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?
$\begingroup$ $\endgroup$
1 
- 1$\begingroup$ en.wikipedia.org/wiki/… $\endgroup$Steve Huntsman– Steve Huntsman2015-02-18 18:44:36 +00:00Commented Feb 18, 2015 at 18:44
Add a comment |
1 Answer
$\begingroup$ 
$\endgroup$
7 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.
answered Feb 18, 2015 at 17:09
Bjørn Kjos-Hanssen
25.3k3 gold badges61 silver badges116 bronze badges
- $\begingroup$ Thanks Bjørn! I particularly like Peirce's notation, and will invoke it in the formalized meta language. Do you have a reference? $\endgroup$Frode Alfson Bjørdal– Frode Alfson Bjørdal2015-02-18 17:11:51 +00:00Commented Feb 18, 2015 at 17:11
- 1$\begingroup$ $\bigwedge$ for $\forall$ and $\bigvee$ for $\exists$ were used as well at some point. $\endgroup$2015-02-18 17:16:24 +00:00Commented Feb 18, 2015 at 17:16
- $\begingroup$ Oops! I now see the Wikipedia entry where I gleaned and missed earlier. $\endgroup$Frode Alfson Bjørdal– Frode Alfson Bjørdal2015-02-18 17:17:04 +00:00Commented 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. $\endgroup$Paul Taylor– Paul Taylor2015-02-19 14:54:08 +00:00Commented 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- $\endgroup$Mauro ALLEGRANZA– Mauro ALLEGRANZA2015-02-20 13:37:48 +00:00Commented Feb 20, 2015 at 13:37