Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence1 / 12 Statements Definition A statement (or proposition) is a sentence (or assertion) that is true or false but not both. We typically use letters like p, q to denote propositions. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence2 / 12 Food for thought Am I saying the truth in the following statement? \I am lying now." Examples Examples 1 Washington, DC, is the capital of United States. 2 Annapolis is the capital of United States. 3 It is snowing. 4 I made a mistake in signing up for this course. 5 1 + 1 = 2 All these statements are simple statements. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence3 / 12 Examples Examples 1 Washington, DC, is the capital of United States. 2 Annapolis is the capital of United States. 3 It is snowing. 4 I made a mistake in signing up for this course. 5 1 + 1 = 2 All these statements are simple statements. Food for thought Am I saying the truth in the following statement? \I am lying now." Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence3 / 12 Compound Statements Definition A combination of two or more simple statements is a compound statement. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence4 / 12 Examples Examples 1 Washington, DC, is the capital of United States and it is snowing. 2 Washington, DC, is the capital of United States or it is snowing. 3 It is not snowing. 4 I did not make a mistake in signing up for this course or 1 + 1 6= 2. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence5 / 12 Examples 1 p: \It is snowing". Then ∼ p: \It is not snowing" 2 q: 1 + 1 = 2. Then ∼ q: 1 + 1 6= 2. The negation symbol Definition The symbol ∼ denotes not. Given a statement p, the sentence ∼ p is read not p. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence6 / 12 The negation symbol Definition The symbol ∼ denotes not. Given a statement p, the sentence ∼ p is read not p. Examples 1 p: \It is snowing". Then ∼ p: \It is not snowing" 2 q: 1 + 1 = 2. Then ∼ q: 1 + 1 6= 2. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence6 / 12 The or and and symbols Definition Let p and q be two statements. 1 The conjunction of p and q is p ^ q and is read \p and q" 2 The disjunction of p and q is p _ q and is read \p or q" Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence7 / 12 Example Example (Kemeny, Shell, Thompson) Let p: \Fred likes George" and q: George likes Fred. Write the following statements in symbolic form: 1 Fred and George like each other. 2 Fred and George dislike each other. 3 Fred likes George, but George does not reciprocate. 4 George is liked by Fred, but Fred is disliked by George. 5 Neither Fred nor George dislike each other. 6 It is not true that Fred and George dislike each other. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence8 / 12 A very convenient way of tabulating this dependency is by means of a truth table. Truth Tables Fact The truth value of a compound statement is determined by the truth value of its component. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence9 / 12 Truth Tables Fact The truth value of a compound statement is determined by the truth value of its component. A very convenient way of tabulating this dependency is by means of a truth table. Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence9 / 12 T Truth table for ∼ p Definition The truth table for negation is: p ∼ p T F F Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence10 / 12 Truth table for ∼ p Definition The truth table for negation is: p ∼ p T F F T Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence10 / 12 T TF F FT F FF F Truth table for conjunction Definition The truth table for conjunction is: p q p ^ q TT Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence11 / 12 F FT F FF F Truth table for conjunction Definition The truth table for conjunction is: p q p ^ q TT T TF Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence11 / 12 F FF F Truth table for conjunction Definition The truth table for conjunction is: p q p ^ q TT T TF F FT Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence11 / 12 F Truth table for conjunction Definition The truth table for conjunction is: p q p ^ q TT T TF F FT F FF Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence11 / 12 Truth table for conjunction Definition The truth table for conjunction is: p q p ^ q TT T TF F FT F FF F Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence11 / 12 T TF T FT T FF F Truth table for disjunction Definition The truth table for disjunction is: p q p _ q TT Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence12 / 12 T FT T FF F Truth table for disjunction Definition The truth table for disjunction is: p q p _ q TT T TF Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence12 / 12 T FF F Truth table for disjunction Definition The truth table for disjunction is: p q p _ q TT T TF T FT Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence12 / 12 F Truth table for disjunction Definition The truth table for disjunction is: p q p _ q TT T TF T FT T FF Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence12 / 12 Truth table for disjunction Definition The truth table for disjunction is: p q p _ q TT T TF T FT T FF F Chapter 2: The Logic of Compound Statements 2.1: Logical Forms and Logical Equivalence12 / 12.