Who invented $\vee$ and $\wedge$, $\forall$ and $\exists$?

Question

I can rather easily imagine that some mathematician/logician had the idea to symbolize "it E xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for A ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it was one and the same person.)

What is hard (for me) to imagine is, how the one who invented $\forall$ could fail to consider the notations $\vee$ and $\wedge$ such that today $(\forall x \in X) P(x)$ must be spelled out $\bigwedge_{x\in X} P(x)$ instead of $\bigvee_{x\in X}P(x)$? (Or vice versa.)

Since I know that this is not a real question, let me ask it like this: Where can I find more about this observation?

Answer 1

See Earliest Uses of Symbols of Set Theory and Logic for this and much more.

Answer 2

I've misplaced my copy of it, but I recall S. C. Kleene in his Mathematical Logic noting that "v" came as an abbreviation of "vel". In Latin "vel" is one of the words which commonly gets translated to the English word "or", and at least people believed that the Latin word "vel" comes closer to alternation (or equivalently, inclusive disjunction) than any of the other words which commonly get translated as "or". Since Russell read Peano, and Peano's book on arithmetic got written in Latin, it does seem at least plausible that Russell first used "v" for alternation, as Bill's reference states.

That "∀" has to get interpreted by "⋀" as you correctly state in at least some cases (though not always), may seem strange at first, I agree. But, one way to think of things here comes as to have the truth set as linearly ordered. When you do this with "0" as the least truth value, and "1" as the greatest truth value, "⋀" most closely corresponds to, if not in fact is, the infimum, while "v" most closely corresponds to the supremum. Both "infimum" and "supremum", at least according to my intuition of them, involve notions just as strange if you don't look at them carefully.

Answer 3

The four types of propositions used in the classical Greek syllogisms were called A, E, I, O. Statements of type A were "All p are q". Statement of type E were "Some p are q". So of course a millennium later, mathematicians (who had a classical education) used A and E for these quantifiers, then later turned them upside down to avoid confusion with letters used for other things.

By the way: I and O were "All p are not q" = "No p are q" and "Some p are not q"="Not all p are q", but I don't remember which is I and which is O.

Answer 4

My understanding of the quantifier symbols $\bigvee$ ("there exists") and $\bigwedge$ ("for all") was that they were supposed to be large versions of $\vee$ ("or") and $\wedge$ ("and"). Then $\bigvee_{x\in X}Fx$ would mean $Fx_1\vee Fx_2\vee Fx_3\vee\dots$ whereas $\bigwedge_{x\in X}Fx$ would mean $Fx_1\wedge Fx_2\wedge Fx_3\wedge\dots$

This is similar to the notation $\bigcup_{x\in X}S_x$ for $S_{x_1}\cup S_{x_2}\cup S_{x_3}\cup\dots$ and $\bigcap_{x\in X}S_x$ for $S_{x_1}\cap S_{x_2}\cap S_{x_3}\cap\dots$

I don't see that these symbols "fail". I can see that $\forall$ could be confused for $\bigvee$, and that would be bad since $\forall$ means the same as $\bigwedge$. However, I don't see that as being a condemnation of one over the other, and there would be no confusion if only one were being used at a time.