Logical Connectives – Concepts and Questions Based on Logical Conditions

Monday, May 7th, 2018


Logical Connectives Concepts and Questions Based on Logical Conditions

What are Logical Connectives?

Logical connectives are basically words or symbols which are used to form a complex sentence from two simple sentences by connecting them. Some Logical Connectives are – If, Only if, When, Whenever, Unless etc. Questions based on statements formed using logical connectives are simpler than other questions of exam so it will be beneficial to take command on this topic.

These type of sentences consist of two parts, one is Cause and one is Effect. Usually Cause is present after the connectives.

Some examples are like-

1. If you are in navy, you have to wear uniform.

Cause – you are in navy, Effect –you have to wear uniform

2. If I am bored, I go for a movie.

Cause- bored, Effect- go for a movie.

Let’s discuss all different types of questions on connectives.

1. Logical Connective (If, then)-

These statements are in the form of( If p, then q) or( q if p).

Examples-

1. If it rains then ground gets wet.

2. Raman will pass if he works hard.

Cause-effect relation – In these statements cause is a sufficient condition for effect to happen but not necessary condition.

Let’s discuss example 1 in detail – If it rains then ground gets wet.

In this statement firstly we’ll identify the cause and effect. “It rains is cause and ground gets wet is effect.

Now we’ll make all possible cases for understanding it clearly.

Case1. If cause is happening, it means it is raining so obviously ground gets wet (given).

So if cause (yes) then effect (yes).

Case2. If cause is not happening, it means its not raining so we have no idea about the condition of ground because it may be wet or not. It can get wet by some other cause also, we don’t know about that cause.

So if cause (No) then effect (may or may not be)

Case3. If effect is happening, it means ground gets wet but we don’t know how it gets wet. We have no idea about exact cause so cause may be or may not be happened.

So if effect (Yes) then cause (may be or may not be)

Case4. If effect is not happening, it means ground does not get wet so we can clearly say that it has not rained because if it had rained the ground would definitely got wet.

So if effect (No) then cause is surely (No).

We can see that in this example we can say surely only in two cases (case1 & case4).

If we draw a conclusion for If and then connectives.

Conclusion (If, then):-

Cause – yes; effect – yes

Cause –no; effect -?

Effect –yes; cause -?

Effect –no; cause –no

In these type of statements we have SINGLE EFFECT and MULTIPLE CAUSES like ground gets wet is one effect causes may be many like rain, bucket of water etc.

Note- sometimes instead of ‘if’ when, whenever, everytime these connectives are used but result will be same in these cases also because meaning of statement doesn’t changes.

2. Logical Connective (only if/when) –

These statements are in the form of (Only If p, then q) or (q, only if p).These type of connectives are very important from exam point of view.

Some examples are like

1. I will meet you only if I come to Delhi.

2. Only if Rohit runs fast, he will catch the train.

Now first we’ll identify the cause and effect

1– Come to Delhi – cause; meet you- effect

2– runs fast – cause; catch the train- effect

Cause-effect relation – In these statements cause is a necessary condition for effect to happen but not sufficient condition.

Let’s discuss example 2 in detail – . Only if Rohit runs fast, he will catch the train.

Case1– cause yes, it means Rohit runs fast so it doesn’t mean that he will suerly catch the train. Cause – yes; effect- may be may not be

Case2 – cause no, it means Rohit doesn’t run fast so according to statement he will catch the train only if he runs fast so surely he will not catch the train.

Cause- no; effect- no

Case3 – effect yes, it means Rohit catches the train so surely we can say he ran fast.

Effect- yes; cause- yes.

Case-4 – effect no, it means he doesn’t catch the train this doesn’t implies that he didnot ran fast may be due to some other reason he was not able to catch the train.

Effect- no; cause – may be may not be

Conclusion (only if):-

Cause – yes; effect- ?

Cause – no; effect- no

Effect – yes; cause- yes

Effect – no; cause-?

In this type of statement there is a SINGLE CAUSE and MULTIPLE EFFECTS like running fast is a cause he can catch the train cannot catch the train these are effects.

Note- Instead of if,( when) also occur sometimes but the situation remains same.

3. Logical Connective (If and Only If) –

These type of sentences are union of sentences having connectives If and only If. These are the simplest in all connectives. It is like necessary and sufficient condition which means cause is a necessary and sufficient condition for effect to take place.

Examples :

1. If and only If he clears the pre exam, he will get selected.

2. Rahul will eat the fruit If and only If it is an apple.

3. A whole number is even If and only If its square is even.

Cause-effect relation – In these statements cause is a necessary and sufficient condition for effect to happen.

Let’s discuss all possible cases for example 1 – If and only If he clears the pre exam, he will get selected.

Cause – clears the pre exam, effect – get selected

Case1. Cause is happening it means he cleared pre exam so surely he will get selected because given is ‘if he clears exam he will get selected’.

Cause –yes; effect – yes

Case2. Cause is not happening it means he has not cleared pre exam so surely he will not get selected.

Cause-no; effect-no

Case3. Effect is happening it means he selected it implies surely he has cleared his pre exam.

Effect – yes; cause – yes

Case4. Effect is not happening it means he has not selected .By the statement ‘only if he clears the pre exam he will get selected’ we can say that he not selected means he has not cleared pre exam.

Effect – No; cause-no

Conclusion (If and only If):-

Cause – yes; effect – yes

Cause – no; effect – no

Effect – yes; cause- yes

Effect – no; cause – no

In these type of statements cause is necessary and sufficient condition for effect.

Note- Instead of (If and only If) connectives (until and unless) and (when and only when) cn also be used for similar conditions. Results will remain same.

4. Logical Connective (unless):-

These statements are in the form of (Unless p, q) or (q, unless p). These type of connectives do not have that much importance for exams but it is better to understand the concept. It is a negative type of connective which gives negative sense in the statement. In these statements there is no proper cause and effect. Only two statements are there in these type of statements.

Examples :-

1. Unless I’m mistaken, John’s house is the second on the left.

2. Unless Virat is the captain, India will lose the match.

Let’s discuss example 2 in detail – . Unless Virat is the captain, India will lose the match.

Here No cause and effects is present because it is negative statement. We can check using statements 1 and 2 directly. Virat is the captain (1) & India lose the match (2).

Case1. If 1 is true it means Virat is the captain it doesn’t mean that India will surely win the match so we can’t say anything.

(1) – yes, (2) – may be may not be

Case2. If 1 is false it means Virat is not the captain it means surely India will lose the match.

(1) – no, (2) – yes

Case3. If 2 is true it means India lose the match so we can’t say anything that Virat is the captain or not. India can also lose the match when Virat is captain.

(2) – yes, (1) – may be may not be

Case4. If 2 is not true it means India won the match so we can surely say that Virat is the captain.

(2) – no, (1) – yes

Conclusion (unless):-

(1) – yes; (2) – ?

(1) – no; (2) – yes

(2) – yes; (1) – ?

(2) – no; (1) – yes

5. Logical Connective (Either or):-

These statements are in the form of (either p or q). These are very simple statements. P and q are two statements. No cause and effect is present in this case.

Examples-

1. I drink either green tea or juice.

2. Either Ronak is intelligent or Raman is foolish.

In these type of statements clearly if one statement is false then second is true and if one is true second is false.

For ex.1

If I drink green tea it means surely I don’t drink juice. And If I drink juice it means surely I don’t drink green tea. No need to make cases here we can directly write conclusion.

Conclusion (Either, or)-

(1) – yes; (2) – no

(1) – no; (2) – yes

(2) – yes; (1) – no

(2) – no; (1) – yes

Note- In these type of statements clearly one of the statements is definitely true and one is definitely false.

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Logical Connectives – Concepts and Questions Based on Logical Conditions

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