Warranted and Unwarranted Consensually Achieved Decisions
The consensus in question arises from the community that produces it. The communities that interest Toulmin are those at which he aims his "critique of collective reason" pertaining to "science or law or art" (TCA1, p. 34). In this enterprise, Toulmin seeks to avoid an a priori definition of what constitutes a valid argument or, for that matter, of what constitutes a given field. He apparently does this by accepting the fields’ own concepts of rationality and impartiality: what he wants to understand about the denizens of such intellectual communities is what they are trying to achieve, basing his insight upon their stated goals, their cultivation of their respective enterprises, and their degrees of success (id.).
Habermas contends that this leaves out "the general pragmatic presuppositions and procedures of the cooperative search for truth" (id.). Impartiality, he says, can be explained only by examining the conditions in which validity claims can be discursively redeemed (p. 35) -- which term seems to imply (see TCA1 p. 19) that a field’s validity claims are not necessarily to be taken as objective without further enhancement. In law, for example, he points out that negotiation of compromises does not at all redeem validity claims discursively. From the standpoint of a philosopher, the latter is superior (though judges, attorneys, and many litigants may not think so) by dint of its appeal to "the force of the better argument" (which, in this sense, is apparently construed as excluding pragmatic considerations pertaining to the preferability of negotiated compromises).
While Habermas himself does not appear to develop Burleson’s "warrant" terminology, Habermas’s central distinction thus appears to be that between "forms of argumentation" and "the institutional differentiation of various rational enterprises" -- which is apparently construed as tantamount to the separation of "conventional claims, which are context-dependent, from universal validity claims" (TCA1 p. 36). As I understand this, context (e.g., the courtroom) provides a setting within which a conventional claim might be asserted, consistent with the institutional differentiation of the relevant rational enterprise (e.g., the legal system); while, by contrast, a discourse (of so far unspecified nature) pertaining to universal validity claims would attend inter alia to the forms of argumentation, as Habermas has been doing in these pages.