3.1 Statements and Logical Connectives 97 History The ancient Greeks were the first people to systematically analyze the way humans thought and arrived at conclusions. Aristotle (384–322 b.c.) organized the study of logic for the first time in a work called Organon. As a result of his work, Aristotle is called the father of logic. The logic from this period, called Aristotelian logic, has been taught and studied for more than 2000 years. Since Aristotle’s time, the study of logic has been continued by other great philosophers and mathematicians. Gottfried Wilhelm Leibniz (1646–1716) had a deep conviction that all mathematical and scientific concepts could be derived from logic. As a result, he became the first serious student of symbolic logic. A self-educated English mathematician, George Boole (1815–1864), is considered to be the founder of symbolic logic because of his impressive work in this area. Mathematician Charles Dodgson, better known as Lewis Carroll, incorporated many interesting ideas from logic into his books Alice’s Adventures in Wonderland and Through the Looking Glass and his other children’s stories. (See the Profile in Mathematics on page 136.) The study of logic is also good preparation for other areas of mathematics. If you preview Chapter 11, on probability, you will see formulas for the probability of A or B and the probability of A and B, symbolized as P A B ( or ) and P A B ( and ), respectively. Special meanings of common words such as or and and apply to all areas of mathematics. The meanings of these and other special words are discussed in this chapter. Logic in Language In reading, writing, and speaking, we use many words such as and, or, and if … then … to connect thoughts. In logic we call these words connectives. How are these words interpreted in daily communication? A judge announces to an individual convicted of a crime, “I hereby sentence you to five months of community service and a fine of $100.” Advertisements often rely on spoken or written statements that are used to favorably portray the advertised product and form a convincing argument that will persuade us to purchase the product. Some familiar advertising statements are: It keeps going and going and going; Once you pop, you can’t stop; Impossible is nothing; and What happens here, stays here. In this section, we will learn how to represent statements using logic symbols that may help us better understand the nature of the statement. We will use these symbols throughout the chapter to analyze more complicated statements. Jenari/Shutterstock Statements and Logical Connectives SECTION 3.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Identify statements and logical connectives. 7 Understand quantifiers and identify the negations of statements that contain quantifiers. 7 Identify compound statements. 7 Identify statements using the words not, and, or, if–then, and if and only if. Why This Is Important Statements appear everywhere in our lives. In addition to statements in advertising, we see statements in legal documents, product instructions, and game rules. By using the symbols introduced in this section, we can represent such statements, and in turn we can better understand these statements. We begin this chapter with a discussion of the history of logic. We also discuss the use of logic in language.
RkJQdWJsaXNoZXIy NjM5ODQ=