A valid categorical syllogism must conform to certain rules. These rules of syllogism are the norms or standard that helps us to test the validity or the invalidity of the moods. If we draw the conclusion in accordance with the rules of syllogism, the argument is valid or else it becomes invalid.
The violation of any of the rule leads to a logical mistake otherwise called a logical fallacy. Let us discuss the rules of syllogism and the corresponding fallacies that are committed when the rules are violated. Mainly, we will deal with the following topics while discussing the rules of syllogism. These are
(A) General Syllogistic Rules.
(B) Special Syllogistic Rules.
(C) Aristotle's Dictum.
(A) General Syllogistic Rules:
General Syllogistics rules are the fundamental and basic rules applicable to all syllogisms in general. These are ten in number. Out of these ten, some are based on the very definition of syllogism and some rules are derivative in nature. Let us discuss them in detail.
Every syllogism must have three and only three terms neither more nor less. This rule can not be regarded as a rule in the strict sense of the term because the very definition of syllogism states that a syllogism must have three propositions and three terms. These terms include the minor term, major term and the middle term. The middle term keeps relationship with the extremes so that a conclusion is drawn. Similarly, we cannot avoid either the major term or the minor term. Thus, in a syllogism, it is necessary to have three and only terms.
If an argument has less than three terms (i.e. two terms), we cannot call it a syllogism, rather it is a case of immediate inference.
All crocodiles are reptiles
Therefore, some reptiles are crocodiles
Here there are two terms and it is a case of immediate inference.
If an argument contains more than three terms (i.e. four terms), it cannot be called a syllogism. We commit the fallacy of four terms. For example,
All cows are quadruped animals.
All dogs are faithful animals.
From this, we cannot draw any conclusion. It is a case of fallacy of four terms.
Sometimes a term is used in different senses in the same argument. In such a case, we commit the Fallacy of Equivocation. This fallacy has three forms. When the major term is used ambiguously, we call it the Fallacy of ambiguous major. For example,
Light is essential to guide our steps
Lead is not essential to guide our steps
Therefore, lead is not light.
The major term 'light' in the above argument has been used in one sense in the major premise, but in another sense in the conclusion.
Similarly, when the minor term is used ambiguously, we commit the fallacy of ambiguous minor.
No man is made of paper.
All pages are men.
Therefore, no pages are made of paper.
In this argument, the minor term 'page' has been used in two different senses.
When the middle term is used ambiguously, we commit the fallacy of ambiguous middle. For example,
Sound travels at the rate of 1120 feet per second.
His knowledge of mathematics is sound
Therefore, his knowledge of mathematics travels at the rate of 1120 feet per second.
Every syllogism must have three and only three propositions. This is also not a rule in the strict sense of the term. Like Rule-I, it states a necessary condition that a syllogism must have three propositions out of which two are called premises and what follows from the premises is called the conclusion. If we take less than three propositions, the argument might become an immediate inference or if we take more than three propositions, we get a train of syllogisms or Sorties.
In a valid syllogism, the middle term must be distributed at least in one of the premises.
The role of middle term in a syllogism is important because it connects both the extremes. In order to establish a relation between the extremes (major and minor terms) in the conclusion, extremes should be shown to be connected in some common part of the middle term. In other
Words, for establishing a connection between the major and minor term in the conclusion, at least one of them must be related to the whole of the middle term, otherwise each of them might be connected only to with a different part of the middle term. If the middle term is not distributed at least once in the premises, both the extremes are not shown to be connected and we commit the fallacy of undistributed middle. For example,
All dogs are quadruped.
All cats are quadruped.
So, all cats are dogs.
In both the premises, the middle term is undistributed (since A proposition doesn't distribute its predicate). No conclusion is possible as the middle term is not properly connected with the extremes. When this rule is violated we commit the, fallacy of undistributed middle.
In a categorical syllogism, if a term is distributed in the conclusion, it must be distributed in the premise.
This rule states a necessary condition of deductive validity. The conclusion of a valid deductive argument cannot be more general than the premises; the conclusion cannot go beyond the premises. The conclusion can only make explicit what is implicitly present in the premises. Syllogistic arguments, being deductive, must abide by this condition.
The conclusion of a syllogism has two terms. These are minor term and major term. Neither the major term nor the minor term should be distributed in the conclusion if it is not distributed in the premise. Of course, the reverse is not a fallacy. A term which is distributed in the premise may remain undistributed in the conclusion.
If the minor term is distributed in the conclusion but not distributed in the minor premise, we commit the fallacy of illicit minor. For example,
AH men are rational.
All men are biped.
Therefore, all bipeds are rational.
Here the minor term 'biped' (subject term of conclusion) is distributed which is not distributed in the minor premise (being the predicate of A proposition). So the fallacy committed in this argument is illicit minor.
Similarly, if the major term is distributed in the conclusion without being distributed in the major premise, we commit the fallacy of illicit major. For example,
All cows are quadruped.
No goats are cows.
Therefore, no goats are quadruped.
Here, the major term is distributed in the conclusion but not in the major premise (since it is the predicate of an A proposition). So the fallacy of illicit major is committed in this argument.
In a categorical syllogism, no conclusion can be obtained from two negative premises.
A negative proposition is one in which the predicate is denied of the subject i.e. the predicate is negatively related with the subject. If both the premises are negative, the middle term will be negatively related to the extremes and no relation can be established between them. So a valid conclusion cannot be drawn. If we draw a conclusion from two negative premises, we commit the fallacy of two negative premises or fallacy of exclusive premises.
No artists are rich persons.
Some rich persons are not theists.
Therefore, some theists are not artists.
Since both the premises are negative, the conclusion (some theists are not artists) is not valid and we commit the fallacy of two negative premises or fallacy of exclusive premises.
In a categorical-syllogism, if either premise is negative, the conclusion must be negative. According to Rule-5 stated above, we cannot draw any valid conclusion from two negative premises. So, if one premise is negative, the other premise must be affirmative. If one premise is affirmative and the other premise is negative, then a relation of inclusion will be asserted between the middle term and one of the extremes in the affirmative premise and the relation of exclusion will be asserted between the middle term and the other extreme.
Thus, if one extreme is included in the middle term and the other excluded then there can be the relation of exclusion between the extremes, and they cannot have affirmative relation in the conclusion. Therefore, the conclusion will be negative. For example,
No poets are scientists.
Some philosophers are poets.
Therefore, some philosophers are not scientists.
This conclusion (negative one) is a valid conclusion. But if we draw any affirmative conclusion (such as "Some philosophers are scientists") from the above premises, it would be a fallacious conclusion. Here, we would have committed the fallacy of drawing an affirmative conclusion from a negative premise. Similarly, we can prove that if the conclusion is negative, one of the premises must be negative
In a categorical syllogism, if both the premises are affirmative, the conclusion must be affirmative.
In an affirmative proposition, the predicate is affirmed of the subject. In other words, in an affirmative premise a relation of inclusion is asserted. If both the premises are affirmative, it is clear that the middle term is affirmatively connected with both the extremes i.e. the minor term and the major term. Thus it is obvious that the minor term and the major term are affirmatively related in the conclusion and the conclusion must be an affirmative proposition.
Similarly, the converse also holds good. If the conclusion is affirmative, both the premises must be affirmative.
In a categorical syllogism, if both the premises are particular, no conclusion follows. As we know, there are two types of particular propositions. These are I and O propositions. If both the premises are particular, then the possible combinations will be I I, I O, O I and OO.
In 11 combination, since no term is distributed in I proposition the middle term is not distributed. So this combination will not yield any conclusion (as per Rule 3) stated above. In OO combination, there will be no conclusion (as per Rule 5) as it leads to the fallacy of two negative premises.
Let us examine the combination of IO and O I. In either of the cases, since one premise is negative, the conclusion will be negative. If the conclusion is negative, the predicate of the conclusion (major term) will be distributed in the conclusion which could not be distributed in the premise because there is only one term distributed in the premises and it is reserved for the middle term to avoid the fallacy of undistributed middle).
So no conclusion follows from any of the combinations when both the premises are particular. In other words, in a categorical syllogism at least one of the premises must be universal.
In a categorical syllogism, if one premise is particular, the conclusion will be particular. If one premise is particular, the other premise will be universal because according to Rule 8, stated above, from two particular premises no conclusion follows. We have also seen that from two negative premises no conclusion can be drawn (See Rule 5 stated above). So we get the following possible combinations.
AI, IA, AO, OA, EI, IE,
Let us examine each pair.
AI and I A:
In this combination, total number of terms distributed is one which is left for the middle term (to avoid the fallacy of undistributed middle). So the conclusion will be a proposition that does not distribute any term (to avoid the fallacy of either illicit major or illicit minor). So the conclusion will be an I proposition which is particular.
AO and OA:
In this combination where one proposition is A and the other is O, the total number of terms distributed in the premises is two, out of which one must be reserved for the middle term to avoid the fallacy of undistributed middle and there is only one term left as distributed. Since one premise is negative, the conclusion is bound to be negative (as per Rule-6). Thus the conclusion will be a negative proposition and it will have only one term distributed. The conclusion, therefore, must be an O proposition which is particular.
EI and IE:
In this combination, total numbers of terms distributed in the premises are two out of which one is reserved for the middle term. So there is only one term left as distributed in the premise. Since one premise is E which is negative, the conclusion will be negative where only one term can be distributed. So it must be an O proposition, which is particular.
Thus we notice that if one premise is particular, the conclusion will be particular.
In a categorical syllogism, if the major premise is particular and the minor premise is negative then no conclusion follows.
If the minor premise is negative, the conclusion becomes negative (Rule 6) and the major premise is bound to be affirmative (Rule 5). Thus the major premise is a particular affirmative (T) proposition. Since the conclusion is negative its predicate (major term) will be distributed in the conclusion which is not distributed in the major premise. So the fallacy of illicit major will be committed.
Therefore, in a syllogism when the major premise is particular and minor premise is negative, no conclusion can be drawn.