1. Rules and fallacies
Formal fallacies

2. Valid syllogisms must conform to certain rules.
If any one of these rules is violated, a specific formal fallacy is committed.
Conversely, if no rules are broken, the syllogism is valid.
The first 2 rules depend on distribution.
The second 2 on the concept of quality.
The last on the concept of quantity.

3. Rule 1: distribute the middle term in at least one premise
Remember? A term is distributed when it refers to all members of a class.
If the middle term is not distributed in at least one premise, the connection required by the conclusion cannot be made.
And the fallacy of undistributed middle is committed.

4. Remember distribution?
Statement type
Terms Distributed A
Subject E
Subject and Predicate I
None O
Predicate

5. example All Russians were revolutionists.
All anarchists were revolutionists. = aaa-2
Therefore, all anarchists were Russians.
The middle term “revolutionists” is not distributed in either premises.
Because “All Russians were revolutionists” does not imply “all revolutionists were Russian.”
And “all anarchists were revolutionists” does not imply “all revolutionists were anarchists.”
So the connection required by the conclusion cannot be made.

6. Another example All sharks are fish. All salmon are fish.
So all salmon are sharks.
The middle term is fish.
In both premises occurs as the predicate.
Fish is Not distributed in either premise.

7. Rule 2: if a term is distributed in the conclusion, then it must be distributed in a premise
if a term that is distributed in the conclusion is not distributed in a premise, then the conclusion does not follow necessarily.
If that’s the case, the fallacies of illicit major or illicit minor are committed.
Look at the conclusion. Then see which term is distributed. Then verify that the same term is distributed in the premises. If it isn’t, a fallacy is committed.

8. Illicit major All dogs are mammals. A: mammals (major) = undistributed
No cats are dogs E: cats (minor) = distributed
Therefore, no cats are mammals E: cats (minor) = distributed / mammals (major) = distributed
Mammals is the major term and is distributed in the conclusion.
But in the major premise, mammals is not distributed!
Consequently, the syllogism commits an illicit process of the major term.
In other words, it commits the illicit major fallacy.

9. Illicit minor All tigers are mammals. A: tigers (major) = distributed
All mammals are animals a: animals (minor) = undistributed
Therefore, all animals are tigers a: animals (minor) = distributed / tigers (major) = undistributed
Animals is the minor term and is distributed in the conclusion.
But when we look at the minor premise where the minor term occurs, we see that the term is undistributed.
Consequently, the syllogism commits the fallacy of illicit minor.

10. Rule 3: two negative premises are not allowed
Any negative proposition e-form or o-form denies class inclusion.
E-form asserts that the subject and the predicate do not share any members.
O-form asserts that some members of the subject class are not members of the predicate class.
Two premises asserting such exclusions cannot give the linkage required by the conclusion.
Therefore the conclusion does not follow from the premise.
This is the fallacy of exclusive premises.

11. Fallacy of exclusive premises
No fish are mammals.
Some dogs are not fish. = eoO - 1
Some dogs are not mammals.
The premises of this syllogism both deny class inclusion.
In the major premise, Fish and mammals are separate. No members of the class of fish are in the class of mammals. And no members of the class of mammals are in the class of fish.
In the minor premise, some members of the class of dogs are not in the class of fish.
Consequently, the conclusion does not follow.
Fallacy of exclusive premises

12. Rule 4: a negative premise requires a negative conclusion, and a negative conclusion requires a negative premise
If the conclusion is affirmative, (A-form or I-form) it asserts that all or some of the members of a class are contained in the other.
If a premise is negative, it denies class inclusion
Consequently, a syllogism with affirmative conclusion and negative premises commits the fallacy of drawing an affirmative conclusion from negative premises.
By the same token, if the conclusion is negative but the premises are affirmative, it commits the fallacy of drawing a negative conclusion from affirmative premises.

13. drawing an affirmative conclusion from negative premises
All pigeons are birds. All triangles are three-angled polygons.
Some dogs are not pigeons. All three-angled polygons are three-sided polygons.
Thus, some dogs are birds. Some three-sided polygons are not triangles.
In the first syllogism, the minor premise is negative but the conclusion is affirmative.
The fallacy consists of going from a premise that excludes some members of the class of dogs from the class of pigeons to a conclusion claiming that some dogs are birds and some birds are dogs.
In the second syllogism, the premises are affirmative but the conclusion negative. Thus, the connection required cannot be made. The fallacy of drawing a negative conclusion from affirmative premises is committed.

14. An alternative formulation of rule 4:
An alternative formulation of rule 4 is that any syllogism having exactly one negative statement is invalid.
Thus, if the conclusion alone is negative, or one of the premises is negative, while the other statements are affirmative, the syllogism is invalid.
From these 4 rules, it turns out that no valid syllogism can have two particular premises.
If a standard-form syllogism has both premises that start with “some” it is invalid.

15. Rule 5: if both premises are universal, the conclusion cannot be particular.
If a syllogism breaks rule 5 it is an invalid syllogism only according to the modern interpretation.
This applies only to the 9 syllogistic forms that are “conditionally valid.”
Because the modern interpretation has no existential assumption, the move from universal premises to a particular conclusion is not warranted. And the existential fallacy is committed.
Figure 1
Figure 2
Figure 3
Figure 4
Required Condition
AAI
EAO
AEO
S exist
M exist
P exist

16. Where do we insert an x? X X X X AAI-2
All P are M. All humans are mammals.
All S are M. = All dogs are mammals.
Some S are P. Some dogs are humans. X X X X
Because all the entities exist, we could place an X inside any circle.
However, if we indicate that S or P exist, we can insert an X in two possible areas, which means we use a bar.
And if we indicate that M exist, we can insert an X in 3 possible areas.
Consequently, the AAI-2 is invalid.

17. Existential fallacy
No politician is honest. No politician is honest.
All bakers are honest. All unicorns are honest.
Thus, some bakers are not politicians. Thus some unicorns are not politicians.
syllogism 1. is an eao-2. - universal premises and particular conclusion.
valid on the condition that the bakers really exist—And they do. Thus it is valid.
Syllogism 2. is an eao-2 as well. The difference is that unicorns do not exist. So the syllogism is invalid.

18. explanation
In The modern interpretation, universal statements a and e, lack an existential assumption. but particular statements I and o do have a built-in existential assumption.
Thus, if a syllogism is made up of universal premises and a particular conclusion, the conclusion asserts that something exists while the premises do not.
This is evident on the modern square of opposition as the relations of superalternation and subalternation are no longer exhibited.
That is to say, the truth of universal statements does not imply the truth of particular statements.
And the falsity of particular statements does not imply the falsity of universal statements.