Every functionally complete $m$-valued logic has a Post-complete axiomatization.
Nuel Belnap, Storrs McCall · Notre Dame Journal of Formal Logic · 1970
Let ^fl be an m-valued functionally complete matrix with truth-values 1, 2, ... ra, and let D be the class of designated values and U the class of undesignated values of β.We shall assume that β is Post-consistent, i.e. that U is not empty.Then the truth-functions definable using ϋ will include the functions Cpq, Np, and Fip, 1 < i < m, with the following properties, where '|α|' denotes the truth-value of a:(1) If, for all assignments of values to the variables of a and β, \a\ e D and I Caβ\ e D, then \ β\ e D for all similar assignments.(2) For all values of p and q, \ CpCNpq] e D.(3) For all values of the variables in a, if \a\ e U then \Na\ e D.(4) The Fip are constant functions such that, for all values of p, \F t p\ = 1, \F 2 p\ = 2,..., \F m p\ =m.