My "specific" perhaps best falls under the heading of "phenomenology" (and "culture" and "politics" as well). The question has been posed: "If we absorb postmodernism, if we recognize the variety and ungroundedness of grounds, but do not want to stop in arbitrariness, relativism, or aporia what comes after postmodernism?" I believe that the beginnings of an "after postmodernism" can clearly be discerned in that philosophical discipline known as phenomenological hermeneutics.
The two most outstanding features of phenomenological hermeneutics from the point of view of the present discussion are (1) that it as "postmodern" as any other form of postmodern thought, but (2) unlike other forms of postmodernism ("poststructuralism," "neopragmatism"), it does not lead into the dead-end of relativism and nihilism (see my submitted paper, "Coping with Nietzsche's Legacy: Rorty, Derrida, Gadamer").
Phenomenological hermeneutics can be viewed as an ongoing attempt to draw out and articulate the radical, postmetaphysical implications of Husserl's phenomenological critique of the Tradition (see Ricoeur, "On Interpretation" in Philosophy in France Today and Madison, The Hermeneutics of Postmodernity). Present-day hermeneutics seeks to rethink many of the traditional notions of philosophy--truth, reason, value, etc.--in strictly phenomenological or experiential terms--in terms, specifically, of communicative praxis (see the work of Calvin Schrag). For this reason, like other forms of postmodernism, it eschews all forms of essentialist and foundationalist thinking, and thus involves, to a significant extent, a work of "deconstruction."
At the same time that hermeneutics embraces the postmodern, deconstructive critique of the Tradition ("metaphysics" [Heidegger], "Platonism" [Rorty]), it also points beyond postmodernism. For what principally serves to differentiate hermeneutical postmodernism from other critiques of modernity is that it does not seek merely to jettison as so much worn-out conceptual baggage the core values of the philosophical tradition and of the Enlightenment; it seeks rather to rearticulate ("reconstruct") these values in such a way as to avoid both metaphysical essentialism and foundationalism and intellectual arbitrariness and cultural relativism (Madison, "Philosophy without Foundations," Reason Papers 16). Hermeneutics seeks to avoid The "philosophers' error" that Nietzsche speaks of:
The philosopher supposes that the value of his philosophy lies in the whole, in the structure; but posterity finds its value in the stone which he used for building, and which is used many more times after that for building--better. Thus it finds the value in the fact that the structure can be destroyed and nevertheless retains value as building material (Genealogy, p. 176).
Can there be values ("truths") that are not merely "local" but genuinely universal? Can there be a philosophical universality that is no longer essentialist, and thus "eurocentric"? Hermeneutics believes that one can get beyond both modernism and postmodernism by using many of the "stones" of a deconstructed modernity to build a genuinely postmetaphysical edifice.
Along with the new rhetoric (Perelman), hermeneutics insists that just because there are no values that are (metaphysically) "groundable," it does not follow that there are not some values that are (philosophically, rationally) justifiable, i.e., universalizable. A deconstructive rejection of metaphysical essentialism, hermeneutics maintains, need not entail cultural "incommensurability." One of the core tasks of phenomenological hermeneutics (building on the work of Merleau-Ponty) is the articulation of a full-fledged theory of communicative rationality in all the domains of social life and endeavor and, along with this, a nondogmatic notion of transcultural universality (Madison, "Hermeneutics, the Lifeworld, and the Universality of Reason," Dialogue and Universalism 7). This theoretical endeavor to articulate a postmetaphysical universalism serves to promote, in the practical sphere, an ethics of mutual recognition and reciprocity and to provide a philosophical justification for the democratic politics of universal human rights.
[After Post-Modernism Conference. Copyright 1997.]