FREE Basic Verbal Concept-04

FREE Basic Verbal Concept-04

Basics of Logic:

If let’s say

L = All mangoes are tasty

G = Non of the mangoes is tasty

M = Some of the mangoes are tasty

O = Some of the mangoes are not tasty

The relationship between these statements are shown in the diagram below:

Two statements are said to be contradictory, if one of the statement is true, the other statement MUST be false, and vice versa. For instance,

If Statement L is true       ⇒       Statement G MUST be false.

If ‘All mangoes are tasty’ is true, then ‘No mango is tasty’ MUST be false.

Similarly,

If Statement G is true       ⇒       Statement L MUST be false.

If ‘No mango is tasty’ is true, then ‘All mangoes are tasty’ MUST be false.

But, it is very important to note that if Statement L is false, then Statement G is not necessarily true. i.e Statement G Could be true, but not MUST be true. For instance,

If Statement L is false       ⇒       Statement G is not necessarily true.

If ‘All mangoes are tasty’ is false, then ‘No mango is tasty’ is not necessarily true.

Similarly,

If Statement G is false       ⇒       Statement L not necessarily true.

If ‘No mangoe is tasty’ is false, then ‘All mangoes are tasty’ is not necessarily true.

Because the opposite of ‘All’ is ‘NOT all’. In other words, the opposite of ‘100%’ is ‘Not 100%’, which not necessarily means 0%. It may be 99%, 98%, 97%, ….. 0%. Therefore, ‘NOT all’ does not only mean 0%.

Sub-contrary Statement:

Two statements are said to be sub-contrary to each others, if both statements are stating opposite points; but when one statement is true, the other statement is not necessarily false. For instance,

If Statement M is true       ⇒       Statement O not necessarily false, but is opposite of Statement M.

If ‘Some mangoes are tasty’ is true, then ‘Some mangoes are not tasty’ is not necessarily false.

Similarly,

If Statement O is true       ⇒       Statement M not necessarily false, but is opposite of Statement O.

If ‘Some mangoes are not tasty’ is true, then ‘Some mangoes are tasty’ is not necessarily false.

But,

If Statement M is false       ⇒       Statement O MUST be true.

If ‘Some mangoes are tasty’ is false, then ‘Some mangoes are not tasty’ MUST be true.

Similarly,

If Statement O is false       ⇒       Statement M Must be true.

If ‘Some mangoes are not tasty’ is false, then ‘Some mangoes are tasty’ MUST be true.

This last one was little it tough for few people. You may ask me if you still have confusion.

Sub-alternative Statement:

Firstly, remember that two statements cannot be Sub-alternative of each other simultaneously. But two statements can be Alternative of each other.

One statement is said to be sub-alternative of the other, if the other statement is true this one statement must also be true, but not hold in reverse scenario (i.e not applicable in vice versa situation). For instance,

If Statement L is true       ⇒       Statement M MUST also be true.

If ‘All mangoes are tasty’ is true, then ‘Some mangoes are tasty’ MUST also be true.

But

If Statement M is true       ⇒       Statement L is not necessarily true.

If ‘Some mangoes are tasty’ is true, then ‘All mangoes are tasty’ is not necessarily true.

Also note that,

If Statement L is false       ⇒       Statement M is not necessarily false.

If ‘All mangoes are tasty’ is false, then ‘Some mangoes are tasty’ is not necessarily false.

But,

If Statement M is false       ⇒       Statement L MUST also be false.

If ‘Some mangoes are tasty’ is false, then ‘All mangoes are tasty’ MUST also be false.

From the diagram above, we can conclude that:

If Statement L is true       ⇒       Both Statement G & Statement O MUST be false.

Similarly,

If Statement G is true       ⇒       Both Statement L & Statement M MUST be false.

Therefore,

(Statement L & Statement G), (Statement L & Statement O), and (Statement M & Statement G), are contradictories.

(Statement M & Statement O) are sub-contraries.

And,

Statement M is sub-alternative to Statement L

Statement O is sub-alternative to Statement G

Important Note: Some means at least one. And ‘at least’ does not mean only equal to, rather it means ‘equal to as well as greater than’.

Thus, if “All people believe that XYZ…”       ⇒       “Some people believe that XYZ…”

But, reverse case not holds true. i.e

if “Some people believe that XYZ…”       does not imply       “All people believe that XYZ…”

Let’s understand it further in terms of numbers,

If there are 100 people. And it is said that “Some people believe XYZ…”

Here, Some people = 1 to 100 people inclusive = At least 1 person = Every possibility except 0.

Similarly,

If there are 100 people. And it is said that “Many people believe XYZ…”

Here, Many people = More than 50 = 51 to 100 inclusive = At least 51 = More than 50%

Similarly,

If there are 100 people. And it is said that “Not all people believe XYZ…”

Here, Not all people = Not 100% = 0 to 99 inclusive = Less than 100%

Let’s understand this through overlapping sets (i.e Venn diagrams), so that you’ll get full grip over these important basics of logic.

All F is A.     (Or you may call it as: All football fans are alcoholic.)

Diagrammatically, we may draw it as follow:

From this statement, we cannot conclude “All A is F, but we can confirm at least “Some A is F.

No F is A.     (Or you may call it as: No football fans is alcoholic.)

Diagrammatically,

From this statement, we cannot conclude “Some A is F, but we can confirm that “Some A is not F. Because as I said, some might be 100% or at least 1%, so “No F is A suggests that “Some A is not F.

The below point is very important that mostly tested in critical reasoning / logical reasoning in exams.

Some F is A.     (Or you may call it as: Some football fans are alcoholic.)

Diagrammatically,

From this statement, we cannot conclude “Some A is not F, because it might be possible that whole of circle A inscribed in Circle F (i.e placed totally in circle F, if circle A is smaller than F). But we can confirm that “Some A is F.

Important Note: Whatever we confirm or conclude MUST be true. In other words, if any statement is identified as COULD be true, rather than MUST be true, it cannot be concluded or confirmed.

Also this below point is extremely important.

Some F is not A.     (Or you may call it as: Some football fans are not alcoholic.)

Diagrammatically,

From this statement, we cannot conclude “Some A is F, but we can confirm that “Some A is not F.

I’m not in a mood to tease you further in such things. ðŸ™‚

So let’s discuss a final scenario where many people confuse.

No non-alcoholic is football fan.     (Generally speaking: No non-A is F)

Diagrammatically, it’s the same as “All football fans are alcoholic”, because it means “no out-sided region of A belongs to F” i.e

I’m sure your commonsense is now getting on the right track, if it wasn’t before. ðŸ™‚

Takeaways:

1. The opposite of ‘All’ is ‘Not all’, which COULD be 0 or 99.

2. The extreme opposite of ‘All’ is ‘Non’, which is 0.

3. ‘Not all’‘Non’

4. All mangoes are tasty = No mango is non-tasty

5. Not all ≠ only 0; Not all means not 100%, therefore, it may be 99% or 0%.

Diagrammatically,

Subscribe
Notify of