chapter 3 introduction to logic © 2008 pearson addison-wesley. all rights reserved

Chapter 3

Introduction to Logic

© 2008 Pearson Addison-Wesley.All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved 3-1-2

Chapter 3: Introduction to Logic

3.1 Statements and Quantifiers

3.2 Truth Tables and Equivalent Statements

3.3 The Conditional and Circuits

3.4 More on the Conditional

3.5 Analyzing Arguments with Euler Diagrams

3.6 Analyzing Arguments with Truth Tables


Chapter 1

Section 3-1Statements and Quantifiers


Statements and Qualifiers

• Statements

• Negations

• Symbols

• Quantifiers

• Sets of Numbers


Statements

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. Various connectives such as and, or, not, and if…then, can be used in forming compound statements.


Example: Compound Statements

Decide whether each statement is compound.a) If Amanda said it, then it must be true.b) The gun was made by Smith and Wesson.


Negations

The sentence “Max has a valuable card” is a statement; the negation of this statement is “Max does not have a valuable card.” The negation of a true statement is false and the negation of a false statement is true.


Inequality Symbols

Use the following inequality symbols for the next example.

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

a ba b

a ba b


Example: Forming Negations

Give a negation of each inequality. Do not use a slash symbol.

a) 3

b) 3 2 12

p

x y


Symbols

To simplify work with logic, we use symbols. 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


Example: Translating from Symbols to Words

Let p represent “It is raining,” and let q represent “It is March.” Write each symbolic statement in words.

a)

b)

p q

p q


Quantifiers

The words all, each, every, and no(ne) are called universal quantifiers, while words and phrases such as some, there exists, and (for) at least one are called existential quantifiers.

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.


Example: Forming Negations of Quantified Statements

Form the negation of each statement.a) Some cats have fleas.b) Some cats do not have fleas.c) No cats have fleas.


Sets of Numbers

Natural (counting) {1, 2, 3, 4, …}Whole numbers {0, 1, 2, 3, 4, …}Integers {…,–3, –2, –1, 0, 1, 2, 3, …} Rational numbers May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333…Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat.Real numbers {x | x can be expressed as a decimal}

and are integers and 0p

p q qq


Example: Deciding Whether the Statements are True or False

Decide whether each of the followingstatements about sets of numbers is true or false.a) Every integer is a natural number.b) There exists a whole number that is not a natural number.


Section 3.1: Statements and Quantifiers

1. Form the negation of “none do.” a) All dob) Some doc) All do not


Section 3.1: Statements and Quantifiers

2. Decide whether or not the following statement is a compound statement. “Jim is a good friend.” a) Yesb) No


Chapter 1

Section 3-2Truth Tables and Equivalent

Statements


Truth Tables and Equivalent Statements

• Conjunctions

• Disjunctions

• Negations

• Mathematical Statements

• Truth Tables

• Alternative Method for Constructing Truth Tables

• Equivalent Statements and De Morgan’s Laws


Conjunctions

The truth values of component statements are used to find the truth values of compound statements.

p qThe truth values of the conjunction p and q, symbolized are given in the truth table on the next slide. The connective and implies “both.”


Conjunction Truth Table

p q

T T T

T F F

F T F

F F F

p qp and q


Example: Finding the Truth Value of a Conjunction

Let p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of .p q


Disjunctions

The truth values of the disjunction p or q, symbolized are given in the truth table on the next slide. The connective or implies “either.”

p q


Disjunctions

p q

T T T

T F T

F T T

F F F

p qp or q


Example: Finding the Truth Value of a Disjunction

Let p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of .p q


Negation

The truth values of the negation of p, symbolized are given in the truth table below.

p

T F

F T

p

,p

not p


Example: Mathematical Statements

Let p represent the statement 4 > 1, q represent the statement 12 < 9, and r represent 0 < 1. Decide whether each statement is true or false.

a)

b)

p q

p r q p


Truth Tables

Use the following standard format for listing the possible truth values in compound statements involving two component statements.

p q Compound Statement

T T

T F

F T

F F


Example: Constructing a Truth Table

Construct the truth table for .p p q


Number of Rows in a Truth Table

A logical statement having n component statements will have 2n rows in its truth table.


Alternative Method for Constructing Truth Tables

After making several truth tables, some people prefer a shortcut method where not every step is written out.


Equivalent Statements

Two statements are equivalent if they have the same truth value in every possible situation.


Example: Equivalent Statements

and p q p q Are the following statements equivalent?


De Morgan’s Laws

For any statements p and q,

( ) and

( ) .

p q p q

p q p q


Example: Applying De Morgan’s Laws

Find a negation of each statement by applying De Morgan’s Law.

a) I made an A or I made a B.b) She won’t try and he will succeed.


Section 3.2: Truth Tables and Equivalent Statements

1. If q is a true statement, then is a) Trueb) False

q q


Section 3.2: Truth Tables and Equivalent Statements

2. A logical statement with 3 component statements will have how many rows in itstruth table?a) 2b) 4c) 8d) 16


Chapter 1

Section 3-3The Conditional and Circuits


The Conditional and Circuits

• Conditionals

• Negation of a Conditional

• Circuits


Conditionals

A conditional statement is a compound statement that uses the connective if…then.

The conditional is written with an arrow, so “if p then q” is symbolized

.p q

We read the above as “p implies q” or “if p then q.” The statement p is the antecedent, while q is the consequent.


Truth Table for The Conditional, If p, then q

p q

T T T

T F F

F T T

F F T

p qIf p, then q


Special Characteristics of Conditional Statements

1. is false only when the antecedent is true and the consequent is false.

2. If the antecedent is false, then is automatically true.

3. If the consequent is true, then is automatically true.

p q

p q

p q


Example: Determining Whether a Conditional Is True or False

Decide whether each statement is True or False (T represents a true statement, F a false statement).a) T (4 2) b) (8 1) F


Tautology

A statement that is always true, no matter what the truth values of the components, is called a tautology. They may be checked by forming truth tables.


Negation of a Conditional

The negation of is .p q p q

p q


Writing a Conditional as an “or” Statement

is equivalent to .p q p q


Example: Determining Negations

Determine the negation of each statement. a) If you ask him, he will come.b) All dogs love bones.


Circuits

Logic can be used to design electrical circuits.

p q

Series circuit

p

q

Parallel circuit


Equivalent Statements Used to Simplify Circuits

p q r p q p r

p q r p q p r

p q q p

p q p q

p p p p p p

p q p q

p q p q


Equivalent Statements Used to Simplify Circuits

T T

F F

T

F.

p

p

p p

p p

If T represents any true statement and F represents any false statement, then


Example: Drawing a Circuit for a Conditional Statement

Draw a circuit for .p q r


1. If q is a true statement and p is a falsestatement, which of the following is false? a)

b)

c)

d)

q p

p q

p p

q q

Section 3.3: The Conditional and Circuits


2. is equivalent to a)

b)

c)

q p

p q

q p

Section 3.3: The Conditional and Circuits

q p


Chapter 1

Section 3-4More on the Conditional


More on the Conditional

• Converse, Inverse, and Contrapositive

• Alternative Forms of “If p, then q”

• Biconditionals

• Summary of Truth Tables


Converse, Inverse, and Contrapositive

Conditional Statement

If p, then q

Converse If q, then p

Inverse If not p, then not q

Contrapositive If not q, then not p

p q

q p

q p

p q


Example: Determining Related Conditional Statements

Given the conditional statement If I live in Wisconsin, then I shovel snow,determine each of the following:a) the converse b) the inverse c) the contrapositive


Equivalences

A conditional statement and its contrapositive are equivalent, and the converse and inverse are equivalent.


Alternative Forms of “If p, then q”

The conditional can be translated in any of the following ways.

If p, then q. p is sufficient for q.

If p, q. q is necessary for p.

p implies q. All p are q.

p only if q. q if p.

p q


Example: Rewording Conditional Statements

Write each statement in the form “if p, then q.”a) You’ll be sorry if I go.b) Today is Sunday only if yesterday was Saturday.c) All Chemists wear lab coats.


Biconditionals

p q

The compound statement p if and only if q (often abbreviated p iff q) is called a biconditional. It is symbolized , and is interpreted as the conjunction of the two conditionals and .p q q p


Truth Table for the Biconditional

p if and only if q

p q

T T T

T F F

F T F

F F T

p q


Example: Determining Whether Biconditionals are True or False

Determine whether each biconditional statement is true or false. a) 5 + 2 = 7 if and only if 3 + 2 = 5.b) 3 = 7 if and only if 4 = 3 + 1.c) 7 + 6 = 12 if and only if 9 + 7 = 11.


Summary of Truth Tables

1. The negation of a statement has truth value opposite of the statement.

2. The conjunction is true only when both statements are true.

3. The disjunction is false only when both statements are false.

4. The biconditional is true only when both statements have the same truth value.


1. Given which of the following is the inverse? a)

b)

c)

q p

p q

q p

,p q

Section 3.4: More on the Conditional


2. Given which of the following is the converse? a)

b)

c)

q p

p q

q p

,p q

Section 3.4: More on the Conditional


Chapter 1

Section 3-5Analyzing Arguments with Euler

Diagrams


Analyzing Arguments with Euler Diagrams

• Logical Arguments

• Arguments with Universal Quantifiers

• Arguments with Existential Quantifiers


Logical Arguments

A logical argument is made up of premises (assumptions, laws, rules, widely held ideas, or observations) and a conclusion. Together, the premises and the conclusion make up the argument.


Valid and Invalid Arguments

An argument is valid if the fact that all the premises are true forces the conclusion to be true. An argument that is not valid is invalid. It is called a fallacy.


Arguments with Universal Quantifiers

Several techniques can be used to check the validity of an argument. One of these is a visual technique based on Euler Diagrams.


Example: Using an Euler Diagram to Determine Validity (Universal Quantifier)

Is the following argument valid? All cats are animals.Figgy is a cat.Figgy is an animal.



Is the following argument valid? All sunny days are hot.Today is not hot Today is not sunny.



Is the following argument valid? All cars have wheels.That vehicle has wheels. That vehicle is a car.


Example: Using an Euler Diagram to Determine Validity (Existential Quantifier)

Is the following argument valid? Some students drink coffee.I am a student . I drink coffee .


1. Are “Valid” and “true” the same?a) Yesb) No

Section 3.5: Analyzing Arguments with Euler Diagrams


2. Premises area) Assumptions b) Rulesc) Lawsd) All of the above

Section 3.5: Analyzing Arguments with Euler Diagrams


Chapter 1

Section 3-6Analyzing Arguments with Truth

Tables


Analyzing Arguments with Truth Tables

• Truth Tables (Two Premises)

• Valid and Invalid Argument Forms

• Truth Tables (More Than Two Premises)

• Arguments of Lewis Carroll


Truth Tables

In section 3.5 Euler diagrams were used to test the validity of arguments. These work well with simple arguments but may not work well with more complex ones. If the words “all,” “some,” or “no” are not present, it may be better to use a truth table than an Euler diagram to test validity.


Testing the Validity of an Argument with a Truth Table

Step 1 Assign a letter to represent each component statement in the argument.

Step 2 Express each premise and the conclusion symbolically.

Continued on the next slide…


Testing the Validity of an Argument with a Truth Table

Step 3 Form the symbolic statement of the entire argument by writing the conjunction of all the premises as the antecedent of a conditional

statement, and the conclusion of the argument as the consequent.

Step 4 Complete the truth table for the conditional statement formed in Step 3. If it is a tautology, then the argument is valid; otherwise it is

invalid.


Example: Truth Tables (Two Premises)

Is the following argument valid? If the door is open, then I must close it.The door is open.I must close it.


Valid Argument Forms

Modus Ponens

Modus Tollens

Disjunctive Syllogism

Reasoning by Transitivity

p q

p

q

~

~

p q

q

p

~

p q

p

q

p q

q r

p r


Invalid Argument Forms (Fallacies)

Fallacy of the Converse

Fallacy of the Inverse

p q

q

p

~

~

p q

p

q


Example: Truth Tables (More Than Two Premises)

Determine whether the argument is valid or invalid.

If Pat goes skiing, then Amy stays at home. If Amy does not stay at home, then Cade will play video games. Cade will not play video games. Therefore, Pat does not go skiing.


Example: Arguments of Lewis Carroll

Supply a conclusion that yields a valid argument for the following premises.

Babies are illogical.Nobody is despised who can manage a crocodile.Illogical persons are despised.

Let p be “you are a baby,” let q be “you are logical,” let r be “you can manage a crocodile,” and let s be “you are despised.”


1. When testing the validity of an argument

and the words “all,” “some,” and “no” are not present you would probably use a) an Euler diagram. b) a truth table.

Section 3.6: Analyzing Arguments with Truth Tables


2. Are the conditional and converse equivalent? a) Yes b) No

Section 3.6: Analyzing Arguments with Truth Tables

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