L.1 - Introduction to Logic Math 166-502 Blake Boudreaux Department of Mathematics Texas A&M University January 16, 2018 Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 1 / 13
Introduction/Definition Logic is the systematic study of arguments For the purposes of this class, we will restrict ourselves to declarative sentences. Definition A statement is declarative sentence that is either true of false but not both. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 2 / 13
Examples Two plus two equals four. Two plus two equals five. These pretzels are making me thirsty. Look at me. Do you enjoy music? What a beautiful sunset! That was a great game. x + 2 = 5. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 3 / 13
Definitions Definition A statement that expresses a single thought is called a simple statement. Example I have money in my pocket. Definition A combination of two or more simple sentences is called a compound statement. Example I have money in my pocket and my gas tank is full. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 4 / 13
Definition Definition A connective is a word or words, such as and or if and only if, that is used to combine two or more simple statements into a compound statement. For the following few definitions, let p and q denote simple statements. Definition A conjunction is a statement of the form p and q and is written symbolically as p q. The conjunction p q is true if both p and q are true, otherwise it is false. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 5 / 13
Example Write the compound statement I have money in my pocket and my gas tank is full in symbolic form. Solution: Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 6 / 13
Definition For the following few definitions, let p and q denote simple statements. Definition A disjunction is a statement of the form p or q and is written symbolically as p q. The disjunction p q is false if both p and q are false and is true in all other cases. Note that we have defined or to mean one or the other, or both. This is known as the inclusive or. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 7 / 13
Example Write the compound statement Mt. Everest is the tallest mountain in the world or the Earth is flat in symbolic form. Solution: Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 8 / 13
Definition For the following few definitions, let p and q denote simple statements. Definition A negation is a statement of the form not p and is written symbolically as p. The negation p is true if p is false and false if p is true. For example, if p is the statement Janet is smart, then p is the statement Janet is not smart. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 9 / 13
Example Let and, be the following statements: : George Bush plays football for the Washington Redskins., : The Dow Jones industrial average set a new record high last week. Write the following statements in symbolic form. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 10 / 13
Example Let and, be the following statements: : George Bush plays football for the Washington Redskins., : The Dow Jones industrial average set a new record high last week. Write the following statements in symbolic form. a. George Bush does not play football for the Washington Redskins, and the Dow Jones industrial average set a new record high last week. Solution: b. George Bush plays football for the Washington Redskins, or the Dow Jones industrial average did not set a new record high last week. Solution: Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 10 / 13
Example Let and, be the following statements: : George Bush plays football for the Washington Redskins., : The Dow Jones industrial average set a new record high last week. Write the following statements in symbolic form. c. George Bush does not play football for the Washington Redskins, and the Dow Jones industrial average did not set a new record high last week. Solution: d. It is not true that George Bush plays football for the Washington Redskins, and the Dow Jones industrial average did not set a new record high last week. Solution: Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 11 / 13
Summary We sum up the previous few definitions in a table: Name Connective Symbol Conjunction and Disjunction or Negation not Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 12 / 13
Practice Let p and q be the following statements: p : Philadelphia is the capital of New Jersey. q : General Electric lost money last year. Write out the statements that correspond to each of the following: Solution: a. b. c. d. a. p q b. p q c. p q d. p q. Blake Boudreaux (Texas A&M University) L.1 - Introduction to Logic January 16, 2018 13 / 13