3.1 Statements and Logical Connectives

Why This Is Important To Study

Studying logic and the meaning of statements will help you be able to:

construct convincing arguments

understand the consequences of making certain decisions

detect when an argument being made is illogical

understand legal documents like wills, leases, and contracts

Simple Statements

Definition Statement:

A statement is a sentence that can be judged either true or false.

A simple statement is any statement that expresses one idea.

EXAMPLE Each statement below is a simple statement. Assign a truth value to each statement (i.e. Label the statement true or false).

a) Triton college is in River Forest, IL.

b) Otters sleep holding hands.

c) The Mississippi River is longer than the Nile River.

Learning Objective:

We will be able to understand the meaning of simple, compound, conditional, and biconditional statements. We will also be able to express these statements using logic symbols.

Negation and Quantifiers

The negation of a statement A is another statement with the opposite meaning of statement A.

The negation of a true statement yields a _______________ statement.

The negation of a false statement yields a _______________ statement.

EXAMPLE Write the negation of each statement below. Indicate the truth value of each statement and its negation.

a) Today is Tuesday.

b) Otters sleep holding hands.

Quantifiers indicate the quantity of the objects that a statement is about. They are words like all, none (or no), and some.

EXAMPLE Write the negation of each statement below. Indicate the truth value of each statement and its negation.

a) All lakes contain fresh water.

b) No birds can swim.

c) Some students have a drivers license. (Note: The use of the word some implies a quantity of at least 1 and possibly all.)

d) Some students do not ride motorcycles.

The negation of statements with quantifiers can be tricky. Here is a summary:

Statement Form Negation Form

All are Some are not None are Some are Some are None are

Some are not All are

Here are two more examples:

e) Some telephones can take photographs.

f) All houses have two stories.

Compound Statements

A compound statement is any statement that consists of two or more simple statements.

In a compound statement, the simple statements are joined together by connectives like and, or, ifthen, and if and only if.

NOTE: The negation of a simple statement is considered to be a compound statement (e.g. Henry is not home. is a compound statement because it is the negation of the simple statement Henry is home.).

EXAMPLE Each statement below is a compound statement. Circle the connective in each compound statement.

a) I sentence you to 5 months of community service and a fine of $100.

b) I sentence you to 6 months in prison or 12 months of community service.

NOTE: Here the connective or is being used as an exclusive or (i.e. exactly one statement can be true, but not both).

c) You may order a cup of soup or a sandwich at a restaurant.

NOTE: Here the connective or is being used as an inclusive or (i.e. either one or both of the statements can be true). Assume the inclusive or unless otherwise indicated.

d) If you score at least a 60% on an exam, then you will pass the exam.

Using Letters and Symbols

Letters are often used to represent simple statements. For example, let:

p represent the statement Dave is in a movie theater.

q represent the statement Dave is at home.

Symbols are often used to represent negation and connectives. The following table summarizes these symbols and their use with statements.

Symbol Symbol Meaning

Verbal Statement Statement using Symbols

Symbols Read

~ negation Dave is not at home. ~ q not q

conjunction Dave is in a movie theater and Dave is at home.

p q p and q

disjunction Dave is in a movie theater or Dave is at home.

p q p or q (inclusive or)

NOTES:

~ (~ )q ________

For example, the negation of Dave is not at home. is _______________________________.

In addition to and, conjunctions are commonly written with but, however, and nonetheless.

EXAMPLE Let r and s represent simple statements.

r: The Counting Crows are on tour.

s: The Counting Crows are recording a new record

Write the following statements in symbolic form.

a) The Counting Crows are not on tour.

b) The Counting Crows are not on tour but they are recording a new record.

c) The Counting Crows are on tour or the Counting Crows are recording a new record.

d) The Counting Crows are not on tour and they are not recording a new record.

e) The Counting Crows are not on tour or they are not recording a new record.

f) The Counting Crows are neither on tour nor are they recording a new record. (This is an example of a neither-nor statement.)

Could the statement in part e) of the preceding problem mean that the Counting Crows are not on tour and they are not recording a new record? Explain your answer.

EXAMPLE Let

p: The house is for sale.

q: We can afford to buy the house.

Write the following symbolic statements in words.

a) p ~ q

____________________________________________________________

____________________________________________________________

b) ~ p ~ q

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c) ~ ( p q )

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Rewrite your answer to part a) in the preceding problem using the word but.

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Conditional Statements

Conditional statements are compound statements written with the ifthen connective.

Let p and q be the simple statements:

p: You score a 95% on your final exam.

q: You earn an A in the course.

The table below shows a conditional statement in verbal and symbolic form:

Symbol Symbol Meaning

Verbal Statement Statement using Symbols

Symbols Read

conditional If you score a 95% on your final exam, then you will earn an A in the course.

qp if p, then q

In the conditional statement qp :

p is the antecedent (sometimes called the hypothesis or premise)

q is the consequent (sometimes called the conclusion)

Sometimes, the then in a conditional statement is omitted. The statement in the table could also be written as:

If you score a 95% on your final exam, you will earn an A in the course.

EXAMPLE Let

p: The portrait is a pastel.

q: The portrait is by Beth Anderson.

Write the following statements symbolically.

a) If the portrait is a pastel, then the portrait is by Beth Anderson.

b) If the portrait is by Beth Anderson, then the portrait is not a pastel.

c) It is false that if the portrait is by Beth Anderson then the portrait is a pastel.

d) If the portrait is not a pastel, then the portrait is by Beth Anderson.

Biconditional Statements

Biconditional statements are compound statements written with the if and only if connective.

Let p and q be the simple statements:

p: The Trojans win the championship.

q: Kirby plays centerfield on the baseball team.

The table below shows a biconditional statement in verbal and symbolic form:

Symbol Symbol Meaning

Verbal Statement Statement using Symbols

Symbols Read

biconditional The Trojans win the championship if and only if Kirby plays centerfield on the baseball team.

qp p if and only if q

NOTES:

the phrase if and only if is abbreviated as iff

qp means the same as )()( pqqp (this is the bi in biconditional)

EXAMPLE Let p and q be the simple statements:

p: Corey Crawford plays goalie on the hockey team.

q: The Blackhawks win the Stanley Cup.

Write the following symbolic statements in words.

a) qp

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____________________________________________________________

b) q ~ p

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____________________________________________________________

c) ~ ( p~ q)

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____________________________________________________________

EXAMPLE Given that qp means the same as )()( pqqp , rewrite the following statement using only conditional statements:

I will graduate from college if and only if I can study hard.

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More Complex Statements

Commas in compound statements tell us how to group the simpler statements that make up the compound statement.

The simpler statements on the same side of the comma should be grouped together using parentheses.

EXAMPLE Let

p: Dinner includes soup.

q: Dinner includes salad.

r: Dinner includes the vegetable of the day.

Write the following statements in symbolic form. Indicate the type of each statement (e.g. negation, conjunction, disjunction, conditional, etc.)

a) Dinner includes soup, and salad or the vegetable of the day.

b) Dinner includes soup and salad, or the vegetable of the day.

EXAMPLE Let

p: Jorge is enrolled in calculus.

q: Jorges major is criminal justice.

r: Jorges major is engineering.

Write the following symbolic statements in words. Indicate the type of each statement (e.g. negation, conjunction, disjunction, conditional, etc.)

a) rpq )~(

____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

b) )(~ rpq

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