SYllogism-Venn Diarams, Euler Circles

Introduction
Syllogism was introduced by Aristotle (a reasoning consisting two premises and a conclusion).Aristotle gives the following definition of syllogism in his fundamental treatise Organon.
“A syllogism is discourse, in which, certain things being stated, something other than what is stated follows of necessity from their being so”. Things that have stated are known as premises and the one that follows from the premises is known as the conclusion of the syllogism.

A categorical syllogism is a type of argument with two premises and one conclusion. Each of these three propositions is one of four forms of categorical proposition.

Type                     Form                                   Example
A                       All S are P                           All monkeys are mammals
E                       No S is P                             No monkeys are birds
I                       Some S are P                       Some philosophers are logicians
O                      Some S are not P               Some logicians are not philosophers
   
These four types of proposition are called A, E, I and O type propositions, the variables S and P are place-holders for terms which represent out a class or category of thing, hence the name “categorical” proposition.
A categorical syllogism contains precisely three terms: the major term, which is the predicate of the conclusion; the minor term, the subject of the conclusion; and the middle term, which appears in both premises but not in the conclusion.
Aristotle noted following five basic rules governing the validity of categorical syllogisms

1. The middle term must be distributed at least once (distributed term refers to all members of the denoted class, as in all S are P and no S is P).
2. A term distributed in the conclusion must be distributed in the premise in which it occurs.
3. Two negative premises imply no valid conclusion.
4. If one premise is negative, then the conclusion must be negative.
5. Two affirmatives imply an affirmative.

John Venn, an English logician, in 1880 introduced a method for analyzing categorical syllogisms, known as the Venn diagram. In a paper entitled “on the Diagrammatic and Mechanical Representation of propositions and Reasoning’s in the “philosophical magazine and journal of science,” Venn shows the different ways to represent propositions by diagrams. For categorical syllogism three overlapping circles are drawn to represent the classes denoted by the three terms. Universal propositions (all S are P, no S is P) are indicated by shading the sections of the circles representing the excluded classes. Particular propositions (some S are P, some S are not P) are indicated by placing some mark, usually an “x”, in the part of the circle representing the class whose members are specified. The conclusion may then be inferred from the diagram.
Venn diagrams has similarity with Euler diagrams, invented by Leonard Euler in the 18th century, but Venn diagrams are visually more complex than the Euler diagrams.
Solving Syllogism problems are usually moderate time consuming by Traditional   methods and considered difficult by most of the students. New Transformed RAVAL’S NOTATION solves Syllogism problems very quickly and accurately. This method solves any categorical syllogism problem with same ease and is as simple as ABC…                          

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

Illustration:
 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

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

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

 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 follows 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

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                        

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 follows because common  linking term (P) multiply.
2. P / TT follows because common   linking term (Q) divides                                                                            
3. T / PP follows 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.


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




                 
SOLVED PROBLEMS
Directions (Q.1-5): 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 bats.         KK - B
c. Some clocks are bats. C – B

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

1) Only 1 and 2 follow      2) Only 2 and 3 follow                      3) Only 1 follows
3) Only 1 follows              4) Only 2 follows
Ans. 5) 1, 2 and 3 follow

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

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

1) Only 1 and 2 follow        2) Only 2 and 3 follow                        4) 1, 2 and 3 follow.
3) Only 1 and 3 follow        4) 1, 2 and 3 follow
Ans:5) None follows

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:
1. Some papers are bags.    P – Ba       x
2. Some books are papers.   B - P
3. Some books are purses.   B - P
1) Only 1 follows            2) Only 2 and 3 follow                            2) Only 2 and 3 follow
3) Only 1 and 3 follow    4) Only 1 and 2 follow
Ans:5) 1, 2 and 3 follow

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

   Conclusions:
1. No car is cloud.           CaCa / CC      x
2. Some cars are birds.     Ca – B       x
3. No bird is car.              BB / CaCa   x
                         
A) Only 3 follows      B) Only either 2 or 3 follows                         E) None of these
C) Only 1 follows      D) Only 1 and either 2 or 3 follow
Ans:E) None of these

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

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


1) Only 1 follows         2) Either 1 or 3 follows                       5) None of these
3) Only 2 and 3 follow   4) Only 1, 2 and 3 follow
Ans:5) None of these



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Great explaination Ravinder kumar

It will be of great help if U can come up with some more questions for our Members to practice.