Possibility and conditional type syllogism Questions for bank P.O Exams

http://www.amazon.com/dp/B012BESC7U for Rapid Reasoning
Method:
In Transformed RAVAL’S NOTATION, each premise and conclusion is written in abbreviated form, and then conclusion is reached simply by connecting abbreviated premises.
NOTATION: Statements (both premises and conclusions) are represented as follows:
Statement                           Notation
a) All S are P                        SS-P
b) Some S are P                  S-P                
c) Some S are not P           S / PP
d) No S is P                         SS / PP
(- implies are and / implies are not)
All is represented by double letters; Some is represented by single letter. Some S are not P is represented as S / PP. No S is P implies No P is S so its notation contains double letters on both sides.
RULES:
(1) Conclusions are reached by connecting Notations. Two notations can be linked only through common linking terms. When the common linking term multiplies (becomes double from single), divides (becomes single from double) or remains double then conclusion is arrived between terminal terms. (Aristotle’s rule: the middle term must be distributed at least once).
(2)If both statements linked are having – signs, resulting conclusion carries – sign (Aristotle’s rule: two affirmatives imply an affirmative).
(3) Whenever statements having – and / signs are linked, resulting conclusion carries / sign. (Aristotle’s rule: if one premise is negative, then the conclusion must be negative).
(4)Statement having / sign cannot be linked with another statement having / sign to derive any conclusion. (Aristotle’s rule: Two negative premises imply no valid conclusion).

Following illustrations will make the above rules very clear

1. Statements                      Notation
a) All S are P                        a) SS - P
b) All P are Q                       b) PP - Q  

Valid Conclusions    
1. All S are Q                       1.SS - Q  
2. Some S are Q                   2.S - Q
3. Some Q are S                   3.Q - S
4. Some P are S                   4.P - S
5. Some Q are P                   5.Q - P
6. Some S are P                   6.S - P
7. Some P are Q                  7.P - Q

Wrong Conclusions                                                                                                                          
1. All Q are S                         1.QQ - S
2. All P are S                         2.PP - S
3. All Q are P                         3.QQ - P
4 Some S are not Q                4. S / QQ
5. Some Q are not S               5.Q / SS
6. Some P are not S               6.P / SS  
                                                   
Explanation: From a) SS – P      
                         b) PP – Q
     Valid Conclusions
1. SS – Q follows, because here common linking term (P) multiplies.
2. S - Q follows, because “Some” is part of All(S is included in SS but not vice versa) and common linking term (P) multiplies.
3. Q - S follows, because here common linking term (P) divides.
4. P- S   follows from the main statement SS – P (by reverse reading).
5. Q – P follows from the main statement PP – Q (by reverse reading, one can isolate P from PP).                  
6. S – P   follows from the main statement SS – P
7. P – Q follows from the main statement PP – Q

Wrong Conclusions
1. QQ – S does not follow because we don’t have any QQ in statement notation.                                       2. PP – S does not follow because there is no common linking term between PP and S.                                 3. QQ – P does not follow because we don’t have any QQ in statement notation.
4. S / QQ is ruled out because we don‘t have any / sign in statement notation.
5. Q / SS is ruled out because we don’t have any / sign in statement notation.
6. P / SS is ruled out because we don’t have any / sign in statement notation.

2. Statements                         Notation
a) All S are P                           a) SS - P
b) Some P are Q                      b) P - Q

Valid Conclusions
1. Some S are P                         1.S - P
2. Some P are S                         2.P - S
3. Some Q are P                        3.Q - P

Wrong Conclusions  
1. All S are Q                              1.SS - Q
2. All P are S                               2.PP - S
3. Some Q are S                          3.Q - S
4. Some Q are not S                   4.Q / SS

Explanation: From a) SS - P
                        b) P – Q
Valid Conclusions
1. S – P follows from statement SS - P                                                                  
2. P – S follows from statement SS – P (reverse reading).
3. Q – P follows from statement P – Q (reverse reading).

Wrong Conclusions
1. SS – Q does not follow because common linking term (P) remains singular.
2. PP – S does not follow because PP is not present in any statement notation.
3. Q - S does not follow because common linking term (P) remains singular.
4. Q / SS is ruled out because we don’t have any / sign in statement notation.


3. Statements:                            Notation
 a) Some S are P                             a) S - P
  b) Some P are Q                           b) P- Q

Valid Conclusions:  
1. Some P are S                              1.P - S
2. Some Q are P                             2.Q –P

Wrong Conclusions
1. Some S are Q                              1.S - Q
2. Some Q are S                              2.Q - S
3. All P are S                                  3.PP - S
4. All Q are S                                 4.QQ - S

Explanation: From a) S- P
                         b) P- Q                                                              
Valid Conclusions
1. P- S follows from statement S- P (reverse reading).
2. Q- P follows from statement P- Q (reverse reading).

 Wrong Conclusions
1. S- Q does not follow because common linking term (P) remains singular.
2. Q- S does not follow because common linking term (P) remains singular.
3. PP- S does not follow because PP is not present in any statement notation.
4. QQ- S does not follow because QQ is not present in any statement notation.

4. Statements:                       Notations:
a) All S are P                            a) SS- P
b) No P is Q                             b) PP / QQ        
                                                 
Valid Conclusions
1. No S is Q                          1. SS / QQ
2. No Q is S                          2. QQ / SS
3. No Q is P                          3. QQ/ PP
4. Some S are not Q              4. S / QQ
5. Some Q are not S               5. Q / SS
6. Some P are not Q               6. P / QQ
7. Some Q are not P               7. Q / PP

Wrong conclusions
 1. Some Q are P                          1. Q - P
 2. All P are S                              2. PP –S

Explanation: From a) SS-P
                         b) PP/QQ                                                                                  
Valid Conclusions
1. SS / QQ follows because common linking term (P) multiplies (Whenever – and / are linked result   carries / sign).
2. QQ / SS follow because common linking term (P) divides.
3. QQ / PP follow from statement b) on reverse reading.
4. S / QQ follows because common linking term (P) multiplies (one can read S from SS).
5. Q / SS follow (by reverse reading) because common linking term (P) divides.
6. P / QQ follow from PP / QQ
7. Q / PP follows from PP/QQ (reverse reading).  

Wrong conclusions
 1. Q –P does not follow because Q is linked with P via / sign.
 2. PP - S does not follow from SS – P

5. Statements:                       Notation
 a) No S is P                           a) SS / PP
 b) No P is Q                          b) PP / QQ

Wrong Conclusions
 1. No S is Q                            1.SS / QQ
 2. No Q is S                            2.QQ / SS
 3. Some P is S                        3. P - S
 4. Some Q is S                        4.Q – S

  Explanation: From a) SS /PP
                           b) PP /QQ
       Wrong Conclusions
1. SS / QQ do not follow because both statements have / sign so statements a) and b) cannot be combined to deduce any conclusion
2. QQ / SS do not follow because both statements have / sign.
3. P- S does not follow because we don’t have – sign between P and S anywhere in statement   notation.    
4. Q- S does not follow because both   statements have / sign.                        

6. Statements:                       Notation
a) Some S are P                    a) S-P
b) No P is Q                           b) PP/QQ
c) All T are Q                         c) TT- Q

Valid Conclusions  
1. Some S are not Q             1.S / QQ
2. Some P are not T              2.P / TT
3. Some T are not P              3.T / PP
4. Some S are not T               4.S / TT

Wrong Conclusions
1. Some Q are not S               1.Q / SS                                                                                             2.Some T are not S                 2.T / SS
                               
Explanation: From a) S – P
                         b) PP / QQ
                         c) TT-Q

Valid Conclusions
1. S / QQ follow because common linking term (P) multiplies.
2. P / TT follow because common   linking term (Q) divides                                                                      
3. T / PP follow because common linking term (Q) multiplies.                                                                   4. S / TT follows because first common linking term (P) multiplies then second linking term (Q) divides
           
Wrong Conclusions
1. Q / SS do not follow because   terminating term S is in single letter.
2. T / SS do not follow because terminating term S is in single letter.


7. Statements:                            Notation
  a) Some S are P                        a) S- P
  b) All P are Q                            b) PP- Q
  c) All Q are T                             c) QQ- T

Valid Conclusions
1. Some S are T                                1. S - T
2. Some T are S                                2. T – S

Explanation: From a) S- P
                         b) PP – Q
                         c) QQ- T
Valid Conclusions
1. S – T follows, when we move from S towards T, P multiplies so we reach Q since Q also   multiplies we get S- T
2. T- S follows, when we move from T towards S, Q divides so we reach PP since PP also divides we get T- S

Directions (Q.1-26): Below are given three or four statements followed by three or four conclusions. You have to take the given statements to be true even if they appear to be at variance with commonly known facts, and then decide which of the conclusions logically follow(s) from the given statements. For each question, mark out an appropriate answer choice that you think is correct.
1. Statements:                         Notations
a. All locks are keys.               LL - K
b. All keys are balls.               KK - B
c. Some clocks are balls.       C – B

  Conclusions:                          Notations
1. Some balls are locks.           B - L
2. Some clocks are keys.         C – K        x
3. All keys are locks.                KK – LL     x

A) Only 1 and 2 follow      B) Only 2 and 3 follow    C) Only 1 follows         D) Only 2 follows     E) 1, 2 and 3 follow
Answer:C) Only 1 follows

2. Statements:                          Notations
a. Some cups are pots.             C - P
b. All pots are toys.                   PP - T
c. All cups are bottles.              CC - B

Conclusions:                              Notations
1. Some bottles are toys.         B - T
2. Some pots are bottles.         P - B
3. Some toys are cups.              T - C

A) Only 1 and 2 follow      B) Only 2 and 3 follow    C) Only 1 and 3 follow       D) 1, 2 and 3 follow   E) None follows
Answer:D) 1, 2 and 3 follow.

3. Statements:                             Notations
a. All papers are books.               PP - B
b. All bags are books.                   BaBa - B
c. Some purses are bags.             Pu - Ba

Conclusions:                                Notations
1. Some papers are bags.            P – Ba       x
2. Some books are papers.         B - P
3. Some books are purses.         B - Pu

A) Only 1 follows          B) Only 2 and 3 follow        C) Only 1 and 3 follow    D) Only 1 and 2 follow   E) 1, 2 and 3 follow
Answer:B) Only 2 and 3 follow

4. Statements:                              Notations
a. No cloud is Bird.                      CC / BB
b. Some goats are birds              G - B
c. All cats are goats                     CaCa - G

Conclusions:                               Notations
1. No cat is cloud.                      CaCa / CC      x
2. Some cats are birds.             Ca – B             x
3. No bird is cat.                        BB / CaCa       x
4. Some clouds are goats.        C – G               x

A) Only 3 follows    B) Only either 2 or 3 follows     C) Only 1 follows   D) Only 1 and either 2 or 3 follow  E) None of these
Answer:E) None of these

5. Statements:                               Notations
a. All grapes are bananas.            GG - B
b. All apples are bananas             PP- B
c. Some bananas are mangoes    B - M

Conclusions:                                     Notations
1. No grape is mango.                       GG / MM       x
2. Some apples are not mangoes.   A / MM         x
3. Some grapes are apples.              G – A              x
4. All mangoes are grapes.               MM – G         x


A) Only 1 follows       B) Either 1 or 3 follows   C) Only 2 and 3 follow   D) Only 1, 2 and 3 follow   E) None of these
E) None of these

6. Statements:                             Notations
a. Some cats are rats.                C – R
b. Some rats are ants.               R - A
c. Some ants are flies.               A - F

Conclusions:                          Notations
1. Some flies are cats.             F – C               x
2. Some flies are not ants.     F / AA              x
3. No rat is fly.                     RR / FF            x
4. No cat is fly.                     CC / FF            x

A) Only 1 and 4 follow       B) Only 2 follows       C) Only 1 and 3 follow        D) Only 1 or 4 follows   E) None of these
Answer:E) None of these

7. Statements:                           Notations
a. Some bags are books.            Ba - Bo
b. All books are boxes.               Bo Bo - Bx
c. No box is board.                      BxBx / BdBd

Conclusions:                               Notations
1. Some bags are not boards.      Ba /Bd Bd
2. Some bags are not boxes          Ba / Bx Bx        x
3. All bags are boxes                  BaBa – Bx         x
4. No bag is board                     BaBa / BdBd      x

A) Only 1 follows         B) 1 and either 2 or 3 follow    C) Only 4 follows     D) Only 2 follows    E) None of these                                                                                        
Answer:B) 1 and either 2 or3 follow (2 & 3 are comp. pair)                                                                  

8. Statements:                              Notations
a. All chalks are dusters.            CC – D
b. Some chalks are boards.       C - B
c. Some dusters are pens.         D -P

Conclusions:                               Notations
1. Some pens are chalks.           P – C      x
2. Some dusters are boards      D - B
3. Some pens are boards           P – B      x
4. All chalks are pens.                CC –P     x

A) Either 1 or 4 follows          B) Only 2 and 3 follow          C) Either 1 or 4 & 2 follow     D) Only 2 follow   E) None of these
Answer:D) Only 2 follow

9. Statements:                            Notations  
a. All bulbs are radios.              BB-R
b. All radios are fans.                RR-F
c. No fans are taps.                   FF/TT

  Conclusions:                            Notations
1. Some fans are bulbs.              F-B
2. No taps are bulbs.                  TT/BB
3. Some radios are bulbs.          R-B
4. Some taps are radios.            T-R        x

A) 1, 2 and 3 follow        B) 2, 3 and 4 follow     C) Only 1 and 2 follow     D) Only 1 and 3 follow   E) 2 and 4 follow
Answer:A) 1, 2 and 3 follow
10. Statements:                            Notations
a. Some lions are tigers.              L - T
b. Some tigers are horses.          G - H
c. Some horses are deers.           H - F

Conclusions:                                 Notations
1. Some lions are horses.           L – H      x
2. Some tigers are deers.           T – D      x
3. Some lions are deers              L – D      x
4. Some horses are lions            H – L      x

A) None follows       B) All follow     C) Only 1 and 4 follow  D) Only 2 and 3 follow  E) None of these
Answer:A) None follows

11. Statements:                                          Notations
a. Some Germans are not Africans.           G /Af Af
b. All Africans are Asians.                            Af Af-As
c. Some Asians are Americans                    As-Am

Conclusions:                                                Notations
1. Some Germans are not Asians.            G/As As       x
2. Some Germans are not Americans.    G/Am Am    x
3. All Africans are Americans.                   Af Af- Am    x
4. Some Americans are Germans.            Am - G        x

A) Only 1 follows          B) Only 2 follows                C) Only 3 follows          D) Only 4 follows   E) Either 2 or 4 follows  Answer:E) Independently each conclusion is incorrect but 2 and 4 makes complementary Pair (4 gives Some Germans are    
Americans)

12. Statements:                                          Notations
a. Only stars are moons.                         MM- S
b. No galaxy is a star.                               GG / SS
c. No planets are moons.                         PP / MM

Conclusions:                                               Notations
1. Some moons are not planets              M / PP          
2. No moon is a glaxy.                               MM / GG    
3. No galaxy is a planet.                           GG / PP      x
4. No stars are planets.                            SS / PP        x

A) 1, 2 and 4 follow    B) 1, 3 and 4 follow   C) 2, 3 and 4 follow    D)3 and 4 follow  E)1 and 2 follow
Answer:E) 1 and 2 follow
13. Statements:                                         Notations
a. Some doctors are lawyers.                    D-L
b. All lawyers are Indians.                          LL- I
c. Some Indians are strong.                        I-S

Conclusions:                                                 Notations
1. Some lawyers are Indians.                        L-I
2. Some doctors are Indians.                        D-I
3. Some doctors are strongs.                       D-S      x
4. Some doctors are not strongs.               D/SS     x

A)1, 2 and 3 follow    B)1,2 and 4 follow    C) 2 and 3 follow     D) 2 and either 3 or 4 follow   E) 1,2 and either 3 or 4 follow
Answer:E)1,2 and either 3 or 4 follow
14. Statements:                                        Notations
a. All chairs are tables.                             CC-T
b. No tables are ships.                              TT/SS
c. Some hammers are ships.                    H-S

Conclusions:                                             Notations
1. Some hammers are not chairs.           H/CC    
2. Some hammers are not tables.          H/TT
3. Some tables are not hammers.          T/HH      x
4. No chair is a ship                            CC/SS

A)1,3 and 4 follow     B)2,3 and 4 follow     C)2 and 4 follow        D)3 and 2 follow   E)1,2 and 4 follow
Answer:E)1,2 and 4 follow
15. Statements:                                      Notations
a. Some books are pens.                        B - Pe
b. All pens are pencils.                           Pe Pe- Pl
c. No pencils are copies.                        Pl Pl /CC
d. Some copies are not calculators.     C / Ca Ca

Conclusions:                                               Notations
1. Some books are not calculators.        B / Ca Ca    x
2. No books are copies.                           BB / CC       x
3. No pens are copies                              Pe Pe / CC
4. Some pencils are not copies              Pl / CC

A) Only 3 follows     B) Only 4 follows      C) Only 3 and 4 follow       D) All follow    E) None of these
Answer:C) Only 3 and 4 follow

16. Statements:                                      Notations
a. No cops are police.                            CC / PP
b. All police are humans.                      PP - H
c. All humans are politicians.               HH – Pc
D. No politicians are doctors.              Pc Pc / DD

Conclusions:                                              Notations
1. Only politicians are police.                   PP - Pc
2. Only police are politicians.                  Pc Pc – P    x
3. No humans are cops.                            HH / CC     x
4. Some cops are not humans.                 C / HH      x

A)All follow           B)Only 1 follows       C)Only 2 follows     D)None follows       E)None of these  
 Answer:D)None follows b + c gives All police arepoliticians.But this can’t be Interepted as either 1 or 2  (Note: If “Only politicians are police”,then”All police are politicians”is true.Butnot vice versa)

17.  Statements: Some tumblers are plates.
Some bottles are tumblers.
All plates are spoons.
Conclusions: I. Some spoons are tumblers
II. Some spoons are plates          
III. Some bottles are plates
IV. No bottle is a plate
(1) Either III or IV follows  
(2) Only I & II follows                        
(3) All follow                      
(4) Only III & II follows
(5) None of these

Statement notations:           Conclusion notations:
T – P                                                      S – T
B – T                                                      S – P
PP – S                                                    B – P      ….x
                                                              BB / PP    …x
Answer: (5) None of these

18.    Statements: All speeches are copies.
All essays are speeches.
No essays are books.
Conclusions: I. Some books are speeches
II. No books are essays
III. No books are copies
IV. No book are speeches
(1) None of these
(2) All follow                      
(3) Either I or II follow.      
(4) Only II, III & IV follows
(5) Either I or IV and II follows

Statement notations:           Conclusion notations:
SS – C                                                  B – S    ….x
EE – S                                                  BB / EE
EE / BB                                                BB / CC   ….x
                                                           BB / SS    ….x
Answer: (5) Either I or IV and II follows

19.    Statements: All roots are stems.
Some branches are trees.
Some stems are branches.
Conclusions: I. Some trees are stems
II. Some trees are branches              
III. All trees are stems
IV. Some trees are not branches
(1) Only II follow        
(2) Only I & II follow        
(3) Only I follow        
(4) Only either I or III & II follow              
(5) Only I, II & III follow
Statement notations:           Conclusion notations:
RR – S                                                T – S   …x
B – T                                                  T – B
S – B                                                  TT – S   …x
                                                          T / BB   …x
Answer: (1) Only II follow        

20.    Statements: All asteroids are stars.
No stars are planets.
Some asteroids are satellites.
Conclusions: I. No planet is asteroid          
II. Some satellites are stars            
III. Some planets are not satellites            
IV. Some satellites are not planets
(1) Only II follows              
(2) Only I & II follows
(3) Only I, II & IV follow  
(4) None follows
(5) All follows  

 
Statement notations:           Conclusion notations:
AA – S                                                  PP / AA
SS / PP                                                St – S
A – St                                                  P / St St   …x
                                                           St / PP
Answer: (3) Only I, II & IV follow  

21.    Statements: Some books are copies.
Some copies are magazines.
No magazines is a dictionary
Conclusions: I. No copies are dictionary            
II. Some copies are dictionary
III. Some copies are not dictionary          
IV. No books are magazines
(1) Only III follows
(2) Only either I or II & III follow
(3) Only I follow  
(4) Only either I or II follows
(5) None of these
Statement notations:           Conclusion notations:
B  – C                                              CC / DD        ….x
C – M                                              C – D            ….x
MM / DD                                        C / DD
                                                        BB / MM     ….x
Answer: (2) Only either I or II & III follow

22.    Statements: No navies are air forces.
All armies are navies.
All air forces are combats.
Conclusions: I. No air forces are navies
II. Some combats are air forces
III. Some combats are not navies            
IV. No armies are air forces
(1) Only either I or II follows        
(2) Only II follows
(3) Only either I or IV follows        
(4) All follows
(5) None follows          
Statement notations:           Conclusion notations:                                                                                           NN / Af Af                         Af Af  / NN
ArAr – N                              C – Af
Af Af  – C                           C / NN
                                        ArAr / Af Af
Answer: (4) All follows

23.    Statements: Some headphones are radios
All radios are telephones.
No telephones are television
Conclusions: I. No radios are television
II. Some headphones are not television
III. Some headphones are telephones
IV. Some telephones are not television
(1) All follow          
(2) Only I, II & III follow
(3) Only II, III & IV follow  
(4) Only I, III & IV follow
(5) None of these
Statement notations:           Conclusion notations:
H – R                                               RR / Tv Tv
RR – T                                              H / Tv Tv
TT / Tv Tv                                        H – T
                                                        T / Tv Tv
Answer: (1) All follow  
     
24.    Statements: All tourists are rich.
Some tourists are leaders.
Some rich are ministers
Conclusions: I. Some leaders are ministers
II. All leaders are rich          
III. Some ministers are not rich
IV. Some rich are not tourists
(1) All follow
(2) None follows              
(3) Only I & II follow          
(4) Only II & III follow
(5) None of these
Statement notations:           Conclusion notations:
TT – R                                                  L – M     …x
T – L                                                    LL – R    ….x
R – M                                                  M / RR  ….x
                                                            R / TT   ….x
Answer: (2) None follows              

25.  Statements: Some pens are keys.
Some keys are locks
All keys are rings.
Conclusions: I. Some pens are locks.
II. Some pens are not locks          
III. Some pens are rings
IV. Some pens are not rings.
(1) Only either III or IV follows
(2) Only either I or II follows
(3) Only either I or II & III follow
(4) None follow                
(5) None of these
Statement notations:           Conclusion notations:
P – K                                                  P – L      …x
K – L                                                   P / LL   ….x
KK – R                                                 P – R  
                                                           P / RR   …x

Answer: (3) Only either I or II & III follow

26.Statements:
No mat is fan.
Some fans are cars.
All cars are buses.
Conclusions: I. All mats are cars            
II. All buses are cars
III. Some buses are fans                
IV. No bus is a mat
(1) Only either II or IV and III follow
(2) Only I and II follow      
(3) Only IV follows
(4) Only III follow              
(5) None of these
Statement notations:           Conclusion notations:
MM / FF                                MM – C      …x
F – C                                      BB  – C         …x
CC – S                                    B  – F
                                              BB  / MM    ...x      
Answer: (4) Only III follow

Possibility type syllogism questions is a regular feature of bank exams. Possibility type syllogism questions are based on the validity of contradictory of the Premise. Both AEIO and Euler circles method are moderate time consuming and not precise. Transformed Raval's notations Solves Any syllogism problems including 'possibility' Type questions at a glance and accurately.After some practicing even no paper work is required and possibility type syllogism questions are solved very speedily say,in 10-12 seconds. Students familiar with Raval's method have a clear and decisive advantage over other competitors.

In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system is logically related ('opposed') to each of the others.
'Contrary' (medieval: contrariae) statements, are such that both cannot at the same time be true. Examples of these are the universal affirmative 'every man is white', and the universal negative 'no man is white'. These cannot be true at the same time.
Every statement has a contradictory opposite, and since a contradictory is true when it’s opposite is false
Since every statement has a contradictory opposite, and since a contradictory is true when its opposite is false, it follows that the opposites of contraries (which the medieval Logicians called subcontraries) can both be true, but they cannot both be false
Square of opposition

In syllogism a conclusion is true, if it is valid in all possible diagrams (When solving with Euler circles method). But if one says possibility of something, it has to occur in any one of the diagrams.
If a premises contradictory is true, it means that given premise is definitely false
If a premises contradictory is false, it means that there exists at least one Euler figure where the given premises is true and that Euler figure represents the possibility of the premises being true.( When premise is derived from the combination of two or more premises).      
In syllogism a conclusion is true, if it is valid in all possible diagrams (When solving with Euler circles method). But if one says possibility of something, it has to occur in any one of the diagrams.
Consider a Example:
Statements:
No door is dog.
All the dogs are cats.
Conclusions:
I. Some cats are dogs.
II. All the cats being dogs is a possibility.
Solution by Euler’s circles

Some cats are dogs is true in all possible figures, so conclusion I follows. All the cats are dogs in Euler figure D, so, All the cats being dogs is a possibility is true conclusion. i.e.  conclusion  II follows.

Solution by Ravals notation
Statements                                  Conclusions

 Do Do / Dg Dg                              1. C – Dg                              
 Dg Dg - C                                       2. C / Dg Dg  …..x   Since Some cats are dogs is false , So, All the cats
                                                                                          being dogs is a possibility. True  
Answer: Both conclusions are true.

Statements:
(a)All TV are LCD
(b)Some LCD are VCR
Conclusions:
(1) Some VCR are TV
(2) All TV being VCR is a possibility

Solution by Raval’s Notations
Statements:
(a)All TV are LCD    TT – Lc
(b)Some LCD are VCR  Lc – V
Conclusions:
(1) Some VCR are TV (false)  V – T    …. x      
(2) All TV being VCR is a possibility (true)   T / V V  …..x
Ans. (2) only  conclusion (2) follows
Statements
(a) Some monkeys are intelligent  
(b)All intelligent are bold
(c) All bold are naughty  
Conclusions
(1) Some monkey are naughty  
(2) Intelligent who are monkeys are naughty

Solution by Raval’s Notations
(a) Some monkeys are intelligent  ( M - I)
(b)All intelligent are bold (II – B)
(c) All bold are naughty  (BB – N)
Conclusions
(1) Some monkey are naughty  (M –N) (true)
(2) Intelligent who are monkeys are naughty ( I – M) – N (true), since (II- N) is true

Ans(5): Both (1) and (2) follow
Statements                                  Notations
All Teachers are lectures              TT- L
All lecturers are professors           LL –P
Conclusions                                                                         Notations
1. All lectures those are professors are also teachers (false)   (L-P) –T, LL- T…x,   PP –T…x

2. All those teachers who are lectures also professors. (true)   (T – L) –P, LL - P


Fallacies in A, E, I, O Method for syllogism
One must note that we have valid conclusions in the following syllogisms
AOO – 2 Baroco                                  By Raval’s notations
All P are M                                        PP - M
Some S are not M                              S  / MM
Concl: Some S are not P                     S / PP

OAO – 3 Bocardo                    By Raval’s notations
Some M are not P                    M  /  PP
All M are S                             MM - S
Concl: Some S are not P         S  / PP

In A, E, I, O Method for syllogism A + O and O + A gives no valid conclusion. But in AOO – 2 Baroco and OAO – 3 Bocardo we have valid conclusions. So A, E, I, O method is not exhaustive and is not applicable in the above cases.

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