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Kurt Gödel in 1931 used $x\Pi a$ where we in contemporary notation would use $(\forall x) A$ or $(x)A$, and $Ex a$ where we would use $(\exists x) A$. I believe that I remember that $\Sigma xA$ has been used with the meaning $\exists x A$. Is my belief correct, and if so by what authors was $\Sigma x$ so used? What, if any, connections are there to Gödel's usage?

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According to Wikipedia,

  • Charles Sanders Peirce used $\Pi_x$, $\Sigma_x$ in 1885;
  • Guiseppe Peano used (x), $(\exists x)$ in 1897;
  • Gentzen introduced $\forall x$ in 1935;
  • $(\forall x)$, $(\exists x)$ became standard in the 1960s.
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  • $\begingroup$ Thanks Bjørn! I particularly like Peirce's notation, and will invoke it in the formalized meta language. Do you have a reference? $\endgroup$ Feb 18, 2015 at 17:11
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    $\begingroup$ $\bigwedge$ for $\forall$ and $\bigvee$ for $\exists$ were used as well at some point. $\endgroup$
    – Asaf Karagila
    Feb 18, 2015 at 17:16
  • $\begingroup$ Oops! I now see the Wikipedia entry where I gleaned and missed earlier. $\endgroup$ Feb 18, 2015 at 17:17
  • $\begingroup$ I seem to recall that $\exists x$ appears in Godel's 1935 paper too; maybe that was implicit in your list. $\endgroup$ Feb 19, 2015 at 14:54
  • $\begingroup$ In Giuseppe Peano, Studi di logica matematica (1896-97) (Engl.trans., page 190) and in the 2nd vol of the Formulaire (1897), page 28, we can find $\exists$. But it seems that the introduction of $(x)$ for the universal quantifier is due to B.Russell, Mathematical Logic as Based on the Theory of Types (1908), page 228- $\endgroup$ Feb 20, 2015 at 13:37

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