Investigations in protothetic (Q1071747)

More than 25 years ago, B. Sobociński published his investigations in protothetic based on the (now standard) single axiom \(A_ n\). In the article under review the author reproves the metarules SI-SV of Sobociński and shows that the usual laws of implication (modus ponens, and equivalences simulating the truth-table of implication) can be derived from \(A_ n\) without using the rule of extensionality. The presented deductions are shorter than those previously known. However, the author has been forced to use a very long and specific definition of implication. Reviewer's remark: The present reviewer has considered systems of protothetic in which the rule of extensionality incorporates also the law of extensionality for propositions and, therefore, expressions shorter than \(A_ n\) may serve as the single axioms [Latv. Mat. Ezheg. 26, 264- 270 (1982; Zbl 0495.03007)]. In these systems, the traditional means of inference were also derived without using higher extensionalities [ibid. 23, 179-193 (1979; Zbl 0438.03018)].