Another form of error more difficult to detect embodies a logical fallacy known as petitio principii, or the circular argument. When a conclusion is circular, usually the methodology has assumed the conclusions implicitly in the premises. In symbolic logic, the form of inference called modus ponens is corrupted in circular reasoning into the tautology:
If p⇒ q and q is true conclude q
The error here is that the truth of q is not entailed by the truth of p. One way of testing for circularity is to see if a methodology generates the same results with random data, as with the real data. This, together with the initial premise that p⇒ q, is equivalent to setting:
p⇒ q and ¬p⇒ q
If the conclusion is still true, i.e. the same results are achieved using random numbers instead of the supposed ‘signal’, then it is highly likely the model entails the result q irrespective of the conditions.