Who was the first to use $\bigwedge$ and $\bigvee$ as universal and existential quantifiers?

394 Views Asked by At

It's not surprised that somebody uses $\bigwedge$ and $\bigvee$ as universal and existential quantifiers since $$ \bigwedge_{x\in A}\,\varphi(x)\Leftrightarrow \varphi(a_0)\wedge\varphi(a_1)\wedge\varphi(a_2)\wedge\cdots\Leftrightarrow\forall x\in A\,\varphi(x) $$ and $$ \bigvee_{x\in A}\,\varphi(x)\Leftrightarrow \varphi(a_0)\vee\varphi(a_1)\vee\varphi(a_2)\vee\cdots\Leftrightarrow\exists x\in A\,\varphi(x). $$

Such uses could be found in

  • (@Mauro ALLEGRANZA 's discovery) P. Lorenzen. Einführung in die operative Logik und Mathematik. Die Grundlehren der mathematischen Wissenschaften, vol. 78. Springer-Verlag, Berlin, 1955. p.3. Writing in German.
  • (@Mauro ALLEGRANZA 's discovery) P. Lorenzen. Formale Logik. Sammlung Goschen, Bd. 1176/1176a. Walter de Gruyter & Co., Berlin, 1958. 165S. Writing in German and for an English translation see Formal Logic. Translated by P Crosson. D. Reidel Publishing Company, 1965. p.74.
  • (@Mauro ALLEGRANZA 's discovery) H. Hermes. Aufzählbarkeit, Entscheidbarkeit, Berechenbarkeit: Einführung in die Theorie der rekursiven Funktionen. Springer-Verlag, Berlin, 1961. Writing in German and for an English translation see Enumerability, Decidability, Computability: An Introduction to the Theory of Recursive Functions. Translated by G. T. Herman and O. Plassmann. Springer-Verlag, Berlin, 1965. p.67.
  • (@Mauro ALLEGRANZA 's discovery) T. Skolem. Investigations on a comprehension axiom without negation in the defining propositional functions. Notre Dame Journal of Formal Logic, 1960, (1-2):13-22.
  • (@Mauro ALLEGRANZA 's discovery) D. Kalish and R. Montague. Logic: Techniques of Formal Reasoning. Harcourt, Brace & World, Inc. 1964. p.86.
  • (My discovery) G. Kreisel and J. L. Krivine. Elements of Mathematical Logic (Model Theory). North-Holland Publishing Company, 1967, pp.17-19.
  • (My discovery) K. Kuratowski and A. Mostowski, Set theory, North-Holland Publishing Company, 1968, pp.45-46.

Furthermore, such uses could be dated back to 1950. In

Lorenzen began to use $\bigwedge_x\varphi(x)$ as "$\varphi(x)$ for all $x$" and $\bigvee_x\varphi(x)$ as "$\varphi(x)$ for some $x$" simultaneously.

But I can't determine whether there are others who made such uses in an earlier time. So my question is: Who was the first to use $\bigwedge$ and $\bigvee$ as universal and existential quantifiers respectively?