# What came first, the $\forall$ or the $\exists$? [closed]

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?

• I think this question is more appropriately asked at History of Science and Mathematics StackExchange. – Joel Reyes Noche Mar 10 '15 at 2:44
• 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. – user50229 Mar 10 '15 at 17:49

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.

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.