Chomsky’s principle of epistemological tolerance says that in theoretical linguistics contradictions between the data and the hypotheses may be temporarily tolerated in order to protect the explanatory power of the theory. The paper raises the following problem: What kinds of contradictions may be tolerated between the data and the hypotheses in theoretical linguistics? First a model of paraconsistent logic is introduced which differentiates between week and strong contradiction. As a second step, a case study is carried out which exemplifies that the principle of epistemological tolerance may be interpreted as the tolerance of week contradiction. The third step of the argumentation focuses on another case study which exemplifies that the principle of epistemological tolerance must not be interpreted as the tolerance of strong contradiction. The reason for the latter insight is the unreliability and the uncertainty of introspective data. From this finding the author draws the conclusion that it is the integration of different data types that may lead to the improvement of current theoretical linguistics and that the integration of different data types requires a novel methodology which, for the time being, is not available.