Editing Lindenbaum–Tarski algebra

{{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>

== Operations ==

The operations in a Lindenbaum–Tarski algebra ''A'' are inherited from those in the underlying theory ''T''. These typically include [[logical conjunction|conjunction]] and [[disjunction]], which are [[well-defined]] on the equivalence classes. When [[negation]] is also present in ''T'', then ''A'' is a [[Boolean algebra (structure)|Boolean algebra]], provided the logic is [[classical logic|classical]]. If the theory ''T'' consists of the [[propositional calculus|propositional tautologies]], the Lindenbaum–Tarski algebra is the [[free Boolean algebra]] generated by the [[propositional variable]]s.

If ''T'' is closed for deduction, then the embedding of ''T/~'' in ''A'' is a [[Filter (mathematics)|filter]]. Moreover, an [[ultrafilter]] in A corresponds to a complete consistent theory, establishing the equivalence between [[Lindenbaum's lemma|Lindenbaum's Lemma]] and the [[Ultrafilter on a set|Ultrafilter Lemma]].

== Related algebras ==

[[Heyting algebra]]s and [[interior algebra]]s are the Lindenbaum–Tarski algebras for [[intuitionistic logic]] and the [[modal logic]] '''S4''', respectively.

A logic for which Tarski's method is applicable, is called ''algebraizable''. There are however a number of logics where this is not the case, for instance the modal logics '''S1''', '''S2''', or '''S3''', which lack the [[rule of necessitation]] (⊢φ implying ⊢□φ), so  ~ (defined above) is not a congruence (because ⊢φ→ψ does not imply ⊢□φ→□ψ). Another type of logic where Tarski's method is inapplicable is [[relevance logic]]s, because given two theorems an implication from one to the other may not itself be a theorem in a relevance logic.<ref name=BP/> The study of the algebraization process (and notion) as topic of interest by itself, not necessarily by Tarski's method, has led to the development of [[abstract algebraic logic]].

==See also==
*[[Algebraic semantics (mathematical logic)]]
*[[Leibniz operator]]
*[[List of Boolean algebra topics]]

==References==
{{reflist}}
*{{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | isbn = 1-56881-262-0}}

{{DEFAULTSORT:Lindenbaum-Tarski algebra}}
[[Category:Algebraic logic]]
[[Category:Algebraic structures]]