Chapter 3 – Introduction to Logic The initial motivation for the study of logic was to learn to distinguish good arguments from bad arguments. Logic is the formal systematic study of the principles of valid inference and correct reasoning. It is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science. Logic examines: (a) general forms which arguments may take, (b) which forms are valid, and (c) which forms are fallacies.

Statements A statement is defined as a declarative sentence that is either true or false, but not both simultaneously. 3.1 – Statements and Quantifiers Compound Statements A compound statement may be formed by combining two or more statements. The statements making up the compound statement are called the component statements. Connectives such as and, or, not, and if…then, can be used in forming compound statements.

Determine whether or not the following sentences are statements, compound statements, or neither. 3.1 – Statements and Quantifiers If Amanda said it, then it must be true. Compound statement (if, then) Today is extremely warm. Statement The gun is made by Smith and Wesson. Statement Compound statement (and) The gun is a pistol and it is made by Smith and Wesson.

Negations 3.1 – Statements and Quantifiers A negation is a statement that is a refusal or denial of some other statement. Max has a valuable card. Max does not have a valuable card. The negation of a true statement is false and the negation of a false statement is true. Statement: Negation: The number 9 is odd. The number 9 is not odd. Statement: Negation: The product of 2 negative numbers is not positive. The product of 2 negative numbers is positive. Statement: Negation:

Negations and Inequality Symbols SymbolismMeaning a is less than b a is greater than b a is less than or equal to b a is greater than or equal to b 3.1 – Statements and Quantifiers Give a negation of each inequality. Do not use a slash symbol. p ≥ 3 p < 3 Statement: Negation: 3x – 2y < 12 3x – 2y ≥ 12 Statement: Negation:

Symbols To simplify work with logic, symbols are used. ConnectiveSymbolType of Statement 3.1 – Statements and Quantifiers Statements are represented with letters, such as p, q, or r, while several symbols for connectives are shown below. and or not Conjunction Negation Disjunction

Translating from Symbols to Words Write each symbolic statement in words. 3.1 – Statements and Quantifiers q represent “It is March.” Let:p represent “It is raining,” p ˅ q It is raining or it is March. ̴ (p ˄ q) it is raining and it is March.It is not the case that

Quantifiers Universal Quantifiers are the words all, each, every, no, and none. Quantifiers are used extensively in mathematics to indicate how many cases of a particular situation exist. 3.1 – Statements and Quantifiers Existential Quantifiers are words or phrases such as some, there exists, for at least one, and at least one. Negations of Quantified Statements StatementNegation All do.Some do not. Some do.None do.

Forming Negations of Quantified Statements 3.1 – Statements and Quantifiers Some cats have fleas. No cats have fleas. Statement: Negation: Some cats do not have fleas. All cats have fleas. Statement: Negation: All dinosaurs are extinct. Not all dinosaurs are extinct. Statement: Negation: No horses fly. Some horses fly. Statement: Negation:

Sets of Numbers 3.1 – Statements and Quantifiers Natural Numbers: {1, 2, 3, 4, …} Whole Numbers: {0, 1, 2, 3, …} Rational Numbers: Any number that can be expressed as a quotient of two integers (terminating or repeating decimal). Irrational Numbers: Any number that can not be expressed as a quotient of two integers (non-terminating and non-repeating). Integers: {…, -3, -2, -1, 0, 1, 2, 3, 4, …} Real Numbers: Any number expressed as a decimal.

True or False 3.1 – Statements and Quantifiers Every integer is a natural number. False: – 1 is an integer but not a natural number. A whole number exists that is not a natural number. True: 0 is the number. There exists an irrational number that is not real. False: All irrational numbers are real numbers.

Conjunctions The truth values of component statements are used to find the truth values of compound statements. The truth values of the conjunction p and q (p ˄ q), are given in the truth table on the next slide. The connective “and” implies “both.” 3.2 – Truth Tables and Equivalent Statements Truth Values Truth Table A truth table shows all four possible combinations of truth values for component statements.

Conjunction Truth Table p q p ˄ qp ˄ q T TT T FF F TF F FF p and q 3.2 – Truth Tables and Equivalent Statements

Finding the Truth Value of a Conjunction If p represent the statement 4 > 1 and q represent the statement 12 < 9, find the truth value of p ˄ q. 3.2 – Truth Tables and Equivalent Statements p q p ˄ qp ˄ q T TT T FF F TF F FF p and q 12 < 9 4 > 1p is true q is false The truth value for p ˄ q is false

Disjunctions The truth values of the disjunction p or q (p ˅ q) are given in the truth table below. The connective “or” implies “either.” 3.2 – Truth Tables and Equivalent Statements p q p ˅ qp ˅ q T TT T FT F TT F FF p or q Disjunction Truth Table

Finding the Truth Value of a Disjunction If p represent the statement 4 > 1, and q represent the statement 12 < 9, find the truth value of p ˅ q. 3.2 – Truth Tables and Equivalent Statements p q p ˅ qp ˅ q T TT T FT F TT F FF p or q 12 < 9 4 > 1p is true q is false The truth value for p ˅ q is true

Negation The truth values of the negation of p ( ̴ p) are given in the truth table below. p ̴ p̴ p TF FT not p 3.2 – Truth Tables and Equivalent Statements

Example: Constructing a Truth Table p q ~ p~ q ~ p ˅ ~ qp ˄ (~ p ˅ ~ q) T T T F F T F F Construct the truth table for: p ˄ (~ p ˅ ~ q) 3.2 – Truth Tables and Equivalent Statements A logical statement having n component statements will have 2 n rows in its truth table. 2 2 = 4 rows

Example: Constructing a Truth Table p q ~ p~ q ~ p ˅ ~ qp ˄ (~ p ˅ ~ q) T TF T FF F TT F FT Construct the truth table for: p ˄ (~ p ˅ ~ q) 3.2 – Truth Tables and Equivalent Statements A logical statement having n component statements will have 2 n rows in its truth table. 2 2 = 4 rows

Example: Constructing a Truth Table p q ~ p~ q ~ p ˅ ~ qp ˄ (~ p ˅ ~ q) T TFF T FFT F TTF F FTT Construct the truth table for: p ˄ (~ p ˅ ~ q) 3.2 – Truth Tables and Equivalent Statements A logical statement having n component statements will have 2 n rows in its truth table. 2 2 = 4 rows

Example: Constructing a Truth Table p q ~ p~ q ~ p ˅ ~ qp ˄ (~ p ˅ ~ q) T TFFF T FFTT F TTFT F FTTT Construct the truth table for: p ˄ (~ p ˅ ~ q) 3.2 – Truth Tables and Equivalent Statements A logical statement having n component statements will have 2 n rows in its truth table. 2 2 = 4 rows

Example: Constructing a Truth Table p q ~ p~ q ~ p ˅ ~ qp ˄ (~ p ˅ ~ q) T TFFFF T FFTTT F TTFTF F FTTTF Construct the truth table for: p ˄ (~ p ˅ ~ q) 3.2 – Truth Tables and Equivalent Statements A logical statement having n component statements will have 2 n rows in its truth table. 2 2 = 4 rows

Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q ̴ p ̴ q T T T F F T F F ̴ p ˄ ̴ q

Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q ̴ p ̴ q T TF T FF T T F F FT ̴ p ˄ ̴ q

Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q ̴ p ̴ q T TF T FF T T F F FT ̴ p ˄ ̴ q F F F T The truth value for the statement is false.

˅ Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q r ̴ p ̴ q ̴ r T T T T T F T F T T F F F T T F T F F F T F F F ̴ p ˄ r ̴ q ˄ p ( ̴ p ˄ r) ˅ ( ̴ q ˄ p)

˅ Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q r ̴ p ̴ q ̴ r T T TF F F T T FF F T T F TF T F T F FF T T T F F F T FT F T F F TT T F F F FT T T ̴ p ˄ r ̴ q ˄ p ( ̴ p ˄ r) ˅ ( ̴ q ˄ p)

˅ Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q r ̴ p ̴ q ̴ r T T TF F F T T FF F T T F TF T F T F FF T T T F F F T FT F T F F TT T F F F FT T T ̴ p ˄ r ̴ q ˄ p F F F F T F T F ( ̴ p ˄ r) ˅ ( ̴ q ˄ p)

˅ Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q r ̴ p ̴ q ̴ r T T TF F F T T FF F T T F TF T F T F FF T T T F F F T FT F T F F TT T F F F FT T T ̴ p ˄ r ̴ q ˄ p FF FF FT FT TF FF TF FF ( ̴ p ˄ r) ˅ ( ̴ q ˄ p)

Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. 3.2 – Truth Tables and Equivalent Statements p q r ̴ p ̴ q ̴ r T T TF F F T T FF F T T F TF T F T F FF T T T F F F T FT F T F F TT T F F F FT T T ̴ p ˄ r ̴ q ˄ p FF FF FT FT TF FF TF FF ( ̴ p ˄ r) ˅ ( ̴ q ˄ p) ˅ F F T T T F T F The truth value for the statement is true.

Equivalent Statements Are the following statements equivalent? p q ~ p ˄ ~ q ̴ (p ˅ q) T T T F F T F F Two statements are equivalent if they have the same truth value in every possible situation. 3.2 – Truth Tables and Equivalent Statements ~ p ˄ ~ q and ̴ (p ˅ q)

Equivalent Statements Are the following statements equivalent? p q ~ p ˄ ~ q ̴ (p ˅ q) T TF T FF F TF F FT Two statements are equivalent if they have the same truth value in every possible situation. 3.2 – Truth Tables and Equivalent Statements ~ p ˄ ~ q and ̴ (p ˅ q)

Equivalent Statements Are the following statements equivalent? p q ~ p ˄ ~ q ̴ (p ˅ q) T TFF T FFF F TFF F FTT Yes Two statements are equivalent if they have the same truth value in every possible situation. 3.2 – Truth Tables and Equivalent Statements ~ p ˄ ~ q and ̴ (p ˅ q)

3.3 – The Conditional A conditional statement is a compound statement that uses the connective if…then. The conditional is written with an arrow, so “if p then q” is symbolized The conditional is read as “p implies q” or “if p then q.” The statement p is the antecedent, while q is the consequent.

Special Characteristics of Conditional Statements for a Truth Table 3.3 – The Conditional When the antecedent is true and the consequent is true, p → q is true. Teacher: “If you participate in class, then you will get extra points." If you participate in class (true) and you get extra points (true) then, The teacher's statement is true. If you participate in class (true) and you do not get extra points (false), then, The teacher’s statement is false. When the antecedent is true and the consequent is false, p → q is false.

Special Characteristics of Conditional Statements for a Truth Table 3.3 – The Conditional If the antecedent is false, then p → q is automatically true. “If you participate in class, then you will get extra points." If you do not participate in class (false), the truth of the teacher's statement cannot be judged. The teacher did not state what would happen if you did NOT participate in class. Therefore, the statement has to be “true”. If you do not participate in class (false), then you get extra points. The teacher's statement is true in both cases. If you do not participate in class (false), then you do not get extra points.

Truth Table for The Conditional p qp → q T TT T FF F TT F FT If p, then q 3.3 – The Conditional A tautology is a statement that is always true, no matter what the truth values of the components are.

Examples: Decide whether each statement is True or False T → (4 < 2) 3.3 – The Conditional (T represents a true statement, F a false statement). T → F F (8 = 1) → F F → F T F → (3 ≠ 9) F → T T

Converse, Inverse, and Contrapositive Conditional Statement 3.4 – More on the Conditional Converse Inverse Contrapositive q → p p → q ̴ p → ̴ q ̴ q → ̴ p If q, then p If not p, then not q If not q, then not p If p, then q

Determining Related Conditional Statements 3.4 – More on the Conditional Given the conditional statement, determine the following: a) the converse, b) the inverse, and c) the contrapositive. If I live in Wisconsin, then I shovel snow, a) Converse If I shovel snow,then I live in Wisconsin. b) Inverse If I do not live in Wisconsin,then I do not shovel snow. c) Contrapositive If I do not shovel snow,then I do not live in Wisconsin.

Equivalences A conditional statement and its contrapositive are equivalent, and the converse and inverse are equivalent. 3.4 – More on the Conditional Alternative Forms of “If p, then q” The conditional p → q can be translated in any of the following ways: If p, then q.p is sufficient for q. If p, q.q is necessary for p. p implies q.All p are q. p only if q.q if p.

Rewording Conditional Statements Write each statement in the form “if p, then q.” 3.4 – More on the Conditional b) Today is Sunday only if yesterday was Saturday. c) All Chemists wear lab coats. a) You’ll be sorry if I go. (q if p) If I go,then you’ll be sorry. (p only if q) If today is Sunday,then yesterday was Saturday. (All p are q) If you are a Chemist,then you wear a lab coat.

Negation of a Conditional 3.4 – More on the Conditional If the river is narrow, then we can cross it. q: we can cross it. Examples: The river is narrow and we cannot cross it. p: the river is narrow. Negation: A Conditional as a Disjunction  p: the river is not narrow.  q: we cannot cross it. The river is not narrow or we can cross it. Disjunction:

Negation of a Conditional 3.4 – More on the Conditional If you are absent, then you have a test. q: you have a test. Examples: You are absent and you do not have a test. p: you are absent. Negation: A Conditional as a Disjunction  p: you are not absent.  q: you do not have a test. You are not absent or you have a test. Disjunction: