\(R\)-reconstruction in open logic (Q1895515)

In order to describe the growth and modification of knowledge and to express the evolution of a hypothesis, \textit{Li Wei} [Sci. China, Ser. A 36, No. 3, 362-375 (1993; Zbl 0771.68097)] built up a theory of an open logic system. \(R\)-reconstruction is an elementary definition of open logic, and is not unique, as pointed out by Li [loc. cit.]. In this note, we give a necessary and sufficient condition for the uniqueness of the \(R\)-reconstruction of \(\Gamma\) for the rejection of \(A\) by facts, where \(\Gamma\) is an arbitrarily given hypothesis, \(A\) is a sentence such that \(\Gamma\models A\) and \(A\) has rejection by facts.