I am looking for a formal way to represent an n-ary relation as a combination of binary relations and logical connectives.

Suppose we have a set [tex]A[/tex], a set [tex]B = \{b: b\subseteq A^2\}[/tex] of binary relations over [tex]A[/tex], and a set of logical connectives [tex]C = \{\neg, \wedge, \vee\}[/tex].

We define a set of propositional variables [tex]V=\{b(a_i, a_j): b \in B, a_i, a_j \in A\}[/tex]. We denote the set of all well-formed formulas over [tex]V \cup C[/tex] as [tex]F[/tex].

Given a propositional function [tex]f \in F[/tex] and using it as an indicator function, we can define an n-ary relation [tex]R=\{(a_0, a_1, ... , a_n) \in A^n | I(f(a_0, a_1, ... , a_n))=1: f \in F\}[/tex].

Does it make any sense?