Lindenbaum–Tarski algebra
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the 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 algebra obtained by factoring the algebra of formulas by this congruence relation.
The algebra is named for
Operations
The operations in a Lindenbaum–Tarski algebra A are inherited from those in the underlying theory T. These typically include
Related algebras
Heyting algebras and interior algebras 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
See also
References
- ISBN 978-0-444-85423-0.
- ^ Jan Woleński. "Lindenbaum, Adolf". Internet Encyclopedia of Philosophy.
- ^ A. Tarski (1983). J. Corcoran (ed.). Logic, Semantics, and Metamathematics — Papers from 1923 to 1938 — Trans. J.H. Woodger (2nd ed.). Hackett Pub. Co.
- ^ a b W.J. Blok, Don Pigozzi (1989). "Algebraizable logics". Memoirs of the AMS. 77 (396).; here: pages 1-2
- Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.