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.

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Rule 2: If a term is distributed in the conclusion, then it must be distributed in a premise.

Fallacy: Illicit major; illicit minor

Examples:
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.

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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.


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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.


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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.


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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.


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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 Md are P
All Sd are M
Some S are P
No Md are Pd
All Md are S
Some S are not Pd
All Pd are M
All Md are S
Some S are P

Hope it helps

Source: http://faculty.bsc.edu/bmyers/Section5.3.htm