My Notes on 7.3 "Rules of Validity for Categorical Syllogisms" by Mark Grannis

Introduction

"The validity of any categorical syllogism depends exclusively on whether the formal structure of the syllogism obeys a very few rules... [it] never depends on whether it happens to come to a true conclusion."

The Syllogism Must Contain Three And Only Three Terms, Each Used Twice

The power of the syllogism comes from relating both subject and predicate to a common middle term. In doing this a revelation between the connection of the subject and predicate is revealed. Therefore, to add in a fourth or more term undermines this process. "There is simply no way any conclusion to the premises that begin argument (9) can relate the major and minor terms by way of the same middle term, because there is no single middle term here." The four term fallacy can also take the shape of using a the same term but in two equivocal ways. Therefore, you end up with four terms again. This is known as the "fallacy of the ambiguous middle." 

The Middle Term Must Be Distributed At Least Once 

The middle term must be fully distributed in either the major or the minor premise. This is because it is the middle term which connects the major and minor terms and therefore if the syllogism is to come to real conclusion then we must know something about the entire distribution of that middle term. Otherwise there could be information about the middle term that is unknown and therefore the major and minor term cannot be related with certainty. "Because if we are using the middle term to relate the major and minor terms to each other, then we need to know that at least one of those comparisons covers the entire extension represented by the middle term." 

Fallacy of the Undistributed Middle - "which is what we get when the middle term of a syllogism is not distributed at least one." Even if the middle term is distributed once we are able to have a valid syllogism because we have a reference for the entire extension of the middle term. 

Any Distributed Term In The Conclusion Must Be Distributed In the Premises

If we come to a conclusion which deals with the entire extension of a concept, then it requires that the same term be distributed in the premises, otherwise we cannot draw a conclusion about the whole of the term. "The gist of the rule is that we can't apply our conclusion to a larger extension than our premises justify." This doesn't mean there has to be distributed terms in the conclusion. 

Fallacy of the Illicit Major - "If the major term is distributed in the conclusion but undistributed in the major premise..." 

Fallacy of the Illicit Minor - "If the minor term is distributed in the conclusion but undistributed in the minor premise..." 

There Must Be An Affirmative Premise 

There must be at least one affirmative premise because if both premises are E propositions then there is no relation between that can be drawn because the middle term isn't relating the major and minor premise together at all. "And note that the problem is not just with the conclusion; there is no conclusion that we could validly draw about how a major and minor term relate to one another if neither of them is ever related affirmatively to the middle term." 

Fallacy of Exclusive (or Negative) Premises - "Nothing validly follows from two negative premises."

The Quality Of The Conclusion Must Follow The Weaker Premise 

"The quality of the conclusion must follow the quality of the weaker premise." In other words, "if either premise is negative, the conclusion must be negative." 

By weaker is meant here a negative statement since an affirmative statement tells more than a negative one. 

Two Corollaries On Particularity 

These are two implied truths that follow from particular propositions. 

1) "No conclusion validly follows form two particular premises." - Every valid syllogism must have one universal premise, you can't have two particular premises. 

2) "A particular premise demands a particular conclusion." - If one premise is particular then the conclusion cannot be universal. 

Validity Matrix 

1) There can be only three terms, each used twice. 

2) The middle term of a categorical syllogism must be distributed at least once. 

3) Any distributed term in the conclusion must be distributed in the premises.

4) Every valid syllogism must have at least one affirmative premise.

5) If either premise is negative, the conclusion must be negative (also if both premises are affirmative, the conclusion must be affirmative). 

6) No conclusion validly follows from two particular premises.

7) A particular premise demands a particular conclusion.