On the single axioms of protothetic. II.
Bolesław Sobociński · Notre Dame Journal of Formal Logic · 1961
In this chapter I propose to present the proofs relevant to the preceding discussion of the single axioms of system ©«-of protothetic.First of all I will prove 1) that thesis A n mentioned in Chapter I, § 3, and also each of the theses A Q , A. and A , can serve as a single axiom of ©c , and 2) that Lesniewski's metatheorem L discussed in § 3, can be replaced by my metatheorem S, whose conditions constitute a relatively small fragment of the original prerequisites set out by Leέniewski.In addition a number of questions closely connected with the topics mentioned above will also be discussed in what follows.It is clear, that if metatheorem S is true, then in order to establish that this or that protothetical thesis in which only equivalence occurs as a constant term, can be adopted as a single axiom of ©^ one must be able to prove that relatively to the rule of procedure of ©^ the thesis under consideration satisfies the requirements of S. And it is evident too, that the truth of metatheorem S depends exclusively on whether or not it is possible, by applying the said rule, to deduce metatheorem L from the assumptions of S. Thus, from the methodological point of view it would appear that thesis A n should be discussed after metatheorem S has been established.There are, however, serious reasons for adopting a reversed order of presentation.For it so happens that the proof concerning A n , though long and complicated, is much more elementary than the deductions required for the proof of S. Hence, it seems to me that in the beginning it is better to show that A n (and also A Q , A. and A ) implies the conditions of S. This will enable one who is not familiar with the methods of deduction in protothetic, to understand better subsequent proofs.Moreover, the proof of the main theorem, due to *The first part of this paper appeared in Notre Dame Journal of Formal Logic, vol.I (I960), pp.52-73 It will be referred to throughout the remaining parts, as [35].See additional Bibliography given at the end of this part.