Why is the + sign commonly used as logic OR operator?
One line of reasoning I always used for logical AND and OR signs is their relation to mathematical operations they represent.

Let's start with logical AND. It's often represented as multiplication sign, for example *. So if you have a long expression like s1*s2*s3*s4.... and one of the variables takes value of 0, or logical false, then the entire expression will take the value of 0, which is quite normal for multiplication, because 1*1*0*1... equals 0.

On the other hand, when we use the + sign, which commonly stands for addition to represent logical OR, we have similar case. If we have several variables that are ORed, then we have again the case of s1+s2+s3+s4... If just one of the variables is non-zero, then the result will be non-zero as well, which is logical (IMHO) when we compare OR to addition. For example 0+0+1+0... equals 1. One point where this breaks is that is we have more ones, the result is still just one. One way of thinking I used for this is to just keep in mind that one represents existence, so of something exists and you add more existence to it, it will still exist.