Note on witnessed Gödel logics with Delta
The paper deals with witnessed Gödel logics with or without the \(\Delta\)-operator (recall that \(\Delta(x) = 1\) if \(x = 1\), \(\Delta(x) = 0\) otherwise, for each \(x \in [0,1]\)). Witnessed Gödel logics are based on the interpretation of the universal (resp. existential) quantifier by minimum (resp. maximum) instead of infimum (resp. supremum). The main results of the paper are: -- An axiomatization of witnessed Gödel logic (with or without \(\Delta\)) over \([0,1]\), and the fact that each uncountable subset of the truth-values set \([0,1]\) determines the same witnessed Gödel logic. -- For each truth-value set \(V \subseteq [0,1]\) a translation from the witnessed Gödel logic with \(\Delta\) over \(V\) to the witnessed Gödel logic (without \(\Delta\)) over \(V\). -- The fact that the sets of all Gödel logics, witnessed Gödel logics, Gödel logics with \(\Delta\), witnessed Gödel logics with \(\Delta\) contain countably many elements.
scientific article; zbMATH DE number 5624901
scientific article