\(Cn\)-definitions of propositional connectives (Q5939995)

``\(Cn\)'' in the title stands for `consequence'. The authors are interested in characterizing connectives, \(\neg\), \(\rightarrow\), \(\vee\), by means of \(Cn\). As the authors comment, the work in the milieu of intuitionistic logic has been accomplished neatly: besides the general condition for \(Cn\), each connective is defined in a natural form. (E.g. \(\alpha\to \beta\in Cn(X)\) iff \(Cn(X,\beta)\subseteq Cn(X, \alpha)\).) They try ``to find an appropriate set of \(Cn\)-definitions for the classical logic''. [Tarski's renowned characterization is not in the definitional form above.] What is interesting \{at least to the reviewer\} is, after examining 7, 8 candidates, they state ``our attempts\dots failed. We conjecture that there is no such set.'' Of course, the authors achieve the purpose by modifying the notion of consequence or \(Cn\)-definability. For instance, by adding: \(p\in Cn(X)\) iff \(Cn(X,\neg p)=\) (the set of all formulas), for propositional variables the set of conditions (D) does characterize the classical logic and connectives. Other possibilities include: (K) Make the definitions contextual, (L) Use Gentzen style sequential consequence, and (M) Narrow the \(Cn\)-relation by an additional second-order requirement. The problem is, as the authors say, how to prove formally the nonexistence of a set of \(Cn\)-definitions in the desired form. The reviewer naturally wonders how the story goes for intermediate logics.