[note 1] In contrast, the inference "Montreal is north of New York, therefore New York is south of Montreal" is materially valid only; its validity relies on the extra-logical relations "is north of" and "is south of" being converse to each other.
[note 2] Classical formal logic considers the above "north/south" inference as an enthymeme, that is, as an incomplete inference; it can be made formally valid by supplementing the tacitly used conversity relationship explicitly: "Montreal is north of New York, and whenever a location x is north of a location y, then y is south of x; therefore New York is south of Montreal".
In contrast, the notion of a material inference has been developed by Wilfrid Sellars[1] in order to emphasize his view that such supplements are not necessary to obtain a correct argument.
While a classical logician could add a ceteris paribus clause to 1. to make it usable in formally valid inferences: However, Brandom doubts that the meaning of such a clause can be made explicit, and prefers to consider it as a hint to non-monotony rather than a miracle drug to establish monotony.
In a similar way, Lewis Carroll's dialogue "What the Tortoise Said to Achilles" demonstrates that the attempt to make every inference fully complete can lead to an infinite regression.