Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 1 Thinking Critically 1

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 1

ThinkingCritically1


Unit 1B

Propositions andTruth Values


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.


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: ~


Example

Find the negation of the proposition Amanda is the fastest runner on the team. Write its negation. If the negation is false, is Amanda really the fastest runner on the team?Solution The negation of the given proposition is Amanda is not the fastest runner on the team. If the negation is false, the original statement must be true, meaning that Amanda is the fastest runner on the team.


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.


Example

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 University of North Carolina (Chapel Hill) was asked about the possibility of a similar association among younger workers at another national laboratory. He was quoted as saying (Boulder Daily Camera): My opinion is that it’s unlikely that there is no association. Does the scientist think there is an association between low-level radiation and cancer among younger workers?


Example (cont)Solution Because of the words “unlikely” and “no association,” the scientist’s statement contains a double negation.

p = it’s likely that there is an association (between low-level radiation and cancer)

The word “unlikely” gives us the statement it’s unlikely that there is an association, which we identify as not p. The words “no association” transform this last statement into the original statement, it’s unlikely that there is no association, which we recognize as not not p.


Example (cont)

Because the double negation has the same truth value as the original proposition, we conclude that the scientist believes it likely that there is an association between low-level radiation and cancer among younger workers.


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.


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)


ExampleEvaluate 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.

Solutiona. The statement contains two distinct propositions: The capital of France is Paris and Antarctica is cold. Because both propositions are true, their conjunction is also true.


ExampleEvaluate 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.b. The statement contains two distinct propositions: The capital of France is Paris and the capital of America is Madrid. Although the first proposition is true, the second is false. Therefore, their conjunction is false.


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.

Two Types of Or


p q p or q

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

The Logic of Or (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:


Example

Consider the statement airplanes can fly or cows can read. Is it true?

Solution The statement is a disjunction of two propositions: (1) airplanes can fly; (2) cows can read. The first proposition is clearly true, while the second is clearly false, which makes the disjunction p or q true. That is, the statement airplanes can fly or cows can read is true.


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.

The Logic of If . . . Then Statements (Conditionals)


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


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 not p, then not q

If not q, then not p

Variations on the Conditional


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

Logical Equivalence

p q n o t p n o t q if p , th en q if q, then p (converse)

i f n o tp , th e n n o tq (inverse)

i f n o t q , th e n n o tp (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


Example

Consider the true statement if a creature is a whale, then it is a mammal. Write its converse, inverse, and contrapositive. Evaluate the truth of each statement. Which statements are logically equivalent?

Solution The statement has the form if p, then q, where p = a creature is a whale and q = a creature is a mammal. Therefore, we findconverse (if q, then p): if a creature is a mammal, then it is a whale. This statement is false, because most mammals are not whales.


Example

inverse (if not p, then not q): if a creature is not a whale, then it is not a mammal. This statement is also false; for example, dogs are not whales, but they are mammals.

contrapositive (if not q, then not p): if a creature is not a mammal, then it is not a whale. Like the original statement, this statement is true, because all whales are mammals.


Example

Note that the original proposition and its contrapositive have the same truth value and are logically equivalent. Similarly, the converse and inverse have the same truth value and are logically equivalent.

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Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 1, Unit 1B, Slide 1 Thinking Critically 1