[Reasoning] Three-Statement Syllogism: technique, method, shortcuts for IBPS, SSC, CSAT, CAT, CMAT

Aptitude218 Comments

Don’t read further, until and unless you’ve mastered the 2-statement syllogism technique explained in previous article (click ME)

  1. Recap of 2-Statement syllogism
  2. Parent Statements
  3. Chain formula
  4. DemoQ: Married Student Dancers (CSAT-2012)
  5. DemoQ: Rich n Sick Air travellers (CSAT-2012)
  6. DemoQ: Drug addict Artists (CSAT-2012)

Recap of 2-Statement syllogism

Before we understand the 3-statement syllogism, let’s recap the 2-statement trick just for refreshing your memory.
What to do when 2-statement syllogism question is given?

  1. They must have only three terms (A, B and C)
  2. Are the question statements in standard format (A to B then B to C)? if no, then refer to following conversion table. (important: priority order for conversion is PP>UN>UP.)
TypeValid Conversion
Universal Positive (UP)Only PP
Universal Negative (UN)PN or UN
Particular Positive (PP)Only PP
Particular Negative (PN)Can’t do.

3. Classify the Question statement (UP, UN, PP, PN)
4. Apply the combo rules on Question statements.

No conclusion scenario

possible conclusion scenario

  1. UP’s politicians hate giving particular statements (both positive and negative). E.g. they donot reveal their clear position on FDI in retail until the 11th hour. (UP+PP/PN=NO)
  2. United Nations hates negativity. (both Universal and particular)(UN+UN/PN=NO)
  3. Pritish Nandy hates everybody. (first statement is PN=NO, Irrespective of second statement.)
  4. Two-negatives=no conclusion.
  5. Two particulars=no conclusion.
  1. If Uttar Pradesh meets Uttar Pradesh, then its size doesn’t increase. (UP+UP=UP)
  2. If Uttar Pradesh meets United Nations then size increases and it becomes United Nations. (UP+UN=UN)
  3. United Nations Secretary Ban Ki Moon is in very positive mood. But he meets another positive person, and his attitude is totally reversed- he becomes particularly negative! (reversed =C to A). (UN+UP/PP=PN)
  4. When Mr.PP observes the universe via NASA telescope, his mood becomes particularly positive or negative depending on the mood of universe.(PP+UP/UN=PP/PN)

5. (rarely required): if no-conclusion and “either or” given in answer, then check for Complimentary case.
Now we’ll see how to solve three-statement syllogism.

Parent Statements

The crux of 2-statement syllogism was

  1. When we’ve Question statements in standard format (A to B then B to C).
  2. We apply some combo rules and may get a conclusion in the form of A to C

(or we may get the conclusion in form of “C to A”, in case the question statements were in the format of UN+(UP/PP). Recall the Ban-ki-Moon’s mood reversal).
In case of three statement syllogism, we accept the conclusion statement (A to C) as valid, then try to find out its parents (those question statements A to B then B to C).
Then, we try to get a valid conclusion out of those two-question statements and see if it matches with the given conclusion state in answer.
No need to get confused, let’s try with a simple scenario.

question statement

conclusion statement

  1. All cats are dogs
  2. some pigs are cats.
  3. All dogs are tigers
  1. some tigers are cats
  2. some pigs are tigers
  3. all cats are tigers
  4. some cats are not tigers

Answer choices

  1. Only 1 and 2
  2. Only 1, 2 and 3
  3. All follow
  4. None Follow

Start with first conclusion statement

i) Some tigers are cats (PP)

Q. if this is a valid conclusion, who’re its parents?
Ans. Thouse question statements with words “tigers”, “cats”, and a common term.
You can see, first and third statement fits the bill.

Q.statementType
  1. All cats(A) are dogs(B)
UP
  1. All dogs(B) are tigers(C)
UP
  • Three terms=Ofcourse yes.
  • Are they in standard format (A to B then B to C)? Yes.
  • Then what are you waiting for? Just apply the combo rules. UP meets UP then its size doesn’t increase (UP+UP=UP) A to C. Hence conclusion will be “All cats are tigers.” (meaning given conclusion statement #3 is valid).
  • If we convert this valid conclusion “All cats are tigers (UP)”, then UP–>PP= Some tigers are cats.

It means the given conclusion statement#1 is also valid.
So far: 1 and 3 are correct.
Now test the second conclusion statement.

Chain formula

ii) some pigs are tigers

if this is a valid conclusion, who’re its parents?
Ans. . Thouse question statements with words “pigs”, “tigers”, and a common term(B).
But I don’t see any such question statements.
Now we’ll have to apply chain formula. Meaning, (A to B1, then B1 to B2, then B2 to C).
Consider this arrangement

Question statements (CHAIN)Chain
  1. some pigs(A) are cats.(B1)
Pig to Cat, cat to dog and finally dog to tiger. Let’s see if we connect pig to tiger.
  1. All cats(B1) are dogs(B2)
  1. All dogs(B2) are tigers(C)

We’ll take two statements at time and try to get an intermediate conclusion.

StatementType
  1. some pigs(A) are cats.(B1)
Particular positive (PP)
  1. All cats(B1) are dogs(B2)
Universal positive (UP)

Three terms = yes
Standard format= yes. (there A to B1 and then B1 to B2, which is just like A to B then B to C)
Apply combo-rule
PP+UP=PP (NASA mood change!) (A to B2)
Hence intermediate conclusion is Some pigs(A) are dogs(B2)
Now take this intermediate conclusion with the next statement in our chain.

Question-statementsType
Some pigs(A) are dogs(B2)Particular positive (PP) (derived)
All dogs(B2) are tigers(C)Universal positive (UP) (given in question).

Again, same standard operating procedure of 2-statement syllogism.
Three terms = yes
Standard format= yes. (there A to B2 and then B2 to C, which is just like A to B then B to C)
Apply combo-rule
Again, PP+UP=PP (NASA mood change!) (A to C)
Therefore conclusion is Some pigs(A) are tigers (C).
Voila! Second conclusion statement is also correct.
So far 1, 2 and 3 are correct.
Let’s check the last statement (IV).

iv) some cats are not tigers

while we were checking the first conclusion statement, we had found that “All cats are tigers (UP)”. Therefore, given conclusion statement is not possible.
Final answer: only 1, 2 and 3 are correct. (option b)

DemoQ: Married Student Dancers (CSAT-2012)

Question statements

  1. None but students are the members of the club.
  2. Some members of the club are married.
  3. All married persons are invited for dance.

Which one of the conclusions can be drawn from the above statements?

  1. All students are invited for dance
  2. All married students are invited for dance
  3. All members of the club are married person
  4. None of the above conclusions can be drawn

Solution and approach

  • first, rephrase the given statements so that processing becomes easier.
  • Recall the “special conversion” rule from previous article on 2 statement.
  • None but Politicians(A) are honest(B)=> All honest(B) are politicians(A) (Universal positive)
  • I’m replacing the word “members of the club” with “club-members”.
  • Thus simplified version of the given question is following
Question statementsAnswer statements
1. All clubmembers are students.2. Some clubmembers are married.
3. All married are invited for Dance.
a) All students are invited for danceb) All married students are invited for dance
c) All club-members are married.
d) None of the above conclusions can be drawn

Ok now what?
We’ve to pick up the answer statement one by one and test them.

a) All students are invited for dance

there are two ways to solve this statement, first the

longcut method

Who’re the parents of this conclusion statement?
No direct parents. We’ve to apply chainrule.

Question statementsChain rule
  1. All clubmembers(B1) are students(A)
We’ll try to link students-clubmembers-married-dance invitation.
  1. Some clubmembers(B1) are married.(B2)
  1. All married(B2) are invited for Dance.(C)

Let’s start. First two statements

  1. All clubmembers(B1) are students(A)
UP
  1. Some clubmembers(B1) are married.(B2)
PP

Standard format? Nope.
Conversion needed: yes. But priority order=PP>UN>UP. Means we’ll convert the second statement (particular positive)

  1. All clubmembers(B1) are students(A)
UP
  1. Some married (B2) are clubmembers(B1).
PP converted to PP.

They’re still not in standard format. So Exchange positions

  1. Some married (B2) are clubmembers(B1).
PP converted to PP.
  1. All clubmembers(B1) are students(A)
UP

Ok now they’re in standard format. Apply the combo rule. PP+UP=PP (NASA Mood change)
Some married are students. This is our intermediate conclusion. Now pair it up with third question statement from the chain rule

Some married(B2) are students(A)PP
All married(B2) are invited for Dance.(C)UP

Standard format? Nope.
Then convert!

Some students(A) are married(B2)PP
All married(B2) are invited for Dance.(C)UP

Ok now in standard format (A to B then B to C)
Apply combo rule PP+UP=PP (NASA mood change again!)
Conclusion = Some students are invited for dance. (PP)
But the given answer statement says “All students are invited for dance”(UP)=impossible.
Hence first answer choice is eliminated.

Shortcut

  • You see the conclusion statement says “all students are invited for Dance.” (univ.positive statement).
  • When do we get “universal positive” statement as conclusion.?
  • Only when UP+UP=UP.
  • If we apply the chain-rule, we’ll encounter one particular positive (PP). And that’ll ruin the mood (because whenever particular positive statement comes, the conclusion is either

1. PP+UP=Particular positive (NASA mood change) OR
2. UP+PP=No conclusion. (UP politicians hate particular statements).

  • Hence we can never get a Universal positive (UP) type of conclusion, in either case! Means this answer choice is invalid by default! No need to manully apply chain rule here. Anyways, Move to the next answer choice

b) All married students are invited for dance

superficially this statement contains three terms.

  1. Married
  2. Student
  3. Dance

Wait a minute! Our syllogism conclusions contain only two terms (e.g. All dogs are cats.)
So, how can we apply syllogism here?
Well, if you observe carefully, the syllogism rules are still applicable in this conclusion statement containing three terms.
From the “longcut” method in previous option, we’ve found that “Some students are invited for dance. (PP)”

  • Question: which students are invited? Well, we eliminated the middle-term (B2) “Married”. Means all married students are invited for dance. (this represents the intersecting area between two Venn Diagram circles).
  • Therefore, we can say “All married students are invited for dance.” Hence Answer is (B).

c) All club-members are married.(UP)

The second question statement says, “Some clubmembers are married.” (PP).
A particular positive statement can be converted into only PP. Hence we cannot say for sure that all club members are married. Hence this answer choice is incorrect.

DemoQ: Rich n Sick Air travellers (CSAT-2012)

Question Statements

  1. None but the rich ran afford air-travel.
  2. Some of those who travel by air become sick
  3. Some of those who become sick require treatment

Conclusion statements

  1. All the rich persons travel by air.
  2. Those who travel by air become sick
  3. All the rich persons become sick.
  4. All those who travel by air are rich

Solution

  • first we will simplify the given statements.
  • Recall the “special conversion” rule from earlier article on 2-statement syllogism.
  • None but Politicians(A) are honest.(B)=> All honest(B) are politicians(A) (Universal positive)
  • Similarly, None but the rich ran afford air-travel=> All air-travellers are rich. (UP).
  • That means, Correct answer is (D). Case is over.
  • But just for concept clarity, let’s test remaining answer choices as well.
Simplified question statements

  1. All air-travellers are rich.
  2. Some air travellers are sick.
  3. Some sick are treatment.
Simplified Answer statements

  1. All rich are air-travellers.
  2. (all) air travellers are sick
  3. All rich are sick.
  4. All air-travellers are are rich.

Now let’s solve

  1. All rich are air-travellers.
The given question statement is “All air-travellers(A) are rich.(B)”(UP). Apply the conversion here, UP=>PP. Hence Some rich(B) are air-travellers(A).Therefore, first answer choice is incorrect.
  1. (all) air travellers are sick
Question statement #2 says “Some air travellers are sick.” (PP). can’t convert to UP.Hence this option is also incorrect.

c) All rich are sick.

Q.If this is the conclusion statement, then who are its parents?
Ans. Those question statements which contain the terms “rich”, “sick”, along with a common middle term.
From the given question statements, following two fit the bill

1. All air-travellers(B) are richUP
2. Some air travellers(B) are sickPP

Approach #1 (shortcut)

Please observe: “All rich are sick.”= Universal positive statement.
When do we get UP conclusion? Only when combo rule UP+UP=UP is applied.
Now in above case, one question statement is PP. so it’ll “kill” the mood. UP conclusion is not possible. (no need to convert any statement.)

Approach #2 (longcut)

1. All air-travellers(B) are rich.(A)UP
2. Some air travellers(B) are sick.(C)PP

As you can see, there is one middle term (air travellers). But the question statements are not in standard format (A to B then B to C).
It means, we must convert anyone statement. But priority for conversion is PP>UN>UP.
So we will convert second statement.
Some air-travellers are sick (PP)==convert==> Some sick are air-travellers(B)

1. All air-travellers(B) are rich.UP
2. Some sick are air-travellers(B)PP

But they are still not in standard format (A to B then B to C). well no problem, just exchange position of question statements

1.Some sick(A) are air-travellers(B)PP
2. All air-travellers(B) are rich.(C)UP

Apply the Combo-rule. PP+UP=PP (NASA Mood change).
Hence Some sick(A) are rich.(C). (PP)
If we convert it then Some rich are sick. (PP convert to PP).
But answer choice says All rich are sick.= this is not possible.
Therefore, third answer choice is also incorrect.

DemoQ: Drug addict Artists (CSAT-2012)

Question statements

  1. All artists are whimsical.
  2. Some artists are drug addicts.
  3. Frustrated people are prone to become drug addicts.

From the above three statements it may be concluded that:

  1. Artists are frustrated
  2. Some drug addicts are whimsical
  3. All frustrated people are drug addicts.
  4. Whimsical people are generally frustrated

Statement 3 says “Frustrated people are prone to become drug addicts.” For our purpose this is a “Particular positive (PP)” statement. Let’s simplify it to “Some frustrated people are drug addicts” Now start with answer (A)

a) (all) Artists are frustrated (UP)

if this is the answer, then what could be the question statements?
The question statements could be those statements where the words Artist and frustrated come along with a common middle-term (B)
Consider these question statements:

2. Some artists are drug addicts.(B)PP
3. Some Frustrated people are drug addicts.(B)PP

Although this in not in “standard format”, but even when we convert one of them (PP->PP), we’ll be left with PP+PP=no conclusion.
Hence move to next option.

b) Some drug addicts are whimsical

  • if this is the answer, then what could be the question statements?
  • The question statements could be those statements where the words drug addict and whimsical come along with a common middle-term (B)
  • Consider these question statements:
1. All artists (B) are whimsical.UP
2. Some artists (B) are drug addicts.PP
  • Three terms = yes.
  • Standard format (A to B then B to C)=No.
  • So maybe to convert anyone know the statement. according to the priority order PP>UN>UP, we must convert second statement.
1. All artists (B) are whimsical.UP
2. Some drug addicts are artists (B).PP->PP converted.

Now interchange position of question statement 1 and 2.

Some drug addicts(A) are artists (B).PP->PP converted.
All artists (B) are whimsical.(C)UP
  • Okay now what? Apply the combo rules
  • PP+UP=PP. (NASA mood change).
  • Thus final answer is “some drug addicts are whimsical.” (option B)

For the archive of all [Aptitude] articles, visit Mrunal.org/aptitude.

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So far 218 Comments posted

  1. Nabojit

    Statements:
    1. Some files are not tubs.
    2. All buckets are tubs.
    3. Some files are mugs.
    Conclusion:
    I. Some buckets are files.
    II. Some files are not buckets.
    III. Some tubs are not mugs.

    1) Only I follows
    2) Only II follows
    3) Only III follows
    4) Only II and III follows
    5) None follows

    Sir, Please solve this.

    1. manoj kumar

      none follows

  2. rajesh

    what should we do if in conclusions definitely is given

  3. Killer Miller

    Statements No Apple is mango
    All mangoes are fruits
    All fruits are vegetarians

    conclusion No Fruit is apple
    No vegetarian is apple
    some vegetarian are mangoes
    some vegetarian are Apple

    Answers
    None follows
    All follows
    Either II or IV and III follows
    Either II or III and I follow
    none of these

    Every1 is saying ansr is EITHER II or IV AND III follows
    bt m getting only III follows bcz there is no option for this thus NONE OF THESE
    PLZ EXPLAIN SM1

    1. Abhi

      Correct Answer EITHER II or IV AND III follows
      We get “Either II or IV” by Complimentary Pair Method

      When we check Conclusion II “No vegetarian is apple”, we need to use all 3 statements.

      From Statement I and II we get “Some Fruits are not Apple”
      From this conclusion and III Statement we get ” No conclusion”

      Since we get No conclusion and Answers has “Either Or”, We apply Complimentry Method for 2 statement Syllogism

      Here Given Either II or IV
      1. II and IV has same subjects and Predicate
      2.PP+UN pair

      So Either II or IV is Correct
      and III is also correct (can be solved with normal method)

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