Basic Verbal Lesson-03
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:
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 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. 🙂
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. 🙂
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,
Cause & Effect:
Let’s say,
Maria is good in logic, so she is good in reasoning.
Cause = Maria is good in logic
Effect = She is good in reasoning.
Similarly,
If rain fall, then soil wet.
Cause = Rain fall
Effect = Soil wet.
At this stage, you should start thinking critically. The above argument is a MUST be true argument, which says If rain fall, then soil wet. Because rain fall always results soil wet.
Can you think “Some rain fall does not result soil wet?” Well, the answer is no, so this statement is false.
Therefore, All rain fall results to soil wet.
This is a General Believe or a fact, which cannot call into question.
But what can be called into question is the reverse scenario.
Note that soil is wet because of raining. But the reverse case is not MUST be true scenario. i.e,
As the soil wet, therefore, the rain must have fallen.
According to the diagrammatic approach, as stated earlier, you must believe that ‘rain fall’ is a smaller circle inscribed (i.e. completely inside) in larger circle ‘soil wet’. Because rain fall always result soil wet. But if somebody see wet soil, it does not necessarily due to the rain fall. The soil wet may be due to some other reason or reasons.
Remember that the game of cause and effect is very complex. In advance level study plan, we’ll discuss this in such a manner that you’ll feel much confident while solving arguments related to cause & effect. So at this stage, we just need to learn basics. So let’s go ahead.
Valid vs Invalid argument:
Let’s understand valid and invalid argument.
A valid argument has a valid conclusion, while an invalid argument has an invalid conclusion. For instance,
Example 1:
All primates have heart.
All humans are primates
⇒ All humans have heart. (Which is valid and true.)
Similarly,
All breathing things are living organisms
All humans are breathing
⇒ All humans are living organisms. (Which is valid and true.)
Valid: Which can be logically and correctly drawn from the premises (or given information).
True: Which is generally believed as true.
On other hand,
Example 2:
All journalists are eloquent
Some politicians are eloquent
⇒ Some politicians are journalist. (Which is invalid.)
This conclusion is invalid because in order to have a valid conclusion, the statement MUST be true, rather than COULD be true.
All journalists are eloquent ⇒ Circle J is inscribed in circle E
Some politicians are eloquent ⇒ Circle P is either inscribed in or at least overlapped with circle E.
This is because Some = partial or total
Based on this information, think critically whether “Some politicians are journalist” MUST be true?
You’ll be right, if your answer is no. Because it is also possible that Circle P do not overlap Circle J at all, but is inscribed or overlapped with Circle E. Therefore, the conclusion is not valid.
Similarly,
All dogs are mammals
Non of the cats is dog
⇒ Non of the cats is mammals (Which is invalid and false.)
This conclusion is also invalid, because the conclusion is not MUST be true. Therefore, we cannot infer ‘Non of the cats is mammals’ from the given information. The reason is same as discussed above. Circle D is inscribed in Circle M, while Circle C and Circle D are disjoint (i.e not overlap). So it might be possible that Circle C overlap Circle M, despite remain disjoint with circle D as shown below:
Similarly,
Some leopards have horns.
Some snakes are leopards.
⇒ Some snakes have horns. (Which is invalid and not true.)
General types of Reasoning:
i. Deductive Reasoning
ii. Inductive Reasoning
Deductive reasoning:
Deductive reasoning moves from general to specific (i.e broad to narrow). This is basically deduced from broad to narrow. Furthermore, in deductive reasoning, conclusion is drawn based on premises that are generally accepted as true. For instance,
Honesty is the best policy
Government is adopting honesty
⇒ Government is adopting the best policy
Inductive reasoning:
Inductive reasoning moves from specific to general i.e it induced from narrow and then conclude to broad. Furthermore, in inductive reasoning, conclusion is drawn based on premises that are based on hypothesis and collected from some previous observations. Here the conclusion might be a prediction, a forecast, or some sort of theory in generalized form. For instance,
In last 2 matches, Mr A got injured while playing football.
Mr A will play all the next matches of football
⇒ Mr A will get injured in all the next matches of football.
Remember that,
Positive premise + Positive premise = Positive conclusion
Positive premise + Negative premise = Negative conclusion
Negative premise + Negative premise = Negative conclusion
For instance,
People of Country A are not US citizens
US citizens are not terrorist
⇒ People of Country A are terrorist. (Which is invalid)
Note that this positive conclusion has drawn from two negative premise. Therefore, the conclusion is invalid.