categorical syllogisms (logic slide 8)

THE SIMPLE CATEGORICAL THE SIMPLE CATEGORICAL SYLLOGISMSYLLOGISM

THE BASIC STRUCTURE

A Simple Categorical Syllogism is composed of three (3) categorical or attributive propositions so put together that the subject (t) and predicate (T) of the conclusion are united or separated through the intermediacy of a middle term (M)

Every animal is mortal;

but every dog is an animal;

therefore every dog is mortal.


The first proposition is the major premise; the second proposition is the minor premise; and the third is the conclusion. “Mortal” the predicate of the conclusion, is the major term; “dog,” the subject of the conclusion, is the minor term; and “animal,” which occurs in both premises but not in the conclusion, is the middle term.

A. Major Term

The major term is the predicate of the conclusion. It must occur in the conclusion and in the premise, generally the first, which is therefore called the major premise. The major term shall be symbolized by T, or, to display the structure of a syllogism more graphically, by a rectangle.


B. Minor Term

The minor term is the subject of the conclusion. It must occur in the conclusion and in the premise in which the term does not occur. It is often introduced by the adversative conjunction “but” (because in controversy it introduces a turn of thought to the expectations of an opponent). Minor term shall be designated by t, or, to display the structure of a syllogism more graphically, by an ellipse.


C. Middle Term

The middle term occurs in each of the premises but not in the conclusion. In the major premise, it occurs in conjunction with the major term; and in minor premise, in conjunction with the minor term. It is the medium through which the major and minor term are united in the affirmative syllogism and separated in the negative syllogism. As opposed to the middle term, the minor and major terms are called extremes.


Example of a graphically marked simple categorical syllogism:

Every animal is mortal;

But every dog is an animal;

Therefore every dog is mortal.


Exercise: First pick out the conclusion of each of the following syllogisms. Then pick out the minor and major terms, then the minor and major premises, and, finally, the middle term.

1. All metals are conductors of electricity;

but copper is a metal;

therefore copper is a conductor of electricity.

2. No insulator is a good conductor of electricity;

but copper is a good conductor of electricity;

therefore copper is not an insulator.


3. George W. Carver was a Negro;

but George W. Carver was an eminent scientist;

therefore some eminent scientist was a Negro.

4. The building on the corner of Grand and Lindell must be a church. Why? Because it is a building having a steeple with a cross on it, and all such buildings are churches.

5. Every X is a Y; therefore, since every Y is a Z, every X must also be a Z.


6. A good leader has the confidence of his followers; Joe Doe, though, does not have the confidence of his followers, and is therefore not a good leader.

7. Since winesaps are apples and apples are fruit, winesaps, too, must be fruit.

8. Some metal are precious; this follows from the fact that gold and silver which are metals, are precious.

9. Diamonds are precious; and since diamonds are stones, it follows that some stones are precious.

10. Some mortal are rather stupid. You can infer this from the fact that all men are mortal, and some men are rather stupid.

GENERAL RULES OF THE GENERAL RULES OF THE CATEGORICAL SYLLOGISMCATEGORICAL SYLLOGISM

A. THE RULES OF THE TERMS

1. Their Number and Arrangement

1.Their Number…

2. Their Quantity, or Extension

3. The Quantity of the Minor and Major Terms: …

2.Their Arrangement…

4. The Quantity of the Middle Term: …


B. THE RULES OF THE PROPOSITIONS

1. Their Quality

5. If both premises are affirmative…

2. Their Quantity (Corollaries of Rules 3 and 4)

8. At least one premise must be…

3. Their Existential Import

10. If the actual real existence of a subject has not been asserted in the premises…

6. If one premise is affirmative and the other negative… 7. If both premises are negative, …

9. If a premise is particular, the conclusion must be…


a. The Rules of the Terms

1. THEIR NUMBER AND ARRANGEMENT

Rule 1. There must be three terms and only three- the major term, the minor term and the middle term.

The necessity of having only three terms follows from the very nature of a categorical syllogism, in which a minor (t) and a major (T) term are united or separated through the intermediacy of a third term, the middle term (M).


The terms must have exactly the same meaning and (except for certain legitimate changes in supposition) must be used in exactly the same way in each occurrence. A term that has a different meaning in each occurrence is equivalently two terms. We must be especially on our guard against ambiguous middle terms.

Example: Every animal is mortal;but every dog is an animal;therefore every dog is mortal.

Violation: Men must eat; but the picture on the wall is a man;therefore the picture on the wall must eat.


Rule 2. Each term must occur in two propositions. The major term must occur in the conclusion, as predicate, and in one of the premises, which is therefore called the major premise. The minor term must occur in the conclusion, as subject, and in the other premise, which is therefore called the minor premise. The middle term must occur in both premises but not in the conclusion. Hence, there must be three propositions.

The necessity of having three terms arranged in this way in three propositions also follows from the very nature of a categorical syllogism. Two propositions (the premises) are required for the middle term to fulfill its function of uniting or separating the minor and major terms and a third proposition (the conclusion) is required to express the union or separation of the minor and major terms.


1. THE QUANTITY, OR EXTENTION, OF THE TERMS

The reason for this rule is that we may not conclude about all the inferiors of a term if the premises have given us information about only some of them. The conclusion is an effect of the premises and must therefore be contained in them implicitly; but all are not necessarily contained in some—at least not by virtue of the form of argumentation alone.

Violation: All dogs are mammals;but no men are dogs;therefore no men are mammals.

Rule 3. The major and minor terms may not be universal (or distributed) in the conclusion unless they are universal (or distributed) in the premises.


Violation of this rule is called either extending a term or an illicit process of a term. There is an illicit process of the major term if the major term is particular in the premise but universal in the conclusion; and an illicit process of the minor term, if the minor term is particular in the premise but universal in the conclusion.

Take note that there is no illicit process if the major or minor term is universal in the premises and particular in the conclusion. To go from a particular to a universal is forbidden, but to go from a universal to a particular is not permissible.


Rule 4. The middle term must be universal, or distributed, at least once.

The reason for this rule is that when the middle term is particular in both premises it might stand for a different portion of its extension in each occurrence and thus be equivalent to two terms, and therefore fail to fulfill its function of uniting or separating the minor and major terms.

Violation: A dog is an animal; but a cat is an animal;therefore a cat is a dog.

Violation of this rule is often called the fallacy of the undistributed middle.


b. The Rules of the Propositions

1. THE QUALITY OF THE PROPOSITIONS

Rule 5. If both premises are affirmative, the conclusion must be affirmative.

The reason for this rule is that affirmative premises either unite the minor and major terms, or else do not bring them into relationship with one another at all—as when there is an undistributed middle. In neither case may the major term be denied of the minor term. Hence, to get a negative conclusion you must have one—and only one—negative premise.

Violation: All sin is detestable;but some pretense is sin;therefore some pretense is not detestable.


As soon as you see that both premises are affirmative but the conclusion negative, you can be sure that your syllogism is invalid. Be on your guard, however, against apparent affirmative or negative propositions.

Example: Animals differ from angels;but man is an animal;therefore man is not an angel.

The syllogism is valid because ‘differ from’ is equivalent to ‘are not’.


Rule 6. If one premise is affirmative and the other negative, the conclusion must be negative.

The reason for this rule is that the affirmative premise unites the middle term with one of the extremes (that is, with either the minor or the major term) and the negative premise separates the middle term from the other extreme. Two things, of which the one is identical with a third thing and the other is different from that same third thing, cannot be identical with one another. Hence, if a syllogism with a negative premise concludes at all, it must conclude negatively.

Violation: Every B is a C;but some A is not a B;therefore some A is a C.


However, there are apparent exceptions to this rule. Keep in mind that many negative propositions are equivalent to affirmative propositions and can be changed into them by one or other kinds of immediate inference.

Example: Dogs are not centipedes;but hounds are dogs;therefore hounds differ from centipedes.

This is a valid syllogism since the conclusion is equivalently negative, since “differ from” is equivalent to “are not”.


Rule 7. If both premises are negative—and not equivalently affirmative—there is no conclusion at all.

To fulfill its function of uniting or separating the minor and the major term, the middle term must itself be united with at least one of them. But if both premises are negative, the middle term is denied of each of the extremes and we learn nothing about the relationship of the extremes towards one another.

Violation: A stone is not an animal;but a dog is not a stone;therefore a dog is not an animal.


2. THE QUANTITY OF THE PROPOSITIONS

Example: Every animal is mortal;but every dog is an animal;therefore every dog is mortal.

Rule 8. At least one premise must be universal.

Rule 9. If a premise is particular, the conclusion must be particular.


3. THE EXISTENTIAL IMPORT OF THE PROPOSITIONS

The reason for this rule is the general principle that nothing may ever be asserted in the conclusion that has not been asserted implicitly in the premises. This rule takes us out of the domain of formal logic, which does not consider existence except incidentally. We mention it only as a practical aid to argumentation.

Rule 10. The actual real existence of a subject may not be asserted in the conclusion unless it has been asserted in the premises.


Exercises:

1. All animals are substances;but all frogs are animals;therefore all frogs are substances.

Apply the general rules of the categorical syllogism to the following examples.

2. All cows are animals;but no horses are cows;therefore no horses are animals.

3. Murder is sinful;but abortion is murder;therefore abortion is sinful.


4. Contradictories are opposites;but black and white are opposites;therefore black and white are contrdictories.

5. All mammals have lungs;but most fish do not have lungs;therefore most fish are not mammals.

6. No dog is a man;but Fido is not a man;therefore Fido is a dog.

7. All mammals are viviparous;but whales are viviparous;therefore whales are mammals.


8. Those who are not sick may go;but Johnny is not sick;therefore Johnny may go.

9. No dog is not an animal;but no hound is not a dog;therefore all hounds are animals.

10.Democracies are free;but some of the governments of the Middle Ages were not democracies;therefore some of the governments of the Middles Ages were not free.

categorical syllogisms (logic slide 8)

Education

major term dog

major t term

themiddle term

major terms

minor premise

middle term themiddle

major term themajor

minor term theminor