Chapter 7 Logic
7.5 Equivalent Statements
Learning Objectives
By the end of this section, you will be able to:
- Determine whether two statements are logically equivalent using a truth table
- Compose the converse, inverse, and contrapositive of a conditional statement
Have you ever had a conversation with or sent a note to someone only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly.
Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience.
In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned persuasive arguments that can be customized to address specific audiences.
Determine Logical Equivalence
Two statements, [latex]p[/latex] and [latex]q[/latex], are logically equivalent when [latex]p \leftrightarrow q[/latex] is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology. To determine whether two statements [latex]p[/latex] and [latex]q[/latex] are logically equivalent, construct a truth table for [latex]p \leftrightarrow q[/latex] and determine whether it is valid. If the last column is all true, the argument is a tautology, it is valid, and [latex]p[/latex] is logically equivalent to [latex]q[/latex] ; otherwise, [latex]p[/latex] is not logically equivalent to [latex]q[/latex].
An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever [latex]p[/latex] is true, [latex]q[/latex] is also true, and whenever [latex]p[/latex] is false, [latex]q[/latex] is also false.
Example 1
Create a truth table to determine whether the following compound statements are logically equivalent.
a) [latex]p\rightarrow q; \sim{p} \rightarrow \sim {q}[/latex]
Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, [latex](p \rightarrow q)\leftrightarrow(\sim p \rightarrow \sim q)[/latex] .
[latex]\begin{array}{|c|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{p}&\sim{q}&\sim{p} \rightarrow \sim{q} &(p \rightarrow q) \leftrightarrow (\sim{p} \rightarrow \sim{q})\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{F}&\text{T}&\text{T}&\textbf{F} \\ \hline \text{F}&\text{T}&\text{T}&\text{T}&\text{F}&\text{F}&\textbf{F} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex]
Because the last column is not all true, the biconditional is not valid and the statement [latex]p \rightarrow q[/latex] is not logically equivalent to the statement [latex]\sim{p} \rightarrow \sim {q}[/latex].
b) [latex]p \rightarrow q; \sim p \vee q[/latex]
Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, [latex](p \rightarrow q) \leftrightarrow (\sim{p} \vee q)[/latex].
[latex]\begin{array}{|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{p}&\sim{p} \vee q &(p \rightarrow q) \leftrightarrow (\sim{p} \vee q)\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{F}&\text{F}&\textbf{T} \\ \hline \text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex]
Because the last column is all true, the biconditional is valid and the statement [latex]p \rightarrow q[/latex] is logically equivalent to the statement [latex]\sim{p} \vee q[/latex]. Symbolically, [latex]p \rightarrow q \equiv \sim{p} \vee q[/latex].
Exercise 1
a) [latex]p\rightarrow q; \sim{q} \rightarrow \sim {p}[/latex]
b) [latex]p \rightarrow q; p \vee \sim{q}[/latex]
Solution
a) [latex]p \rightarrow q[/latex] is logically equivalent [latex]\sim{q} \rightarrow \sim {p}[/latex]. [latex]\begin{array}{|c|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{q}&\sim{p}&\sim{q} \rightarrow \sim{p} &(p \rightarrow q) \leftrightarrow (\sim{q} \rightarrow \sim{p})\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{T}&\text{F}&\text{F}&\textbf{T} \\ \hline \text{F}&\text{T}&\text{T}&\text{F}&\text{T}&\text{T}&\textbf{T} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex]
b) [latex]p \rightarrow q[/latex] is not logically equivalent [latex]p \vee \sim{q}[/latex]. [latex]\begin{array}{|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{q}&p\vee \sim{q} &(p \rightarrow q) \leftrightarrow (p \vee \sim{q})\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{T}&\text{T}&\textbf{F} \\ \hline \text{F}&\text{T}&\text{T}&\text{F}&\text{F}&\textbf{F} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex]
Compose the Converse, Inverse, and Contrapositive of a Conditional Statement
The converse, inverse, and contrapositive are variations of the conditional statement, [latex]p \rightarrow q[/latex].
- The converse is if [latex]q[/latex], then [latex]p[/latex], and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse.
- The inverse is if [latex]\sim{p}[/latex], then [latex]\sim{q}[/latex], and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse.
- The contrapositive is if [latex]\sim{q}[/latex], then [latex]\sim{p}[/latex], and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional.
The table below shows how these variations are presented symbolically.
[latex]\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline & & & & \text{Conditional}&\text{Contrapositive} &\text{Converse}&\text{Inverse}\\ \hline p&q&\sim{p} & \sim{q} & p\rightarrow q& \sim{q}\rightarrow \sim{p}&q \rightarrow p &\sim{p} \rightarrow \sim{q}\\ \hline \text{T}&\text{T}&\text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{T}&\text{F}&\text{F}&\text{T}&\text{T} \\ \hline \text{F}&\text{T}&\text{T}&\text{F}&\text{T}&\text{T}&\text{F}&\text{F} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\text{T}&\text{T} \\ \hline \end{array}[/latex]
Example 2
Use the statements [latex]p[/latex]: Harry is a wizard and [latex]q[/latex]: Hermione is a witch to write the following statements:
a) Write the conditional statement, [latex]p \rightarrow q[/latex], in words.
The conditional statement takes the form, “if [latex]p[/latex], then [latex]q[/latex],” so the conditional statement is “If Harry is a wizard, then Hermione is a witch.” Remember the if … then … words are the connectives that form the conditional statement.
b) Write the converse statement, [latex]q \rightarrow p[/latex], in words.
The converse swaps or interchanges the hypothesis, [latex]p[/latex], with the conclusion, [latex]q[/latex]. It has the form “if [latex]q[/latex], then [latex]p[/latex].” So the converse is “If Hermione is a witch, then Harry is a wizard.”
c) Write the inverse statement, [latex]\sim{p} \rightarrow \sim{q}[/latex], in words.
To construct the inverse of a statement, negate both the hypothesis and the conclusion. The inverse has the form “if [latex]\sim{p}[/latex], then [latex]\sim{q}[/latex]” so the inverse is “If Harry is not a wizard, then Hermione is not a witch.”
d) Write the contrapositive statement, [latex]\sim{q} \rightarrow \sim{p}[/latex], in words.
The contrapositive is formed by negating and interchanging both the hypothesis and the conclusion. It has the form “if [latex]\sim{q}[/latex], then,[latex]\sim{p}[/latex],” so the contrapositive statement is: “If Hermione is not a witch, then Harry is not a wizard.”
Exercise 3
Use the statements [latex]p[/latex]: Elvis Presley wore capes and [latex]q[/latex]: Some superheroes wear capes to write the following statements:
a) Write the conditional statement, [latex]p \rightarrow q[/latex], in words.
b) Write the converse statement, [latex]q \rightarrow p[/latex], in words.
c) Write the inverse statement, [latex]\sim{p} \rightarrow \sim{q}[/latex], in words.
d) Write the contrapositive statement, [latex]\sim{q} \rightarrow \sim{p}[/latex], in words.
Solution
a) If Elvis Presley wore capes, then some superheroes wear capes.
b) If some superheroes wear capes, then Elvis Presley wore capes.
c) If Elvis Presley did not wear capes, then no superheroes wear capes.
d) If no superheroes wear capes, then Elvis Presley did not wear capes.
Example 4
Use the conditional statement “If all dogs bark, then Lassie likes to bark” to identify the following.
a) Write the hypothesis of the conditional statement and label it with a [latex]p[/latex].
The hypothesis is the phrase following the if. The answer is [latex]p[/latex]: All dogs bark. Notice that the word if is not included as part of the hypothesis.
b) Write the conclusion of the conditional statement and label it with a [latex]q[/latex].
The conclusion of a conditional statement is the phrase following the then. The word then is not included when stating the conclusion. The answer is [latex]q[/latex]: Lassie likes to bark.
c) Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs bark.”
“Lassie likes to bark” is [latex]q[/latex] and “All dogs bark” is [latex]p[/latex]. So “If Lassie likes to bark, then all dogs bark” has the form “if [latex]q[/latex], then [latex]p[/latex],” which is the form of the converse.
d) Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then some dogs do not bark.”
“Lassie does not like to bark” is [latex]\sim{q}[/latex] and “Some dogs do not bark” is [latex]\sim{p}[/latex]. The statement “If Lassie does not like to bark, then some dogs do not bark” has the form “if [latex]\sim{q}[/latex], then [latex]\sim{p}[/latex],” which is the form of the contrapositive.
e) Which statement is logically equivalent to the conditional statement?
The contrapositive [latex]\sim{q} \rightarrow \sim{p}[/latex] is logically equivalent to the conditional statement [latex]p \rightarrow q[/latex].
Exercise 4
Use the conditional statement “If Dora is an explorer, then Boots is a monkey” to identify the following:
a) Write the hypothesis of the conditional statement and label it with a [latex]p[/latex].
b) Write the conclusion of the conditional statement and label it with a [latex]q[/latex].
c) Identify the following statement as the converse, inverse, or contrapositive: “If Dora is not an explorer, then Boots is not a monkey.”
d) Identify the following statement as the converse, inverse, or contrapositive: “If Boots is a monkey, then Dora is an explorer.”
e) Which statement is logically equivalent to the inverse?
Solution
a) [latex]p[/latex]: Dora is an explorer.
b) [latex]q[/latex]: Boots is a monkey.
c) Inverse
d) Converse
e) Converse
Example 5
Assume the conditional statement, [latex]p \rightarrow q[/latex], “If Chadwick Boseman was an actor, then Chadwick Boseman did not star in the movie Black Panther,” is false, and use it to answer the following questions.
a) Write the converse of the statement in words and determine its truth value.
The only way the conditional statement can be false is if the hypothesis, [latex]p[/latex]: Chadwick Boseman was an actor, is true and the conclusion, [latex]q[/latex]: Chadwick Boseman did not star in the movie Black Panther, is false. The converse is [latex]q \rightarrow p[/latex], and it is written in words as “If Chadwick Boseman did not star in the movie Black Panther, then Chadwick Boseman was an actor.” This statement is true, because false [latex]\rightarrow[/latex] true is true.
b) Write the inverse of the statement in words and determine its truth value.
The inverse has the form “[latex]\sim{p} \rightarrow \sim{q}[/latex]. ” The written form is “If Chadwick Boseman was not an actor, then Chadwick Boseman starred in the movie Black Panther.” Because [latex]p[/latex] is true and [latex]q[/latex] is false, [latex]\sim{p}[/latex] is false and [latex]\sim{q}[/latex] is true. This means the inverse is false [latex]\rightarrow[/latex] true, which is true. Alternatively, from Question 1, the converse is true, and because the inverse is logically equivalent to the converse, it must also be true.
c) Write the contrapositive of the statement in words and determine its truth value.
The contrapositive is logically equivalent to the conditional. Because the conditional is false, the contrapositive is also false. This can be confirmed by looking at the truth values of the contrapositive statement. The contrapositive has the form “[latex]\sim{q}\rightarrow \sim{p}[/latex].” Because [latex]q[/latex] is false and [latex]p[/latex] is true, [latex]\sim{q}[/latex] is true and [latex]\sim{p}[/latex] is false. Therefore, [latex]\sim{q}\rightarrow \sim{p}[/latex] is true [latex]\rightarrow[/latex] false, which is false. The written form of the contrapositive is “If Chadwick Boseman starred in the movie Black Panther, then Chadwick Boseman was not an actor.”
Exercise 5
Assume the conditional statement [latex]p \rightarrow q[/latex]: “If my friend lives in Shreveport, then my friend does not live in Louisiana” is false and use it to answer the following questions.
a) Write the converse of the statement in words and determine its truth value.
b) Write the inverse of the statement in words and determine its truth value.
c) Write the contrapositive of the statement in words and determine its truth value.
Solution
a) If my friend does not live in Louisiana, then my friend lives in Shreveport. True.
b) If my friend does not live in Shreveport, then my friend lives in Louisiana. True.
c) If my friend lives in Louisiana, then my friend does not live in Shreveport. False.
Two expressions that have the same truth value for all possible combinations of truth clauses for all variables in the expressions.
A logical statement that is always true.
The statement formed by interchanging the hypothesis and the conclusion of a conditional statement.
The logical statement formed by negating both the hypothesis and the conclusion.
The statement formed by interchanging and negating both the hypothesis and the conclusion of a conditional statement.