I argue that examining the explanatory power of the hypothesis of extended cognition (HEC) offers a fruitful approach to the problem of cognitive system demarcation. Although in the discussions on HEC it has become common to refer to considerations of explanatory power as a means for assessing the plausibility of the extended cognition approach, to date no satisfying account of explanatory power has been presented in the literature. I suggest that the currently most prominent theory of explanation in the special (...) sciences, James Woodward’s contrastive-counterfactual theory, and an account of explanatory virtues building on that theory can be used to develop a systematic picture of cognitive system demarcation in the psychological sciences. A major difference between my differential influence (DI) account and most other theories of cognitive extension is the cognitive systems pluralism implied by my approach. By examining the explanatory power of competing traditions in psychological memory research, I conclude that internalist and externalist classificatory strategies are characterized by different profiles of explanatory virtues and should often be considered as complementary rather than competing approaches. This suggests a deflationary interpretation of HEC. (shrink)