Suppose we have a cryptosystem homomorphic for addition (say Paillier's). Is there a way to perform a logical OR operation between two binary values (with a binary result).
We can, of course, obtain the encoded result of addition between two values (by multiplying the encoded values):
$Enc(x + y) = Enc(x) \cdot Enc(y)$
and that gives us something close to a logical OR, except if both $x$ and $y$ have value $1$, in which case the result will be $2$, when it should be $1$.
Therefore another way to solve the problem might be to figure a homomorphic function: $F: \{1, 2\} \rightarrow 1; 0 \rightarrow 0$.
Does such a function exist?
Conversely, is there any way to prove that a true logical OR operation would break some fundamental properties/limitations of (non-full) homomorphic cryptosystems (and that I am wasting my time looking for one)?