Every functionally complete $m$-valued logic has a Post-complete axiomatization.

Abstract

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 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, \Ftp = 1, \F2p = 2 , . . . , \Fmp =m. Consider now the deductive system M having as theses all tautologies, and having among its rules of inference the rule of substitution of wffs for proposition variables and the rule of modus ponens for C in the sense that is derivable from a and Caβ. We shall show that adding any nonβ -tautology γ to M as a thesis makes M Post-inconsistent. Since γ is not an -tautology, there will be an assignment of values to its variables such that |y | e U. Let y be the result of substituting appropriate constant functions Fip for the variables of γ so that y is itself a constant function and |y τ | e U. We shall write t-a' to denote that a is a thesis of M. Since \-γ, we obtain hy f by substitution. Also, since | y r | e ( J , |Afy r |eD, and hence \-Nγ Using the appropriate substitution of \CpCNpq and two applications of the rule of modus ponens we obtain \-q. Hence M plus \-γ is Post-inconsistent, and M i s Post-complete.