“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 2-8-17

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

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.

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)

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.

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)

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.

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

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

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.

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.”

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