Slide 3-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION

Slide 3-1 Copyright © 2005 Pearson Education, Inc.

SEVENTH EDITION and EXPANDED SEVENTH EDITION

Copyright © 2005 Pearson Education, Inc.

Chapter 3

Logic


3.1

Statements and Logical Connectives


HISTORY—The Greeks:

Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle (384-322 B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.


Mathematicians

Gottfried Wilhelm Leibniz (1646-1716) 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.


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.


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.


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…


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.


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.”


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.


Not Statements

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


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.


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

.p q


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:

p q


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.


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)p q p q


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.”


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.

q p p q


3.2

Truth Tables for Negation,Conjunction, and Disjunction


Truth Table

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


Negation Truth Table

TCase 2 F

FCase 1 T

~p p


Conjunction Truth Table

The conjunction is true only when both p and q are true.

FFFCase 4

FTFCase 3

FFTCase 2

TTTCase 1

qp p q


Disjunction

The disjunction is true when either p is true, q is true, or both p and q are true.

FFFCase 4

TTFCase 3

TFTCase 2

TTTCase 1

qp p q


3.3

Truth Tables for theConditional and Biconditional


Conditional

The conditional statement is true in every case except when p is a true statement and q is a false statement.

TFFCase 4

TTFCase 3

FFTCase 2

TTTCase 1

qp p q

p q


Biconditional

The biconditional statement, means that and or, symbolically

5647231order of steps

FTFTFTFFFcase 4

FFTFTTFTFcase 3

TTFFFFTFTcase 2

TTTTTTTTTcase 1

p)(qq)(pqp

p q p q ,q p . p q q p


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.


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.


Implication

An implication is a condition statement that is a tautology.

The consequent will be true whenever the antecedent is true.


3.4

Equivalent Statements


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.

Symbols: or


De Morgan’s Laws

~ ( ) ~ ~p q p q

~ ( ) ~ ~p q p q


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.

~ p q p q


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.


“if not q, then not p”~p~qContrapositive of the conditional

“if not p, then not q”~q~pInverse of the conditional

“if q, then p”pqConverse of the conditional

“if p, then q”qpConditional

ReadSymbolic FormName

Variations of the Conditional Statement


3.5

Symbolic Arguments


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.


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.


Law of Detachment

Also called modus ponens. The argument form:

p q

p

q


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


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.


Valid Arguments

Law of Detachment

Law of Syllogism

Law of Contraposition

Disjunctive Syllogism

p q

p

q

~

~

p q

q

p

p q

q r

p r

~

p q

p

q


Invalid Arguments

Fallacy of the Converse Fallacy of the Inverse

p q

q

p

~

~

p q

p

q


3.6

Euler Diagrams and Syllogistic Arguments


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.


Euler Diagrams

One method used to determine whether an argument is valid or is a fallacy.

Uses circles to represent sets in syllogistic arguments.


Symbolic Arguments Versus Syllogistic Arguments

Euler diagramsall 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

Documents

Slide 3-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION