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What is the "etymology" of the symbol ∀ (for all), the universal quantifier. I can't find anything here, but I suspect it was adopted in the same way our numerals were adopted.


As simchona has pointed out, the symbol appears to have come from a shorthand (see here) from Gerhard Gentzen's publication "Investigations into Logical Deduction" in the The American Philosophical Quarterly, a symbol which shares an interesting likeness to the symbol in the link above. Perhaps the symbol had an earlier meaning, which gave reason to its usage in the Gentzenian way. Thank you, simchona.

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closed as off topic by Jim, FumbleFingers, Andrew Leach, MετάEd, tchrist Mar 9 '13 at 21:01

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Is this an English question or a Math question? – Jim Mar 9 '13 at 19:51
It seems fairly obvious that it's a variant of A for "all", made into a symbol to distinguish it from a normal letter. But I'm not sure this is an English question either. – Andrew Leach Mar 9 '13 at 20:01
Wikipedia says "The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for "for all" or "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for "there exists" or "exists"." – Alex B. Mar 9 '13 at 20:57
@Trancot Actually, no, the monetary symbol you linked to is actually rendered "V" on old coins. "∀" was not borrowed from old Roman coinage. – MετάEd Mar 9 '13 at 21:00
I mean that when it comes to writing new symbols during book imprintation, it's easier to rotate an old symbol instead of minting a new one – simchona Mar 9 '13 at 21:05

See the end of the fourth paragraph in the History section of wikipedia's Quantification article:

In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s.

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Yes, but from what alphabet--or from what source--did Gentzen "introduce" this symbol? – Trancot Mar 9 '13 at 20:19
was canonical when I studied logic in the early 60's. As I put it in the Logic guide (p.8), "There are many different kinds of quantifier in natural language, but logic uses only two abstract varieties: the universal quantifier (each, every, any, all; symbolized by ) and the existential quantifier (some, there exists, at least one; symbolized by )". – John Lawler Mar 9 '13 at 20:24
@jwpat7 Consider this source. – Trancot Mar 9 '13 at 20:47
@Trancot You might as well argue that logicians borrowed it from ancient Phoenician: fileformat.info/info/unicode/char/10900/index.htm But the reality is that the symbol is a rotated "A". – MετάEd Mar 9 '13 at 21:09
I never made an "argument" that the symbol I linked to was an attested primitive of the modern form. It was only there to demonstrate that other potentialities exists. All the sources that people in this discussion are drawing from are a bit dilute. So am I to believe that ∀ was introduced by Gentzen after his own personal inversion of the letter A, and this is a concrete original fact? What I am afraid of is that this notion of it being this way--how MετάEd, jwpat7, simchona, and Alex describe it--is a folk etymological artifact. Perhaps there are sources discussion members are not aware of. – Trancot Mar 9 '13 at 21:47

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