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**SEVENTH EDITION and EXPANDED SEVENTH EDITION**

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Chapter 3 Logic

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**Statements and Logical Connectives**

3.1 Statements and Logical Connectives

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HISTORY—The Greeks: Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle ( B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.

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Mathematicians Gottfried Wilhelm Leibniz ( ) believed that all mathematical and scientific concepts could be derived from logic. He was the first to seriously study symbolic logic. In this type of logic, written statements use symbols and letters. George Boole (1815 – 1864) is said to be the founder of symbolic logic because he had such impressive work in this area.

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**Logic and the English Language**

Connectives - words such as and, or, if, then Exclusive or - one or the other of the given events can happen, but not both. Inclusive or - one or the other or both of the given events can happen.

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**Statements and Logical Connectives**

Statement - A sentence that can be judged either true or false. Labeling a statement true or false is called assigning a truth value. Simple Statements - A sentence that conveys only one idea. Compound Statements - Sentences that combine two or more ideas and can be assigned a truth value.

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Quantifiers Negation of a statement - the opposite meaning of a statement. The negation of a false statement is always a true statement. The negation of a true statement is always false. Quantifiers - words such as all, none, no, some, etc…

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**Example: Write Negations**

Write the negation of the statement. Some candy bars contain nuts. Since some means “at least one” this statement is true. The negation is “No candy bars contain nuts,” which is a false statement.

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**Example: Write Negations continued**

Write the negation of the statement. All tables are oval. This is a false statement since some tables are round, rectangular, or other shapes. The negation could be “Some tables are not oval.”

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Compound Statements Statements consisting of two or more simple statements are called compound statements. The connectives often used to join two simple statements are and, or, if…then…, and if and only if.

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Not Statements The symbol used in logic to show the negation of a statement is ~. It is read “not”.

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**And Statements is the symbol for a conjunction and is read “and.”**

The other words that may be used to express a conjunction are: but, however, and nevertheless.

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**Example: Write a Conjunction**

Write the conjunction in symbolic form. The dog is gray, but the dog is not old. Solution: Let p and q represent the simple statements. p: The dog is gray. q: The dog is old. In symbol form, the compound statement is

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**Or Statements: The disjunction is symbolized by and read “or.”**

Example: Write the statement in symbolic form. Carl will not go to the movies or Carl with not go to the baseball game. Solution:

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**If-Then Statements The conditional is symbolized by**

and is read “if-then.” The antecedent is the part of the statement that comes before the arrow. The consequent is the part that follows the arrow.

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**Example: Write a Conditional Statement**

Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. If Nathan goes to the park, then he will swing. If Nathan does not go to the park, then he will not swing. Solutions: a) b)

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**If and Only If Statements**

The biconditional is symbolized by and is read “if and only if.” If and only if is sometimes abbreviated as “iff.”

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**Example: Write a Statement Using the Biconditional**

Let p: The dryer is running. q: There are clothes in the dryer. Write the following symbolic statements in words. a) b) Solutions: The clothes are in the dryer if and only if the dryer is running. It is false that the dryer is running if and only if the clothes are not in the dryer.

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**Truth Tables for Negation, Conjunction, and Disjunction**

3.2 Truth Tables for Negation, Conjunction, and Disjunction

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Truth Table A truth table is used to determine when a compound statement is true or false.

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Negation Truth Table T Case F F Case T ~p p

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**Conjunction Truth Table**

The conjunction is true only when both p and q are true. F Case 4 T Case 3 Case 2 Case 1 q p

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Disjunction The disjunction is true when either p is true, q is true, or both p and q are true. F Case 4 T Case 3 Case 2 Case 1 q p

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**Truth Tables for the Conditional and Biconditional**

3.3 Truth Tables for the Conditional and Biconditional

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Conditional The conditional statement is true in every case except when p is a true statement and q is a false statement. T F Case 4 Case 3 Case 2 Case 1 q p

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**Biconditional 5 6 4 7 2 3 1 F T case 4 case 3 case 2 case 1 p) (q q)**

The biconditional statement, means that and or, symbolically 5 6 4 7 2 3 1 order of steps F T case 4 case 3 case 2 case 1 p) (q q) (p q p

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Self-Contradiction A self-contradiction is a compound statement that is always false. When every truth value in the answer column of the truth table is false, then the statement is a self-contradiction.

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**Tautology A tautology is a compound statement that is always true.**

When every truth value in the answer column of the truth table is true, the statement is a tautology.

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Implication An implication is a condition statement that is a tautology. The consequent will be true whenever the antecedent is true.

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**Equivalent Statements**

3.4 Equivalent Statements

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**Equivalent Statements**

Two statements are equivalent if both statements have exactly the same truth values in the answer columns of the truth tables. In a truth table, if the answer columns are identical, the statements are equivalent. If the answer columns are not identical, the statements are not equivalent.

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De Morgan’s Laws

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To change a conditional statement into a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same. To change a disjunction statement to a conditional statement, negate the first statement, change the disjunction symbol to a conditional symbol, and keep the second statement the same.

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**Variations of the Conditional Statement**

The variations of conditional statements are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional.

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**Variations of the Conditional Statement**

“if not q, then not p” ~p ~q Contrapositive of the conditional “if not p, then not q” Inverse of the conditional “if q, then p” p q Converse of the conditional “if p, then q” Conditional Read Symbolic Form Name

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3.5 Symbolic Arguments

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Symbolic Arguments An argument is valid when its conclusion necessarily follows from a given set of premises. An argument is invalid (or a fallacy) when the conclusion does not necessarily follow from the given set of premises.

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Valid or Invalid? If the truth table answer column is true in every case, then the statement is a tautology, and the argument is valid. If the truth table answer column is not true in every case then the statement is not a tautology, and the argument is invalid.

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Law of Detachment Also called modus ponens. The argument form:

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**Determining Whether an Argument is Valid**

Write the argument in symbolic form. Compare the form with forms that are known to be either valid or invalid. If the argument contains two premises, write a conditional statement of the form [(premise 1) (premise 2)] conclusion

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**Determining Whether an Argument is Valid continued**

Construct a truth table for the statement in step 3. If the answer column of the table has all trues, the statement is a tautology, and the argument is valid. If the answer column of the table does not have all trues, the argument is invalid.

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**Valid Arguments Law of Detachment Law of Syllogism**

Law of Contraposition Disjunctive Syllogism

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Invalid Arguments Fallacy of the Converse Fallacy of the Inverse

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**Euler Diagrams and Syllogistic Arguments**

3.6 Euler Diagrams and Syllogistic Arguments

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**Syllogistic Arguments**

Another form of argument is called a syllogistic argument, better known as syllogism. The validity of a syllogistic argument is determined by using Euler diagrams.

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Euler Diagrams One method used to determine whether an argument is valid or is a fallacy. Uses circles to represent sets in syllogistic arguments.

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**Symbolic Arguments Versus Syllogistic Arguments**

Euler diagrams all are, some are, none are, some are not Syllogistic argument Truth tables or by comparison with standard forms of arguments and, or, not, if-then, if and only if Symbolic argument Methods of determining validity Words or phrases used

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