I've moved this over to HSM.
It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (quite reasonably) the "meet product", but write that as $A\vee B$.
I'm aware the answer might be "it just turned out that way", but was this choice of symbol motivated by a natural way to think of $A\wedge B$ as a meet in some lattice or other?
If not, which historical figure should I blame for this situation? I understand Grassmann wrote $A\wedge B$ as $[A, B]$, which would let him off the hook.
EDIT: Intuitively, if I have two vectors $a$ and $b$, $a\wedge b$ is the smallest object that contains both $a$ and $b$. If you order elements of the exterior algebra by inclusion, which seems very natural if you're doing geometry, you get a lattice where the join operation is the exterior product.
This mostly seems to come to the fore in Clifford-algebra-related approaches to geometry. Examples of people mentioning this relationship and complaining about / altering the standard notation:
While digging up these references I also turned up a claim that Peano wrote the exterior product using $\vee$, whereas Cartan used $\wedge$, but it's not clear to me whether Cartan originated (or popularised) this notation or whether he had a particular reason for choosing it [or indeed whether this claim is correct -- it was an offhand remark with no citation].