copyright © 2011 pearson education, inc. thinking critically class notes bb 1-7 p.20 1-20 1 web 41....

Copyright © 2011 Pearson Education, Inc.

Thinking Critically

Class Notes BB 1-7p.20 1-201 web

41. Campaign Site42. Gun Control Debate43. Death penalty Debate44. Health Care Debate45. Fallacy Web Sites

1 world46. Editorial Fallacies47. Fallacies in Advertising48. Fallacies in Politics49. Personal Fallacies

Copyright © 2011 Pearson Education, Inc. Slide 1-3

Unit 1B

Propositions andTruth Values

1-B


Definitions

A proposition makes a claim (either an assertion or a denial) that may be either true or false. It must have the structure of a complete sentence.

Any proposition has two possible truth values:T = true or F = false.

A truth table is a table with a row for each possible set of truth values for the propositions being considered.

1-B


Negation (Opposites)

p not p

T F F T

The negation of a proposition p is another proposition that makes the opposite claim of p.

← If p is true (T), not p is false (F).

← If p is false (F), not p is true (T).

Symbol: ~

1-BNegationCN 1a-b

1.Consider the proposition – Betsy is the fastest runner on the team.

a. Write its negation. b. If the negation is false, is Betsy really the

fastest runner on the team?


1-B


Double Negation

p not p not not p

T F T F T F

The double negation of a proposition p, not not p, has the same truth value as p.

1-BRadiation and HealthCN 2

After reviewing data showing an association between low-level radiation and cancer among older workers at the Oak Ridge National Laboratory, a health scientist from the U of North Carolina was asked about the possibility of a similar association among younger workers at another national laboratory. He was quoted as saying: My opinion is that it’s unlikely that there is no association.

2. Does the scientist think there is an association between low-level radiation and cancer?


1-BThe 2000 Miranda RulingCN 3

In a June 2000 decision, the US Supreme Court voted 7-2 to uphold the basic requirements of the 1966 Miranda decision. That decision required that suspects taken into custody be informed of their constitutional rights, such as the right to remain silent and the right to legal counsel. In his majority opinion, Chief Justice William Rehnquist wrote: …legal principles weigh heavily against overruling Miranda now.

3. According to this statement, did the Chief Justice feel that legal principles support or oppose the original Miranda decision?


1-B


Propositions are often joined with logical connectors—words such as and, or, and if…then.

Example:p = I won the game.q = It was fun.

Logical Connectors

Logical Connector

and

or

if…then

New Proposition

I won the game and it was fun.

I won the game or it was fun.

If I won the game, then it was fun.

1-B


p q p and q

T T T T F F F T F F F F

Given two propositions p and q, the statement p and q is called their conjunction. It is true only if p and q are both true.

Symbol:

And Statements (Conjunctions)

1-BAnd StatementsCN 4

4. Evaluate the truth value of the following two statements:

a. The capital of France is Paris and Antarctica is cold.

b. The capital of France is Paris and the capital of America is Madrid.


1-B


An inclusive or means “either or both.”

An exclusive or means “one or the other, but not both.”

The word or can be interpreted in two distinct ways:

In logic, assume or is inclusive unless told otherwise.

Or Statements (Disjunctions)

1-BInclusive or ExclusiveCN 6

Kevin’s insurance policy states that his house is insured for earthquakes, fire, or robbery. Imagine that a major earthquake levels much of his house, the rest burns in a fire, and his remaining valuables are looted in the aftermath.

6. Would Kevin prefer that the or in his insurance policy be inclusive or exclusive? Why?


1-B


p q p or q

T T T T F T F T T F F F

Or Statements (Disjunctions)Given two propositions p and q, the statement p or q is called their disjunction. It is true unless p and q are both false.

Symbol:

1-BSmart Cows?CN 7

Consider the statement: Airplanes can fly or cows can read.

7. Is it true?


1-BKey Word SearchCN 8

You are doing a research report for which you need to find articles about the federal debt that appeared in either Time or Newsweek between 2008 and 2010. You have access to a database that allows you to search on date, magazine, and title.

8. How should you structure the search?


1-B


A statement of the form if p, then q is called a conditional proposition (or implication). It is true unless p is true and q is false.

p q if p, then q

T T T T F F F T T F F T

Proposition p is called the hypothesis. Proposition q is called the conclusion.

If… Then Statements (Conditionals)

1-BConditional TruthsCN 9

9. Evaluate the truth of the statement:

If 2 + 2 = 5, then 3 + 3 = 4


1-B


Truth Table Practice

Note: ~ signifies NEGATION signifies AND signifies OR

Practice by writing the truth values of each row in the table above.

p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T

T F

F T

F F

1-B





p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F

F T

F F

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p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F F T F T T

F T

F F

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p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F F T F T T

F T T F F T T

F F

1-B





p q ~ p ~ q p q ~ p ~ q ~ ( p q)

T T F F T F F

T F F T F T T

F T T F F T T

F F T T F T T

1-B


p is sufficient for q

p will lead to q

p implies q

The following are common alternative ways of stating if p, then q:

q is necessary for p

q if p

q whenever p

Alternative Phrasings of Conditionals

1-BRephrasing Conditional PropositionsCN 10a - c

Recast each of the following statements in the form if p, then q. Discuss whether the statement is true of false.

a. A rise in seas level will devastate Florida b. A red tag on an item is sufficient to mean it’s on

sale. c. Being male is necessary for being president of

the United States.


1-B


Conditional:

Converse:

Inverse:

Contrapositive:

If it is raining, then I will bring an umbrella to work.

If I bring an umbrella to work, then it must be raining.

If it is not raining, then I will not bring an umbrella to work.

If I do not bring an umbrella to work, then it must not be raining.

If p, then q

If q, then p

If ~p, then ~q

If ~q, then ~p

Variations on the Conditional

1-B


Two statements are logically equivalent if they share the same truth values.

Logical Equivalence

p q ~p ~ q if p , th en q if q, then p (converse)

i f ~ p , t h e n ~ q (inverse)

if ~q, then ~p (contrapositive)

T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T

logically equivalent

logically equivalent

1-BLogical EquivalenceCN 11a-c

Consider the statement

If a creature is a whale, then it is a mammal.

a. Write its converse, inverse, and contrapositive. b. Evaluate the truth of each statement. c. Which statements are logically equivalent?


1-B1B Homework

1A Discussion Paragraph (DP) Class Notes (CN) 1-11 P.31 Quick Quiz 1-10 p. 3: 1-12 1 web

103. Web Searches 1 world

104. Logical Or 105. Multiple Negation 106. Conditional News


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