Chiralkine is an apparatus for counting distinct objects which treats the relationships distinguishing the objects symmetrically.

It treats counting as a process that records the clearing of perceived distinctions drawn between what is and what is not a distinct object using both XOR and its mirror opposite XNOR.

It records the clearing of the perceived distinctions between what is and what is not a distinct object using interpenetrating truth tables for both XOR and its mirror opposite XNOR. As it records the clearing of the perceived distinction, it draws new distinctions.

It comprises a pair of mirror opposite rings each consisting of six rows of two 1s and two 1s, the arrangement being such that any adjacent four rows contain the truth tables for both XOR and XNOR.

It records the clearing of a distinction by rotating each of the rings by one row. This exchanges one 1 for one 1 in each of the mirror opposite rings. Arithmetic and accounting operations, in which distinct objects are counted from one category across to another, are thus treated as paired rotations.

Conventional counting of distinct objects does not treats the relationships distinguishing the objects symmetrically. It conflates that which is not a distinct object with a cleared distinction. Thus if 1 represents that which is a distinct object and 0 represents that which is not a distinct object, then placing a 1 over the 0 produces 1 and 1, which is a cleared distinction. Additionally placing a 0 over the 1 during clearing is ignored, because the 1 is privileged over the 0. When a chiralkine apparatus is used in counting, the clearing of a distinction treats that which is and that which is not a distinct object symmetrically, which means that placing a 1 over a 0 produces a 1 and placing a 0 over a 1 produces a 0. Neither side of the relationship defining a distinction is privileged. The 0 which is an integral component of the relationship defining a distinction is not conflated with a cleared distinction. This results from using both XOR and XNOR to draw and clear distinctions, instead of just one of these as in conventional counting.

A pair of rings can be constructed by recursive distinctioning, starting with any two inputs for the truth tables for XOR and XNOR and applying XOR to them in one order and XNOR in the other to produce four rows consisting of the two inputs and two outputs, then repeating the process treating each output and its adjacent input as inputs.

The rows can be represented symbolically with letters arranged in a ring, for example i, j , k, i, j, k. 

The relationships of the letters in the ring are non-commutative, since for example i is turned by j into k while j  is turned by i into k. This reflects the mirror opposite symmetry of the truth tables for XOR and XNOR from which the rings are constructed.

In a chiralkine system, there is no absolute zero or nothingness. Every value is defined relative to some other value. Every value and operation has a mirror opposite dual.

The apparatus has many potential applications, including:

a solution to the double co-incidence of wants problem inherent in barter that treats the two sides of a relationship symmetrically. It could be implemented on a large scale using blockchain technology and smart contracts. It could provide a secure, efficient and transparent way for individuals to exchange goods and services without the need for money or traditional financial institutions. The use of smart contracts would ensure that both parties fulfil their obligations in the exchange and prevent any cheating and fraud;

a voting system that allows voters to look at a list of options from the viewpoint of which they like the best and which they dislike the most, including the option of supporting or rejecting the vote itself. For example they could vote to reject a referendum that offers yes and no as the only candidates, or vote to block the election of the most popular amongst many candidates;

a system for resolving disputes and conflicts;

a system for conducting consumer surveys, which separately identifies and evaluates the features that a consumer likes and dislikes about a product.

a system for coding the three primary additive and subtractive colours, black and white;

a system for coordinating opposing forces, for example in the co-ordination of movement effected by muscle antagonist pairs or their robotic equivalents;

a model for bistable perception in a visual system, as experienced for example in the Necker cube effect;

cryptography; and

a logical basis for the construction of artificial intelligence (AI) which is safer than that currently used, because it clears and draws distinctions symmetrically in both XOR and XNOR. It looks at relationships from both sides of the relationship.