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I imagine that these symbols originated in one of the following ways:

"I will declare a symbol for "for all." I will just use the letter "A" flipped upside-down. Yes, let $\forall$ represent "for all."

Hmmm, now I need a similar notation for "there exists." Well, I will do the same thing with the letter "E". But wait! The letter "E" has horizontal symmetry, so this won't work! I must flip it backwards instead. $\exists$ it is!"

OR

"I will declare a symbol for "there exists." I will just use the letter "E" flipped backwards. Yes, let $\exists$ represent "there exists."

Hmmm, now I need a similar notation for "for all." Well, I will do the same thing with the letter "A". But wait! The letter "A" has vertical symmetry, so this won't work! I must flip it upside-down instead. $\forall$ it is!"

So which is it? Did we flip our A's or our E's first?

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    $\begingroup$ I think this question is more appropriately asked at History of Science and Mathematics StackExchange. $\endgroup$ – Joel Reyes Noche Mar 10 '15 at 2:44
  • $\begingroup$ Isn't E backwards the same as E upside-down regardless of the horizontal symmetry? I prefer to think of both as upside-down letters, as this is how they would have been produced on a printing press. You cannot make any letter backwards with the movable type, but you can place letters upside-down. $\endgroup$ – user50229 Mar 10 '15 at 17:49
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The $\forall$ (for all, universal quantifier) symbol first appeared in the 1935 publication Untersuchungen ueber das logische Schliessen ("Investigations on Logical Reasoning") by Gerhard Gentzen.

The $\exists$ (there exists, existential quantifier) symbol was first used in the 1897 book Formulaire de mathematiqus by Giuseppe Peano.

$\exists$ came first.

(source)

Untersuchungen euber das logische Schliessen page 178 appears to be the first use of the symbol $\forall$ by Gentzen in a publication.

Gentzen's work

A googletranslate of the footnote at the bottom reads:

We undertake the characters $\vee$, $\supset$, $\exists$ from Russell. The Russell's character for "and", "equivalent", "NOT", "all", namely, $\cdot$, $\equiv$, $\sim$, (), are already used in mathematics with a different meaning. We therefore take the Hilbert &, whereas for Hilbert equivalence, space and non-character $\sim$, (), $~^\overline{~~}$, also has other meanings are customary. The non-character represents also represents a deviation from the linear array of characters that is uncomfortable for some purpose. We therefore use for equivalence and negation signs of Heyting, and an all-characters corresponding to the $\exists$ character.

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    $\begingroup$ For a long time they used (∃x) for existential and (x) for universal. $\endgroup$ – GEdgar Mar 10 '15 at 2:50
  • $\begingroup$ That's a fun PDF you used as a source! $\endgroup$ – Kevin Mar 10 '15 at 2:51
  • $\begingroup$ Excellent links, thank you. Now I just need to learn German. :) $\endgroup$ – Jonathan Hebert Mar 10 '15 at 2:59
  • $\begingroup$ Corrected the page number after trying to read it and realizing what it was talking about. It was in Getzen's first publication in that journal, not in the second one. (Interesting to note, Erdos had something published in this volume as well. That was only a year or two after he completed his phd) $\endgroup$ – JMoravitz Mar 10 '15 at 3:19

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