[Reasoning] Logical Connectives (if, unless, either or) for CSAT, CAT shortcuts formulas approach explained

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Economic Survey
  1. Difference: Syllogism vs Logical connectives
  2. Standard format: logical connectives
  3. Logical connective: if then
  4. Logical connective: Only IF
  5. Logical Connective: UNLESS
  6. Logical connective: otherwise
  7. Logical connective: When, Whenever, every time
  8. Logical Connective: Either OR
  9. Demo Q: Only if: bored TV brother (CSAT 2012)
  10. Demo Q (If, then) Professor Headaches  (CAT’98)
  11. Demo Q: Either or: derailed/late train (CAT’97)

Difference: Syllogism vs Logical connectives

Syllogism (all cats are dog) is a common and routinely appearing topic in most of the aptitude exams (Bank PO, LIC, SSC etc). But Logical connectives is rare. However, in UPSC CSAT 2012 the topic was asked, therefore, you’ve to prepare it.

Syllogism

Logical connectives

Contains words like “all, none, some” etc. Can be classified into UP, UN,PP and PN. Already explained in previous articles. Contains words like “if, unless, only if, whenever” etc. can be classified into 1, ~1, 2, ~2 (we’ll see in this article)
Have to mugup more formulas, takes more time than logical connective questions. Less formulas and quicker than syllogism.
Question Statements:

  1. All cats are dogs
  2. some pigs are cats
  3. no dogs are bird

Conclusion choices:

  1. Some cats are dogs
  2. No birds are cats
  3. some pigs are birds
  4. Some pigs are not birds
Question statements:

  1. I watch TV only if I am bored
  2. I am never bored when I have my brother’s company.
  3. Whenever I go to the theatre I take my brother along.

Conclusion choices:

  1. If I am bored I watch TV
  2. If I am bored, I seek my brother’s company.
  3. If I am not with my brother, than i’ll watch TV.
  4. If I am not bored I do not watch TV.

Standard format: logical connectives

  • If, unless, only if, whenever, every time etc. are examples of Logical connectives.
  • Whenever you’re given a question statement, first rule is: question statement must be in the standard format.
  • The standard format is
  • ****some logical connective word *** simple statement#1, simple statement #2.
  • It means, the question statement must start with a logical connective word, otherwise exchange position. For example
Given question statement Exchange position?
If you’re in the army, you’ve to wear uniform
  • no need because the simple statement containing “IF” is given in the beginning. This is already in the standard format.
You’ve to wear uniform, if you’re in the army
  • We need to exchange position because the part containing “IF” is not given in the beginning of this statement, given statement is not in standard format.
  • Therefore, Rewrite given statement as
  • If you’re in the army, you’ve to wear uniform.
You’ve to salute, whenever Commanding Officer comes in your cabin.
  • Need to exchange position. Because statement doesn’t start with the logical connective “whenever”.
  • Therefore rewrite the given statement as
  • Whenever CO comes in your cabin, you have to salute.

Now let’s derive valid inferences for various logical connectives.

Logical connective: if then

Consider these two simple statements

  1. You’re in army
  2. You’ve to wear uniform.

These are two simple statements. Now I’ll combine these two simple statements (#1 and #2) to form a complex statement.

  • If you’re in army(#1), you have to wear uniform.(#2)

What about its reverse?

  • You’ve wearing uniform (#2)—> that means you’re in the army.(#1)
  • But there is possibility, you’re in navy—-> you’ll still have to wear a uniform. It means,
  • if 1=>2, then 2=>1 is not always a valid inference.
  • Let’s list all such scenarios in a table.
Given statement:If you’re in army(#1), you have to wear uniform.(#2)
Inference? Valid / invalid?
  1. If #2, then #1
If you’ve to wear uniform, you’re in army. you’ve to wear uniform in navy, air force, BSF etc. so this inference is not always valid.
  1. If not #1, then not #2
if you’re not in army, you don’t have to wear uniform. you’ve to wear uniform in navy, air force, BSF etc. so this inference is not always valid.
  1. if not #2, then not #1
If you don’t have to wear uniform, you’re not in army. Always valid.
  • In the exam, you don’t have to think ^that much. Just mugup the following rule:
  • Given statement =“If #1 then #2”, in such situation the only valid inference is “if Not #2, then not #1”.
  • In other words, “if 1st happens then 2nd happens”, in such situation, the only valid inference is “if 2nd did not happen then 1st did not happen”.
  • Now I want to construct a short and sweet reference table for the logical connective problems. So I’ll use the symbol ~= negative.

~1=meaning NOT 1 ( or in other words, negative of #1)

Given Valid inference
If 1, then 2 If not 2, then not 1
If 1=>2 ~2=>~1
  • In some books, material, sites, you’ll find these rules explained as using “P” and “Q” instead of 1 and 2.
  • But in our method, you first make sure the given (complex) statement starts with a logical connective (or you exchange position as explained earlier)
  • We denote the first simple sentence as #1 and second simple sentence as #2.
  • The reason for using 1 and 2= makes things less complicated and easier to mugup.

Logical connective: Only IF

  • In such scenario, you’ve to rephrase given statement into “if then” and then apply the logical connective rule for “if then”.
  • For example: given statement: he scores a century, only if the match is fixed.
  • The “standard format”= only if the match is fixed(1), he scores a century(2).
  • In case of “only if”, we further convert it into an “if” statement, by exchanging positions. That is
  • if he scores a century(#2), the match is fixed(#1).
  • Then apply the formula for “if then” and get valid inference.
  • Here we’ve “if 2=>1” as per our formula for “if then”, the valid inference will be ~1=>~2. Don’t confuse between 1 and 2. Because essentially the valid inference is “negative of end part => negative of starting part”.
  • Therefore “if 2=>1 then ~1=~2”
  • similarly “if 98=>97, then valid inference will be ~97=>~98”
  • Similarly “if p=>q, then valid inference will be ~q=>~p”,
  • similarly “if b=>a, then valid inference will be ~a=~b”) .
  • Update our table
Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=>~1 Negative of end part=> negative of start part
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.

Logical Connective: UNLESS

  • Given statement: Unless you bribe the minister(#1), you will not get the 2G license.(#2)
  • Unless = if…..not.
  • So, I can re-write the given statement as
  • (new) Given statement: If you don’t bribe the minister(#1), you’ll not get the 2G license.(#2)

How to come up with a valid inference here?

#1 You don’t bribe the minister
#2 You’ll not get the 2G license.
  • For “if..then”, We’ve mugged up the rule:  1=>2 then only valid inference is ~2=>~1. (in other words, negative of end part => negative of starting part).
  • let’s construct the valid inference for this 2G minister.
  • we want ~2 => ~1
  • Negative of (2) => negative of (1)
  • Negative of (you’ll not get the 2G license)=>negative of (you don’t bribe the minister)
  • You’ll get the 2G license => you bribe the minister.
  • In other words, If I see a 2G license in your hand, then I can infer that you had definitely bribed the minister.
  • This is one way of doing “unless” questions = via converting it into “if…not” type of statement.
  • The short cut is to mugup another formula: unless1=>2 then ~2=>1.
  • How did we come up with above formula?

Deriving the formula for unless

  • Unless 1=>2 (given statement)
  • if not 1=>2  (because unless=if not)
  • if ~1=>2 (I’m using symbol ~ instead of “not”)
  • ~2=> ~(~1) (because we already mugged up the rule “if 1=>2, then valid inference is ~2=>~1)
  • ~2=>1 (because ~(~1) means double negative and double negative is positive hence ~(~1)=1)

This is our second rule: Unless1=>2 then ~2=>1

Table

Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=>~1 Negative of end part=> negative of start part
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.
Unless Unless 1=>2 ~2=>1 Negative of end part=>start part unchanged.

Logical connective: otherwise

  • Suppose given statement is: 1, otherwise 2.
  • you can write it as unless 1 then 2.  (unless1=>2)
  • Then use the formula for “unless.”

Logical connective: When, Whenever, every time

  • Given statement: he scores century, when match is fixed.
  • This is not in standard format of “**logical connective word**, simple statement #1, simple statement #2.”
  • So first I need to exchange the positions: “when match is fixed (#1), he scores century (#2)”.
  • In case of when and whenever, the valid inference is= same like “If, then”. That means negative of end part=>negative of starting part.
  • Same formula works for “whenever” and “Everytime”.
  • Update the table
Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=~1 Negative of end part=> negative of starting part
When When 1=>2
Whenever Whenever 1=>2
Everytime Everytime 1=>2
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.
Unless Unless 1=>2 ~2=>1 Negative of end part=>starting part unchanged.

Logical Connective: Either OR

Given statement: Either he is drunk(1) or he is ill(2).

In such cases, if not 1 then 2. And if not 2 then 1.

Meaning,

  1. if he is not drunk then he is definitely ill
  2. if he is not ill, then he is definitely drunk

both are valid. Update the table

Logical connective Given statement Valid inference using symbol Valid inf. In words
If If 1=>2 ~2=~1 Negative of end part=> negative of starting part
When When 1=>2
Whenever Whenever 1=>2
Everytime Everytime 1=>2
Only if Only if 1=>2 ~1=>~2 Negative of start part=>negative of end part.
Unless Unless 1=>2 ~2=>1 Negative of end part=>starting part unchanged.
Otherwise 1 otherwise 2=> rewrite as Unless1=>2.
Either or Either 1 or 2
  • ~2=>1
  • ~1=>2
Negative of any one part=> remaining part remains unchanged.
  • Now let’s solve some questions from old CSAT and CAT papers
  • Please note: in the exam, actual wording / meaning of the simple statement doesn’t matter. Just apply the formulas as given in above table.
  • For example, “if you’re in army, you have to wear uniform.” Then valid inference is ~2=>~1 (you don’t have to wear uniform, then you’re not in army).
  • Now ofcourse there would be exceptional situation when army officer/jawan doesn’t need to wear uniform, for example during espionage mission behind the enemy lines. In that case you don’t have to wear uniform, but you’re still in the army.
  • But keep in mind, while solving logical connective question under the “aptitude/reasoning” portion you don’t have to surgically dissect or nitpick the meaning every statement. Just “if 1=>2” then “~2=>~1”.

Demo Q: Only if: bored TV brother (CSAT 2012)

Examine the following statements:

  1. I watch TV only if I am bored
  2. I am never bored when I have my brother’s company.
  3. Whenever I go to the theatre I take my brother along.

Which one of the following conclusions is valid in the context of the above statements?

  1. If I am bored I watch TV
  2. If I am bored, I seek my brother’s company.
  3. If I am not with my brother, then I’ll watch TV.
  4. If I am not bored I do not watch TV.

Approach

First we’ll construct valid inferences from the question statements

Given Question Statement #1:

  • Given =I watch TV only if I am bored
  • This is not in standard format. So first exchange position
  • Only if I’m bored (1), I watch TV(2)
  • What is the valid inference? Just look at the formula table
  • Only if 1=>2 then ~1=~2
  • Valid inference= if I’m not bored, I do not watch TV.
  • Look at the statements given in the answer choices, (D) matches. Therefore, final answer is (D).

Demo Q (If, then) Professor Headaches  (CAT’98)

You’re given a statement, followed by four statements labeled A to D. Choose the ordered pair of statements where the first statement implies the second and two statements are logically consistent with the main statement.

Given statement: If I talk to my professors(1), then I didn’t need to take a pill for headache.(2)

Four Statements

  1. I talked to my professors
  2. I did not need to take a pill for headache
  3. I needed to take a pill for headache
  4. I did not talk to my professor.

Answer choices

  1. AB
  2. DC
  3. CD
  4. AB and CD

Approach

Given statement is in standard format already

#1 I talk to my professors
#2 I didn’t need to take a pill for headache.

Let’s classify the four statements

Classification Four statements
1
  1. I talked to my professors
2
  1. I did not need to take a pill for headache
~2
  1. I needed to take a pill for headache
~1
  1. I did not talk to my professor.

Answer choice (i) AB

If you observe the answer choice (I): AB= I talked to my professors, I did not need to take a pill for headache. This is valid because if 1=>2 is already given in the question statement itself.

Answer choice (ii) DC

  • I did not talk to my professor (~1), I needed to take a pill for headache (~2). Meaning ~1=>~2.
  • This is invalid because as per our table, if 1=>2, then valid inference is ~2=>~1.

Answer choice (iii) CD

I needed to take pill for headache (~2), I did not talk to my professor (~1). Meaning ~2=>~1. This is valid as per our table. Therefore final answer is (IV) AB and CD

Demo Q: Either or: derailed/late train (CAT’97)

Given statement: either the train is late (1) or it has derailed (2)

Four statements

  1. Train is late = 1
  2. Train is not late = ~1
  3. Train is derailed =2
  4. Train is not derailed =~2

(^note: I’ve classified the statements in advance)

Answer choice

  1. AB
  2. DB
  3. CA
  4. BC

Approach

As per our table, the valid inferences for either or are

~2=>1 If the train is not derailed, it is late. DA
~1=>2 If the train is not late, it is derailed BC

Correct answer is (III): BC
For more articles on reasoning and aptitude, visit Mrunal.org/aptitude

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197 Comments on “[Reasoning] Logical Connectives (if, unless, either or) for CSAT, CAT shortcuts formulas approach explained”

  1. MRUNAL SIR PLS TELL ME FROM WHERE SHOULD I PRACTISE THESE TYPE OF QUESTIONS.
    THANKS MRUNAL SIR FOR SUCH A GREAT SIMPLIFICATION.

  2. mrunal sir i have doubt in these syllogisms ques pl help..
    que 1: some tools are not hammers
    all tools are made of iron
    conclusions : a)some hammers are made of iron
    b)some hammers are not made of iron
    c) some things made of iron are not hammers
    answer is c.

    que2 all cigaretes are cigars
    some cigaretes are not good for health
    concl a)some cigars are not good for health
    b)some cigars are good for health
    c) both
    d) none
    correct ans a

    all true are false
    some true are not lies
    conc: a)some false are lies
    b)some false are not lies
    answer is b

    some tangles are not hair
    all strings are hair
    conc a) some tangles are strings
    b) some tangles are not strings
    c) both
    d) none
    correct is b

    plz explain according to ur method u taught us….

    1. answers
      q1 a and c
      q2 a
      q3 b
      q4 b

      1. hey matt how did u get the answers whether by following mrunal sir’s approach or by some other method?? pl explain for 1 or 2 ques

        1. hi HJ
          i solve such question by drawing vein diagram…..not by mrunal method

          1. ya got it by venn diagram because these questions by sirs method are not answerable or else i might have made some mistakes ….

          2. yes

  3. @ahamed.
    dont use the rules exclusively..use logic alongside too… the answer to ur question is D. read the answer choices carefully corelating with question and voila.. ull have the answer…

  4. Sir, how can i prepare for Public Admn as my optional subject

    1. hey matt how did u get the answers whether by following mrunal sir’s approach or by some other method?? pl explain for 1 or 2 ques..

      1. sorry posted here by mistake..

  5. friends,
    hers’s a qn from previous papers..pls explain d science behind it..

    ‘A hunter aims his gun at a point between the eyebrows of a monkey sitting on the branch of a tree. Just as he fires the monkey jumps down.
    The bullet will

    a)Hit the monkey at the point aimed
    b)Hit the monkey below the point aimed
    c)Hit the monkey above the point aimed
    d)miss the monkey altogether

    given ans ‘a’

    1. all objects fall with the same constant acceleration = 9.8 m/s^2 Irrespective of the object’s weight.
      Also , horizontal motions and vertical motions are independent: gravity acts only upon an object’s vertical velocity, not upon its velocity in the horizontal direction.
      The hunter’s bullet, therefore, falls with the same acceleration as the monkey and hence hits the monkey at the point aimed.
      Its a question from Projectile Motion from Class XI.

      When monkey jumps down it falls freely with 9.8 ms-2 and the bullet makes a parabolic distance under gravity field and hence they fall the same distance in the same time: the monkey falls from the tree branch, and the bullet falls the same distance from the straight-line path it would have taken in the absence of gravity. Therefore, the bullet will always hit the monkey, no matter the initial speed of the bullet.

      So basically the Science is Gravity.

      1. Well monkey is no in free fall in here…monkey jumps down ie he must have some initial velocity in vertically downward direction… so either in question it should be that monkey falls down or the answer will be above the point aimed or miss altogether…..

        1. I had the same doubt as Mr.vips’..yes its projectile motion..regards Mr.Jha, Mr.vips

  6. this is very very good Mrunal bhai.
    Can you or someone please provide 10-15 practice problems.
    this is the area i was most concerned in reasoning.
    thank you

  7. hi mrunal,
    thanks a lot
    now i can guarantee you i m going to crack IAS PRE this year.

  8. THANK YOU SIR.. THIS METHOD HELPED ALOT.

    CAN YOU PLEASE PROVIDE EXPLANATION FOR:

    1.CRITICAL REASONING(STRENGTHEN N WEAKEN ARGUMENTS)

    AND

    2.DECISION MAKING

    I ALWAYS GETS CONFUSED BETWEEN CHOOSING LINE OF DUTY OR HUMANISTIC ANSWERS..

    THANK YOU!

  9. thanks a lot sir jee

  10. I want to download all articles on economy but there is no common link. Whether i need to save each individually…. ? Plz guide me. Thanks in advance….

  11. Sample Question

    Q) Consider the following statements:
    1)King mocks only when courtiers laugh 2)The queen weeps only if courtiers laugh
    3) Only if the queen weeps does the king mock
    Which of the following can be definitely concluded from the above statements
    a) Courtiers laugh only if queen weeps b) King does not mock if queen does not weep
    c) If courtiers laugh, queen weeps d) If queen weeps, king laughs

    Answer to this question is: option (b)

  12. SIR i am really confused, as i am not able to find the solution to many questions.please help .e.g

    1) If it rains there will be strong wind.
    2)Only if it is humid, SAM goes for a walk.
    3)It is humid if there is a strong wind.

    Conclusions:
    1) SAM goes for a walk whenever it is humid
    2) SAM does not go for a walk when there’s a strong wind
    3) SAM does not go for a walk if it does not rain.
    4) NONE OF ABOVE.
    KINDLY HELP WITH COMPLETE SOLUTION.I HAVE TRIED IT MY SELF,BUT NOT ABLE TO UDERSTAND.

    1. Hi k,
      I did some analysis and found ans is D-None of the above. Please let me know the correct answer and then I will try to explain in detail.

      1. YES. Correct answer is D. kindly explain pls

        1. An analysis of staements first and then rejection of option one by one
          1) If rains occur>> There will be strong wind (>> means implies)
          2) If it is not humid >> Sam Does not go for a walk or it can be written as:
          2-a) If Sam Goes for a walk >> It is humid
          3) If there is a strong wind >> it is humid

          Now Analysis options
          1) First option says that If(Whenever) it is humid >> Sam goes for a walk, which is inconsistent with 2-a) because reverse may not be true

          2) Second option says that If (when) there is a strong wind>> Sam does not go for a walk. When one combines 3 and 2-a) you will find that no conclusion can be drawn and it turns to be same as first option and hence not true

          3) If it does not rain >> Sam does not go for a walk. Now it is the easiest exclusion as we don’t know what happens or what it leads to ‘when it does not rain’ by three statements.

          I hope, I tried to make it clear.

    2. The answer must NONE, since none of the conclusion follows anypossible outcome

      1.rains==>stronwind
      2.sam walks==>if it is humid
      3.strong wind==>humid

      Possible outcome: rains==>strongwinds==>humid , ther is no similar conclusions in the given answer.

      1. What’s wrong with option 3
        Not humid = not strong wind= not rains
        Not humid = not walk

        So option 3 _ not walk= not rain is right? Help me.

  13. can some one upload questions from this part?i am not able to find in the book..kindly some one do it..

  14. Superb. All concepts cleared!

  15. sir my name is prathyusha. I filled my Upsc prelims 2013 application successfully. but i didn’t received registration id so as to download myHall ticket of CSP2013. I forgot registration id. please help me so as to get my registration id. pls

  16. Please provide some material on decision making and interpersonal and communication skills.

  17. Thanks Mrunal…..Before this publication, this section (logical connectives) was making me nervous. Now I am feeling confident. If possible, please provide some sample questions. Thanks once again.

  18. This is of great help .Thank U sir!!

  19. Sir thank you very much

  20. mrunal sir what wil we do if word DESPITE comes..plz reply

  21. Thanks Mrunal
    in ur article….last line should be
    Correct Ans is IV BC
    Anyways
    Can u help me with this

    1)The aircraft takes off if the light blinks
    2)Only if there is no storm the aircraft takes off
    3)If there is a storm the light does blink

    options
    a. if the light blinks there is no storm
    b. the aircraft takes off if there is no storm

    Solution
    storm==>not take off===>not blink
    blink ===>no storm====>…….

    here cant we add “take off” which will make option b also true.(can see option a is true)
    if not why cant we do that
    Plz answer!!!!

    1. option A cannt be correct
      3)If there is a storm(1) the light does blink(2)
      n as per rules
      -(2)=>-(1) 0r (1)=>(2)
      and a. if the light blinks there is no storm
      this is contradictory

      1. shani ji here both statements a & b are wrong na

        1. yupp both r wrng

        2. Statement B is correct. It is rephrase of given statement “Only if there is no storm the aircraft takes off” -> The aircraft takes off (only) if there is no storm.

          1. A is wrong, right?

      2. what is wrong in option b??

    2. actually u tiped it wrong… 3)it must be “doesnt”
      if its not, then 1 and 3 contradicts each other.
      In that case option A is wrong and B is right.

  22. sir post some practice questions too or refer to some book from where we may get some good questions 4 practice.

  23. First time i understand these thing in a very lucid manner…..thanks to u mrunal!!!!

  24. thank you so much !! you are really a true teacher or much more then that….awesome explanation man !!

  25. Sir u are great, u explain tough things with ease. I like all your articles. U are a great teacher. U are doing a great job, God has to bless u.

  26. Dear Mrunal, how many months of study is necessary to clear upsc cse to become an IAS or IPS ? Thank you.

  27. Hi Mrunal
    thanks a lot for all ur help
    Can u please give some explanation on type of Syllogism questions where we have to find which of statements are “both can be true but both cannot be false ” or ” both can be false but cannot be true” or ” both can be true and both can be false”
    Pls throw some light on this kind of questions , it was asked in 2011 paper

  28. Pl help me with following ques
    When ever shyam hears of an exam, he losses sleep
    a) shyam heard of an exam
    b) shyam didnt heard of an exam
    c) shyam lost slepp
    d) shyam didnt lost losse sleep
    Option —
    1) CA
    2) BD
    3) DB
    4) AD
    Ans according to TMH is – 1,

    1. CA cnnt be answer as per rules
      When ever shyam hears of an exam(1), he losses sleep(2)
      so as per rules
      -(2)=>-(1) OR (1)=>(2) n CA IS contradictory so it cnnt be

    2. please apply ur mind… don’t blindly belief on any guide………… or material. answer is (DB)

    3. please visit takshzila shikshak on youtube the ans ca is correct

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