An alternative generalisation of the concept of duality (Q775346)

The symbol \([Y, X_1, X_2, \ldots, X_m, Y]\) was introduced in an earlier paper of the author [Math. Ann. 123, 76--78 (1951; Zbl 0042.00702)]. As the generalization of the dual of a formula in the two-valued propositional calculus, there is defined a relation between two formulae \(\Lambda\), \(\Phi\) of the \(m\)-valued propositional calculus, namely that \(\Lambda\) is an \(m\)-al of type \(i\) of \(\Phi\) (the definition is inductive according to \(i)\). There are proved four theorems each of which asserts or denies the existence of a functor of the following nature: the \(m\)-valued propositional calculus with the functor (as the only primitive) is functionally complete and the functor satisfies various conditions expressed in terms of \([\ldots]\) and \(m\)-ality. The last theorem is a generalization of a result of \textit{A. Church} [Port. Math. 7, 87--90 (1948; Zbl 0034.29101)]; it states that certain \(m-1\) functors form a complete set of independent primitives for the \(m\)-valued propositional calculus.