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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.

REMARK ABOUT A COMMENT

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.

Trancot
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    Is this an English question or a Math question? – Jim Mar 09 '13 at 19:51
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    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 09 '13 at 20:01
  • @AndrewLeach I don't think I'll ever accept any of your simple opinions. CLICK HERE – Trancot Mar 09 '13 at 20:43
  • @Trancot That's not the same sign. It's an upside-down A, because of typesetting limitations –  Mar 09 '13 at 20:53
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    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 09 '13 at 20:57
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    @Trancot Actually, no, the monetary symbol you linked to is actually rendered "V" on old coins. "∀" was not borrowed from old Roman coinage. – MetaEd Mar 09 '13 at 21:00
  • When you say "typesetting limitations" do you just mean that the ∀ should be seen as a symbol created so it is not to be seen as an expression (P or Q), so, for example, a postfix expression won't get read wrong; that is, AQ reads differently then ∀Q. – Trancot Mar 09 '13 at 21:03
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    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 –  Mar 09 '13 at 21:05

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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.

James Waldby - jwpat7
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  • Yes, but from what alphabet--or from what source--did Gentzen "introduce" this symbol? – Trancot Mar 09 '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 09 '13 at 20:24
  • @jwpat7 Consider this source. – Trancot Mar 09 '13 at 20:47
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    @Trancot You might as well argue that logicians borrowed it from ancient Phoenician: http://www.fileformat.info/info/unicode/char/10900/index.htm But the reality is that the symbol is a rotated "A". – MetaEd Mar 09 '13 at 21:09
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    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 09 '13 at 21:47
  • Umm...yes? Read the footnotes from Gentzens work "Investigations into logical deduction", where he talks about creating the symbol after the rotated E. –  Mar 09 '13 at 22:01
  • @simchona You must admit that the inverted form of the English A is quite unlike the one in Gentzen's footnote. – Trancot Mar 15 '13 at 19:39
  • Where did you reproduce that footnote from? There is a reproduction of the body of the article here which does not have the funny looking "∀" you are asking about. – MetaEd Mar 15 '13 at 21:30