Mrunal [Reasoning] Logical Connectives (if, Unless, Either or) for CSAT, CAT Shortcuts Formulas Approach Explained » Mrunal

6/13/2014 Mrunal [Reasoning] Logical Connectives (if, unless, either or) for CSAT, CAT shortcuts formulas approach explained » Mrunal

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

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 ofthe 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 andPN. Already explained in previousarticles.

Contains words like “if, unless, onlyif, whenever” etc. can be classifiedinto 1, ~1, 2, ~2 (we’ll see in thisarticle)

Have to mugup more formulas, takesmore time than logical connectivequestions.

Less formulas and quicker thansyllogism.

Question Statements:

I. All cats are dogsII. some pigs are cats

III. no dogs are bird

Conclusion choices:

A. Some cats are dogs

Question statements:

1. I watch TV only if I am bored2. 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:

A. If I am bored I watch TVB. If I am bored, I seek my

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B. No birds are catsC. some pigs are birdsD. Some pigs are not birds

brother’s company.C. If I am not with my brother, than

i’ll watch TV.D. 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 Logicalconnectives.Whenever you’re given a question statement, first rule is: question statementmust be in the standard format.The standard format is****some logical connective word *** simple statement#1, simplestatement #2.It means, the question statement must start with a logical connective word,otherwise exchange position. For example

Given questionstatement

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 inthe standard format.

You’ve to wear uniform,if you’re in the army

We need to exchange position because the partcontaining “IF” is not given in the beginning ofthis statement, given statement is not in standardformat.Therefore, Rewrite given statement asIf you’re in the army, you’ve to wear uniform.

You’ve to salute,whenever CommandingOfficer comes in yourcabin.

Need to exchange position. Because statementdoesn’t start with the logical connective“whenever”.Therefore rewrite the given statement asWhenever CO comes in your cabin, you have tosalute.

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

Logical connective: if then

Consider these two simple statements



1. You’re in army2. 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 wearuniform, you’re in army.

you’ve to wear uniform in navy, air force,BSF etc. so this inference is not alwaysvalid.

2. If not#1,thennot #2

if you’re not in army,you don’t have to wearuniform.

you’ve to wear uniform in navy, air force,BSF etc. so this inference is not alwaysvalid.

3. if not#2,thennot #1

If you don’t have towear uniform, you’renot in army.

Always valid.

In the exam, you don’t have to think ^that much. Just mugup the followingrule: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 logicalconnective 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 startswith 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 tomugup.

Logical connective: Only IF

In such scenario, you’ve to rephrase given statement into “if then” and thenapply 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 exchangingpositions. That isif 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 willbe ~1=>~2. Don’t confuse between 1 and 2. Because essentially the validinference 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

Logicalconnective

Givenstatement

Valid inferenceusing symbol

Valid inf. In words

If If 1=>2 ~2=>~1Negative of end part=>negative of start part

Only ifOnly if1=>2

~1=>~2Negative of startpart=>negative of end part.

Logical Connective: UNLESS

Given statement: Unless you bribe the minister(#1), you will not get the 2Glicense.(#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 the2G 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 => ~1Negative of (2) => negative of (1)Negative of (you’ll not get the 2G license)=>negative of (you don’t bribe theminister)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 haddefinitely 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 validinference is ~2=>~1)~2=>1 (because ~(~1) means double negative and double negative is positivehence ~(~1)=1)

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

Table

Logicalconnective

Givenstatement


Valid inf. In words

If If 1=>2 ~2=>~1Negative of end part=>negative of start part

Only ifOnly if1=>2

~1=>~2Negative of startpart=>negative of end part.

UnlessUnless1=>2

~2=>1Negative of end part=>startpart 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**, simplestatement #1, simple statement #2.”So first I need to exchange the positions: “when match is fixed (#1), he scorescentury (#2)”.In case of when and whenever, the valid inference is= same like “If, then”. Thatmeans negative of end part=>negative of starting part.Same formula works for “whenever” and “Everytime”.Update the table

Logicalconnective

Givenstatement


Valid inf. In words

If If 1=>2

~2=~1Negative of end part=> negativeof starting part

WhenWhen1=>2

WheneverWhenever1=>2

EverytimeEverytime1=>2

Only ifOnly if1=>2

~1=>~2Negative of start part=>negativeof end part.

UnlessUnless1=>2

~2=>1Negative of end part=>startingpart 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 ill2. if he is not ill, then he is definitely drunk

both are valid. Update the table

Logicalconnective

Given statementValidinferenceusing symbol

Valid inf. In words

If If 1=>2

~2=~1Negative 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=>~2Negative of startpart=>negative of end part.

Unless Unless 1=>2



Otherwise1 otherwise 2=>rewrite asUnless1=>2.

~2=>1 Negative of end part=>startingpart unchanged.

Either or Either 1 or 2~2=>1~1=>2

Negative of any one part=>remaining part remainsunchanged.

Now let’s solve some questions from old CSAT and CAT papersPlease note: in the exam, actual wording / meaning of the simple statementdoesn’t matter. Just apply the formulas as given in above table.

For example, “if you’re in army, you have to wear uniform.” Then validinference is ~2=>~1 (you don’t have to wear uniform, then you’re not inarmy).Now ofcourse there would be exceptional situation when army officer/jawandoesn’t need to wear uniform, for example during espionage mission behindthe enemy lines. In that case you don’t have to wear uniform, but you’re still inthe army.But keep in mind, while solving logical connective question under the“aptitude/reasoning” portion you don’t have to surgically dissect or nitpick themeaning 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 bored2. 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 abovestatements?

A. If I am bored I watch TVB. If I am bored, I seek my brother’s company.C. If I am not with my brother, then I’ll watch TV.D. 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 boredThis is not in standard format. So first exchange positionOnly 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=~2Valid 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 theordered pair of statements where the first statement implies the second and twostatements 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 forheadache.(2)

Four Statements

A. I talked to my professorsB. I did not need to take a pill for headacheC. I needed to take a pill for headacheD. I did not talk to my professor.

Answer choices

I. ABII. DC

III. CDIV. 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 A. I talked to my professors

2 B. I did not need to take a pill for headache

~2 C. I needed to take a pill for headache

~1 D. 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 needto take a pill for headache. This is valid because if 1=>2 is already given in thequestion 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

A. Train is late = 1B. Train is not late = ~1C. Train is derailed =2D. Train is not derailed =~2

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

Answer choice

I. ABII. DB

III. CAIV. 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): BCFor more articles on reasoning and aptitude, visit Mrunal.org/aptitude

http://mrunal.org/aptitude



URL to article: http://mrunal.org/2013/04/reasoning-logical-connectives-if-unless-either-or-for-csat-cat-shortcuts-formulas-approach-explained.html

Posted By Mrunal On 30/04/2013 @ 13:47 In the category Aptitude

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Mrunal [Reasoning] Logical Connectives (if, Unless, Either or) for CSAT, CAT Shortcuts Formulas Approach Explained » Mrunal