# Material inference

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
is considered, respectively.

## Examples

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 ^{[1]} in order to emphasize his view that such supplements are not necessary to obtain a correct argument.

## Brandom on material inference

### Non-monotonic inference

Robert Brandom adopted Sellars' view,^{[2]} 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.^{[3]}

## See also

Material inference should not be confused with the following concepts, which refer to *formal*, not **material** validity:

- Material conditional — the logical connective "→" (i.e. "formally implies")
- Material implication (rule of inference) — a rule for formally replacing "→" by "¬" (negation) and "∨" (disjunction)

## Notes

- ↑ 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)
**in**valid 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*"

## References

Stanford Encyclopedia of Philosophy on Sellars view

- ↑ 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.