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\end{array}</math>Completeness requires any generated hypothesis ''<span class="texhtml mvar" style="font-style:italic;">h</span>'' to explain all positive examples <math display="inline">E^+</math>, and consistency forbids generation of any hypothesis ''<span class="texhtml mvar" style="font-style:italic;">h</span>'' that is inconsistent with the negative examples <math display="inline">E^{-}</math>, both given the background knowledge ''<span class="texhtml mvar" style="font-style:italic;">B</span>''.
In Muggleton's setting of concept learning,<ref name="setting2">{{cite journal |last1=Muggleton |first1=Stephen |year=1999 |title=Inductive Logic Programming: Issues, Results and the Challenge of Learning Language in Logic |journal=Artificial Intelligence |volume=114 |issue=1–2 |pages=283–296 |doi=10.1016/s0004-3702(99)00067-3 |doi-access=}}; here: Sect.2.1</ref> "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "''Necessity''", which postulates that ''{{mvar|B}}'' does not entail <math display="inline">E^+</math>, does not impose a restriction on ''{{mvar|h}}'', but forbids any generation of a hypothesis as long as the positive facts are explainable without it. . "Weak consistency", which states that no contradiction can be derived from <math display="inline">B\land H</math>, forbids generation of any hypothesis ''{{mvar|h}}'' that contradicts the background knowledge ''{{mvar|B}}''. Weak consistency is implied by strong consistency; if no negative examples are given, both requirements coincide. Weak consistency is particularly important in the case of noisy data, where completeness and strong consistency cannot be guaranteed.<ref name="setting2" />
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