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Since the validity of a categorical syllogism depends solely upon its logical form, it is relatively simple to state the conditions under which the premises of syllogisms succeed in guaranteeing the truth of their conclusions.
Relying heavily upon the medieval tradition, Copi & Cohen provide a list of six rules, each of which states a necessary condition for the validity of any categorical syllogism.
Violating any of these rules involves committing one of the formal fallacies, errors in reasoning that result from reliance on an invalid logical form.
In every valid standard-form categorical syllogism . . .
A careful application of these rules to the 256 possible forms of categorical syllogism (assuming the denial of
existential import) leaves only 15 that are valid.
Medieval students of logic, relying on syllogistic reasoning in their public disputations, found it convenient to assign a unique name to each valid syllogism.
These names are full of clever reminders of the appropriate standard form: their initial letters divide the valid cases into four major groups, the vowels in order state the mood of the syllogism, and its figure is indicated by (complicated) use of m, r, and s.
Although the modern interpretation of categorical logic provides an easier method for determining the validity of categorical syllogisms, it may be worthwhile to note the fifteen valid cases by name:
The most common and useful syllogistic form is "Barbara", whose mood and figure is AAA-1:
All M are P. All S are M. Therefore, All S are P.Instances of this form are especially powerful, since they are the only valid syllogisms whose conclusions are universal affirmative propositions.
All P are M. Some S are not M. Therefore, Some S are not P.The valid form OAO-3 ("Bocardo") is:
Some M are not P. All M are S. Therefore, Some S are not P.
Four of the fifteen valid argument forms use universal premises (only one of which is affirmative) to derive a universal negative conclusion:
One of them is "Camenes" (AEE-4):
All P are M. No M are S. Therefore, No S are P.Converting its minor premise leads to "Camestres" (AEE-2):
All P are M. No S are M. Therefore, No S are P.Another pair begins with "Celarent" (EAE-1):
No M are P. All S are M. Therefore, No S are P.Converting the major premise in this case yields "Cesare" (EAE-2):
No P are M. All S are M. Therefore, No S are P.
Syllogisms of another important set of forms use affirmative premises (only one of which is universal) to derive a particular affirmative conclusion:
The first in this group is AII-1 ("Darii"):
All M are P. Some S are M. Therefore, Some S are P.Converting the minor premise produces another valid form, AII-3 ("Datisi"):
All M are P. Some M are S. Therefore, Some S are P.The second pair begins with "Disamis" (IAI-3):
Some M are P. All M are S. Therefore, Some S are P.Converting the major premise in this case yields "Dimaris" (IAI-4):
Some P are M. All M are S. Therefore, Some S are P.
Only one of the 64 distinct moods for syllogistic form is valid in all four figures, since both of its premises permit legitimate conversions:
Begin with EIO-1 ("Ferio"):
No M are P. Some S are M. Therefore, Some S are not P.Converting the major premise produces EIO-2 ("Festino"):
No P are M. Some S are M. Therefore, Some S are not P.Next, converting the minor premise of this result yields EIO-4 ("Fresison"):
No P are M. Some M are S. Therefore, Some S are not P.Finally, converting the major again leads to EIO-3 ("Ferison"):
No M are P. Some M are S. Therefore, Some S are not P.Notice that converting the minor of this syllogistic form will return us back to "Ferio."