(As this is a critique, I will be trying to illustrate the limitations of logic positively. A critique is not a rejection, but an attempt to understand something better.)
All λόγος(logos, discourse) is a kind of gathering.
For many, logic is the standard that determines the success(or failure) of discourse. It is entirely suited to logic to remain with this determination. However, success and failure are not the only way of evaluating discourse; we can also consider discourses in terms of the nature of what is gathered. Logic as a standard of gathering that succeeds is only possible where gathering has not only already occurred in some way, but also has gathered people in agreement with what is gathered. Gathering itself does not depend upon logic, but first makes a standard such as logic possible through the gathering-in-agreement of individuals through a discourse that gathers.
Logic, which is founded on the agreement of what is gathered in discourse, is the establishing of a standard discourse agreed upon. The effects of logic being established can be seen in any human co-operation.
Humans did not need to wait for Aristotle to set down his categories or analytics. Aristotle's texts on 'logic' - called the Organon (literally, tool)by his followers - was a discourse that formulated logic concerning assertions in a particularly general way. Assertions describe only one way of gathering through discourse.
We may organize our discourse around logic, and the standard for agreement it dictates. In this way we can begin to overlook the origin of the agreement in the original gatheringthat set the standard of logic. I will repeat this with greater precision: we may gather through our discourse in a way that restricts itself to what was gathered in an original discourse that gathered people in agreement and was established as logic. Such a restriction of our own gathering in discourse to what has been gathered by the discourse that established logic can begin to obscure the original gathering, since the nature of what was gathered cannot enter into the original gathering as a part.
A total restriction of discourse to logic does not keep us 'rational', but obscures the original gathering and cuts discourse off from considering radically, that is considering its roots, even though these roots are still assumed.