# A Class-Room Introduction to Logic

## May 4, 2009

### Unit-X: Rules and Fallacies for Categorical Syllogisms

Filed under: Logic,Rules for validity of syllogism — Dr. Desh Raj Sirswal @ 8:09 am
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Aristotle and other traditional logicians provided certain rules which determine the validly/invalidity of syllogism. Here are some rules to check the validity of a syllogism.

Rule 1: Avoid Four Terms

Fallacy: Fallacy of four terms (A formal mistake in which a categorical syllogism contains more than three terms.)

Example:

All men are rational animal.

All chalks are white.

Therefore, ————————.

Justification: A valid standard-from categorical syllogism must contain exactly three terms, each of which is used in the same sense throughout the argument. If there is more terms than, it cannot be in standard-form syllogism, we cannot call it syllogism.

Rule 2: The middle term must be distributed at least once.

Fallacy: Undistributed middle( A formal mistake in which a categorical syllogism contains a middle term that is not distribute in either premise.)

Example:

 All sharks are fish.All salmon are fish.Therefore, 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 3: If a term is distributed in the conclusion, then it must be distributed in a premise.

Fallacy: Illicit major (A formal mistake in which the major term of a syllogism is undistributed in the major premise, but is distributed in the conclusion.)

Illicit minor (A formal mistake in which the minor term of a syllogism is undistributed in the minor premise, but is distributed in the conclusion.)

Examples:

 And: All horses are animals.Some dogs are not horses.Therefore, Some dogs are not animals. All tigers are mammals. All mammals are animals. Therefore, 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 4: No conclusion drawn from two negative premises.

Fallacy: Exclusive premises (A formal mistake in which both premises of a syllogism are negative)

Example:

 No fish are mammals.Some dogs are not fish.Therefore, 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 5: 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. (A formal mistake in which one premise of a syllogism is negative but the conclusion is affirmative.)

Example:

 All crows are birds.Some wolves are not crows.Therefore, 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 6: If both premises are universal, the conclusion cannot be particular. And also there is no conclusion from two particular premises.

Fallacy: Existential fallacy (As a formal fallacy, the mistake of inferring a particular conclusion from two universal premises.)

Example:

 All mammals are animals.All tigers are mammals.Therefore, 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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