# Origins of conjunction and disjunction symbols ($\wedge$ and $\vee$) in formal logic

I hope this question isn't off-topic: I'm just curious. In logic, we commonly use $$\wedge$$ and $$\vee$$ to represent conjunction and disjunction respectively. I often find myself confused about why we chose these particular symbols. On some level, perhaps I should be satisfied with the answer "Choice of symbols is arbitrary; we could have chosen any symbols, but we chose these." But there are reasons why I think this particular choice wasn't completely arbitrary.

First of all, we also use $$\wedge$$ and $$\vee$$ to represent the meet and join operators. That's, of course, no accident: The set of equivalence classes (defined by mutual entailment) of first-order sentences forms a Boolean algebra. So, I'm guessing the use of these symbols developed simultaneously within logic and algebra as the connections between algebra and logic were being discovered. That would be a satisfactory explanation, but I still find myself puzzled.

In the Boolean algebra described above, $$\top$$ ("top") is the equivalence class of first-order sentences that contains all tautologies and, likewise, $$\perp$$ ("bottom") is the equivalence class of first-order sentences that contains all contradictions. The meet operator ($$\wedge$$) is rather suggestive: it seems to visually invite one to map two points in the lattice to some point which is located above it. And similarly, the appearance of $$\vee$$ seems to invite us to map points to a point lower in the lattice. But of course, if "top" is at the top and "bottom" is at the bottom of the lattice, this is not what we do.

I suppose we can solve this problem if we just draw our lattices with $$\top$$ at the bottom and $$\perp$$ at the top. But it just seems to me that the combination of symbols and words used to describe them creates a confusing mess. Perhaps someone with knowledge of the history can explain how we came to be here.

• They're analogous to $\cup$ and $\cap$ in set theory. I always supposed that $\cup$ stood for the letter U in union. – Angina Seng Aug 20 '20 at 15:28
• I thought the disjunctive symbol $\lor$ stands for vel, which is Latin for or; consider asking this at History of Science and Mathematics Stack Exchange – J. W. Tanner Aug 20 '20 at 15:29
• @AnginaSeng: And indeed $\cap$ and $\cup$ were used first; according to this page, they were introduced in $1888$ by Peano. The same source says that Russell introduced $\lor$ in $1908$, and Heyting introduced $\land$ in $1930$. – Brian M. Scott Aug 20 '20 at 17:34