Peano introduced a number of logical symbols still used today:

  • $∨$ (from Latin vel)
  • $∧$ (inverted $∨$)
  • $∃$

This inversion of Latin letters as symbols (and inversion of symbols to signify their 'opposite' operation) was followed by later logicians:

  • $∀$ (Gentzen, 1935: inverted A from "All-Zeichen" / "Für Alle", by analogy to $∃$)
  • $⊥$ (inverted $⊤$)

I had always assumed that ∃ stood for "E" in "Existential" / "there Exists", but Peano introduced this symbol in a French text, not using any words beginning 'e':

Mais nous préférons l'indiquer par la nouvelle notation


qu'on peut lire « il y a des $a$ ».

So why did he choose an inverted "E"?

Earliest Uses of Symbols of Set Theory and Logic
Is the symbol for set membership $\in$ derived from greek letter $\epsilon$?
Math SE: What came first, the ∀ or the ∃?

  • 6
    $\begingroup$ Peano was well familiar with Greek and Latin, and previously attributed the element symbols ϵ and ε to Latin est and Geek ἐστι, both meaning "is", see Is the symbol for set membership ∈ derived from greek letter ϵ? So he did not need E to appear in French. $\endgroup$ – Conifold Jan 14 at 10:57
  • $\begingroup$ @Conifold I don't quite understand what you're saying $\endgroup$ – ukemi Jan 14 at 15:32
  • 1
    $\begingroup$ "Is" or "exists" starts with variations on "E" in multiple languages that Peano knew, none of them in particular had to be "the origin". With the element symbol he explicitly attributed it to Latin in one place and to Greek in another. And Russell might have adopted Peano's symbol because it matched the English word. It is from Principia that this notation spread afterwards, not the Formulaire. $\endgroup$ – Conifold Jan 14 at 19:59
  • $\begingroup$ @Conifold "none of them in particular had to be "the origin"" I really don't understand your point - I'm asking if there is any evidence shedding light on why he chose an inverted E, I'm not claiming it must have had only one origin/motivating factor. My point about the French was he introduced the symbol in a document written in French, and none of the words he used in the explanation of said symbol begin with an 'E', so it wasn't 'obvious' from the text what the origin was (to me). $\endgroup$ – ukemi Jan 14 at 20:20
  • $\begingroup$ But why does it matter that it was written in French? Mathematicians routinely use Greek and Latin letters for symbols, and rarely comment on why specifically they chose them. In the case of Peano we even know that he commented on the use of Greek and Latin ϵ in other contexts, also written in French or Italian, and continued to use it in the Formulaire. $\endgroup$ – Conifold Jan 14 at 20:25

When introducing the older terminology in the previous sentence, Peano describes it thus:

... signifie "il y a des a", "les a existent"...

It seems likely this is the source of the inverted "E".


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