I read another chapter, on cockfighting in Bali, and I begin to understand him better.

When he dismisses cognitivism, he's probably thinking of Levi-Strauss. The introductory chapter which seemed illogical to me was probably not intended to be a logical argument of any kind, but a summary of his views. Geertz seems to have the common non-hard-science mystical view of human thought, as something not amenable to logical analysis, and a view of logic as reductionist. He seems, like I suspect Wittgenstein does, to think that a logical analysis of language means analyzing individual words, and that the information provided by context and by the ways the words are put together is somehow beyond the grasp of logic, and that language and human activity just can't be explained that way.