Editing Lindenbaum–Tarski algebra (section)

{{distinguish|Jónsson–Tarski algebra}}
In [[mathematical logic]], the '''Lindenbaum–Tarski algebra''' (or '''Lindenbaum algebra''') of a [[model theory#Definition|logical theory]] ''T'' consists of the [[equivalence class]]es of [[Sentence (mathematical logic)|sentence]]s of the theory (i.e., the [[Equivalence class|quotient]], under the [[equivalence relation]] ~ defined such that ''p'' ~ ''q'' exactly when ''p'' and ''q'' are provably equivalent in ''T''). That is, two sentences are equivalent if the theory ''T'' proves that each implies the other. The Lindenbaum–Tarski algebra is thus the [[quotient (universal algebra)|quotient algebra]] obtained by factoring the algebra of formulas by this [[congruence relation]].

The algebra is named for [[logician]]s [[Adolf Lindenbaum]] and [[Alfred Tarski]]. 
Starting in the academic year 1926-1927, Lindenbaum pioneered his method in [[Jan Łukasiewicz]]'s mathematical logic seminar,<ref>{{cite book|author=S.J. Surma|title=Logic, Methodology and Philosophy of Science VI, Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science |chapter=On the Origin and Subsequent Applications of the Concept of the Lindenbaum Algebra|series=Studies in Logic and the Foundations of Mathematics |year=1982|volume=104 |pages=719–734 |doi=10.1016/S0049-237X(09)70230-7|isbn=978-0-444-85423-0 }}</ref><ref>{{cite web|title=Lindenbaum, Adolf|author=Jan Woleński|website=Internet Encyclopedia of Philosophy|url=https://iep.utm.edu/lindenba/#H4}}</ref> and the method was popularized and generalized in subsequent decades through work 
by Tarski.<ref>{{cite book| author=A. Tarski| title=Logic, Semantics, and Metamathematics — Papers from 1923 to 1938 — Trans. J.H. Woodger| year=1983| publisher=Hackett Pub. Co.| editor=[[John Corcoran (logician)|J. Corcoran]]| edition=2nd}}</ref>
The Lindenbaum–Tarski algebra is considered the origin of the modern [[algebraic logic]].<ref name=BP>{{cite journal| author=[[Wim Blok|W.J. Blok]], Don Pigozzi| title=Algebraizable logics| journal=[[Memoirs of the American Mathematical Society|Memoirs of the AMS]]| year=1989| volume=77| number=396| url=http://orion.math.iastate.edu/dpigozzi}}; here: pages 1-2</ref>