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Understanding why "some are not" is the negation of "all"
dcdcdcdcdc
Alum Member
In the existential quantifiers lessons, JY explains how to negate statements with the universal quantifier "all." The conclusion was that "some are not" was the negation and that the new set contained 0-99 items, whereas the original "all" represented 100 items.
In the comments section there was some confusion about why the "some are not" statement included 0 items in the new set and some contributors suggested the statement encompassed only 1-100 items.
After diagramming the all statement and its negation, I think I see where some (myself included) may have become confused. The important distinction is that the new set of 0 to 99 items is comprised of items with the same property mentioned in the all statement. My reasoning is below and I welcome any input on its accuracy. Thanks!
Example: All cats (C) are pretentious (P)
For simplicity, let us assume that there are only four cats in the world. The total number of cats which are pretentious and not pretentious must add up to 4.
P | /P
4 | 0 <-- every cat is P; the all statement we negate
---- <-- the binary cut
3 | 1 <-- min. condition to contradict our all statement
2 | 2
1 | 3
0 | 4 <-- often thought of as negation of all; "No cats are P"
In the above table we see that in the 5 possible groupings based on our 4 cats, one represents the all statement and the other 4 cases together represent the negation of that all statement. The set which represents 0 through 3 inclusive (comparable to 0 - 99) is the set of pretentious cats. I believe this is where many became confused and thought the set of 0-99 was made up of unpretentious (that is /P) cats. However, above we see that our unpretentious set always contains at least 1 cat and therefore follows our definition of some (it is comprised of one, possibly all cats, but not 0).
In the comments section there was some confusion about why the "some are not" statement included 0 items in the new set and some contributors suggested the statement encompassed only 1-100 items.
After diagramming the all statement and its negation, I think I see where some (myself included) may have become confused. The important distinction is that the new set of 0 to 99 items is comprised of items with the same property mentioned in the all statement. My reasoning is below and I welcome any input on its accuracy. Thanks!
Example: All cats (C) are pretentious (P)
For simplicity, let us assume that there are only four cats in the world. The total number of cats which are pretentious and not pretentious must add up to 4.
P | /P
4 | 0 <-- every cat is P; the all statement we negate
---- <-- the binary cut
3 | 1 <-- min. condition to contradict our all statement
2 | 2
1 | 3
0 | 4 <-- often thought of as negation of all; "No cats are P"
In the above table we see that in the 5 possible groupings based on our 4 cats, one represents the all statement and the other 4 cases together represent the negation of that all statement. The set which represents 0 through 3 inclusive (comparable to 0 - 99) is the set of pretentious cats. I believe this is where many became confused and thought the set of 0-99 was made up of unpretentious (that is /P) cats. However, above we see that our unpretentious set always contains at least 1 cat and therefore follows our definition of some (it is comprised of one, possibly all cats, but not 0).
Comments
https://7sage.com/discussion#/discussion/comment/33079
I hope you're having fun with logic. I never miss a chance, however impertinent, to encourage someone's interest in it. You'd be surprised how much discovery and active research is still happening in the field.