Chapter 3 – Introduction to Logic 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. The initial motivation for the study of logic was to learn to distinguish good arguments from bad arguments.

3.1 – Statements and Quantifiers A statement is defined as a declarative sentence that is either true or false, but not both simultaneously. 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.

3.1 – Statements and Quantifiers Determine whether or not the following sentences are statements, compound statements, or neither. 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 The gun is a pistol and it is made by Smith and Wesson. Compound statement (and)

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

Negations and Inequality Symbols 3.1 – Statements and Quantifiers Negations and Inequality Symbols Symbolism Meaning 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 Give a negation of each inequality. Do not use a slash symbol. Statement: p ≥ 3 Statement: 3x – 2y < 12 Negation: p < 3 Negation: 3x – 2y ≥ 12

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

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

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

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

3.1 – Statements and Quantifiers Sets of Numbers Natural Numbers: {1, 2, 3, 4, …} Whole Numbers: {0, 1, 2, 3, …} Integers: {…, -3, -2, -1, 0, 1, 2, 3, 4, …} 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). Real Numbers: Any number expressed as a decimal.

3.1 – Statements and Quantifiers True or False 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.