After having reached the unanimous decision that there was no way we could find a half-adder in the Bob/Ike/Qui set of rules, we began the search for a new approach. Knowing that computers (the primary users of logic like this) operate on a basis of negative and positive voltage, we decided to drop the 2 from the system and replace it with -1.
Suddenly, patterns galore leapt off the page, grabbed us in a head lock, and began screaming epithets about AC currents and massive, million-ohm resistors. Fearing for our lives, we decided to listen to their cries and speedily wrote a BASIC program which creates said truth table, using just two multi-line if-then-else statements, one for the sum and one for the carry digit. The table essentially depends on the absolute values of the input digits being the same. (We hope none of that made sense. If it did, please consider IMMEDIATE psychiatric assistance.)
Ok, this is it in more reasonable terms. Our possible inputs are -1, 0, and 1. As we already mentioned, the half-adder is all-important in a computer logic system. Here's the truth table.
Notice the patterns? In the Sum column, a 0 is output if and only if the absolute value of the two inputs is the same. If the two inputs are not the same, the non-zero input is the output.
In the carry column, if the two inputs are identical, then the output is their value. Otherwise, the output is 0.
Fascinating, eh? Upon seeing these patterns, we were sure that we had struck gold (or at least oil). But then we tried to figure out specific gates that would lead to this marvelous network, as well as a way to convert any set of numbers using 1's and -1's to base 10. And again we planted our faces in a brick wall, watering periodically.
Still more discouraged, we decided to search for yet another path to luminesence.