### 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 ([13], 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 ([13], Proposition 7). We make use of Lemmas 20 and 22. Given Lemma 23, the proof of Proposition 9 is immediate.