Valid or Invalid? - Six Rules for the Validity of Syllogisms

Each of the following rules constitutes a necessary condition for the validity of syllogisms. If a syllogism violates one of these rules, then it commits a formal fallacy, and it's not valid.

Rule 1: Exactly three categorical terms

To be valid, a syllogism must have exactly three categorical terms, and their sense mustn't vary over the course of the syllogism. A fallacy of equivocation occurs when a term is used in a different way within the course of an argument. So, for example

All lovers are horny
God is love
Therefore, God is horny

commits the fallacy of equivocation, because the word "love" is being used in different senses in the first two premises (and indeed arguably has no precise meaning at all in the second premise).

Rule 2: A distributed middle term

The middle term of a valid syllogism is distributed in at least one of the premises. The fallacy of the undistributed middle occurs when this doesn't happen. For instance, the middle term (furry animals) in this syllogism

All dogs are furry animals
Some furry animals are cats
Therefore, dogs are cats

isn't distributed, and the argument is clearly fallacious.

Rule 3: If a term is distributed in the conclusion, it must be distributed in the premises

A conclusion that states something about a whole class must be supported by a premise that does the same thing. For example:

All Protestants are Christians
No Catholics are Protestants
Therefore, no Catholics are Christians

doesn't work, because the term "Christians" is distributed in the conclusion, but not in the (major) premise.

The fallacy of illicit major occurs (as above) when the major term is distributed in the conclusion, but not in the (major) premise. The fallacy of illicit minor occurs when the minor term is distributed in the conclusion, but not in the (minor) premise.

Rule 4: A valid syllogism can't have two negative premises

The fallacy of exclusive premises occurs when a syllogism has two premises that are negative. A negative premise is either an "E" statement ("No S are P") or an "O" statement ("Some S are not P"), and if you've got two of them in your premises, your syllogism isn't valid.

Rule 5: The conclusion of a syllogism must be negative, if either premise is negative

The fallacy of drawing an affirmative conclusion from a negative premise occurs if this rule is violated. Similarly, if a conclusion is negative, then one of the premises must be negative (which rule, if broken, constitutes the fallacy of drawing a negative conclusion from an affirmative premise).

Rule 6: No particular conclusion can be drawn from two universal premises

This is arguably the most counterintuitive of the rules for validity. An existential fallacy occurs whenever a particular conclusion appears with two universal premises (for example, All M are P, All S are M, Therefore, some S are P).

It's a fallacy because universal statements do not imply members of a class exist, whereas particular statements do. Arguably, though, categorical syllogisms that are invalid on these grounds can be seen as conditionally valid - that is, their validity is conditional upon the existence of the particular under consideration.