Rule 1: All syllogisms must contain three terms: a major term, a minor term and a transitory (or middle) term.
 
Any argument that uses more than three terms lacks a proper basis for comparing the major and the minor terms. Thus, if there are more than three terms in the argument there can be no single transitory (i.e., middle) term to logically connect the two remaining major and minor terms. To illustrate Rule 1, consider the following two statements:
 
(1)   All men are mortal (major term).

(2)   Socrates (minor term) is a Greek.

  greek guy holding apple


Even though both the major premise and the minor premise are true, there is no way to logically connect these two statements together. The reason for this is because there is no single “middle” term. In this example that “middle term” might be the word “mortal” (used in statement 1, supra) or it could be the word “Greek” (used in statement 2, supra). But, without a MUTUAL connecting term, there is not any way to logically construct a single, unifying syllogism.

One way to fix this problem would be to construct two separate, but related, syllogisms, as in the following illustration:
 
Syllogism 1:

(1)   All men are mortal (major term).

(2)   All Greeks (minor term) are men.

(3)   Therefore, all Greeks are mortal.
 
The conclusion in Syllogism 1, supra, is derived from the unifying middle term, “men,” that is contained in both the major and the minor premise statements.
 


Syllogism 2, infra, starts with the conclusion deduced from Syllogism 1, supra:

(1)   All Greeks are mortal (major term).

(2)   Socrates (the minor term) is a Greek.

(3)   Therefore, Socrates is mortal.
 
The conclusion in Syllogism 2, supra, is derived from a different unifying middle term, “Greek(s),” that is contained in both the major and the minor premise statements in the previous two syllogisms, supra.
 
Depending upon what you were trying to prove (i.e., either that “all Greeks are mortal,” or that “Socrates is mortal”) only one of these arguments would be appropriate, and not the other. Moreover, notice that both Syllogisms are now logically valid because they each do contain three, but only three, terms.