The present paper is based on the author's Ph.D. thesis (The University of California, Berkeley, 1963). The author is most grateful to his thesis adviser Professor Craig for the initiation to this field and the subsequent warm encouragement. It was written while the author was a post-doctorate fellow of N.R.C. of Canada.

¹ The ordinary interpretation is the 1-interpretation in our sense.

² In a relational system of power less than κ, a formula beginning with is never κ-satisfiable and a formula beginning with is always κ-valid according to the phrases (b) and (c). Hence such a degenerate case is excluded from the consideration of κ-satisfiability.

³ More rigorously, we should extend the language L by introducing new individual symbols corresponding to the elements of the universe under consideration and should understand A[X] to be the result of substitution of these constants to free variables. But we use a sloppy notation, its being convenient.

⁴ Cf. Robinson, A., Complete Theories, Amsterdam, 1956, p. 6.Google Scholar

⁵ The cofinality cf(κ) of κ is the least cardinal λ such that there is a sequence of ordinals cofinal in κ.

⁶ From the set [CE], one sees that the exception for the case κ = No in Theorems 6 and 8 is essentially due to the fact that there are denumerably many symbols in L. One may ask if results similar to Theorems 4, 5, 6 and 8 hold for languages having more than denumerably many symbols. The author does not find any reason to the contrary, but he has not and does not intend to look into this matter.

⁷ Refer: Fuhrken, G., Skolem-type normal forms for first order languages with a generalized quantifier, Fundamenta Mathematicae LIV, (1964)Google Scholar Theorem 3.4.

⁸ Our new paper, An axiomatic system for the first order languages with an equicardinality quantifier, (this Journal, Vol. 31 (1966), pp. 633–640.) may not be completely uninteresting in this connection. There, the structure of the V₊^κ is studied to some extent.

⁹ The recursive enumerability of VI follows from a result of Vaught that the set of valid formulae of the usual first-order language strengthened by introducing and under the N₁-interpretation is recursively enumerable and the Corollary to Theorem 6 of the last section. Our deduction system is, however, more explicit and straightforward for our purposes and furnishes a workable tool for the proofs of the cut-elimination theorem and of the non-recursiveness. For the result of Vaught, refer to: Vaught, R., The completeness of logic with the added quantifier “there are uncountably many”, Fundamenta Mathematicae, LIV (1964), pp. 303CrossRefGoogle Scholar, 304.

¹⁰ Refer: Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, Band 39, (1934)Google Scholar.

¹¹ Refer: Surányi, J., Reduktionstheorie des Entscheidungsproblems, Budapest, 1959.Google Scholar Satz XIV.

¹² For instance, and are in π(1, 2, 3) B.

¹³ Since the universal quantifier is not available in L, this theorem is formulated in the open formula form.

¹⁴ For instance, centering our attention on V₅, V_i are subsets of V₅ for all i > 5 exept i = 6, 7, 8; 11, 12; 16. For these i, V₅ and V_i are incomparable. Also V₅ is a subset of V₁, V₂, and V₃, but is incomparable with V₄.