The Categorical Syllogism

A SYLLOGISM is typically a three-proposition deductive argument—that is, a mediate inference that consists of two premises and a conclusion. In a categorical syllogism,all the propositions used are categorical statements, hence the label ‘categorical.’

The three categorical propositions contain a total of three different terms, each of which appears twice in distinct propositions. The following is an example of a categorical syllogism:

All amphibians are cold-blooded vertebrates.

All frogs are amphibians.

Therefore, all frogs are cold-blooded vertebrates.

As you would notice from our example, acategorical syllogism consists of three and only three propositions and three and only three terms.

Components of categorical syllogism

The terms used in our example are amphibians, cold-blooded vertebrates, and frogs. In a categorical syllogism, the predicate term of the conclusion is called the major term (usually represented by P) while the subject term is the minor term (S). In our example therefore, the major term is cold-blooded vertebrates whereas the minor term is frogs.

            The third term in the syllogism does not appear in the conclusion, but is employed in both premises. It is the term that connects, relates, or mediates the two other terms, hence called middle term (M).What do you think is the middle term in our example?

Notice that aside from the conclusion, we have two premises in a categorical syllogism. One of them contains the major term and is thus called the major premise. The other premise, which contains the minor term, is referred to as the minor premise. “All amphibians are cold-blooded vertebrates” is the major premise in our example. Can you identify the minor premise in our example?

Standard form

One thing about categorical syllogism is that its validity depends solely upon its structure or logical form, not upon its contents. Thus, in determining whether or not a particular categorical syllogism is valid, restating the argument in standard form is vital.  A categorical syllogism is in standard form if itmeets the following four conditions:

1. All three statements are standard-form categorical propositions. This means that each statement in the argument has a proper quantifier, subject term, copula, and predicate term (Q-S-C-P).

2. Each term appears twice in the argument. The major term appears once in the conclusion and once in the major premise; the minor term, in the conclusion and the minor premise; and the middle term, in both premises.

3. Each term is used in the same sense throughout the argument. The possibility of equivocation must be ruled out.

4. The major premise is listed first, the minor premise second, and the conclusion last.

Note that our example about ‘amphibians’ is already in standard form since it meets all these conditions.

Figure and mood

The variety of structures in which a standard categorical syllogism may occur can be labeled by stating its form. The form of a categorical syllogism is the combination of its figure and mood. The mood simply refers to the types of categorical propositions (A, E, I, or O) used in the syllogism, listed in the order in which they appear in standard form. For instance, a syllogism with an E proposition as its major premise, an I proposition as its minor premise, and O proposition as its conclusion has the mood EIO. So, what’s the mood of our example (about ‘amphibians’)?

            Now, you would also notice that each syllogistic mood, say EIO, can have four distinct versions depending on the arrangement of the major, minor, and middle terms in the premises—for a particular term can either be the subject or the predicate of the proposition. Thus, we need to supplement the labeling system with mentioning its figure, that which identifies the four distinct ways the middle terms are arranged in the syllogism:

Figure 1: the middle term is the subject term of the major premise and the predicate term of the minor premise

Figure 2: the middle term is the predicate term of both premises

Figure 3: the middle term is the subject term of both premises

Figure 4:  the middle term appears as the predicate term of the major premise and the subject term of the minor premise

            If we let S represent the minor term, P the major term, and M the middle term, and leave out the quantifiers and copulas, the four figures may be illustrated as follows:

        M   P        P   M        M   P        P   M

     1    \       2      |     3  |         4    /

        S   M        S   M        M   S        M   S

            Going back to our example, its logical form (mood and figure) is therefore AAA-1.

            Considering all possible moods and figures, there are exactly 256 distinct forms of categorical syllogism (four types of major premise multiplied by four kinds of minor premise multiplied by four kinds of conclusion multiplied by four possible figures).

Validity, rules, and fallacies

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 certain rules we are about to study. 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. For every rule that is violated, a specific formal fallacy is committed.

The four fundamental syllogistic rules

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.

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 A proposition and the middle term is used as the predicate term in both premises, then the middle term isnever 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/men has 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 conductoris 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 personsis 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 (E and 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 aconclusion can follow is if bothpremises 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 subject class is separate either wholly orpartially 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.

The Residual Syllogistic Rules

If you have consulted many references, you may be wondering why the syllogistic rules vary in quantity depending on the author of the lecture. One reason is that some authors split the fundamental Rule 4 into two. Concerning Rule 2, some texts list one rule as regards minor term and another for major term.

            Some ‘rules’ are not really rules but are actually warnings against creating a non-standard categorical syllogism like, “The middle term must not appear in the conclusion.” Moreover, other listings include in the rules those conclusions which are derivable from the fundamental ones. For instance, as a result of the interaction of the four rules, we can conclude that no valid syllogism can have two particular premises. Thus, some texts include in the rules, “No conclusion can be drawn from two particular premises.” (Violation of this rule accordingly results in committing thefallacy of two particulars or “fallacy of two particular premises”).

           

What others put as the rule, “If one premise is particular, the conclusion must also be particular” is also an offshoot of the interplay of the four fundamental rules. This came from the realization that a syllogism with a particular premise and a universal conclusion commits at least one of the four aforementioned rules, usually Rule 2.

            Furthermore, what others submit as seemingly distinct rules are actually mere paraphrases of the fundamental ones. Notice, for example that, “Only an affirmative conclusion can be drawn from two affirmative premises” is a just a rewording of “A negative conclusion requires a negative premise” (Rule 4).“At least one premise must be affirmative”is similar to“Two negative premises are not allowed” (Rule 3).Likewise, the rule“If either premise is negative, the conclusion must also be negative” is just a restatement of the first part of Rule 4, “A negative premise requires a negative conclusion.”

            These rules we call residual can be helpful nonetheless as they provide other ways of understanding the fundamental rules and aid us in identifying as invalid various syllogisms. Other residual rules of these kinds are the following:

“At least once, the middle term must be universal.”

“The major and the minor terms cannot have greater extension in the conclusion than in the premise.”

“The conclusion follows the weaker premise.” (If one premise is affirmative and the other is negative, the conclusion must be negative. If one premise is particular and the other is universal, the conclusion must be particular.)

Four terms

There is a residual rule that is not a mere derivative much less a mere translation of the fundamental ones. This states, “There must be three and only three terms to be used in the same sense throughout the argument.” The following example violates this rule, and is thus deemed to commit the fallacy of four terms(quaternio terminorum):

All stars are heavenly bodies.

Jessica Alba is a star.

Therefore, Jessica Alba is a heavenly body.

The term star in the first premise means a cosmic material (mass of gas in space) while it means celebrity (popular performer) in the second. Since the use of an ambiguous term in two dissimilar senses amounts to the use of two distinct terms, the argument contains a total of four terms, and thereby, the premises fail to interrelate the terms in the conclusion.

            We nonetheless did not include this rule in the fundamental ones since this requirement is included as part of the requisite of standard-form categoricalsyllogism and is thus incorporated into the definition of, not in the rules concerning, categorical syllogism. Moreover, the issue in this presumed rule is not more of the syllogistic form (figure, mood, and distribution) but of the content of the argument, which is the domain of informal logic. (In fact, the corresponding fallacy here is equivalent to fallacy of equivocation under informal fallacies).

           

Boolean standpoint

There’s a rule which states, “If both premises are universal, the conclusion cannot be particular.” This rule originated from the standpoint of British mathematician and logicianGeorge Boole(1815-1864). Boole submits that universal statements do not contain existential import, i.e. they do not claim whether or not the classes they refer to exist. For him, to say that All gremlins are mortal is like saying that “if something is a gremlin, then it is mortal”, without asserting that gremlin exists. On the other hand, to say that “some gremlins are mortal” (particular) necessarily involves that “gremlins exist” or “there are gremlins”.

            Thus, the following argument, in Boolean standpoint, is invalid and commits the existential fallacy:

All trees are plants.

All pines are trees.

Therefore, some pines are plants.

The example has two universal premises and a particular conclusion. It is invalid in Boolean standpoint because the conclusion asserts that pines exist, whereas the premises are interpreted as making no such claim. Notice that our example, though invalid in Boolean view, is definitely valid from the Aristotelian standpoint, especially that we know that pines do exist.

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.

UNCONDITIONALLY VALID FORMS
Figure 1	Figure 2	Figure 3	Figure 4
AAA	AEE	AII	AEE
AII	AOO	IAI	IAI
EAE	EAE	OAO	EIO
EIO	EIO	EIO

CONDITIONALLY VALID FORMS
Figure 1	Figure 2	Figure 3	Figure 4	Required condition
EAO AAI	EAO AEO		AEO	The minor term(S) exists
		EAO AAI	EAO	The middle term (M) exists
			AAI	The major term (P) exists

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 calledCelarent.

   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, andFresison.

Structures of conditionally valid syllogisms

For practical purposes, we can say that the following nine (9) forms are 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.

Significance and some applications

Categorical syllogism, for one thing, provides us a conclusive yardstick in distinguishing correct from incorrect reasoning. For instance, someone who has a bias against celebrities may submit an argument which virtually amounts to this:

      All politicians are people who want power.

      All politicians are popular persons.

      Therefore, all popular persons are people who want power.

For some, it may be difficult to assess the validity of this argument especially at first glance. But having learned categorical syllogism, we know that this argument has the form AAA-3. This syllogistic form is not among the list of valid forms, hence is certainly invalid. The invalidity of this argument is perfectly the same as the incorrectness of the following syllogism with the same form AAA-3:

   All Dalmatians are dogs.

   All Dalmatians are mammals.

   Therefore, all mammals are dogs.

This argument is obviously erroneous. And since its form is identical with that of the argument about ‘popular persons’, then both are not valid. This method of demonstrating the invalidity of syllogisms we call logical analogy is useful in many contexts and can be appreciated even by those who have not undergone formal training in Logic.

            As a way of improving your reasoning skill, you may also try to provide example for each of the valid forms we have listed. For Barbara (AAA-1) for instance, we can submit this example:

      All God’s laws are just decrees.

      Respecting one’s parents is God’s law.

      Therefore, respecting one’s parents is a just decree.

If you are a writer or a public speaker and you wish to make a persuasive essay or piece, the valid forms of categorical syllogism can serve as your outline. For example, if your topic is about the value of respecting parents, you may use our example as your framework.

You may center the first part of the piece on the first premise by mentioning God’s laws which are unquestionably righteous. In the middle part, you may zero in on proving the second premise, perhaps by quoting verses from Holy Scriptures. The last part may be devoted to stressing the conclusion and perhaps stating the benefits of obeying God’s command on respecting parents. (Copyright 2013 by Jensen DG. Mañebog)