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

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

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