Final answer:
The negation of a proposition denies the original statement. For example, the negation of 'Mei has an MP3 player' is 'Mei does not have an MP3 player.' The law of noncontradiction states that a statement and its negation cannot both be true, which leads to the law of the excluded middle where either the proposition or its negation must be true.
Step-by-step explanation:
The negation of a proposition is formed by denying the original statement, such that if the original statement is true, the negation is false, and vice versa. Here are the negations for the given propositions:
- A. Mei has an MP3 player. Negation: Mei does not have an MP3 player.
- B. There is no pollution in New Jersey. Negation: There is pollution in New Jersey.
- C. 6≠12. Negation: 6=12.
- D. The summer in Maine is hot and sunny. Negation: The summer in Maine is not hot and sunny.
Example of a Statement and Its Negation
Statement: The voters elected Mr. Thompson mayor. Negation: The voters did not elect Mr. Thompson mayor.
Law of Noncontradiction and Excluded Middle
The law of noncontradiction states that a statement and its negation cannot both be true at the same time. The law of the excluded middle implies that for any proposition, either that proposition is true, or its negation is true. Logically, if a statement and its negation cannot both be true, then there are no other options than for one of them to be true, thereby implying the law of the excluded middle.