This is a preview of subscription content, log in via an institution to check access.
References
The term “quantifier expression” is used here in a different sense than in the paperOn Proper Quantifiers.
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.
The term “Sm” belongs to the semantic category of quantifiers of two arguments.
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.
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.
Cf.A. N. Whitehead — B. Russell:Principia Mathematica, v. II, p. 210, Df.*120.03.
It is obvious if we take into account such antinomies asRussell's antinomy, the antinomy of the class of all things, the antinomy of the class of all classes and the antinomy of the greatest cardinal number. But a question may be raised whether the antinomy of the class of all cardinal numbers cannot be recostructed in our system, since all expressions determining cardinal numbers belong here to the same semantic category, namely to the semantic category of quantifiers of one argument. However, we cannot define in our system the class of all cardinal numbers if we confine ourselves to such definitions as D2. Suppose, we define:\(\begin{gathered} Cl'f(g) \equiv \prod\limits_{g(k)} {f(k)} \hfill \\ \mathop {nc_1 }\limits_g \left\langle F \right\rangle G(g) \equiv \mathop {Sm}\limits_g G(g)F(g) \hfill \\ Nc_1 (n) \equiv \mathop \Sigma \limits_F \mathop \Pi \limits_G \mathop {[n}\limits_g G(g) \equiv \mathop {Sm}\limits_g G(g)F(g)] \hfill \\ \end{gathered} \) Then, although the symbols “Nc” and “Nc 1” belong to the same semantic category, they correspond to different classes of cardinal numbers. We can namely prove:Nc 1 (nc1 〈Cl'f〉) but we cannot prove:Nc (nc1 〈Cl'f〉), since the variables “f” and “g” occurring on the right side of D2 do not represent such expressions as “Cl'f”, which belong to the higher type. For each type we can define the corresponding „Nc” and all such expressions belong to the same semantic category (to the semantic category of proposition — forming functors of quantifier arguments). But we cannot define by means of object deflnition (definition belonging to the object language of the system) the sum of all such „Nc” since in the object language we do not have to our disposal the variables representing type — indices and quantifiers binding such variables. We must however introduce some restriction concerning definitions formulated in the metalanguage, in order to exclude the definition:
For this purpose it is sufficient to assume one of the following restrictions (a), (b). (a). In the definitions formulated in the metalanguage we can use variables representing names of expressions of an arbitary type only if we define an expression belonging to the semantic category of quantifiers. (b). The principle of gnasi-homogeneousness. If variables (free or bound) representing names of expressions of an arbitrary type occur on one side of a definition formulated in the metalanguage, then such variables must occur also on the other side of this definition. The restriction (a) or (b) excludes the definition (I) but allows us to introduce such definition as e. g. the definitions D6 or D1′ given in the present paper. This question will be discussed in more detail in the paper in which the general arithmetic of cardinal numbers will be presented.This result will be presented in another paper.
For this purpose it is sufficient to assume one of the following restrictions (a), (b). (a). In the definitions formulated in the metalanguage we can use variables representing names of expressions of an arbitary type only if we define an expression belonging to the semantic category of quantifiers. (b). The principle of gnasi-homogeneousness. If variables (free or bound) representing names of expressions of an arbitrary type occur on one side of a definition formulated in the metalanguage, then such variables must occur also on the other side of this definition. The restriction (a) or (b) excludes the definition (I) but allows us to introduce such definition as e. g. the definitions D6 or D1′ given in the present paper. This question will be discussed in more detail in the paper in which the general arithmetic of cardinal numbers will be presented.Additional information
Allatum est die 26 Junii 1958
Rights and permissions
About this article
Cite this article
Borkowski, L. Reduction of arithmetic to logic based on the theory of types without the axiom of infinity and the typical ambiguity of arithmetical constants. Stud Logica 8, 283–295 (1958). https://doi.org/10.1007/BF02126760
Issue Date:
DOI: https://doi.org/10.1007/BF02126760