10 Apr 2010
30 Sep 2013

Categorical Logic



Categorical Syllogism


Categorical syllogism is an ancient, yet still extremely useful, form of deductive reasoning.  We've already taken a cursory look at an ages-old example, to consider how each of its three statements links to the other two.

All men are mortal.
Socrates is a man.
So, Socrates is mortal.

In this lesson, we'll examine the parts, structure, and function in greater detail.  Again, we'll be breaking the statements into subject and predicate components, but this time we'll carry the analysis a bit further.  The purpose of this exercise is not to lead the reader to believe that Socrates died from boredom, but to acquire a firmer grasp of precisely how logical argumentation works.  With such knowledge, we can then figure out how categorical arguments can (and cannot) be legitimately formulated and interpreted.

 Subject and predicate  Syllogisms can take a variety of forms.  But the classic categorical syllogism is marked by a specific, compact, well-defined form, and therefore serves as a very handy model for the study of reasoning in general.  To begin, we note that the classic form has exactly three statements: two premises—All men are mortal and Socrates is a man—and one conclusion—Socrates is mortal.

 Terms of a syllogism  As we learned before, each statement comprises a subject and a predicate.  The subject and predicate each contain one term of the syllogism, thus each statement poses a relationship between two terms..

  • In the first statement of our example, men is the subject term, and mortal is the predicate term.
  • In the second statement, Socrates is the subject term, and a man is the predicate term.
  • In the third, Socrates is again the subject term, and mortal is the predicate term.

Thus, the classic categorical syllogism contains three terms (in this case man / men, mortal, and Socrates).  Each of the three statements of the syllogism contains two of these three terms.  The conclusion's predicate term (mortal) is called the major term of the syllogism, and the conclusion's subject term (Socrates) is called the minor term of the syllogism.  The remaining term (man / men), which is not found in the conclusion, but (in this case) is the subject term of the first premise and the predicate term of the second premise, is called the middle term of the syllogism.

Working from this, we can further differentiate the syllogism's two premises.  The premise whose predicate term is the same as the syllogism's major term (mortal) is identified as the major premise.  The other premise, then, becomes the minor premise of the the syllogism.

 Sequence of statements  As far as logic goes, the statements of a syllogism can be arranged in any order—
major premise, minor premise, conclusion (
All men are mortal [and] Socrates is a man, [so] Socrates is mortal);
minor premise, major premise, conclusion (
Socrates is a man [and] all men are mortal, [so] Socrates is mortal);
major premise, conclusion, minor premise (
All men are mortal, [and] Socrates is mortal, [because] Socrates is a man);
minor premise, conclusion, major premise (
Socrates is a man, [and] Socrates is mortal, [because] all men are mortal);
conclusion, major premise, minor premise (
Socrates is mortal, [because] all men are mortal [and] Socrates is a man);
conclusion, minor premise, major premise (
Socrates is mortal, [because] Socrates is a man [and] all men are mortal)
—and the sequence makes no functional difference as to the reasoning.  However, to avoid confusion in the study process, it's customary always to place the major premise first, the minor premise second, and the conclusion last.

With the statements arranged in the standard order, it's easy to detect the linkages among them.  And we can further illustrate these linkages by representing the various terms visually.

In the major premise, All men are mortal, all things belonging to the category specified by the middle term men is determined also to be included in the category defined by the major term as [things that are] mortal.  If we represent the category of men as a circle, and the larger category of things that are mortal as a square, we show that the first is contained within the second, thus:

The minor premise, Socrates is a man, signifies that the single thing in the category identified by the minor term Socrates is included within the category of the middle term men.  If we continue to use the circle to represent the category of men, and represent the category of things that are Socrates as a single dot, we can illustrate the relationship described in this premise thus:

We obtain the conclusion, Socrates is mortal, by combining the information provided in the two premises.  When we do this, we can then bypass the middle term, men, which has already served its purpose by providing the logical link between the major and minor terms.  In this way, we're able to conclude that the category of the minor term Socrates is entirely contained within the category of the major term [things that are] mortal.

Using classical categorical syllogism as a well developed example, we're now beginning to see and appreciate how some of the logical relationships and connections we routinely take for granted actually work to hold reasoning solidly together.



In this lesson, we've used the quantifier ALL extensively.  However, we aren't always concerned with every member of a group, or with groups into which every individual fits.  In many cases, we must deal with partial or empty groups, and with excluded individuals or subgroups.  In the next lesson, we'll round out our list of quantifiers with SOME and NO.

Next: Categorical Statements