Since the groundbreaking work of Boole, De Morgan, Pierce, and Frege in the later 19th century, logic has been treated as either the foundation of mathematics — as Russell argued — or as a branch of mathematics, as suggested by contemporary type theory and category theory. This all builds on the “formal” view of logic that has been dominant in the West since the later middle ages.
In fact, the place of formalism in the practice of mathematics is debated by mathematicians. A century after Hilbert and Bourbaki, the complete systematic formalization of mathematics remains an unrealized ideal, although new work in homotopy type theory seems the most promising development yet for this (see New Approaches to Modality).
Plato and Aristotle never thought that reasoning should be “value free”. On the contrary, they treated it as an essential part of ethical life. Aristotle pioneered formal reasoning by composition, but justified the principle of non-contradiction in unmistakably ethical terms. Plato and Aristotle reasoned mainly by examining meanings, whereas in the formal view of logic, all that matters are formal rules for mechanical manipulation of arbitrary symbols. (See also Formal and Informal Language.)
Taking up Kant’s thesis of the primacy of practical (ethical) reason, Hegel took what he called “logic” in a very different direction from that of the modern formalists, focusing like Plato and Aristotle on the development of concrete meanings rather than rules for formal, meaning-agnostic operators.
Within the tradition of modern analytic philosophy, Wilfrid Sellars and Robert Brandom have revived interest in non-formal approaches to logic that are closer to the reasoning we employ in everyday life. Brandom has also written extensively on the ethical content of Hegel’s work and its connections to Hegelian logic. He has always acknowledged that his earlier work on inferential semantics is deeply indebted to Hegel. Brandom’s “inferentialism” puts reason and the interpretation of meaning in relations of reciprocal dependence, in this respect recovering what I think is the perspective of Plato and Aristotle as well as Hegel.
The suggestion here — also supported, I believe, by Harris’ commentary on the Phenomenology — is that “logic” is most fundamentally concerned with what we ought to conclude from what, within the open philosophical perspective of what Hegel somewhat confusingly called “pure negativity”, where our view of the world is “inferential all the way down”. At the level of practical application with real-world meanings that I want to say is most important, logical “laws” are neither tautologies nor some strange kind of abstract facts, but rather a kind of best practices that themselves require interpretation to be applied.