- Algebraic logic
mathematical logic, algebraic logic formalizes logic using the methods of abstract algebra.
Logics as models of algebras
In algebraic logic:
* Variables are tacitly universally quantified over some
universe 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;
Formulas, built from terms in the usual way, can be equated if they are logically equivalent. To express a tautology, equate a formula with a truth value;
* The rules of proof are the substitution of equals for equals, and uniform replacement.
Modus ponensremains 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 structures shown on the right in the same row. These structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically models of what are called "Boolean algebras with operators."
Algebraic formalisms going beyond
first-order logicin at least some respects include:
Combinatory logic, having the expressive power of set theory;
Relation algebra, arguably the paradigmatic algebraic logic, can express Peano arithmeticand most axiomatic set theories, including the canonical ZFC.
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 by
George Boole, and of relation algebra, begun by Augustus DeMorgan, extended by Charles Peirce, and taking definitive form in the work of Ernst Schröder;
Abstract algebraic logic, a branch of contemporary mathematical logic.
Perhaps surprisingly, algebraic logic is the oldest approach to formal logic, arguably beginning with a number of memoranda
Leibnizwrote in the 1680s, some of which were published in the 19th century and translated into English by Clarence Lewisin 1918. But nearly all of Leibniz's known work on algebraic logic was published only in 1903, after Louis Couturatdiscovered it in Leibniz's Nachlass. 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 Schroderand Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski, the founder of set theoretic model theoryas a major branch of contemporary mathematical logic, also:
*Wrote the 1940 paper that revived
relation algebra, and that can be seen as the starting point of abstract algebraic logic.
mathematical logicbegan in 1847, with two pamphlets whose respective authors were Augustus DeMorganand George Boole. They, and later Charles Peirce, Hugh MacColl, Frege, Peano, Bertrand Russell, and A. N. Whiteheadall shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy. Relation algebrais arguably the culmination of Leibniz's approach to logic. With the exception of some writings by Leopold Loewenheimand Thoralf Skolem, algebraic logic went into eclipse soon after the 1910-13 publication of " Principia Mathematica", not to revive until Tarski's 1940 reexposition of relation 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 in
logicand metaphysicscan draw inspiration from, and shed light on, Leibniz's thought, see [http://mally.stanford.edu/Papers/leibniz.pdf Zalta (2000).]
Abstract algebraic logic
Boolean algebra (logic)
Boolean algebra (structure)
Monadic Boolean algebra
Predicate functor logic
* 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.
Stanford Encyclopedia of Philosophy: " [http://plato.stanford.edu/entries/consequence-algebraic/ Propositional Consequence Relations and Algebraic Logic] " -- by Ramon Jansana.
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