Chapter 20 Simple and Compound Statements Compound Statements

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Chapter 20: Simple and Compound Statements

Chapter 20: Simple and Compound Statements

Compound Statements (pp. 202 -207) A compound statement contains another statement as a proper

Compound Statements (pp. 202 -207) A compound statement contains another statement as a proper part. • Nontruth-functionally compound statements – A nontruth-functionally compound statement is a compound statement whose truth or falsehood is independent of the truth-values (truth or falsehood) of its component statement. • Example: “It is my opinion that this course is easy” is compound. It contains the statement “This course is easy. ” The statement describes the speaker. The truth or falsehood of the statement is independent of its component proposition. • Example: “John believes that Juanita likes opera” is true or false regardless of Juanita’s musical tastes.

Compound Statements (pp. 202 -207) • Truth-functionally compound statements – The truth value of

Compound Statements (pp. 202 -207) • Truth-functionally compound statements – The truth value of a truth-functionally compound statement is wholly dependent upon the truth values of its component statements. – There is a symbolic apparatus that has been developed to accompany truth-functionally compound statements. The symbolic representations are included here, as they are in the book.

Compound Statements (pp. 202 -207) • Types of truth-functionally compound statements • Negation: not,

Compound Statements (pp. 202 -207) • Types of truth-functionally compound statements • Negation: not, it is not the case that, no (~p) – The negation of a true statement is a false statement. The negation of a false statement is a true statement. • Conjunction, conjuncts: and, yet, but, although, however, nevertheless, even though (p & q) – A conjunction is true if and only if both of its conjuncts are true.

Compound Statements (pp. 202 -207) • Disjunction, disjuncts: or, either … or …, unless

Compound Statements (pp. 202 -207) • Disjunction, disjuncts: or, either … or …, unless (p v q) – A disjunction is true except when both of its disjuncts are false. • Conditionals, antecedents and consequents: if … then …, … only if …, provided that, on the condition that (p q) – The if-clause of a conditional is the antecedent; then-clause if the consequent. – A conditional is true except when its antecedent is true and its consequent is false.

Compound Statements (pp. 202 -207) • Biconditionals: … if and only if …, …

Compound Statements (pp. 202 -207) • Biconditionals: … if and only if …, … just in case that … (p q) – A biconditional is true if and only if the statements flanking the ‘if and only if’ have the same truth value: both true or both false. • Grouping Indicators – Commas and semicolons group in English; parentheses, square brackets ([ ]), and braces ({ }) group in symbolese.

Compound Statements (pp. 202 -207) • Determining truth values of compound statements – In

Compound Statements (pp. 202 -207) • Determining truth values of compound statements – In determining the truth value of a complex statement, you begin with statements most deeply embedded in the complex statement. You work out the truth value of the whole by working out the truth values of the proper parts.

Compound Statements: Examples of Determining Truth Values • George W. Bush was president in

Compound Statements: Examples of Determining Truth Values • George W. Bush was president in 2002 and Texas is in the United States. – Both component statements (conjuncts) are true. So the whole statement — the conjunction — is true. • Either Richmond is in Virginia or Richmond is in Norway. – The first disjunct is true, so the whole statement is true. • If Honolulu is in Hawaii, then George W. Bush was elected president in 1832. – The antecedent is true. The consequent is false. So the statement is false.

Compound Statements: Examples of Determining Truth Values • If the moon is made of

Compound Statements: Examples of Determining Truth Values • If the moon is made of apple pie, then Rush Limbaugh talks on the radio. – The antecedent is false, so the statement is true. – Also, the consequent is true, so the statement is true. • If the moon is not made of apple pie, then Cedar Rapids is in Iowa. – The statement, “The moon is made of apple pie” is false, so its denial is true. – The antecedent is true and the consequent is true. So, the statement is true.

Compound Statements: Examples of Determining Truth Values • If Miami is in Florida, then

Compound Statements: Examples of Determining Truth Values • If Miami is in Florida, then the South won the Civil War; and either Washington, D. C. is the home of the Chicago Cubs or Green Bay is the home of the Packers. – The first conjunct – the conditional – is false, since its antecedent is true and its consequent is false. – The second conjunct is true, since the second disjunct is true, even though the first disjunct is false. – Since the compound statement is a conjunction and one of its conjuncts is false, the entire statement is false. • Since the first conjunct was false, it wouldn’t have been necessary to examine the truth value of the second conjunct.

Compound Statements: Examples of Determining Truth Values • Yale is in Connecticut if and

Compound Statements: Examples of Determining Truth Values • Yale is in Connecticut if and only if the University of Hawaii is in North Dakota; or motley moose munch mice meditatively only if George W. Bush was president in 2002. – The first disjunct – the biconditional – is false, since it is true that Yale is in Connecticut and it’s false that the University of Hawaii is in North Dakota. – The second disjunct could be restated as, “If motley moose munch mice meditatively, then George W. Bush was president in 2002. ” You probably don’t know a lot about the personal habits of motley moose – although I would question the truth of the antecedent. If it’s false, the conditional is true. But the consequent is true, so regardless of the truth value of the antecedent, the conditional is true. So, the second disjunct is true. – Since at least one of the disjuncts is true, the disjunctive statement is true.