### Towards Rational Closure for Fuzzy Logic: The Case of Propositional Gödel Logic [chapter]

Giovanni Casini, Umberto Straccia
2013 Lecture Notes in Computer Science
In the field of non-monotonic logics, the notion of rational closure is acknowledged as a landmark and we are going to see whether such a construction can be adopted in the context of mathematical fuzzy logic, a so far (apparently) unexplored journey. As a first step, we will characterise rational closure in the context of Propositional Gödel Logic. Proof. It follows the proof of the classical counterpart (, Proposition 6). It is sufficient to reformulate the set S Now we can prove the main
more » ... lemma. Lemma 23. Let M = S, , ≺ be a minimal ranked model of K = T , D s.t. for every Gödel valuation v compatible with T , D there is a state s ∈ S s.t. (s) = v. For every formula C, rank(C) = i iff k M (C) = i. Proof. It follows the proof of the classical counterpart (, Proposition 7). We make use of Lemmas 20 and 22. Given Lemma 23, the proof of Proposition 9 is immediate.