Notes: Logic: Rules of Categorical Syllogisms

Rules of Categorical Syllogism 

A. Introduction

Recall that Categorical Syllogism is a syllogism consisting of three categorical propositions, and containing three distinct terms, each of which appears in exactly two of the three propositions.

Classical Logic lists eight rules of valid categorical syllogism:

·       four concern the terms, and

·       four concern the propositions.

These rules are not provable. They have to be either accepted or rejected. If they are rejected, syllogism is not possible. Therefore, what is given is only an explication of the rules.

Classical logic classified these rules under:

·       rules of structure,

·       rules of distribution of terms,

·       rules of quality, and

·       rules of quantity.

B. Rules of Structure

1. Syllogism must contain three, and only three, propositions

Syllogism is defined as a kind of mediate inference consisting of two premises which together determine the truth of the conclusion.

This definition shows that if the number of propositions is more than two, then it ceases to be syllogism.

Therefore, by definition syllogism must consist of two premises and one conclusion.

2. Syllogism must consist of three terms only

A proposition consists of two terms. However, three propositions consist of only three terms because each term occurs twice.

Suppose that there are four terms. Then there is no middle term, a term common to two premises. In such a case the violation of rule results in a fallacy called fallacy of four terms.

Such a fallacy is never committed knowingly because knowing fully well the fixed number of terms, we do not choose four terms. But we do it unknowingly.

It happens when an ambiguous word is used in two different senses on two different occasions. Then there are really four terms, not three terms. If an ambiguous word takes the place of middle term, then the fallacy committed is known as fallacy of ambiguous middle.

Similarly, if an ambiguous term takes the place of the major or the minor term, then the fallacy of ambiguous major or ambiguous minor, as the case may be, is committed.

The following argument illustrates the fallacy of ambiguous middle.

All charged particles are electrons.

Atmosphere in the college is charged.

Atmosphere in the college is an electron.

The word in italics is ambiguous.

The other two fallacies are hardly committed. Therefore, there is no need to consider examples for them.

The moral is that all sentences in arguments must be unambiguous. This is possible only when all terms are unambiguous in the given argument.

We must also consider the inversion of ambiguous middle. Suppose that synonymous words are used in place of middle term. Then apparently there are four terms. But, in reality, there are three terms. For example starry world and stellar world are not two terms. Such usages also are uncommon. Hence they deserve to be neglected.

C. Rules of distribution of terms

1. Middle term must be distributed at least once in the premises.

If this rule is violated, then the argument commits the fallacy of undistributed middle. One example will illustrate this rule.

All circles are geometrical figures.

All squares are geometrical figures.

All circles are squares.

2. In the conclusion, no term may be taken in a more ‘extensive’ sense than in the premises.

It also means that a term which is distributed in the conclusion must remain distributed in the respective premise. This rule can be stated this way also.

A term which is undistributed in the premise must remain undistributed in the conclusion.

However, it is not necessary that a term, which is distributed in the premise, must be distributed in the conclusion.

Suppose that the major term violates this rule. Then the argument commits the fallacy of illicit major. E.g.,

All philosophers are thinkers.

No ordinary men are philosophers.

No ordinary men are thinkers.

When the minor term violates this rule, fallacy illicit minor is committed. E.g.,

All aquatic creatures are fish.

All aquatic creatures swim.

All those which swim are fish.

D. Rules of quality

1. From two negative premises, no conclusion can be drawn. It only means that at least one premise must be affirmative.

2. If both premises are affirmative, the conclusion cannot be negative. Negatively, it only means that a negative conclusion is possible only when one premise is negative.

E. Rules of quantity

1. If both premises are particular, no conclusion can be drawn or the conclusion must always follow the weaker part. Here weaker part is particular. This rule shows that at least one premise must be universal.

2. If one premise is particular, then the conclusion must be particular only. It means that universal conclusion is possible only when both premises are universal.

In practice, last three sets of rules play an important role in determining the validity of categorical syllogism.