A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. Questions under this topic contain two or more statements which may be followed by conclusions. You have to find out which of the conclusions logically follows from the given statements.
There are four ways in which the relationship can be made.
All A are B – Means the whole circle representing A lies within the circle representing B.
Here we can also make a conclusion: Some B are A. Some A are B.
For example, All boys are men.
Here we can also make a conclusion: Some men are boys. Some boys are men.
All apples are fruits.
Here we can also make a conclusion: Some fruits are apples. Some apples are fruits.
No A is B – means that circles representing A and B does not intersect at all.
For example No ball is bat.
No door is wall.
Some A are B
Means that some part of the circle represented by A is within the circle represented by B.
This type of (category 3) statement gives the following conclusions:
(i) Some A are B also indicates that – Some A are not B
(ii) Some A are B also indicates that – All A are B.
(iii) Some A are B also indicates that – All B are A.
(iv) Some A are B also indicates that – All A are B and All B are A.
For e.g.: Some mobiles are phones.
Some A are not B
Means that some portion of circle A has no intersection with circle B while the remaining portion of circle A is uncertain whether this portion touches B or not.
(i) Some A are not B also indicates that – Some A are B.
(ii) Some A are not B also indicates that – No A is B.
Important Points –
1. At least statement – At least statement is same as some statement.
E.g: Statement: All kids are innocent.
Here we can make a conclusion: At least some innocent are kids (Some innocent are kids).
2. Some not statement: Some not statement is opposite to “All type” statement. If All being true then Some not being false
E.g.1. Statement: Some pens are pencils. No pencils are jug. Some jug is pens.
Here we can make conclusions: Some pens are not pencils, which is true. In the above figure, green-shaded part shows; some pens are not pencils, because in the statement it is already given, "No pencils are jug".
Complementary Pairs: (Either & or) – Either and or cases only takes place in complementary pairs.
Conclusions: (i) Some A are B. (ii) No A are B.
From the given above conclusions, it is easy to understand that one of the given conclusions must be true, which is represented by option either (i) or (ii). These types of pairs are called complementary pairs.
Note: ‘All A are B’ & ‘Some A are not B’ are also complementary pairs.
It is important to note that, in complementary pairs, one of the two conclusions is true and the other will be false.
For example –Statement: All A are B. Some B are C.
Conclusion: I. All C are A. II. Some C are not A.
Here we can make a conclusion, either I or either II follows.
Possibility cases in Syllogism – In possibilities cases, we have to create all possibilities to find whether the given conclusion is possible or not. If it is possible and satisfies the given statement than given conclusion will follow otherwise conclusion will not follow.
1. E.g. Statement: All A are B. Some B are C.
Conclusion: All A being C is a possibility.
Conclusion is true.
Possibility figure –
2. E.g.Statements: No stone is white. Some white is papers.
Conclusions: I. All stones being paper is a possibility.
Conclusion is true.
3. E.g.Statements: Some mouse is cat.
All mouse are pets. No pet is animal.
Conclusions: I. All mouse being animal is a possibility.
The conclusion is false because the possibility figure is not possible.
If we say all mouse being an animal is a possibility is true, then given statement "No pet is animal" will be wrong. Here in the statement, it is given "No pet is animal" and "All mouse is pet". So we can make also conclusion here that "no mouse are animal" is true.
Re-statement is not a conclusion – Conclusion has to be different from the statement.
E.g. Statement – All A are B
Conclusion – All are B. (invalid) Conclusion does not follow.
Conclusion – Some A are B (follow) Conclusion follows.
Note: If the statement and conclusion are the same then, the conclusion does not follow. This rule also follows in possibilities case
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