Difference is not a univocal concept. X and Y may be orthogonally different like “day” and “raining”, or they may be relationally different like “black” and “white”. Things of whatever sort that are relationally different from each other are materially incompatible; things that are orthogonally different from each other are not materially incompatible.

Aristotle and Hegel both emphasize the importance of what I just called relational difference as the principal source of meaning and intelligibilty. Information theory, arithmetical subtraction, and the Euclidean logos or ratio between two magnitudes are all purely concerned with relational as opposed to orthogonal difference.

I’d like to point out that Saussurean phonological difference — say, the distinction between a “b” sound and a “p” sound — is also a relational difference, not an orthogonal one. Interpreting the sound as “b” is materially incompatible with interpreting the sound as “p”. (Brandom’s reference to Saussure as pre-Kantian and pre-Fregean on the ground that the latter worked with subsentential units of analysis in what was actually phonology is an unfortunate mistake.)

The famous 20th century “structuralism”, for which Saussurean difference was widely considered to have been a launching point, did not seem to be explicitly much concerned with inference, but it was very much concerned with the relational kind of difference, and in this way should be considered a potential ally of inferentialism rather than an opponent. Popular accounts do not much mention the role of 20th century French epistemological rationalism in the structuralist ferment, but I think it was significant, and that this could support additional connections to the inferentialist project. Synchronic structure is an expressive metaconcept, in no way inherently conflicting with a simultaneous recognition of the importance of diachronic process.

Writers like Deleuze and Badiou, on the other hand, and perhaps even someone like Rorty, while making valid points against our culture’s obsession with identity, have unfortunately chosen to valorize nonexclusive difference. This is not the answer. Ironically, an exclusive focus on nonexclusive, orthogonal difference leads back to undifferentiated sameness, via incommensurability. Deleuze and Badiou actually celebrate this, with slogans like “pluralism = monism” or “generic multiplicities”. This is precisely the night in which all cows are black. Even Kant’s point about the infinity of each person tends in this direction.

As Hegel saw clearly and pointed out in the Encyclopedia Logic, the polemic of Reason against Understanding should not lead us to try to throw out determinateness. Understanding wants to lock everything down under Identity, which is ultimately disastrous. The indiscriminate valorization of orthogonal Difference, on the other hand, ultimately destroys meaning and intelligibility. We should be looking for an Aristotelian mean (outside of, rather than between) these one-sided, shallow, and unattractive extremes.

I want to say that difference, when unbounded, ceases to be what I wanted to mean by difference. A thoughtful dwelling on relational difference, with due attention to real-world contingency and ambiguity, would be my candidate for the mean. (See also Determinate Negation; Conceptual, Representational.)