In logic, inference is the process of deriving logical conclusions from premises known or assumed to be true. In checking a logical inference for formal and material validity, the meaning of only its logical vocabulary and of both its logical and extra-logical vocabulary [clarification needed] is considered, respectively.
For example, the inference "Socrates is a human, and each human must eventually die, therefore Socrates must eventually die" is a formally valid inference; it remains valid if the nonlogical vocabulary "Socrates", "is human", and "must eventually die" is arbitrarily, but consistently replaced. [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]
Material inferences vs. enthymemes
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  in order to emphasize his view that such supplements are not necessary to obtain a correct argument.
Robert Brandom adopted Sellars' view, arguing that everyday (practical) reasoning is usually non-monotonic, i.e. additional premises can turn a practically valid inference into an invalid one, e.g.
- "If I rub this match along the striking surface, then it will ignite." (p→q)
- "If p, but the match is inside a strong electromagnetic field, then it will not ignite." (p∧r→¬q)
- "If p and r, but the match is in a Faraday cage, then it will ignite." (p∧r∧s→q)
- "If p and r and s, but there is no oxygen in the room, then the match will not ignite." (p∧r∧s∧t→¬q)
Therefore, practically valid inference is different from formally valid inference (which is monotonic - the above argument that Socrates must eventually die cannot be challenged by whatever additional information), and should better be modelled by materially valid inference. While a classical logician could add a ceteris paribus clause to 1. to make it usable in formally valid inferences:
- "If I rub this match along the striking surface, then, ceteris paribus,[note 3] it will inflame."
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.
Moreover, the "match" example shows that a typical everyday inference can hardly be ever made formally complete. 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.
- A completely fictitious, but formally valid inference obtained by consistent replacement is e.g. "Buckbeak is a unicorn, and each unicorn has gills, therefore Buckbeak has gills".
- A completely fictitious, but materially (and formally) invalid inference obtained by consistent replacement is e.g. "Hagrid is younger than Albus, therefore Albus is larger than Hagrid". Consistent replacement doesn't respect conversity.
- literally: "all other things being equal"; here: "assuming a typical situation"
- Wilfrid Sellars (1980). J. Sicha, ed. Inference and Meaning. pp. 261f.
- Robert Brandom (2000). Articulating Reasons: An Introduction to Inferentialism. Harvard University Press. ISBN 0-674-00158-3.; Sect. 2.III-IV
- Carroll, Lewis (Apr 1895). "What the Tortoise Said to Achilles" (PDF). Mind, n.s. 4 (14): 278–280.