Logic Gates

Counting in mathematics is constructed from two mutually exclusive symbols each tethered to one of two mutually exclusive meanings: distinct (countable), not same and not distinct (not countable), same. Each symbol is tethered to one absolute meaning: the symbols and meanings do not get exchanged.

The meanings of the symbols are constructed from the outputs of their interactions in accordance with one set of rules. Like symbols (black and black or white and white) interact to output a white symbol. Distinct symbols (black and white or white and black) interact to output a black symbol.

Chiralkine counting is constructed from two mutually exclusive symbols each tethered to one of two mutually exclusive meanings: distinct, not same (countable) and not distinct, same (not countable). Each symbol is tethered to each meaning: the symbols and meanings get exchanged.

Logic is constructed from two mutually exclusive symbols each tethered to one of two mutually exclusive meanings: true, not false and not true, false. Each symbol is tethered to one absolute meaning: the symbols and meanings do not get exchanged.

The meanings of the symbols are constructed from the outputs of their interactions in accordance with sets of rules set out in what are known as truth tables for logic gates. These rules are embodied in different orderings of the symbols. There are six logic gates: OR, AND, XOR (exclusive OR), XNOR (not exclusive OR), NAND and NOR.

In the truth table for XOR, the symbol meaning true, not false also means distinct, not same (countable) and the symbol meaning not true, false also means not distinct, same (not countable).

The six truth tables form into three mirror opposite pairs. Hence in the truth table for XNOR, the symbol meaning true, not false also means not distinct, same (not countable) and the symbol meaning true, not false also means distinct, not same (countable). Switching between XOR and XNOR exchanges the meanings of the symbols. Chiralkine counting follows the rules of both XOR and XNOR and so relies on this switching to ensure that the two sides of a relationship defining a distinct object are treated symmetrically.

In the orderings of the symbols in two of the truth tables (XOR and XNOR), the symbols mean same, not different and not same, different. In the other four truth tables, these meanings are mixed up. The symbols in the six truth tables are ordered such that in the outputs, black always means true, not false and white always means not true, false.

Because the truth tables for the six logic gates form into mirror opposite pairs, exchanging the meanings of black and white as between pair members having mirror opposite truth tables would have the effect of exchanging their meanings: OR would become AND as AND would become OR; XOR would become XNOR as XNOR would become XOR and NAND would become NOR as NOR would become NAND. Note however that it is NOR, not AND that has the mirror opposite meaning of OR, and it is NAND, not OR that has the mirror opposite meaning of AND. The truth tables for AND and NAND and the truth tables for OR and NOR have mirror opposite meanings, but are not mirror opposites.

This can be visualised in colour, where the overlap of the meanings true and false with same and different in XOR and XNOR is shown in green and the separation of the meanings true and false from same and different in AND, OR, NOR and AND is shown in blue and yellow.

When the symbols in the truth tables are ordered in mirror pairs as shown above, the truth tables for OR, XOR and NAND differ only in the outputs in their second and third columns. In three of the four columns, black means distinct, not same and white means same, not distinct, but in the remaining column of OR and NAND they have the mirror opposite meanings. Hence the distinct and same meanings of black and white are mixed up in OR and NAND. The truth tables for AND, XNOR and NOR also differ only in their second and third columns. In three of the four columns, white means distinct, not same and black means same, not distinct, but in the remaining column of AND and NOR they have the mirror opposite meanings. Hence the distinct and same meanings of white and black are mixed up in AND and NOR.

The truth tables for the six logic gates can behave as a counting ring like the six chiralkine numbers. There are two (XOR and XNOR) in which the meanings of distinct, countable and same, not countable for the ordered component symbols are cleanly separated, and four in which they are mixed up (coupled). The four in which they are mixed up appear as different forms of nothing (like passive and active abstention in voting, looking from the separate perspectives of like and dislike). Over a complete counting cycle, the meanings of black and white as distinct, countable, and same, not countable are treated symmetrically.