branch of contemporarymathematical logic .Perhaps surprisingly, algebraic logic is the oldest approach to formal logic, arguably beginning with a number of memoranda

Leibniz wrote in the 1680s, some of which were published in the 19th century and translated into English byClarence Lewis in 1918. 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) and Loemker (1969) translated selections from Couturat's volume into English.Brady (2000) discusses the rich historical connections between algebraic logic and

model theory . The founders of model theory,Ernst Schroder andLeopold Loewenheim , were logicians in the algebraic tradition.Alfred Tarski , the founder of set theoreticmodel theory as a major branch of contemporarymathematical logic , also:

*Co-discoveredLindenbaum-Tarski algebra ;

*Inventedcylindric algebra ;

*Wrote the 1940 paper that revivedrelation algebra , and that can be seen as the starting point ofabstract algebraic logic .Modern

mathematical logic began in 1847, with two pamphlets whose respective authors wereAugustus DeMorgan andGeorge Boole . They, and laterCharles Peirce ,Hugh MacColl ,Frege ,Peano ,Bertrand Russell , andA. N. Whitehead all shared Leibniz's dream of combiningsymbolic logic ,mathematics , andphilosophy .Relation algebra is arguably the culmination of Leibniz's approach to logic. With the exception of some writings byLeopold Loewenheim andThoralf Skolem , algebraic logic went into eclipse soon after the 1910-13 publication of "Principia Mathematica ", not to revive until Tarski's 1940 reexposition ofrelation algebra .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 the logician stems mainly from the work of Wolfgang Lenzen, summarized in [

*http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf Lenzen (2004).*] To see how present-day work inlogic andmetaphysics can draw inspiration from, and shed light on, Leibniz's thought, see [*http://mally.stanford.edu/Papers/leibniz.pdf Zalta (2000).*]**ee also***

Abstract algebraic logic

*Algebraic structure

*Boolean algebra (logic)

*Boolean algebra (structure)

*Cylindric algebra

*Lindenbaum-Tarski algebra

*Mathematical logic

*Model theory

*Monadic Boolean algebra

*Predicate functor logic

*Relation algebra

*Universal algebra **References*** Brady, Geraldine, 2000. "From Peirce to Skolem: A neglected chapter in the history of logic". North-Holland.

*Ivor Grattan-Guinness , 2000. "The Search for Mathematical Roots". Princeton Univ. Press.

*Lenzen, Wolfgang, 2004, " [*http://www.philosophie.uni-osnabrueck.de/Publikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf 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.

*

*Roger Maddux , 1991, "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations," "Studia Logica 50": 421-55.

* Parkinson, G.H.R., 1966. "Leibniz: Logical Papers." Oxford Uni. Press.

*Willard Quine , 1976, "Algebraic Logic and Predicate Functors" in "The Ways of Paradox". Harvard Univ. Press: 283-307.

* Zalta, E. N., 2000, " [*http://mally.stanford.edu/leibniz.pdf A (Leibnizian) Theory of Concepts*] ," "Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3": 137-183.**External links***

Stanford Encyclopedia of Philosophy : " [*http://plato.stanford.edu/entries/consequence-algebraic/ Propositional Consequence Relations and Algebraic Logic*] " -- by Ramon Jansana.

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