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 . . .
- . . . there must be exactly three unambiguous categorical terms. The use of exactly three categorical terms is part of the definition of a categorical syllogism, and we saw earlier that the use of an ambiguous term in more than one of its senses amounts to the use of two distinct terms. In categorical syllogisms, using more than three terms commits the fallacy of four terms (quaternio terminorum).
- . . . the middle term must be distributed in at least one premise. In order to effectively establish the presence of a genuine connection between the major and minor terms, the premises of a syllogism must provide some information about the entire class designated by the middle term. If the middle term were undistributed in both premises, then the two portions of the designated class of which they speak might be completely unrelated to each other. Syllogisms that violate this rule are said to commit the fallacy of the undistributed middle.
- . . . any term distributed in the conclusion must also be distributed in its premise. A premise that refers only to some members of the class designated by the major or minor term of a syllogism cannot be used to support a conclusion that claims to tell us about every menber of that class. Depending which of the terms is misused in this way, syllogisms in violation commit either the fallacy of the illicit major or the fallacy of the illicit minor.
- . . . at least one premise must be affirmative. Since the exclusion of the class designated by the middle term from each of the classes designated by the major and minor terms entails nothing about the relationship between those two classes, nothing follows from two negative premises. The fallacy of exclusive premises violates this rule.
- . . . if either premise is negative, the conclusion must also be negative. For similar reasons, no affirmative conclusion about class inclusion can follow if either premise is a negative proposition about class exclusion. A violation results in the fallacy of drawing an affirmative conclusion from negative premises.
- . . . if both premises are universal, then the conclusion must also be universal. Because we do not assume the existential import of universal propositions, they cannot be used as premises to establish the existential import that is part of any particular proposition. The existential fallacyviolates this rule.
Although it is possible to identify additional features shared by all valid categorical syllogisms (none of them, for example, have two particular premises), these six rules are jointly sufficient to distinguish between valid and invalid syllogisms.
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.
A syllogism of the form AOO-2 was called “Baroco”:
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.”