A Mathematical Approach to Syllogism - Vikas Saini

  • 1. General cases

    General cases consists statements of  All, Some, No.

    Let's we put assume some Value

    For All

    Object 1 = 10

    Object 2 = 5

    For Some

    object 1 = 5

    object 2 = 5

    For No

    object 1 = 10

    object 2 = 10.

    Some rules

    1. One positive & another  positive statement make a positive conclusion.

    2. One positive & one negative statement make one negative conclusion.

    3. Two negative statement will yield no conclusion.

    4. One  common element value must be 10.

    Q. CAT 1996 question

    (A) Some stone is bone.

    (B ) All slipper is bone.

    (C) All bone is stone.

    (D) No stone is slipper.

    (E) No bone is stone.

    (F) Some bone is slipper.

    (a) BCA (b) AFE (c) DEC (d) CEA

    Let us solve option by option.

    (a) BCA

    B - All slipper (10) is bone (5).

    C - All bone(10) is stone(5).

    A - Some stone(5) is bone (5).

    Here A is conclusion where stone (5) and stone is (5).

    By statement B & C it can be conclusion.

    Option a is the answer but let's check other option too.

    (b) AFE

    A - Some stone(5) is bone(5).

    F - Some bone(5) is slipper(5).

    E - No bone is stone.

    Here E is a negative conclusion and statements are positive.

    Not possible.

    (c) DEC

    D - No stone(10) is slipper(10).

    E - No bone (10) is stone(10).

    C - All bone(10)is stone(5).

    2 negative statements never make any conclusion.

    (d) CEA

    C - All bone (10) is stone (5).

    E - No bone(10) is stone(10).

    A - Some stone(5) is bone(5).

    Not possible because one positive & one negative statement make a negative statement.


    (A) No rice is a pest.

    (B ) All pest is lice.

    (C) Some rice is lice.

    (D) All rice is pest.

    (E) All rice is lice.

    (F) No rice is lice.

    (a) BEF

    B - All pest(10) is lice(5).

    E - All rice(10) is lice (5).

    F - No rice is lice.

    Not possible because statements are positive and conclusion is negative.

    (b) FCB

    F - No rice is lice.

    C - Some rice is lice.

    B - All pest is lice.

    Not possible because

    +ve + (-ve) = -ve


    A - No rice(10) is a pest(10).

    B - All pest(10) is lice(5).

    F - No rice(10) is lice (10).

    It's also not possible because in statement lice (5) but in conclusion (10).

    (d) BDE

    B - All pest(10) is lice (5).

    D - All rice(10) is pest (5).

    E - All rice(10) is lice.(5)

    It is the answer,because pest is common in both statements and one pest is 10 & value of rice (10) & lice (5).

    Hence option d.

    (A) All tigers lay eggs.

    (B ) All cats lay leggs.

    (C) Some cats can fly.

    (D) All tigers can't fly.

    (E) All tigers are cats.

    (F) All tigers can't swim.

    (a) BEA (b) ABE (c) DEC (d) ECB

    (a) BEA

    B - All cats(10) lay eggs(5).

    E - All tigers(10) are cats(5).

    A - All tigers(10) lay eggs (5).

    It must be answer.Satisfy all answer,need not to check other options.

    Hence option A.

    (A) Some curtains are cloth.

    (B ) All cloth is wood.

    (C) All that is wood is cloth.

    (D) All cloth are curtains.

    (E) All curtains are wood.

    (F) All cloth are curtains.

    (a) BED (b)BDF  (c)FAB  (d)FBA

    (a) BED

    B - All cloth(10) is wood (5).

    E - All curtains(10) are wood (5).

    D - All cloth (10) are curtains (5).

    It can't be answer because wood is common but value of it 5 both times.

    It's value must be 10 at least once.

    (b) BDF

    B - All cloth(10) is wood(5).

    D - All cloth (10) is curtain (5).

    F - Some curtains(5) are wood (5).

    Option b is the answer.

    (c) FAB

    F - Some curtains(5) are wood(5).

    A - Some curtains(5) is cloth(5).

    B - All cloth (10) is wood (5).

    Not possible at all.

    (D) FBA

    F - Some curtains (5) are wood (5).

    B - All cloth(10) is wood (5).

    A - Some curtains (5) are cloth(5)

    Not possible.

    (A) All balls are talls.

    (B ) Some talls are dolls.

    (C) Some dolls are balls.

    (D) Some talls are not balls.

    (E) All talls are dolls.

    (F) No talks are dolls.

    (a) EDC (b) BCD (c) ABC (d) EAC

    (a) EDC

    E -All talls (10) are dolls (5).

    D - Some talls (5) are not balls (5)

    C - Some dolls(5) are balls (5).

    Not possible.

    (b) BCD

    B - Some talls(5) are dolls(5).

    C - Some dolls(5) are balls (5).

    D -Some talls (5) are not balls(5).

    (c) ABC

    A - All balls(10) are talls (5).

    B - Some talls(5) are dolls (5).

    C - Some dolls (5) are balls (5).

    not possible

    (d) EAC

    E - All talls (10) are dolls (5).

    A - All balls(10) are talls (5).

    C - Some dolls(5) are balls (5).

    Option d is the answer.

    2nd case - Either / Or case

    Rules :-

    1. Elements must be same.

    2. One conclusion must be positive & another must be negative.

    3. Both statement must be false.

    If 1 & 2 rules are not satisfied by conclusion then it's obvious these are not either/or case.



    1. Some books are table.

    2..Some tables are mirror.


    1. Some mirrors are books.

    2. No book is a mirror.

    Let's put a glance on 3 rules

    1. Elements must be same(mirror & books in both conclusion)

    2. One conclusion should be positive & one should be negative.

    3. Both conclusion must be false as per rules.

    So all the three rules are satisfying here.

    So either 1 or 2 follow.



    (A) Engineers marry only fair girls.

    (B ) Ankita is very fair.


    (1) Ankita was married to an engineer.

    (2) Ankita was not married to an engineer.

    Let's put a glance

    1. Both element are same.(Ankita & Engineer)

    2.One is positive & one is negative.

    3. Both conclusion false too.

    So either 1 or 2 conclusions follow.



    (a) All cats are dogs.

    (b) All dogs are brown.


    (1) All cats are brown.

    (2) All brown are dogs.

    glance on 3 rules

    1. Both elements are not same.

    2. Both are positive.

    3. 1st conclusion is true here & 2nd conclusion is false.

    Only conclusion 1 follow.



    (A) Some rods are sticks.

    (B ) Some scales are rods.


    (1) Some sticks are rods.

    (2) Some scales are sticks.

    Glance on rules

    1. Elements are not same.

    2. Both are positive.

    3. Both conclusion are false.

    So neither 1 nor 2 follow.



    (A) Rohit Sharma is a good batsman.

    (B ) Batsman are physically powerful.


    (1) All physically powerful are batsman.

    (2) Rohit Sharma is physically powerful.

    Put a glance on rules

    1. Both Elements are not same.

    2. Both are positive.

    3. 1st conclusion is false.2nd is true.

    So only 2nd conclusion follows.


  • E - All talls (10) are dolls (5).
    A - All balls(10) are talls (5).
    C - Some dolls(5) are balls (5).
    Option d is the answer.

    @vikas_saini sir, in the above triplet (example that you had solved above in your post) in the conclusion, balls has a value of 5, while in the statement 'A' it has a value of 10 - so there's a mismatch, then how could it be the answer? please spare time & explain

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