**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:

**Contradictory Statements:**

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*. Because as I said, some might be 100% or at least 1%, so

**A**is not**F**“*“No*suggests that

**F**is**A**“*“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,

Very difficult to understand this concept. Plz if any easy way to understand

You have stated in the beginning that: ‘if some people believe that XYZ…’ DOES NOT IMPLY ‘all people believe that XYZ…’ Now, Referring to ‘No F is A’, You conclude that ‘no F is A’ suggests that ‘some A is not F’. ‘no F is A’ clearly means that a person, who is alcoholic, cannot be a football fan. It is crystal clear. But, the statement ‘some A is not F’ is not as strong and clear as ‘no F is A’. In my opinion, the word ‘some’ refer to anything between 0% and 100%; it does not refer to… Read more »

Yeah you are correct, But the point which I like people to know is that if you can infer that “All A is not F” and from this we can further infer that some A is not F. For instance, if I have 100 mangoes in a basket, and all mangoes in the basket are sweet. Can I say some mangoes are sweet? Of course, as some means at least 1. i.e if all mangoes are sweet, then it is logically correct that at least 1 mango is sweet. At beginning, as it’s mentioned if “All people believe XYZ…”, then… Read more »