Various methods to solve syllogism questions

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A syllogism is a type of a logical argument in which a conclusion is derived from two other statements (premises) of a certain form. In questions on syllogisms, we try to find the relation between two or more terms (conclusion) based on the relation given between another set of terms (premises).

For e.g.,    Premises: All humans must die. I am a human.

Conclusion: I must die.

Each part (premise or conclusion) is composed of two categorical terms (A or B), linked in the form "Some/all A is/are [not] B.”

There are multiple methods of solving syllogism questions.

  • Verbal method
  • Venn diagram method
  • Tick and Cross method

 

Verbal method of solving syllogism questions

The verbal method of solving syllogism questions depends on a person’s ability to understand the set of premises verbally and derive a conclusion. Although this method can be used to easily solve syllogism problems in the best case scenarios, using it to solve problems in the worst case scenario is relatively difficult.

Example:

All tigers are cats.

All cats are animals.

Solution:

Since all tigers are necessarily cats and all cats are necessarily animals, it can be concluded that 'All tigers are animals'.

 

Venn diagram method of solving syllogism questions

In the Venn diagram method, questions on syllogisms are solved by drawing Venn diagrams. These Venn diagrams involve circles which represent the various sets given in the question. The orientation of the circles represents the relation between the various sets. The following figures show ways of representing the premises.

Although this method is effective in solving questions on syllogisms, drawing a lot of circles for the purpose becomes cumbersome and time-consuming and hence is not recommended in aptitude tests.

 

Ticks and Crosses method of solving syllogism questions

This method is named the ‘Ticks and Crosses’ method because ticks are used to denote sets that are defined and crosses to denote sets that are not defined. A set is said to be defined (distributed) if all the elements of that set have to be known in order to define a particular premise. If all the elements of a set need not be known in order to make a particular statement, then the set is said to be not defined (distributed).

All syllogism premises and conclusions fall in one of the following four statements:

Universal Affirmative

All As are Bs.

Universal Negative

No As are Bs.

Particular Affirmative

Some As are Bs.

Particular Negative

Some As are not Bs.

The first two statements are known as universal statements as they specify all the elements in the set. The last two statements are known as particular statements as they specify only a particular portion of the set. The ticks and crosses for these four statements can be indicated as follows:

All A’s are B’s

E.g., A = {1, 2, 3} B = {1, 2, 3, …} The box indicates that some numbers are hidden. These numbers could be of any random nature. All the terms in set A have to be specified in order to state the premise and hence it is said to be distributed. This is indicated with a tick. In set B, only terms that are present in set A need to be specified and not all need to be specified and hence set B is not distributed. This is indicated with a cross.

 

No A’s are B’s

E.g., A = {1, 2, 3} B = {4, 5, 6} All the terms in set A have to be specified in order to state the premise and hence it is said to be distributed. This is indicated with a tick. We need to ascertain that terms in A are absent in B and hence all terms in B need to be specified. Hence, set B is distributed too. This is indicated with a tick.

 

Some A’s are B’s

E.g., A = {4, 2, 3, …} B = {3, 2, 1, …} The box indicates that some numbers are hidden. These numbers could be of any random nature. Not all terms in set A have to be specified in order to state the premise and hence it is said to be not distributed. This is indicated with a cross. In set B, at least one term that is present in set A needs to be specified and not all need to be specified and hence, set B is not distributed. This is indicated with a cross.

 

Some A’s are not B’s

E.g., A = {4, 2, 3, …} B = {3, 2, 1, …} The box indicates that some numbers are hidden. These numbers could be of any random nature. Not all terms in set A have to be specified in order to state the premise and hence it is said to be not distributed. This is indicated with a cross. In set B, at least one term that is present in set A needs to be absent in set B and hence all the terms in set B has to be specified. This is indicated with a cross. The above results can be summarized as follows:

Now that the background of the tick and cross method has been elucidated, the following rules have to be followed in order to arrive at the conclusion.

  1. There should be exactly three terms in the premises and conclusion put together.
  2. If both the premises are negative (has the words NO or NOT), then no conclusion is possible.
  3. If both the premises are particular (premise talks only about a few elements), then no conclusion is possible.
  4. The middle term (the term commonly present in both the premises) must be distributed at least once.
  5. If one of the premises is negative, then the conclusion has to be negative.
  6. If one of the premises is particular, then the conclusion has to be particular.
  7. A term that is not distributed in the premises cannot be distributed in the conclusion.

 

The first 4 rules mentioned above will help us determine if a conclusion is possible and the last 3 rules will help in determining the nature of the conclusion. In the conclusion based on considering both premises, the middle term will be absent from the conclusion. Additionally, do not rely on factual knowledge while answering deductive logical questions. The information in the statements may be fictitious. There may be circumstances where the ticks and crosses method will have to be coupled with basic levels of verbal ability in order to arrive at the solution. Statements may have to be converted to the form in which ticks and crosses method can be used and then solved.


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