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{{clarify|reason=The distinction between logical and extra-logical vocabulary should be explained. In a first approach, a notion is called 'logical' if it applies to sentences only. E.g. the connective 'x and y' joins two sentences x, y and hence is logical vocabulary, while 'x is human' applies to a real-world object x and hence is extra-logical. However, a quantifier like 'each x satisfies y' is usually considered as logical vocabulary although it applies to a real-world object x and a sentence y.|date=August 2013}} 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.<ref group="note">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''".</ref>

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.<ref group="note">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.</ref>

==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<ref>{{cite book| author=Wilfrid Sellars| author-link=Wilfrid Sellars| title=Inference and Meaning| year=1980| pages=261f| editor=J. Sicha}}</ref> in order to emphasize his view that such supplements are not necessary to obtain a correct argument.

==Brandom on material inference== {{See also|Counterfactual conditional#Classic puzzles|label 1 = Sobel sequences}} ===Non-monotonic inference=== Robert Brandom adopted Sellars' view,<ref>{{cite book| author=Robert Brandom| author-link=Robert Brandom| title=Articulating Reasons: An Introduction to Inferentialism| year=2000| publisher=Harvard University Press| isbn=0-674-00158-3}}; Sect. 2.III-IV</ref> 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,<ref group="note">literally: "''all other things being equal''"; here: "''assuming a typical situation''"</ref> 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.<ref>{{cite journal |author=Carroll, Lewis |title=What the Tortoise Said to Achilles |journal=Mind |series=New Series |volume=4 |issue=14 |date=Apr 1895 |pages=278–280 |url=http://courseweb.stthomas.edu/kwkemp/logic/R/Tortoise.pdf}}</ref>

== 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== {{Reflist|group="note"}}

==Citations== {{Reflist}}

==References== * [http://plato.stanford.edu/entries/sellars#3.1 Stanford Encyclopedia of Philosophy on Sellars view]

Category:Non-classical logic Category:Inference