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On proper quantifiers I

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  1. I express here my sincere gratitude to Prof. Dr.J. Słupecki for his numerous remarks and suggestions.

  2. Cf.J. Łukasiewicz:Elementy logiki matematycznej, 1929, pp. 154–155, 158.

  3. Cf.J. Łukasiewicz:Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, 1951, p. 84.

  4. Cf.A. Mostowski:Logika matematyczna, 1948, p. 49.

  5. Cf.W. V. Quine:Mathematical Logic, 1951, p. 102.

  6. Cf. e. g.A. Tarski:O logice matematycznej i metodzie dedukcyjnej, 1936, pp. 47–8,R. Carnap:Logical Syntax of Language, 1937, p. 143,W. V. Quine:Methods of Logic, 1953, p. 231.

  7. Cf.W. Dubislav:Elementarer Nachweis der Widerspruchlosigkeit des Logik-Kalküls, „Journal für die reine und angewandte Mathematik”, Bd. 161 (1929), p. 111.

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  8. Cf.W. V. Quine:Methods of Logic, pp. 101–107.

  9. Cf.K. Kuratowski andA. Mostowski:Teoria mnogości, 1952, p. 22.

  10. The term „matrix” is used here in a different sense than e. g. in the paper:J. Łukasiewicz andA. Tarski:Untersuchungen über den Aussagenkalkül, 1930. I use this term instead of the term „value function”. This deviation from the ordinary sense is caused by the fact that we shall need such terms as „submatrix of a given matrix” etc.

  11. Cf.H. Reichenbach:Elements of Symbolic Logic 1947, p. 87.

  12. Cf.K. Ajdukiewicz:Die syntaktische Konnexität, „Studia Philosophica” I, 1935, pp. 16–17.

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  13. Cf.R. Carnap:Logical Syntax of Language, p. 190. E. g. we write then\(\mathop \Pi \limits_2 \) 2=2 instead of\(\mathop \Pi \limits_n \) n=n.

  14. Cf. below.R. Carnap:Logical Syntax of Language, p. 101. E. g. we write then\(\mathop \Pi \limits_2 \) 2=2 instead of\(\mathop \Pi \limits_n \) n=n.

  15. Cf. below.R. Carnap:Logical Syntax of Language, p. 110. E. g. we write then\(\mathop \Pi \limits_2 \) 2=2 instead of\(\mathop \Pi \limits_n \) n=n.

  16. This is a modification of the notation introduced byK. Ajdukiewicz in the paperDie syntaktische Konnexität, p. 18. This author does not consider however the semantic category of the quantifier (in the sense of this word used here) but the semantic category of the expression composed of the quantifier and the quantifier variable.

  17. This result will be presented in another paper.

  18. Cf.K. Ajdukiewicz:Die syntaktische Konnexität, „Studia Philosophica” I, 1935, p. 5.

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  19. Cf.K. Ajdukiewicz:Die syntaktische Konnexität, „Studia Philosophica” I, 1935, p. 26.

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  20. This symbolism is a certain modification of the symbolism introduced byS. Leśniewski in the paperGrundzüge eines neuen Systems der Grundlagen der Mathematik, 1929, p. 33. S. Leśniewski used for truth-functors of one argument the symbols: ―,⊢,⊣,|―|, and for truth-functors of two arguments the symbols obtained from the basic symbol: (in such a manner that e. g. the symbol: is a sign of equivalence, the symbol: is a sign of alternation, the symbol: is a sign of conjunction, etc.)

  21. It corresponds to the diagram given byH. Reichenbach for the definienses of these quantifiers inElements of Symbolic Logic, p. 93.

  22. According to the definition given byR. Carnap inLogical Syntax of Language p. 193, Ω is a universal quantifier if for each Φ and ζ the expression Φ( αζ ) is a consequence of the expression ⌈ΩΦ⌉. Then the class of the universal quantifiers consists of the quantifiers „Π” and „F”. It seems that our definition, according to which this class consists of the quantifiers „Π″, „ΠN″ and „ΠZ″, is more intuitive.

  23. Cf. also my paper:Systems of the Propositional and of the Functional Calculus Based on one Primitive Term, „Studia Logica”, VI, pp. 13, 15–17.

  24. Cf.K. Ajdukiewicz:Die Definition, 1936, p. 1.

  25. Cf. my paper:Ueber analytische und synthetische Definitionen, „Studia Logica”, IV, p. 10.

  26. Cf. e. g.A. Mostowski:Logika matematyczna, p. 49,K. Kuratowski andA. Mostowski:Teoria mnogości, p. 39.

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Borkowski, L. On proper quantifiers I. Stud Logica 8, 65–128 (1958). https://doi.org/10.1007/BF02126736

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