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.

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