  Justification terms are built from justification variables  \(x , y , z\), … and justification  constants \(a , b , c\), … (with indices  \(i\) = 1, 2, 3, … which are omitted whenever it is safe) by means of the operations ‘\(\cdot\) ’ and ‘+’.  More elaborate logics considered below also allow additional  operations on justifications. Constants denote atomic justifications  which the system does not analyze; variables denote unspecified  justifications. The Basic Logic of Justifications,  \(\mathsf{J}_{0}\) is axiomatized by the  following.