A sequence of formulas {\displaystyle \phi _{i}(x;a_{i})} is said to be {\displaystyle k}-inconsistent if for every {\displaystyle \{i_{1},\ldots ,i_{k}\}} of size {\displaystyle k}, {\displaystyle \bigwedge _{j=1}^{k}\phi _{i_{j}}(x;a_{i_{j}})} is inconsistent. That is, a sequence of formulas is {\displaystyle k}-inconsistent if any {\displaystyle k} of the formulas in the sequence is jointly inconsistent. For example, 2-inconsistency is equivalent to pairwise inconsistency.

Typically, {\displaystyle k}-inconsistency is only considered when the {\displaystyle \phi _{i}(x;y)} are all the same formula.

This notion is rigged to behave very well with respect to indiscernible sequences. Specifically:

{\displaystyle k}-inconsistency plays a basic role in the definitions of dividing, forking, and their variants (such as thorn-forking). ::: ::: :::