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.

  • $\begingroup$ They're analogous to $\cup$ and $\cap$ in set theory. I always supposed that $\cup$ stood for the letter U in union. $\endgroup$ – Angina Seng Aug 20 '20 at 15:28
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    $\begingroup$ 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 $\endgroup$ – J. W. Tanner Aug 20 '20 at 15:29
  • $\begingroup$ @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 $1930$. $\endgroup$ – Brian M. Scott Aug 20 '20 at 17:34

This is not an answer but more like a comment. I decided to write as an answer because I wanted to include an image.

If you insist on writing your top at the top of the picture and your bottom at the bottom, you can (as I do, most of the time*) picture disjunction as a pushout and conjunction as a pullback, and think about the symbols as the markings one does for these special commutative squares. Here is an image of what I mean:

After I came up with this I could finally sleep well at night.

(*). There are certain situations when it is easier for me to think about the total space as being in the bottom: e.g. thinking about local sections of a sheaf or the closed sets of the spectrum of a ring. One may argue that in these two examples things are "flipped over" by some contravariant functor somewhere (a sheaf in the first case and the correspondence between ideals and closed sets in the second).


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