1. The term “quantifier expression” is used here in a different sense than in the paperOn Proper Quantifiers.
2. I have introduced this conception in the paperOn Proper Quantifiers, Ch. I. §. 2, “Studia Logica”, Vol. VIII, where this conception and other principles mentioned above are discussed. Prof. Dr.J. Słupecki called my attention to the possibility of applying this conception to the arithmetic of natural numbers on the ground of logic based on the theory of types — formulating thus the problem whose solution is presented in this paper. The main result of this paper was presented for the first time in my lectureReduction of arithmetic to logic based on the theory of types without axiom of infinity delivered at the meeting of the Institute of Logic of the Polish Academy of Sciences in Wrocław on April 26, 1958.
3. The term “Sm” belongs to the semantic category of quantifiers of two arguments.
4. The notation manner of suppositional proofs applied here is explained in the article:L. Borkowski, J. Słupecki:A Logical System Based on Rules and Its Application in Teaching Mathematical Logic, “Studia Logica”, v. VII.
5. With the difference that Frege takes into account the natural numbers from 1 ton, instead of the natural numbers from 0 ton-1. Cf.G. Frege:Grundgesetze der Arithmetik, v. I. p. 217. In this article I used the formulation given in the script ofH. Scholz:Logistik, 1934.