Rules
and Fallacies for Categorical Syllogisms
Hurley,
Section 5.3
Rule 1: The middle term must be distributed at least once.
Fallacy: Undistributed middle
Example:

All sharks are fish All salmon are fish All salmon are sharks 
Justification: The middle term is what connects the major and the
minor term. If the middle term is never distributed, then the major and minor
terms might be related to different parts of the M class, thus giving no common
ground to relate S and P.
Rule 2: If a term is distributed in the conclusion, then it must be
distributed in a premise.
Fallacy: Illicit major; illicit minor
Examples:
And: 
All horses are animals Some dogs are not horses Some dogs are not animals All tigers are mammals All mammals are animals All animals are tigers 
Justification: When a term is distributed in the conclusion, let’s
say that P is distributed, then that term is saying something about every
member of the P class. If that same term is NOT distributed in the major
premise, then the major premise is saying something about only some members of
the P class. Remember that the minor premise says nothing about the P class.
Therefore, the conclusion contains information that is not contained in the
premises, making the argument invalid.
Rule 3: Two negative premises are not allowed.
Fallacy: Exclusive premises
Example:

No fish are mammals Some dogs are not fish Some dogs are not mammals 
Justification: If the premises are both negative, then the
relationship between S and P is denied. The conclusion cannot, therefore, say
anything in a positive fashion. That information goes beyond what is contained
in the premises.
Rule 4: A negative premise requires a negative conclusion, and a negative
conclusion requires a negative premise. (Alternate rendering: Any syllogism
having exactly one negative statement is invalid.)
Fallacy: Drawing an affirmative conclusion
from a negative premise, or drawing a negative conclusion from an affirmative
premise.
Example:

All crows are birds Some wolves are not crows Some wolves are birds 
Justification: Two directions, here. Take a positive conclusion
from one negative premise. The conclusion states that the S class is either
wholly or partially contained in the P class. The only way that this can happen
is if the S class is either partially or fully contained in the M class
(remember, the middle term relates the two) and the M class fully contained in
the P class. Negative statements cannot establish this relationship, so a valid
conclusion cannot follow.
Take a negative conclusion. It asserts that
the S class is separated in whole or in part from the P class. If both premises
are affirmative, no separation can be established, only connections. Thus, a
negative conclusion cannot follow from positive premises.
Note: These first four rules working together
indicate that any syllogism with two particular premises is invalid.
Rule 5: If both premises are universal, the conclusion cannot be particular.
Fallacy: Existential fallacy
Example:

All mammals are animals All tigers are mammals Some tigers are animals 
Justification: On the Boolean model, Universal statements make no
claims about existence while particular ones do. Thus, if the syllogism has
universal premises, they necessarily say nothing about existence. Yet if the
conclusion is particular, then it does say something about existence. In which
case, the conclusion contains more information than the premises do, thereby
making it invalid.
The Aristotelian Standpoint
Any syllogism that violates any of the first
four rules is invalid from either standpoint. If a syllogism, though, violates
only rule 5, it is then valid from the Aristotelian standpoint, provided that
the conditional existence is fulfilled. Thus, in the example above, since
tigers exist, this syllogism is valid from the Aristotelian point of view.
On the other hand, consider this substitution
instance:

All mammals are animals All unicorns are mammals Some unicorns are animals 
Since "unicorns" do not exist, the
condition is not fulfilled, and this syllogism is invalid from either
perspective.
In order to determine the needed condition,
you can simply consult the chart (but not on the exam!). But there are two
other ways. First, as we learned in section 5.2, you can draw a Venn diagram
and find the circle with only one open area. The term that that circle
represents is the required existent thing. Second, you can check the
distributions and, in these cases, there will always be one term that is
superfluously distributed. That is, there will be one term that is distributed
more than is necessary to insure the validity of the syllogism.
Examples:
All M^{d}^{
}are P All S^{d}
are M Some S are P 
All M^{d}
are S Some S are not P^{d} 
All P^{d} are M All M^{d}
are S Some S are P 