CHAPTER 3 Logic.

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Presentation transcript:

CHAPTER 3 Logic

Statements, Negations, and Quantified Statements 3.1 Statements, Negations, and Quantified Statements

Objectives Identify English sentences that are statements. Express statements using symbols. Form the negation of a statement. Express negations using symbols. Translate a negation represented by symbols into English. Express quantified statements in two ways. Write negations of quantified statements.

Statements Examples of statements: London is the capital of England. William Shakespeare wrote the television series Modern Family.

Not Statements Commands, questions, and opinions, are not statements because they are neither true or false. Examples: Titanic is the greatest movie of all time. (opinion) Read pages 23 – 57. (order or command) If I start losing my memory, how will I know? (question)

Using Symbols to Represent Statements In symbolic logic, we use lowercase letters such as p, q, r, and s to represent statements. Here are two examples: p: London is the capital of England q: William Shakespeare wrote the television series Modern Family. The letter p represents the first statement. The letter q represents the second statement.

Negating Statements The negation of a true statement is a false statement and the negation of a false statement is a true statement.

Example: Forming Negations Form the negation of each statement. a. Shakespeare wrote the television series Modern Family. Shakespeare did not write the television series Modern Family. It is not true that Shakespeare wrote the television series Modern Family.

Example (cont) Form the negation of each statement. b. Today is not Monday. It is not true that today is not Monday. Today is Monday.

Example: Expressing Negations Symbolically Let p and q represent the following statements: p: Shakespeare wrote the television series Modern Family. q: Today is not Monday. Express each of the following statements symbolically: a. Shakespeare did not write the television series Modern Family. b. Today is Monday. ~p ~q

Quantified Statements Quantifiers: The words all, some, and no (or none). Statements containing a quantifier: All poets are writers. Some people are bigots. No math books have pictures. Some students do not work hard.

Equivalent Ways of Expressing Quantified Statements

Negation of Quantified Statements The statements diagonally opposite each other are negations.

Example: Negating a Quantified Statement The mechanic told me, “All piston rings were replaced.” I later learned that the mechanic never tells the truth. What can I conclude? Solution: Let’s begin with the mechanic’s statement: All piston rings were replaced. Because the mechanic never tells the truth, I can conclude that the truth is the negation of what I was told.

Example continued The mechanic told me, “All piston rings were replaced.” I later learned that the mechanic never tells the truth. What can I conclude? Solution: The negation of “All A are B” is “Some A are not B”. Thus, I can conclude that Some piston rings were not replaced. I can also correctly conclude that: At least one piston ring was not replaced.