logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

, 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.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''". 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.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.

Material inferences vs. enthymemes

Classical formal logic considers the above "north/south" inference as an

enthymeme An enthymeme ( el, ἐνθύμημα, ''enthýmēma'') is a form of rational appeal, or deductive argument. It is also known as a rhetorical syllogism and is used in oratorical practice. While the syllogism is used in dialectic, or the art of log ...

, 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 Wilfrid Stalker Sellars (May 20, 1912 – July 2, 1989) was an American philosopher and prominent developer of critical realism, who "revolutionized both the content and the method of philosophy in the United States". Life and career His father ...

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 Robert Boyce Brandom (born March 13, 1950) is an American philosopher who teaches at the University of Pittsburgh. He works primarily in philosophy of language, philosophy of mind and philosophical logic, and his academic output manifests both sys ...

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 A match is a tool for starting a fire. Typically, matches are made of small wooden sticks or stiff paper. One end is coated with a material that can be ignited by friction generated by striking the match against a suitable surface. Wooden matc ...

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 Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as ...

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,

literally ''Literally'' is an English adverb. It has been controversially used as an intensifier for figurative statements. History The first known use of the word ''literally'' was in the 15th century, or the 1530s, when it was used in the sense of "in ...

: "''all other things being equal''"; here: "''assuming a typical situation''" 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 Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are '' Alice's Adventures in Wonderland'' (1865) and its sequ ...

's dialogue "'' What the Tortoise Said to Achilles''" demonstrates that the attempt to make every inference fully complete can lead to an infinite regression.

Notes

Citations

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References

Stanford Encyclopedia of Philosophy on Sellars view
Non-classical logic Inference