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Syllogism (Logical Deductions)

NOTE:

* Restatements always false.

* If any conclusion is false in any one diagram (B.D + P.D), that conclusion is always false.

* If any conclusion is true in all the diagrams (B.D + P.D), that conclusion is always true.

* If statement/ statements are in +VE, conclusions are in –VE: then that –VE conclusions are always false (VISE-VERSA).

ALL
Statements: All A are B.
                     All B are C.

Conclusions:

1. All A are B (False)

2. All A are C (True)
3. All B are C (False)
4. All C are A (False)
5. All C are B (False)
6. All B are A (False)


Conclusions:
1. Some A are B (True)
2. Some A are C (True)
3. Some B are C (True)
4. Some C are A (True)
5. Some C are B (True)
6. Some B are A (True)


Conclusions:

1. Some A are not B (False)
2. Some A are not C (False)
3. Some B are not C (False)
4. Some C are not A (False)
5. Some C are not B (False)
6. Some B are not A (False)


Conclusions:

1. No A are B (False)
2. No A are C (False)
3. No B are C (False)
4. No C are A (False)
5. No C are B (False)
6. No B are A (False)


SOME
Statement: Some A are B
Basic diagram:


Possible diagrams:


Conclusions:

1. All A are B (False)
2. All B are A (False)
3. Some A are B (False)
4. Some B are A (True)
5. No A is B (False)
6. No B is A (False)
7. Some A are not B (False)
8. Some B are not A (False)


NO
Statement: No A is B.


Conclusions:

1. All A are B (False)
2. All B are A (False)
3. Some A are B (False)
4. Some B are A (False)
5. No A is B (False)
6. No B is A (True)
7. Some A are not B (True)
8. Some B are not A (True)


SOME NOT
Statement
: Some A are not B


Conclusions:

1. All A are B (False)
2. All B are A (False)
3. Some A are B (False)
4. Some B are A (False)
5. No A is B (False)
6. No B is A (False)
7. Some A are not B (False)
8. Some B are not A (False)


EITHER OR CASE
Conditions:

* Given 2 conclusions must be false individually.
* Elements must be same in both the conclusions.
* Complimentary pair must be available in both the conclusions.


SPECIAL CASES



Conclusions:

I. No A is C (True)
II. No C is A (True)
III. Some A are not C (True)
IV. Some C are not A (True)
3. Statements:

All A are B.
No A is C.


Conclusions:

I. Some B are not C (True)
II. Some C are not B (False)


POSSIBILITY CASE
1. Statements:
 All A are B.
                          Some B are C.
                          No C is D.


Conclusions:
I. Some B’s being D’s is a possibility (True)
II. All A’s being C’s is a possibility (True)
III. All D’s being A’s is a possibility (True)
IV. Some C’s being B’s is a possibility
(False)
Note: Existing relations or Established
relations are always FALSE in Possibility
conclusions.


2. Statements: No A is B.
                         All B are C.
                         Some C are D.


Conclusions:

I. All C’s can be A’s (False)
II. All A’s can never be D (False)
III. Some B ‘s can be D’s (True)
IV. There is possibility that Some C’s are A’s (True)
V. There is possibility that Some D are C (False)


Model Questions
Directions (Q.1-3):
In each of the questions below are given three statements followed by two conclusions numbered I and II. You have to take the given statements to be true even if they seem to be at variance from commonly known  facts. Read all the conclusions and then decide  which of the given conclusions logically follows  from the given statements disregarding commonly  known facts.


Give Answer:
A) If only Conclusion I follows
B) If only Conclusion II follows
C) If either Conclusion I or Conclusion IIfollows
D) If neither Conclusion I nor Conclusion IIfollows
E) If both Conclusion I and II follows


1. Statements: Some Dares are Dream.
                         All Dreams are Real.
                         No Real is Fake.
Conclusions:

I. Some Dreams are Fake.
II. All dream is not fake.


2. Statements:

No Rain is Game.
Some Games are Chain.
No Chain is Lane.
Conclusions:

I. Some Chains are not Rain.
II. Some Games are Lane.


3. Statements:   Some Movie are Ticket.
                           No Ticket is Popcorn.
                           Some Popcorn is Burger.
Conclusions:

I. Some Movies are Burger.
II. Some Burgers are not Movie.


4. Statements:  Some Bag is Hot.
                          No Hot is Cake.
                          All Cakes are Milk.
Conclusions:

I. Some Bag is not Cakes.
II. Some Hots can be Milk.
III. Some Milk is not Hot.


Give Answer:
A) If only conclusion II follows
B) If both conclusions II and III follow
C) If both conclusions I and III follow
D) If both conclusions I and II follow
E) If all conclusions follows

Key With Explanations

Directions (1 - 10): In each questions below are given some statements followed by two conclusions I and II. You have to take given statements to be true even if they seem to be at variance from commonly known facts and then decide which of the given conclusions logically follow/s from the given statements, disregarding commonly known facts. Read both the conclusions and give answer as -
1) If only conclusion I follows.
2) If only conclusion II follows.
3) If either conclusion I or II follows.
4) If neither conclusion I nor II follows.
5) If both conclusions I and II follow.
 

1. Statements: All mobiles are androids.
                          No android is a phone.
Conclusions: I. Some mobiles are not phones.
                      II. No phone is a mobile.
Ans: Both conclusions I and II follow.
 

Explanation:
                             
 

2. Statements: Some nails are pins.
                          Some pins are fingers.
Conclusions: I. No finger is a nail.
                      II. Some nails are fingers.
Ans: Either I or II follows.
 

Explanation:
                Basic diagram                          Alternate diagram                   
      

3. Statements: Some cranes are ducks.
                           No duck is a peacock.
Conclusions: I. No peacock is a crane.
                      II. Some peacocks are cranes.
Ans: Either I or II follows.
 

Explanation:
                     Basic diagram                                                  Alternate diagram
     

4. Statements: No chair is a glass.
                          All glasses are tables.
Conclusions: I. All tables are chairs.
                      II. Some tables are not chairs.
Ans: Only conclusion II follows.
 

Explanation:
                         
               When no table is a chair - definitely 'Some tables are not chairs' can be concluded. But this negative can conclusion in this question cannot have its complementory Conclusion (i.e. - All tables are chairs) because the first statement conditions get violated.
 

5. Statements: All cinemas are films.
                           All films are movies.
Conclusions: I. Some movies are cinemas.
                      II. All cinemas are movies.
Ans: Both conclusions I and II follow.
 

Explanation:
                                  
 

6. Statements: All friends are officers.
                          Some relatives are friends.
    Conclusions: I. No officer is a relative.
                         II. Some relatives are officer.
Ans: Only conclusion II follows.
 

Explanation:

 7. Statements: Some girls are engineers.
                          Some engineers are teachers.
                          All teachers are politicians.
Conclusions: I. No politician is a girl.
                      II. Some politicians are engineers.
Ans: Only conclusion II follows.
 

Explanation:
              
                  In this question for 'No politician in a girl' gets its Complementary Conclusion i.e. Some Politicians are girls. Since it is not given, the given negative Conclusion become invalid.

                      see the alternate picture.
 
 

8. Statements: All fingers are rings.
                           No ring is a chain.
                          Some bags are rings.
Conclusions: I. No bag is a chain.
                      II. Some bags are chains.
Ans: Either I or II follows.
 

Explanation:
                     Basic diagram                                                     Alternate diagram
          

9. Statements: All numbers are digits.
                           Many digits are dots.
                           No dot is a light.
Conclusions: I. Some dots are numbers.
                      II. Some lights are digits.
Ans: Neither conclusion I nor II follows.
 

Explanation:
                     Basic diagram                               
              

10. Statements: All songs are smiles.
                             All smiles are lives.
                             No life is a cinema.
Conclusions: I. No cinema is a song.
                      II. No smile is a cinema.
Ans: Both conclusions I and II follow.
 

Explanation:
   

*  Syllogism (Logical Deductions) is one of the important topics of competitive examinations. In the reasoning (Mental ability) part of every bank examination, questions from 'Syllogism' topic are frequently asked. These type of questions are generally solved by using two methods.
i) Through Syllogism rules.
ii) Through drawing Venn diagrams.
              The first method (Syllogism rules) is preferred, when the question has only two statements, but for the questions which have more than two statements, the second method (venn diagrams) is the best. The task of the students for this type of questions is 'drawing the appropriate conclusions', from the given statements. To draw the appropriate conclusion one must think logically to understand the logical implications of the given statements and verify the conclusions for their truthfulness, strictly from the conditions of the statements.
              Each statement has two words and they are related with each other with some kind of relationship. The relationship in based upon the number (All (or) some) and the nature (positive (or) negative). So, overall we get four types of statements. Study the following examples.
a) All Teachers are Professors
b) No Teacher is a Professor
c) Some Teachers are Professors
d) Some Teachers are not Professors.
              (words like 'few', 'most', 'many', 'more' have the same relation as the word 'some' in the statement, and the words like 'Not all' 'All not' have the same relation as the word 'some not' in the statement, the word 'Never' has the same relation as the word 'No' in the statement.)
              As we have discussed above, that 'venn diagram' method is the preferable method for solving questions, then it is necessary to know how to draw a venn diagram for the given statements. There is a possibility to draw more than one diagram (i.e. different diagrams) to represent the relation between the terms of the statements.
For example- Statement: 'All Teachers are Professors.'

  Statement: 'Some Teachers are Professors'

     
           So, the most important part of Syllogism statements is knowing how to draw a 'basic diagram' and an alternate diagram'.
           For instance, study the following diagrams for the given statements.
Statements: All fruits are mangoes.
                      Some mangoes are apples.
                         diagram-I                                                       diagram-II

Here diagram-I is called 'a basic diagram'. diagram-II is called 'an alternate diagram.'
           So, what is the difference we have observed, let us discuss.
           In diagram-I (basic diagram), 'the fruit circle' and 'the apple circle' were kept away from each other.
           In diagram-II (alternate diagram), 'the fruit circle' and 'the apple circle' both were intersected with each other. Because of this difference we get two opposite relations between the 'fruits and apples' in the conclusions, i.e. according to basic diagram
- 'No fruit is an apple', according to alternate diagram- 'Some fruits are apples'.
   Hence, these two conclusions together form ' a complementary pair'. So, we mark the answer as 'either conclusion I or conclusion II follows'.
So, let us discuss now -
i) How to draw a basic diagram?
ii) What is an alternate diagram and how do we draw it?
iii) Do we need to draw a basic diagram compulsorily for every given statements?
iv) Do we need to draw an alternate diagram for every given statements?
v) Do we get more than one basic diagram?
vi) Do we get more than one alternate diagram?
The following are the answers for the above questions.

 

Question No.1.
'How to draw a basic diagram'?
Answer: When we are representing the given relation between terms of the statements in the form of "circle representation ' (or venn diagram), we shall take the minimum over laping area in the basic diagram. And another the most important point that, we should always remember is - 'we shall try to keep the circles of the terms away from each other/ one another for which, the specific relation is not given in the statements.

 

Study the following example-
Statements: All bags are purses.
                          Some purses are oranges

    Here in the first statement the relation is given between bags and purses (i.e All bags are purses), so the individual diagram is
                                       
     In the second statement the relationship is given between purses and oranges (i.e., some purses are oranges). So, the individual diagram is
               
    But there is no specific relation was given between 'the bags and the oranges' so, they could be kept separated from each other.
 However it is not possible in all cases, in few, even though a specific relation is not given between the words some time they may get intersected or one may come in the other circle. Because some indirect relation flows between/ among the words.
Let us study the following examples:
Example: 1
  Statements : All Pens are Books.
                             Some Markers are Pens
Basic diagram
                                        
    Here also, there is no specific relation was given between Books or Markers, inspite they got intersected because the ralation is given between Pens (which is inside the Books) and Markers.
As the relation in 'Some Markers are Pens' 'The Markers' Should intersect with 'Pens' and it is possible only through intersecting 'The Books'. So the relation is established between 'Books and Markers'
Example: 2    Statements: All Doctors are Men
                                                     All Men are Experts
Basic diagram
                                                    
          Here also, there was no specific relation between 'Doctors and Experts' since the 'Doctor's circle is inside of the Men circle, which is inside the Experts circle so, automatically the Doctor's circle is falling inside of the Experts circle' and the relation is established between 'the Doctors and the Experts.  So, from the above examples we understand that in a few cases due to indirect relationship we establish some kind of relationship between those terms, which have not given any spicific relation between them in the statements.

 

Question No. 2:
What is an alternate diagram and how do we draw it?
Ans: In the alternate diagram we try to overlap the maximum area of the circle. Hence the terms, which have not given any specific relation between them in the statements, we establish the relation (i.e by overlaping or intersection etc.,) in the diagram, without   violating the conditions of the statements.
Look at the following example:
    Statements: All animals are birds
                                Some tigers are birds

    so we have now understood the difference between the basic diagram and an alternate diagram
 

Question No. 3:
"Do we need to draw a basic diagram compulsorily for every given statements"?
Ans: Yes, the basic diagram is compulsory for any given statements of the premises.
3 (a) What is premises?
Ans:
The statements for which we draw the venn diagram and from them we draw the conclusions.

 

Question No 4:
"Do we need to draw an alternate diagram for every given statements of the premises"?
Ans: No, only when the nagative conclusion follows from the basic diagram, to confirm do we get the positive complementary conclusion for the negative conclusion (which is proved right or follow from basic diagram). We require to draw an alternate diagram. Now study the following example carefully.
Statements: All fans are lights
                           Some lights are fruits
conclusions: i) No fan is a fruit
                             ii) Some fans are fruits
Basic diagram for the above example
                                          
    From this diagram, the first conclusion -'No fan is a fruit' which is a negative conclusion follows.
So, to get its positve complementary conclusion.
    - 'Some fans are fruits' we need to draw an alternate diagram.

 

Alternate diagram
                              
    So, we clearly see here 'fans and fruits' circles got intersecting without violating  any condition of the statements of the premises. So, the negative conclusion from a basic diagram and positive conclusion from an alternate diagram
           i.e. i) No fan is a fruit (negative)
                 ii) Some fans are fruits (positive)
    together are complementary to each other. Hence we mark the answer as 'either conclusion I or conclusion II follows!
        Here we get two doubts.
4(a): For every negative conclusion do we get a positive complementary conclusion?
4(b): What happens, if in the set of given conclusions for negative conclusion, there is not any its positive complementary conclusion?
         Now, Follow the following discussion-

 

4(a) For every nagative conclusion do we get its positive complementary conclusion?
Ans: Not necessarily.
Example: Statements: All Teachers are Professors
                                               No Professor is a Student
                  Conclusions: I) No Teacher is a Student
                                               II) Some Teachers are Students
Basic diagram:
                                 
     From this basic diagram we can prove, the given negative conclusion i.e., 'No Teacher is a Student' follows
Alternative diagram
                                            
     From this also we are proving only the negative conclusion follows. Because we are not getting a different alternative diagrams than the basic diagram. (Because if 'the students circle' has approach (intersect), 'the teacher circle' it should go via 'professors circle', then the condition of the statement i.e., 'No professor is a student' gets violated, hence we have to keep "professor and student" circles apart from each other, therefore "the teacher circle and the student circle" are also remain apart from each other. Hence there is not any positive complementary conclusion possible for the given negative conclusion.
         Hence only the conclusion  I  follows.
         So, the answer is only  conclusion I follows.Now you can ask 'already in the set of given conclusions we have given a complementary pair i.e., 
                   I. No Teacher is a Student
                 II. Some Teachers are Students

 

     So, do we need to check in the alternate diagram, whether they are complementary to each other or not?
 

Ans: Ofcourse, even though the given pair is a complementary pair, whether they are complementary to each other or not, we have to confirm, by drawing the alternate diagram. Because some times they may not be complementary to each other, which we have proved in the above example.
Now the other doubt "Now what we should do with the second conclusion i.e., 'some teachers are students'?
Ans: We are not able to get this conclusion in the alternative diagram (because in the basic diagram any how it cannot be proved so we try in the alternative diagram) also. Hence the second conclusion does not follow.
           So, dear friends let us now go back to our doubt 4 (b).


                                   

Posted Date : 22-01-2021

 

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.

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