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Concept
Abstract algebraic logic
Résumé
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems
arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.
The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and then further developed and refined by others, especially C. S. Peirce and Ernst Schröder, from the 1870s to the 1890s. This work culminated in Lindenbaum–Tarski algebras, devised by Alfred Tarski and his student Adolf Lindenbaum in the 1930s. Later, Tarski and his American students (whose ranks include Don Pigozzi) went on to discover cylindric algebra, whose representable instances algebraize all of classical first-order logic, and revived relation algebra, whose models include all well-known axiomatic set theories.
Classical algebraic logic, which comprises all work in algebraic logic until about 1960, studied the properties of specific classes of algebras used to "algebraize" specific logical systems of particular interest to specific logical investigations. Generally, the algebra associated with a logical system was found to be a type of lattice, possibly enriched with one or more unary operations other than lattice complementation.
Abstract algebraic logic is a modern subarea of algebraic logic that emerged in Poland during the 1950s and 60s with the work of Helena Rasiowa, Roman Sikorski, Jerzy Łoś, and Roman Suszko (to name but a few). It reached maturity in the 1980s with the seminal publications of the Polish logician Janusz Czelakowski, the Dutch logician Wim Blok and the American logician Don Pigozzi.
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