3.1 Statements and Logical Connectives 99 of the statement is “Some birds can swim” or “At least one bird can swim,” each of which is a true statement. Now let’s consider statements involving the quantifier some, as in “Some students have a driver’s license.” This statement is true, meaning that at least one student has a driver’s license. The negation of this statement must therefore be false. The negation is “No student has a driver’s license,” which is a false statement. Consider the statement “Some students do not ride motorcycles.” This statement is true because it means “At least one student does not ride a motorcycle.” The negation of this statement must therefore be false. The negation is “All students ride motorcycles,” which is a false statement. The negation of quantified statements is summarized as follows: Form of statement Form of negation All are. Some are not. None are. Some are. Some are. None are. Some are not. All are. The following diagram might help you to remember the statements and their negations: Did You Know? Playing on Words George Boole, Augustus De Morgan, and other mathematicians of the nineteenth century were anxious to make logic an abstract science that would operate like algebra but be applicable to all fields. One of the problems logicians faced was that verbal language could be ambiguous and could easily lead to confusion and contradiction. Comedians Bud Abbott and Lou Costello had fun with the ambiguity of language in their skit about the baseball players: “Who’s on first, What’s on second, I Don’t Know is on third— Yeah, but who’s on first?” All are. None are. Some are. Some are not. The quantifiers diagonally opposite each other are the negations of each other. Example 1 Write Negations of Statements Involving Quantifiers Write the negation of each statement. a) All apples are fruit. b) Some insects are mammals. c) Some days are not Mondays. d) No coins have the image of Abraham Lincoln. Solution a) The statement “ All apples are fruit” is a true statement and is of the form “All are.” Referring to the text above, to negate a statement of the form “All are,” we write a statement of the form “Some are not.” Thus, the negation of the statement “ All apples are fruit” is “ Some apples are not fruit,” which is a false statement. b) The statement “ Some insects are mammals” is a false statement and is of the form “Some are.” To negate a statement of the form “Some are,” we write a statement of the form “None are.” Thus, the negation of the statement “ Some insects are mammals” is “ None of the insects are mammals” or “No insects are mammals,” which are true statements. c) The statement “Some days are not Mondays” is a true statement and is of the form “Some are not.” To negate a statement of the form “Some are not,” we write a statement of the form “All are.” Thus, the negation of the statement “Some days are not Mondays” is “All days are Mondays,” which is a false statement. d) The statement “No coins have the image of Abraham Lincoln” is false because pennies have the image of Abraham Lincoln. Although the statement “No coins have the image of Abraham Lincoln” does not exactly fit any of the forms discussed in the text above, it can be considered to be of the form “None are.” To negate a statement of the form “None are,” we write a statement of the form “Some are.” Thus, the negation of the statement “No coins have the image of Abraham Lincoln” is “Some coins have the image of Abraham Lincoln,” which is a true statement. 7 Now try Exercise 7
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