Now, it seems that we were not explicit enough in pointing out that we were only entertaining weak cognitivism for the sake of argument. We really do stick by strong cognitivism. That's why Adams & Aizawa, 2010, defend it over the course of several pages, namely, pp. 582-589.Weak cognitivism: cognition involves the processing of representations.... Adams and Aizawa are inconsistent in the strength of their claims as they also assert that cognitive content must be underived (in this world at least). If they hold to that stipulation—that cognitive content must be underived—then they cannot, as they do in this issue, allow the possibility of weak cognitivism (in this world).
Strong cognitivism: all cognition involves the processing of representations with underived content.
I am not sure why Menary thinks that if we hold what he calls "strong cognitivism" that we cannot allow the possibility of weak cognitivism. As I understand the positions, the strong position entails the weak position, so if we hold the strong position (which we do) then we had better maintain the weak position, hence allow the possibility of the weak position.
Adams, F., & Aizawa, K. (2010). The value of cognitivism in thinking about extended cognition, Phenomenology and the Cognitive Sciences. 9:579–603