In the ordering $\preceq$ of nonnegative integers by divisibility, 1 is the least element and 0 is the greatest, and we have for instance $$ 1\preceq 2\preceq 6\preceq 12\preceq\dots\preceq 0.$$ In this ordering, gcd is the same as meet (greatest lower bound), which is dual to least upper bound, which is what boolean OR is for $\{0,1\}$.
So it makes sense if you think of numbers as "degrees of truth", where multiplicative factors are evidence of falsehood.
See also: What is gcd(0,0)?
