Homomorphic encryption of logical OR/AND
One practical (if not theoretically satisfying) way would be to use Pallier, with the mapping "any encrypted nonzero value is an encrypted TRUE; any encrypted zero value is an encrypted FALSE".  To encrypt a TRUE value, you pick a random value and encrypt that, and the logical OR operation is the standard Pallier homomorphic addition.

Now, obviously, the OR of two TRUE values has a tiny probability of becoming FALSE; however it is far more likely that your computer is hit by a meteor...