logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

inference Inferences are steps in logical reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinct ...

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 (, ''enthýmēma'') is an argument with a hidden premise. Enthymemes are usually developed from premises that accord with the audience's view of the world and what is taken to be common sense. However, where the general premise of a s ...

, 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". His work has had a profou ...

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 s ...

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 An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...

, then it will not ignite." (''p''∧''r''→¬''q'') # "If ''p'' and ''r'', but the match is in a

Faraday cage A Faraday cage or Faraday shield is an enclosure used to block some electromagnetic fields. A Faraday shield may be formed by a continuous covering of conductive material, or in the case of a Faraday cage, by a mesh of such materials. Faraday cag ...

, then it will ignite." (''p''∧''r''∧''s''→''q'') # "If ''p'' and ''r'' and ''s'', but there is no

oxygen Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...

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 ' (also spelled ') (Classical ) is a Latin phrase, meaning "other things equal"; some other English translations of the phrase are "all other things being equal", "other things held constant", "all else unchanged", and "all else being equal". ...

clause to 1. to make it usable in formally valid inferences: # "If I rub this match along the striking surface, then, ceteris paribus, literally: "''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, mathematician, photographer and reluctant Anglicanism, Anglican deacon. His most notable works are ''Alice ...

's dialogue "''

What the Tortoise Said to Achilles "What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal ''Mind'', is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achill ...

''" demonstrates that the attempt to make every inference fully complete can lead to an infinite regression.

Notes

Citations

{{Reflist

References

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