Gödel logic

In mathematical logic, a Gödel logic, sometimes referred to as Dummett logic or Gödel–Dummett logic,^[1] is a member of a family of finite- or infinite-valued logics in which the sets of truth values V are closed subsets of the unit interval [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics. The concept is named after Kurt Gödel.^[2][3]

In 1959, Michael Dummett showed that infinite-valued propositional Gödel logic can be axiomatised by adding the axiom schema

${\displaystyle (A\rightarrow B)\lor (B\rightarrow A)}$

to intuitionistic propositional logic.^[1][4]

See also

• Intermediate logic

References

1. von Plato, Jan (2003). "Skolem's Discovery of Gödel-Dummett Logic". Studia Logica. 73 (1): 153–157. doi:10.1023/A:1022997524909.
2. Baaz, Matthias; Preining, Norbert; Zach, Richard (2007-06-01). "First-order Gödel logics". Annals of Pure and Applied Logic. 147 (1): 23–47. arXiv:math/0601147. doi:10.1016/j.apal.2007.03.001. ISSN 0168-0072.
3. Preining, Norbert (2010). "Gödel Logics – A Survey". Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science. Vol. 6397. pp. 30–51. doi:10.1007/978-3-642-16242-8_4. ISBN 978-3-642-16241-1. Retrieved 2 March 2022.
4. Dummett, Michael (1959). "A propositional calculus with denumerable matrix". The Journal of Symbolic Logic. 24 (2): 97–106. doi:10.2307/2964753. ISSN 0022-4812. JSTOR 2964753.