The Rules in 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, a categorical syllogism consists of three and only three propositions and three and only three terms.

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 fundamental syllogistic rules

Filipino Philosophy professor and book author Jensen DG. Mañebog submits that there are just four (4) fundamental syllogistic rules. (This is contrary to the belief of many other Logic professors that the rules involving the validity of categorical syllogisms are more than four. Later in this lecture, we will discuss how Prof. Jensen explains those other "rules" commonly mentioned by other authors.)

In his Logic books, Prof. Jensen DG. Mañebog mentions the following four (4) rules. The  first two fundamental rules depend on the concept of distribution of terms.

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.

Prof. Jensen explains that 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.

            Prof. Jensen also explains the reason behind Rule 1--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 the 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).

Prof. Jensen elucidates that 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, according to Prof. Jensen, 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 indeed 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, according to the author, 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. Prof. Jensen Mañebog explains that 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 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 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

Prof. Jensen Mañebog admits that if we would consult many other references, we would be wondering why the syllogistic rules vary in quantity depending on the author of the lecture. Prof. Mañebog elucidates that one reason is that some authors split the fundamental Rule 4 into two. Concerning Rule 2, some accordingly texts list one rule as regards minor term and another for major term.

            Prof. Mañebog also observes that 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.”

          Other listings, he continues, 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, he observes, include in the rules, “No conclusion can be drawn from two particular premises.” (Violation of this rule accordingly results in committing the fallacy of two particulars or “fallacy of two particular premises”).

For the Filipino professor, 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, the book author proposes that what others submit as seemingly distinct rules are "actually mere paraphrases of the fundamental ones." He explains: "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 which Prof. Jensen Mañebog calls residual can be helpful nonetheless, according to the professor, 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

Prof. Jensen Mañebog explains that 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.

            Prof. Jensen Mañebog nonetheless does not include this rule in the fundamental ones since this requirement is included as part of the requisite of standard-form categorical syllogism and is thus incorporated into the definition of, not in the rules concerning, categorical syllogism. Moreover, Prof. Jensen Mañebog believes that 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, he says, the corresponding fallacy here is equivalent to fallacy of equivocation under informal fallacies.

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