Reasoning Test Syllogism Notes with Examples
Syllogism is a noun which means form of reasoning in which a
conclusion is drawn from two statements, i.e., deductive reasoning. In more clear terms, Syllogism is a
mediate deductive inference in which two propositions are given in such an
order that they jointly or collectively imply the third. Thus, Syllogism can be
defined as ‘a form of reasoning in which the conclusion establishes a relation
between two terms on the basis of both terms being related to the same third
term as derived in the premises’.
For example
Statements: All human beings are mortal.
[A]
Socrates is a human
being. [A]
Conclusions: Socrates is mortal.
The conclusion is reached
through the medium of a middle term, i.e., ‘human being’. with both subject
Socrates and the predicate (mortal). Therefore, in a Syllogism two premises are
necessary to arrive at a conclusion.
Points to Remember
Proposition: A
proposition is a sentence which comprises a subject, a predicate and a copula.
Subject is that about which something is said. Predicate is a term which states
something about a subject and copula is that part of proposition which denotes
the relation between the subject and the predicate.
A proposition also known
as a premises.
Examples:
1. All cows(subject) are(copula)
white(predicate)
2. Some flowers(subject) are(copula)
red(predicate)
Categorical Proposition: A
categorical proposition makes a direct assertion. It has no conditions attached
with it.
For examples, ‘All S are
P’, ‘Some S are P’, ‘No S is P’ etc.
But ‘Either S or P’, ‘If
S, then P’ are not categorical proposition.
Immediate Inference and
Mediate Inference: Immediate inference is drawn from a
single statement whereas the mediate inference is drawn from two statements.
- Major term: The predicate of the
conclusion is called major term.
- Minor term: The subject of the
conclusion is called minor term.
- Middle term: The common term in the
premises is called the middle term.
Types of Categorical Proposition
Categorical proposition
has been classified on the basis of quality and quantity of proposition.
Quantity represents whether the proposition is universal or particular and
quality denotes whether the proposition is affirmative or negative.
Hence there are four
types of categorical propositions:
1. Universal affirmative (A)
2. Universal negative (E)
3. Particular affirmative (I)
4. Particular negative (O)
Universal Affirmative
Proposition (denoted by A):
A proposition of the form
‘All S are P’ is called a Universal Affirmative Proposition i.e., Universal
Affirmative Proposition fully include the subject. Universal affirmative
propositions begin with All, Every etc.
Universal Negative
Proposition (denoted by E):
Universal Negative
Proposition fully exclude the subject. Therefore, a proposition of the form ‘No
S is P’ is called a Universal Negative Proposition. It begins with ‘No’, ‘None
of the’, ‘Not a single’ etc.
Particular Affirmative
Proposition (denoted by I):
Particular affirmative
proposition partly include the subject. Hence, a proposition of the form ‘some
S are P’ is called a Particular affirmative proposition. It begins with ‘Some’
Particular Negative
Proposition (denoted by O):
A proposition of the form
‘some S are not P’ is called Particular Negative Proposition. Particular
Negative Proposition partly exclude the subject.
Methods for Immediate Inference
Implication: In
implication, the quantity of a given proposition are changed. The subject,
predicate and the quality of proposition remain unchanged. Thus, A will be
changed to I and E will be changed to O.
Example 1:
- Statement: All tables are trees. (A)
- Conclusion: Some tables are trees. (I)
Example 2:
1. Statement: No table is tree. (E)
2. Conclusion: Some tables are not tree. (O)
Conversion: In
conversion, the subject becomes the predicate and the predicate becomes the
subject. The quantity of the proposition remains unchanged.
Thus,
- A-type proposition can be
converted into I-type.
- E-type proposition can be
converted into E-type.
- I-type proposition can be
converted into I-type.
- But O-type proposition
cannot be converted.
Example 1:
- Statement: All tables are trees. (A)
- Conclusion: Some trees are tables. (I)
Example 2:
- Statement: No table is tree. (E)
- Conclusion: No tree is table. (E)
Example 3:
- Statement: Some tables are trees. (I)
- Conclusion: Some trees are tables. (I)
Methods for Mediate Inference
Format of the Conclusion: The
conclusion is itself a proposition whose subject is the subject of the first
statement and whose predicate is the predicate of the second statement and the
common term disappears.
Example:
- Statement: All dogs are cats.
- Statement: All cats are bats.
- Conclusion: All dogs are bats.
Steps to find the
Conclusion
Step I: Aligning: Two
propositions are said to be aligned if the common term is the predicate of the
first proposition and the subject of the second one.
If the sentences are not
already aligned then they can be aligned by changing the order of the sentences
or converting the sentences.
Example:
Here, common term is
‘birds’ and it is the predicate of the first proposition and the subject of
second proposition.
Step II: After aligning the two
sentences properly, use the following table to draw
Here, O* mean that the
conclusion or inference is of type O but the subject of inference is the
predicate of the second statement and the predicate of the inference is the
subject of the first statement i.e. its format is opposite to the normal format
of the conclusion
Points to Remember
There are only 6 cases
where a conclusion can be drawn. In other cases, no conclusion can be drawn.
A + I → No
conclusion;
|
E + O → No
conclusion.
|
A + O → No
conclusion;
|
I + I → No
conclusion.
|
E + E → No
conclusion;
|
I + O → No
conclusion.
|
- If two propositions have no
common term then no conclusion could be drawn.
- In ‘Syllogism’ a conclusion has
to be drawn from two propositions.
Solved Examples:
Example 1:
- Statement: I. Some cars are roads.
- Statement: II. Some roads are buses.
Solution: Since, both statements are I-type,
therefore, no mediate conclusion follows. But immediate conclusions can be
followed from conversion of statements (I) and (II).
- Conversion of statement I: Some
roads are cars.
- Conversion of statement II: Some
buses are roads.
Example 2:
- Statement: I. Some men are lions.
- Statement: II. All lions are foxes.
Solution: Here, some men are lions. … I-type
All lions are foxes. …
A-type.
Conclusion:I + A = I-type
∴ Some men
are foxes.
Also, conversion of
statement I : Some lions are men.
Conversion of statement
II : Some foxes are lions.
and Implication of
statement II : Some lions are foxes.
Example 3:
- Statement: I. All birds are books.
- Statement: II. All books are cars.
Solution: Here, both statements are of A-type.
and A + A =A-type
conclusion.
All birds are books.
All books are cars.
Conclusions: All birds
are cars.
Some birds are books.
(Implication of statement I)
Some books are cars.
(Implication of statement II)
Some books are birds.
(Conversion)
Some cars are books.
(Conversion)
Example 4:
- Statement: I. Some dogs are cats.
- Statement: II. No cat is cow.
Solution: Since I + E = O-type
conclusion.
Conclusions:Some dogs are
not cow.
Some cats are dogs.
(conversion of I)
Some cats are not
cow.(Implication of II)
No cow is cat.
(conversion of II)
Example 5:
- Statement: I. All fathers are sons.
- Statement: II. No son is educated.
Solution: Here, A + E = E-type
conclusion.
Conclusions:No father is
educated.
Some fathers are
sons.(Implication of I)
Some sons are fathers.
(conversion)
Some sons are not
educated.(Implication of II)
Example 6:
- Statement: I. No magazine is cap.
- Statement: II. All caps are cameras.
Solution: Since E + A = O*-type
conclusion.
Conclusions:Some cameras
are not magazine.
Some caps are cameras.
(Implication of II)
Some magazines are not
cap. (Implication of I)
Some cameras are caps.
(conversion of II)
No cap is magazine.
(conversion of I)
Example 8:
- Statement: I. No table is water.
- Statement: II. Some water are
clothes.
Solution: Here, E + I = O*-type
conclusion.
Conclusions:Some clothes
are not tables.
Some tables are not
water. (Implication of I)
No water is table.
(conversion of I)
Some clothes are water.
(conversion of II)
