# To Solve Syllogism Questions with the help of Rules

Hello Aspirants,
In all of your exams, 5 questions come from the topic Syllogism in Reasoning Section. In these questions you are given some statements and conclusions, and based on the statements you are asked to tell the correctness of the conclusions.

First identify the type of statements as:

 Statement Type A All A are B E No A is B I Some A are B O Some A are not B

When the subject and predicate of a conclusion are present in a single statement, the rules are:

A(All A are B)              →    I(Some A are B)
E(No A is B)                →    O(Some A are not B)
I(Some A are B)          →    A(All A are B) with Possibility & O(Some A are not B) with Possibility
O(Some A are not B)   →    E(No A is B) with Possibility & I(Some A are B) with Possibility

Examples –

1. Statements: All trees are plants. Some plants are flowers.
Conclusions: Some trees are plants
Conclusion’s subject (trees) and predicate (plants) present in a single statement (All trees are plants). Now since in the rules we have A → I. This gives that if ‘All trees are plants’, then ‘Some trees are plants’ is true
2. Statements: All trees are plants. Some plants are flowers.
Conclusions: All plants are flowers. All plants are flowers is a possibility.
Since in the rules we have I → A with possibility. So ‘All plants are flowers is a possibility’ is true. And ‘All plants are flowers’ is not true.

When two or more statements are required to answer the correctness of conclusion, the rules are:

First, check whether all statements are aligned i.e. the subject of one statement should be same as the predicate of previous statement
For example: Consider statements –
All A are B. Some B are C. All C are D. No D is E
These statements are aligned. But if the statements are given like
All A are B. Some C are B. All C are D. No D is E
We see that the second statement (Some C are B) is not aligned with first statement (All A are B) and then third statement is also not aligned with second statement. So we must align the second statement.
During the alignment or conversion of statements, the subject and the predicate changes. Rules for alignment or conversion of statements are
A(All A are B)              →    I(Some B are A)
E(No A is B)                →    E(No B is A)
I(Some A are B)          →    I(All B are A)
O(Some A are not B)   →    not convertible
So, in the statements – All A are B. Some C are B. All C are D. No D is E.
Some C are B (I type) will be converted to Some B are C. and the statements will be now written as
All A are B. Some B are C. All C are D. No D is E.

After the alignment is done, the conclusions can be checked as:
First see the subject and predicate of the conclusion, then see where that subject and that predicate is present and then apply following rules to check the correctness of that conclusion:

A(All A are B) + A(All B are C)                           →         A(All A are C)
A(All A are B) + E(No B is C)                             →         E(No A is C)
A(All A are B) + I(Some B are C)                       →         No conclusion
A(All A are B) + O(Some B are not C)                →         No conclusion

E(No A is B) + A(All B are C)                              →         O reverse(Some C are not A)
E(No A is B) + E(No B is C)                                →         No conclusion
E(No A is B) + I(Some B are C)                          →         O reverse(Some C are not A)
E(No A is B) + O(Some B are not C)                   →         No conclusion

I(Some A are B) + A(All B are C)                        →         I(Some A are C)
I(Some A are B) + E(No B is C)                          →         O(Some A are not C)
I(Some A are B) + I(Some B are C)                    →         No conclusion
I(Some A are B) + O(Some B are not C)             →         No conclusion

O(Some A are not B) + A(All B are C)                 →         No conclusion
O(Some A are not B) + E(No B is C)                   →         No conclusion
O(Some A are not B) + I(Some B are C)             →         No conclusion
O(Some A are not B) + O(Some B are not C)      →         No conclusion

O reverse(Some B are not A) + A(All B are C)               →         O reverse(Some C are not A)
O reverse(Some B are not A) + E(No B is C)                 →         No conclusion
O reverse(Some B are not A) + I(Some B are C)           →         No conclusion
O reverse(Some B are not A) + O(Some B are not C)    →         No conclusion

Example:
Statements: Some horses are goats. All goats are dogs. All dogs are cats. Some cats are tigers.
Conclusions:
I. Some tigers are horses.
II. Some cats are goats.
III. Some dogs are horses.
IV. No tigers is goats.
All the statements are aligned. The statement type is respectively I, A, A, I
To check:
1st conclusion – for this conclusion, see that tigers is present in last statement and horses is present in first statement. So we have to see all statements. Now I (Some horses are goats) + A (All goats are dogs) → I (Some horses are dogs).
I (Some horses are dogs) + A (All dogs are cats) → I (Some horses are cats).
I (Some horses are cats) + I (Some cats are tigers) → No conclusion
Since we have got no conclusion, the conclusion ‘Some tigers are horses’ does not follow.
2nd conclusion – this can be checked by only 2nd and 3rd statement.
A (All goats are dogs) + A (All dogs are cats)→ A (All goats are cats).
So we get that ‘All goats are cats’ is true. Remember from the conversion rules A → I. So ‘All goats are cats’ → ‘Some cats are goats’
So the conclusion ‘Some cats are goats’ follows.
3rd conclusion – this can be checked by only 1st and 2nd statements.
I (Some horses are goats) + A (All goats are dogs) → I (Some horses are dogs)
So we get that ‘Some horses are dogs’ is true. Remember from the conversion rules I → I. So ‘Some horses are dogs’ → ‘Some dogs are horses’
So the conclusion ‘Some dogs are horses’ follows.
4th conclusion – this can be checked by 2nd , 3rd and 4th statements.
A (All goats are dogs) + A (All dogs are cats)→ A (All goats are cats).
A (All goats are cats) + I (Some cats are tigers) → No conclusion
Since we have got no conclusion, the conclusion ‘No tigers is goats’ does not follow.

If two conclusions contain same subject and predicate, then check:

• If both conclusions are incorrect applying the above rules.
• If one conclusion is positive and the other is negative.
• If there is complementary pair – A,O      I,O       I,E

If all of these conditions are true in case of two conclusions with same subject and predicate, the answer will be either of the two conclusions is correct.

Example:
Statements: All A are B, Some B are C, All C are D, No D is F.
Conclusions:
I. No B is F
II. Some B are F
To solve for these 2 conclusions, we will see the last three statements as:
Some B are C (I) + All C are D (A) → I
I + No D is F (E) → O
This means O (Some B are not F) will be correct.
This means both conclusions I and II are incorrect. This satisfies the 1st condition.
2nd condition says that one should be positive and the other negative. In this, I conc. is negative and II is positive. This satisfies the 2nd condition.
Third condition is also satisfied, the conclusions I and II make pair E,I which satisfies the 3rd condition.

Since all the three conditions satisfies, so answer will be either I or II is correct.

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