A syllogism is valid when its conclusion follows logically from its premises. A syllogism is true when the premises are consistent with the facts. To be sound, a syllogism must be both valid and true.

Consider the following valid syllogism:

In the valid syllogism above, both the major premise and the minor premise are factual statements. If both these premises are true, then the conclusion must also be true. Because the syllogism is both valid and true, it is also sound.

However, a syllogism can be valid without being true. For example, look at the following syllogism:

As illogical as it may seem, this syllogism is valid: its conclusion follows logically from its premises. The major premise states that recipients of support services—all such recipients—are wealthy. However, this premise is clearly false: some recipients of support services may be wealthy, but more are probably not. For this reason, even though the syllogism is valid, it is not true.

Keep in mind that validity is a test of an argument’s structure, not of its soundness. Even if a syllogism’s major and minor premises are true, its conclusion may not necessarily be valid.

Consider the following examples of invalid syllogisms.

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Syllogism with an Illogical Middle Term

A syllogism with an illogical middle term cannot be valid. The middle term of a syllogism is the term that occurs in both the major and minor premises but not in the conclusion. (It links the major term and the minor term together in the syllogism.) A middle term of a valid syllogism must refer to all members of the designated class or group—for example, all dogs, all people, all men, or all women.

Consider the following invalid syllogism:

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Even though the statements in the major and minor premises are true, the syllogism is not valid. Mammals is the middle term because it appears in both the major and minor premises. However, because the middle term mammal does not refer to all mammals, it cannot logically lead to a valid conclusion.

In the syllogism that follows, the middle term does refer to all members of the designated group, so the syllogism is valid:

Syllogism with a Key Term Whose Meaning Shifts

A syllogism that contains a key term whose meaning shifts cannot be valid. For this reason, the meaning of a key term must remain consistent throughout the syllogism.

Consider the following invalid syllogism:

In the major premise, man refers to mankind—that is, to all human beings. In the minor premise, however, man refers to males. In the following valid syllogism, the key terms remain consistent:

Syllogism with Negative Premise

If either premise in a syllogism is negative, then the conclusion must also be negative.

The following syllogism is not valid:

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Because one of the premises of the syllogism above is negative (“No students are senators”), the only possible valid conclusion must also be negative (“Therefore, no students can vote on legislation”).

If both premises are negative, however, the syllogism cannot have a valid conclusion:

In the syllogism above, both premises are negative. For this reason, the syllogism cannot have a valid conclusion. (How can Jen deserve special help if she is not a disabled student?) To have a valid conclusion, this syllogism must have only one negative premise: