First we define and construct the extensional valuation function , (intuitively, it can be thought of as the generating function for the classical truth table). The valuation function θ will have dimensions (k = 2i rows by i columns), and the value of each binary entry is given by the following formula, as can easily be checked.
This map, i.e. θ can be generalized to m-ary truth values. The explicit formula is given below:
Those explicit solutions to the recursive definitions (truth tables) make computation significantly more efficient. They also turn out to be handy (they're both elegant and compact) when writing algorithms for k combinations on n sets, or k partitions on n sets. They should prove useful to anyone working in writing algorithms for systems based on multiple value assignments.
Proof (sketch) --- a detailed proof is given in a later post.
First we express the truth tables as recurrence relations of congruences modulo the number of truth values in that given logic, then solve them by iteration, i.e. see the pattern. Finally give the solutions an elegant form (ceiling notation) using some useful number theoretic identities. QED