As for logical operators, the ones in common use are ! (NOT), && (AND) and || (OR). I have already addressed NOT in other posts, so we can focus on the two-input logical operators here.
Here are the equivalent kleene versions of those boolean logical operators:
&& (AND) Min
|| (OR) Max
By the way, here's a really good explanation of why logical XOR (and by extension XNOR) don't exist in C. http://c-faq.com/misc/xor.dmr.html
I'll skip over the obvious uses of the boolean logical operators. Suffice it to say, they regard each operand as a boolean, so a non-zero value is considered TRUE and a zero value is considered FALSE. The result of the operation is returned as a boolean value.
As for ternary logical operators, it first needs to be pointed out that there are 19,683 of them. Of course the vast majority wouldn't be of much use and it would be a complete joke to expect a language to have vocabulary for all of them. However, I suppose it could be possible for a language to compute the truth table for any logical operator if there were a way to describe it in the source code. Say, with the heptavintimal name for the two-input gate for example:
if (x <A6R> y == NEUT) then...
This way, theoretically any of the possible two-input gates could be used as a logical operator. I don't have a clue how (or why) that would be implemented, but it's an interesting idea.
Putting that aside for the moment, let's address the use of the ternary logical operators that correspond to existing logical operators. They evaluate each operand for its kleene value (FALSE for a negative, NEUT for a zero and TRUE for a positive) and they they return kleene values. Min returns the value of the lesser of the two inputs. Max returns the greater of them. Simple.
The thing to keep in mind is that they can return a NEUT and that this can be used for more expressive control statements. More on that in a later post.