CATEGORICAL SYLLOGISMS

Categorical syllogisms are a special type of argument which has been studied for more than two thousand years.   now since the time of Aristotle. It is the central piece of Aristotelian logic, and is still the most visible type of argument in logic courses and textbooks today. I

Categorical syllogisms, no matter what they are about, have a rigorous structure:

  1. There are exactly three categorical propositions.
  2. Two of those propositions are premises; the other is the conclusion.
  3. There are exactly three terms, each appearing only twice.

Given these structural requirements. categorical syllogisms are rather cumbersome and unnatural. The structure, however, is  transparent, and the structural properties of the syllogism (the relationships asserted between the three terms) determine whether an argument is valid or not.

Consider the following argument:

     (1) All men are mortal.  (All P are M)
     (2) Socrates is a man.  (Some S are M)
     (3) Therefore, Socrates is mortal.  (Some S are P)
Each of the three terms in a categorical syllogism occurs in exactly two of the propositions in the argument.  For ease of identification and reference, these terms are called the major, minor and middle terms of the argument.  The term which occurs in both premises is called the middle term and is usually represented by the letter M. The term which occurs as the predicate term in the conclusion is called the major term and it is usually represented by the letter P. The premise which has the major term is called the major premise. The subject of the conclusion is called the minor term, represented by the letter S, and the premise with the minor term is called the minor premise.  So, in the example above, 'Socrates' is the minor term, 'mortals' is the major term, and 'men' is the middle term.

In the example above, it appears that the conclusion follows from the premises.  But how can we be sure?  Fortunately, there are two distinct methods available to us for testing the validity of a categorical syllogism.  The first of the methods relies on an understanding of properties of categorical propositions :  quantity, quality, and the distribution of terms.  

Four rules apply to all valid categorical syllogisms:

Rule 1:  In a valid categorical syllogism, the middle term must be distributed in at least one premise.

Rule 2:  In a valid categorical syllogism, any term that is distributed in the conclusion must be distributed in the premises.

Rule 3:  In a valid categorical syllogism, the number of negative premises must be equal to the number of negative conclusions.

Rule 4:  In a valid categorical syllogism a particular conclusion cannot be drawn from exclusively universal premises unless one assumes existential import.   We do not assume existential import, and we will refer to arguments that would be valid if we did as traditionally valid.

All and only those arguments that pass each of these tests are valid.  Failure to satisfy one or more of the rules renders the argument non-valid.  Applying these rules to our argument, we see that the middle term, 'men', is distributed in the first premise (the subject of an A proposition is distributed), so the argument passes the first test.  Neither of the terms in the conclusion is distributed (both terms in an I proposition are undistributed), so the argument passes the second test.  There are no negative premises and no negative conclusions, and 0 = 0, so the argument passes the third test.  Finally, the second premise is particular, so the argument passes the fourth test even though the conclusion is particular.

 Consider another example:

     (1) Some logicians wear earrings.
     (2) Some persons who wear earrings are rational.
     (3) Therefore, some logicians are not rational.

This, too, is a categorical syllogism. However, this argument is not valid. The reason is that even though some logicians wear earrings and some persons who wear earrings are not rational, it does not necessarily follow that some logicians are not rational. In fact the premises could be all true while the conclusion is false. In terms of the four rules, this argument violates Rule 1, the middle term, 'those who wear earrings', is not distributed in either of the premises.

Fallacies and Rule Violations

Categorical syllogisms that violate one or more of the rules commit a fallacy in reasoning.  Different violations are given specific names.  An argument that violates rule 1 commits the fallacy of the undistributed middle.  If the minor term is distributed in the conclusion but not in the minor premise, the argument commits the fallacy of an illicit minor.  If the major term is distributed in the conclusion but not in the major premise, the argument commits the fallacy of an illicit major.  An argument with 2 negative premises commits the fallacy of 2 negatives,  any other violation of rule 3 is called the fallacy of negative terms.  Finally, an argument that violates rule 4 commits the existential fallacy.

 

Quiz yourself of applying the rules to arguments to test for validity.

 

Many people find the rule tests for validity unnatural and cumbersome.  Fortunately, there is another method for testing categorical syllogisms for validity that involves Venn diagrams.

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