I have few queries from one of the session posted by E-gmat
. Query 1
– I am not able to understand how come SOME can include 100 of the test takers.Query 2
– I am not able to understand how come NOT ALL can include 0 of the test takers.
I would say this is a little extreme. If there is a population of 100 individuals, "some" definitely excludes zero and "not all" definitely excludes 100 --- that much is clear. I would say: it would be a bit deceitful if someone said "some" meant "all", or said "not all" and meant "none" ---- I disagree with those statements. If the GMAT talks about "some" in any context in which the meaning of this term matters, I would say you could count on it not meaning "all". I am particularly doubtful about this numerical scale leading folks to an understanding of the negation terms.
For the below mentioned queries the above mentioned equation is the basis.
Query 3 – I am not able to understand how come logical opposite of ALL is NOT ALL (and not NONE) because if ALL includes all the 100 students then as per equation
Query 4 – I am not able to understand how come logical opposite of SOME is NONE because if SOME includes only 1 (Worst Case Scenario) student then as per equation
Query 5 – I am not able to understand how come logical opposite of NOT ALL is ALL because if NOT ALL includes 99 (Worst Case Scenario) students then as per equation
Query 6 – I am not able to understand how come logical opposite of NEVER is SOMETIMES.
Waiting eagerly for experts valuable comments & insights. Regards, Fame
Many students have misunderstanding about this. For the first, consider this statement:
(1) All triangles are equilateral.
That's a false statement. If we put in a word that means the opposite of "all", we can change this from a false statement to a true statement. If I use "not all", I get a true statement:
(2) Not all triangles are equilateral
Because this substitution changes the sentence from true to false, "not all" is a true opposite of "all". By contrast, look what happens if we use "no" or "none"
(3a) No triangles are equilateral.
(3b) In the set of triangles, none are equilateral.
These are false, as (1) is. If I start with a false statement, make a substitution, and the sentence stays false, then whatever I substituted wasn't a true opposite. A true opposite will change the truth value of the sentence.
The opposite of a "some" statement depends on context. I would say there's a not a one-size-fits all rule here. For example,
(4) Some prime numbers have integer square roots.
That is a false statement. Here, substituting the word no/not/none creates a true statement
(5) No prime numbers have integer square roots.
That is a bonafide true statement.
I would say, though, that most people would consider this a false statement also:
(6) Some even numbers are divisible by two.
Here, to get the correct statement, we need the word "all" ---
(7) All even numbers are divisible by two.
This is an unambiguously correct sentence.
In some ways, I handled "not all" vs. "all" in #1-3 above, but some more examples....
(8) Not all even numbers are divisible by two.
(9) All even numbers are divisible by two.
The substitution, "not all" to "all", changes the true value, so these are true opposites.
Finally, the opposite of "never". Consider this false statement.
(10) The Dow Jones Industrial Average never goes up.
That's a false statement. Substitute in "always".
(11) The Dow Jones Industrial Average always goes up.
That's another false statement. That substitution does not change the truth value, so those two, "never" and "always", are not true opposites. By contrast, substitute "sometimes"
(12) The Dow Jones Industrial Average sometimes goes up.
That's an absolutely true sentence. This substitution shows that "sometimes" is actually the logical opposite of both "never" and "always", because substituting it for either of them (in #10 or #11) produces a true statement (#12)
Does all this make sense?
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