On fuzzification of propositional logics (Q1808920)

The paper elaborates the idea of fuzzifying arbitrary crisp logics. The author proposes two different procedures. Due to the first one, the language of the propositional logic \(L\) is extended by a family of unary propositional operators and \(L\) is extended by the list of axioms related to the basic properties of the measure of fuzziness. The new formulas \(\Phi_r A\), \(\Phi^r A\) represent two kinds of propositions, namely ``the measure of fuzziness of a formula \(A\) is at least \(r\)'' and ``\(\ldots\) at most \(r\)'', respectively. The semantics is constructed using Kripke models. The deduction and completeness theorems are proved for such extensions. The second procedure is more abstract considering an algebra of extensions of a crisp logic \(L\). This algebra is proved to be a complete Heyting algebra and taken as fuzzification of \(L\). The extracted membership function is then connected with the deduction relation.