1. “Some”s and “All”s By Linda Perel 10-02, last updt 2-8-17
Logic Section 3.1
“Some”s and “All”s
By Linda Perel
10-02, last updt

2. Negation of quantified statements
Quantifiers – describe quantities:
UNIVERSAL quantifiers:
ALL, EVERY, NO
PARTIAL quantifiers:
Some
Example: All birds can fly.

3. Negation of quantified statements
The Negation of :
All A are B. is Some A are not B.
and vice versa
No A are B. is Some A are B.

4. 2 things change in taking the negation
universal changes to a partial quantifier (or vice versa)
“Positive” to a “negative” statement (or vice versa) (A “negative” statement contains no or not.)
Ex. The neg. of All A are B. Is Some A are not B.
(pos + universal quantifier) (neg+ partial quantifier)

5. EQUIValent Statements:
No A are B is equiv. to All A are not B.
Some A are B. is equiv. to At least one A is B.

6. An illustration of WHY Why is the negation of “No A are B.” NOT
“All A are B.”
(answer: Because it’s “Some A are B.” …see why)

7. Joey the liar. Joey the liar said, “No circles are X’ed.” Or in a
picture he said:
Since Joey lies, “No circles are X’D.” is false.

8. Aaron knows that Joey is a liar.
So Aaron concludes that the opposite (the negation) must be true, and that is 3 possibilities…

9. So Aaron concludes:
If is false, then either: , or must be true.

10. Aaron says furthermore …
And: , or , or is the same as saying 1 or 2 or 3 are X’ED, which is the same as saying AT LEAST ONE is X’ED, which is the same as SOME CIRCLES ARE X’ED.

11. Back to the question:
WHY is the negation of “No A are B.” not the same as “All A are B.” ? ANSWER: Because the negation of “No A are B.” is “Some A are B.” Specifically in our example: the negation of “No circles are X’ed.” is “Some circles are X’ed.”

12. Back to the question:
That is… The negation of “No circles are X’ed.” Is “Some circles are X’ed.”: , or .