The Wikipedia entry on ternary numeral systems notes:
A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers (as alternative for the Misbaha). The mnemonic benefit is that counting within this system then reduces distraction since the counter needs only to divide Tasbihs into groups of three.
use of ternary numbers conveniently to convey self-similar structures like a Sierpinski Triangle or a Cantor set. The ternary representation is useful for defining the Cantor Set and related point sets, because of the way the Cantor set is constructed.
ternary as being the integer base with the highest radix economy, followed closely by binary and quaternary. It has been used for some computing systems because of this efficiency. Rarely mentioned is the existence of ternary computers (notably defining a tryte to be 6 trits, analogous to the binary byte).
use in the representation of 3 option trees, such as phone menu systems, which allow a simple path to any branch.
Of further relevance to the pattern of argument here is the role of ternary valued logic. Such a three-valued or trivalent logic is one in which there are three truth values indicating true, false and some third value. This is contrasted with the more common bivalent logics (mentioned above) which provide only for true and false. or guilty and not-guilty. An exception occurs in the Scottish legal system providing additionally for not-proven (a distinction which would seem to be of considerable current significance with respect to many detained in Guantanamo Bay).
Conceptual form and basic ideas were initially created by Jan Lukasiewicz, C. I. Lewis and Sulski. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics. In the argument here, the question is whether the pattern in the diagram above holds a meaningful relationship with a range of multi-valued logic systems.