The Validity of Categorical Syllogism

No argument can be both invalid and valid. That is, if it is valid, then it cannot be invalid, and conversely. A categorical syllogism is valid if it conforms to the four fundamental syllogistic rules discussed in passing below. Meaning to say, if an argument violates at least one of these rules, it is invalid. On the other hand, if none of the rules is broken, the syllogism is valid.

The first two fundamental rules depend on the concept of distribution of terms. To comprehend them, you thus have to be familiar with the distribution of terms in each type of categorical proposition. (Since this book is primarily about Debate, we will no longer discuss here in detail the so-called distribution of terms. Though if you want to review the topic, you may check the discussion in Chapter 3 of our other book “Logic: A Foundation of Critical Thinking” [2013]. Other pertinent topics—like ‘Boolean standpoint,’ ‘Residual Syllogistic Rules,’ and ‘Fallacy of Four Terms’—are as well discussed in the book under the lecture “The Categorical Syllogism.”)

The four fundamental syllogistic rules are as follows:

Rule 1: The middle term must be distributed at least once.

The following syllogism violates the rule:

All ministers are men.

Lloyd is a man.

Therefore, Lloyd is a minister.

This example commits the fallacy of undistributed middle as it violates Rule 1. The middle term which is man/men is not distributed in its two occurrences. Logically, singular statements are treated as universal, thus the minor premise “Lloyd is a man”is an A proposition. Now, since both premises are Aproposition and the middle term is used as the predicate term in both premises, then the middle term is never distributed. Thus, the syllogism is invalid.

The reason behind Rule 1 is that the middle term is supposed to provide a satisfactory common ground between the subject and predicate terms of the conclusion, something which is not fulfilled if none of the middle terms in the syllogism is distributed. In our example for instance, not the totality of men are ministers, and obviously not all men are Lloyd. Thus, to relate the terms Lloyd and ministers in the conclusion is unwarranted since the middle term man/menhas not sufficiently and necessarily linked them in the premises.

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

The following syllogisms violate the rule:

All metals are electric conductors.

Mercury is a metal.

Therefore, Mercury is not an electric conductor.

Some boxers are college graduates.

Some boxers are rich persons.

Therefore, all rich persons are college graduates.

In the first argument, the major term electric conductor is distributed in the conclusion (E-predicate term) but not in the major premise (A-predicate term). Thus, the syllogism commits the fallacy of illicit major (also called “illicit process of the major term”).

On the other hand, the second example commits the fallacy of illicit minor (or “illicit process of the minor term”). The minor term rich persons is distributed in the conclusion (A-subject term) but not in the minor premise (I-predicate term).

The logic behind Rule 2 is that the conclusion cannot validly give more information than is contained in the premises. An argument that has a term distributed in the conclusion but not in the premises has more in the conclusion than it does in the premises and is therefore invalid. (Logically, it is permissible to have more in a premise than what appears in the conclusion, so Rule 2 is not transgressed if a term is distributed in a premise but not in the conclusion. Keep also in mind that if no terms are distributed in the conclusion, Rule 2 cannot be violated.)

Rule 3: Two negative premises are not allowed.

The following example violates the rule:

No horses are dogs.

No dogs are cats.

Therefore, no cats are horses.

Since this argument has two negative premises (Eand E), it commits the fallacy of exclusive terms (or “fallacy of exclusive premises”). Any argument whose premises are both negative is invalid since it fails to establish any connection between the terms of the argument. Having both premises negative means that the middle term disagrees with the minor and major terms, thereby failing to mediate or relate the two terms. This precludes us from making a statement about the agreement or disagreement between the two terms in the conclusion.

Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.

The following examples do not conform to the rule:

All headhunters are barbarians.

Some Africans are not barbarians.

Therefore, some Africans are headhunters.

All dogs are mammals.

All mammals are mortals.

Therefore, some mortals are not dogs.

As an aside, these two syllogisms exemplify the principle that the validity of an argument is not equivalent to the truth of its premises and conclusion. It is possible for the statements composing an argument to be regarded all true (as in the above examples) and yet for the argument to be invalid.

It should be noted that both examples satisfy the previous rules (rules 1 to 3). Neither of them is valid nonetheless.

The first example is invalid as it commits the fallacy of drawing an affirmative conclusion from a negative premise. The logic behind it is that an affirmative conclusion expresses that the subject class is contained either wholly or partially in the predicateclass. The only way that such a conclusion can follow is if both premises are affirmative. Remember that only the occurrence of two affirmative premises can establish the connection between the subject and predicate terms through the middle term. So if one of the premises is negative, there is a missing link between the terms in the conclusion, hence, an affirmative conclusion is not warranted.

The second example commits the fallacy of drawing a negative conclusion from affirmative premises. Argument like this is invalid because a negative conclusion asserts that the subjectclass is separate either wholly or partially from the predicate class. But if both premises are affirmative, they assert class inclusion rather than separation. Thus, a negative conclusion cannot be drawn from affirmative premises.

Testing validity by ‘logical analogy’

Without really mastering all the concepts about categorical syllogism, we can still benefit from this topic as far as our wish to enhance our reasoning skill is concerned.

The principles discussed (figure, mood, distribution, validity, rules, etc.) imply that the validity or invalidity of a categorical syllogism can be identified by mere looking at its logical form (mood and figure). Of the 256 distinct syllogistic forms, some are necessarily valid and some are not, no matter what their contents happen to be. Every argument of the form AAA-1 is valid, for example, while all syllogisms of the form OEE-3 are invalid.

We have therefore a clear-cut and practical method of demonstrating the validity (and invalidity) of any syllogism by “logical analogy.” Since logicians had already listed for us the syllogistic forms which are valid, all we have to do is identify the mood and figure of a specific argument and check it against the list. Of all the possible syllogistic forms, there are exactly 15 forms that are unconditionally valid. Thus, if an argument’s logical form exemplifies any of these 15 forms, then it is valid. If not, then it is invalid.

Aside from the 15 unconditionally valid forms, there are 9 that are conditionally valid. They are conditional in the sense that they are valid provided that certain existential assumptions are made.

The following are the tables for valid forms.

Names and structures of unconditionally valid syllogisms

For easy recall, medieval students of logic assigned a unique name to each of the 15 unconditionally valid forms. Various elements of the names serve as reminders of the different aspects of valid syllogisms, but the most obvious of which is the use of the vowels which corresponds to the mood of the syllogism. For instance, the valid form AAA-1 is named Barbara, the highlighted vowels of which (a-a-a) noticeably stand for the mood AAA.

As the 15 forms are necessarily valid, it may be worthwhile to note them by name. And since these forms serve as barometers or standard through which we determine whether a syllogism is valid or not, let us provide here the structure or template of each forms. The letters P, S, and M refer to major term, minor term, and middle term respectively.

1. AAA-1 is called Barbara.

All M are P.

All S are M.

Therefore, All S are P.

2. AII-1 is called Darii.

All M are P.

Some S are M.

Therefore, Some S are P.

3. EAE-1 is called Celarent.

No M are P.

All S are M.

Therefore, No S are P.

4. EIO-1 is called Ferio.

No M are P.

Some S are M.

Therefore, Some S are not P.

5. AEE-2 is called Camestres.

All P are M.

No S are M.

Therefore, No S are P.

6. AOO-2 is called Baroco.

All P are M.

Some S are not M.

Therefore, Some S are not P.

7. EAE-2 is called Cesare.

No P are M.

All S are M.

Therefore, No S are P.

8. EIO-2 is called Festino.

No P are M.

Some S are M.

Therefore, Some S are not P.

9. AII-3 is called Datisi.

All M are P.

Some M are S.

Therefore, Some S are P.

10. IAI-3 is called Disamis.

Some M are P.

All M are S.

Therefore, Some S are P.

11. OAO-3 is Bocardo.

Some M are not P.

All M are S.

Therefore, Some S are not P.

12. EIO-3 is Ferison.

No M are P.

Some M are S.

Therefore, Some S are not P.

13. AEE-4 is Camenes.

All P are M.

No M are S.

Therefore, No S are P.

14. IAI-4 is Dimaris.

Some P are M.

All M are S.

Therefore, Some S are P.

15. EIO-4 is Fresison.

No P are M.

Some M are S.

Therefore, Some S are not P.

Notice that there is just one mood which is valid regardless of its figure. It is the mood EIO which thus appears under all figures in the first table above. This means that any categorical syllogism with this mood is necessarily valid for it automatically typifies one of these valid forms: Ferio, Festino, Ferison, and Fresison.

Structures of conditionally valid syllogisms

For practical purposes, we can say that the following nine (9) forms are also automatically valid as long as we talk about existing matters in the arguments we make. Thus, if the arguments we examine certainly involve nothing but existent concepts, then we add these nine forms among the 15 above-mentioned standards or patterns through which we determine the validity of syllogisms:

1. EAO-1

No M are P.

All S are M.

Therefore, some S are not P.

2. AAI-1

All M are P.

All S are M.

Therefore, some S are P.

3. EAO-2

No P are M.

All S are M.

Therefore, some S are not P.

4. AEO-2

All P are M.

No S are M.

Therefore, some S are not P.

5. EAO-3

No M are P.

All M are S.

Therefore, some S are not P.

6. AAI-3

All M are P.

All M are S.

Therefore, some S are P.

7. AEO-4

All P are M.

No M are S.

Therefore, some S are not P.

8. EAO-4

No P are M.

All M are S.

Therefore, some S are not P.

9. AAI-4

All P are M.

All M are S.

Therefore, some S are P.

Take note that the mood EAO appears four times in this list. This means that aside from EIO, the mood EAO is also an indication of a valid categorical syllogism as long as the argument talks about terms which are existent.

Also Check Out:
Reasoning and Debate: A Handbook and a Textbook by Jensen DG. Mañebog

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