Same, Different, Construction and Distinction

In language and mathematics, we use ordered combinations of words and symbols to convey meaning between us. The mechanism by which this process works depends upon each word or symbol having a settled meaning, such that an overall meaning of an ordered combination of words or symbols can be constructed. Mathematicians and lawyers invest huge amounts of their time to master the art of construction, aided by dictionaries and precedents. The continued functioning of communities is believed to depend on this.

Underlying all of language and mathematics are simple distinctions.

Same and different are words used to convey meanings. As a pair, being mutually exclusive, they define a distinction.  Each is composed of an ordered arrangement of symbols we call letters. These ordered arrangements are themselves symbols. The words same and different can be replaced with single symbols, such as 0 and 1, – and +, or { and }. If 0 means same, 0 turns 0 into 0; 0 turns 1 into 1; 1 turns 0 into 1; and 1 turns 1 into 0. 1 is the same as 1, so it turns itself into 0. This is the truth table for the logic gate XOR. If the meanings for same (0) and different (1) are exchanged, then the table mirror opposite to XOR, XNOR is obtained: 1 turns 1 into 1; 1 turns 0 into 0; 0 turns 1 into 0 and 0 turns 0 into 1.

Since the identities and meanings of same and different are mutually exclusive, it appears appropriate to assign absolute meanings to them. With so many symbols in use, the anomaly that different (1) compared with different (1) is same (0) becomes lost from view. In accounting, it is fudged by allowing a debit entry (-1) and a credit entry (+1) to cancel down to zero (0). The debit (1) and credit (1) entries on each side of a T-bar account, although each different (1) from (0) are the same (0) as one another.

But the anomaly is the tip of an iceberg. There is a tunnel through which symbols can exchange their meanings: a tunnel between XOR and XNOR. At one end of the tunnel, 0 means same and 1 means different. At the other end of the tunnel, 0 means different and 1 means same. In the tunnel, the meanings of 0 and 1 can invert independently. They become entangled. In this tunnel, it is as if a coin is being flipped from head to tail, but the head is turned into a tail independent of the tail being turned into a head.

This spooky behaviour seems counter-intuitive, but it is actually very easy to understand if the truth tables for XOR and XNOR are placed side by side and one on top of the other, so that the two run continuously in a cycle. The spooky states are (this writer believes!) nothing more than the truth tables for the other four logic gates: AND, OR, NAND and NOR.

Far from being a threat to all of civilized life on earth, the possibility of symbols being able to exchange meanings could be exploited to construct new and better kinds of control system. These systems would control from both sides of a relationship, ensuring that each side is held to the performance of mirror opposite obligations. Letting go of absolute concepts like nothing (zero) and embracing the duality of distinction could turn out to be a great favor that mankind could do for itself.