I learnt boolean algebra as part of a computer hardware course where the focus was very much on using it as a foundation for creating digital logic blocks out of gates, so there was very early on introduced a mapping between the boolean values True and False with the numerals 1 and 0.
In a ring consisting of the set $\{0,1\}$ and operations $\cdot$ and $\oplus$ as modulu 2 product and sum respectively, there's a correspondence between the behavior of $\{0,1\}$ under $\cdot$ and $\oplus$ with the logical values $\{\text{False, True}\}$ under the logical connective $\text{AND}$ and $\text{XOR}$ respectively.
I can see why we might feel comfortable mapping the boolean values to "binary digits", and map $\text{AND}$ to the product $\cdot$, and $\text{XOR}$ to the sum $\oplus$. I'm failing to see any connection between the logical $\text{OR}$ with anything resembling the arithmetic sum $+$? In what setting would we be able to have $1+1=1$, so that it corresponds to $\text{TRUE}\lor\text{TRUE}=\text{TRUE}$?
Is it just by convention that $\text{OR}$ is usually denoted with $+$ in boolean algebra, or is there some way of motivating it?
Logical disjunction ($\lor$) is naturally associated with the union ($\cup$) of sets. If there's any way to define $+$ on sets, it would be $\cup$.