Copyright © Cengage Learning. All rights reserved.
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Logic
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Copyright © Cengage Learning. All rights reserved.
1.4
More on Conditionals
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Objectives
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More on Conditionals
Conditionals differ from conjunctions and disjunctions with
regard to the possibility of changing the order of the
statements. In algebra, the sum x + y is equal to the sum
y + x; that is, addition is commutative.
In everyday language, one realtor might say, “The house is perfect and the lot is priceless,” while another says, “The lot is priceless and the house is perfect.” Logically, their meanings are the same, since (p q) ≡ (q p).
The order of the components in a conjunction or disjunction makes no difference in regard to the truth value of the statement. This is not so with conditionals.
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Variations of a Conditional
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Variations of a Conditional
Given two statements p and q, various “if . . . then . . .” statements can be formed.
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Example 1 – Translating Symbols into Words
Using the statements
p: You are compassionate.
q: You contribute to charities.
write an “if . . . then . . .” sentence represented by each of
the following:
a. p → q b. q → p c. ~p → ~q d. ~q → ~p
Solution:
a. p → q: If you are compassionate, then you contribute to � charities.
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Example 1 – Solution
b. q → p: If you contribute to charities, then you are � compassionate.
c. ~p → ~q: If you are not compassionate, then you do not� contribute to charities.
d. ~q → ~p: If you do not contribute to charities, then you� are not compassionate.
cont’d
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Variations of a Conditional
Each part of Example 1 contains an “if . . . then . . .” statement and is called a conditional. Any given conditional has three variations: a converse, an inverse, and a contrapositive.
The converse of the conditional “if p then q” is the compound statement “if q then p”. That is, we form the converse of the conditional by interchanging the antecedent and the consequent; q → p is the converse of p → q.
The statement in part (b) of Example 1 is the converse of the statement in part (a).
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Variations of a Conditional
The inverse of the conditional “if p then q” is the compound statement “if not p then not q.” We form the inverse of the conditional by negating both the antecedent and the consequent; ~p → ~q is the inverse of p → q.��The statement in part (c) of Example 1 is the inverse of the statement in part (a).��The contrapositive of the conditional “if p then q” is the compound statement “if not q then not p.” We form the�contrapositive of the conditional by negating and interchanging both the antecedent and the consequent;�~q → ~p is the contrapositive of p → q.
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Variations of a Conditional
The statement in part (d) of Example 1 is the contrapositive�of the statement in part (a). The variations of a given conditional are summarized in Figure 1.57.
As we will see, some of these variations are equivalent, and some are not. Unfortunately, many people incorrectly treat them all as equivalent.
Variations of a conditional.
Figure1.57
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Equivalent Conditionals
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Equivalent Conditionals
We have seen that the conditional p → q has three variations: the converse q → p, the inverse ~p → ~q, and the contrapositive ~q → ~p.
Do any of these “if . . . then . . .” statements convey the same meaning? In other words, are any of these compound statements equivalent?
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Example 3 – Determining Equivalent Statements
Determine which (if any) of the following are equivalent: a conditional p → q, the converse q → p, the inverse ~p → ~q, and the contrapositive ~q → ~p.
Solution:
To investigate the possible equivalencies, we must construct a truth table that contains all the statements. Because there are two letters, we need 22 = 4 rows.
The table must have a column for ~p, one for ~q, one for the conditional p → q, and one for each variation of the conditional.
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Example 3 – Solution
The truth values of the negations ~p and ~q are readily entered, as shown in Figure 1.58.
An “if . . . then . . .” statement is false only when the�antecedent is true and the consequent is false.
cont’d
Required columns in the truth table.
Figure1.58
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Example 3 – Solution
Consequently, p → q is false only when p is T and q is F; enter an F in row 2 and Ts elsewhere in the column under p → q.
Likewise, the converse q → p is false only when q is T and p is F; enter an F in row 3 and Ts elsewhere.
In a similar manner, the inverse ~p → ~q is false only when ~p is T and ~q is F; enter an F in row 3 and Ts elsewhere.
Finally, the contrapositive ~q → ~p is false only when ~q is T and ~p is F; enter an F in row 2 and Ts elsewhere.
cont’d
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Example 3 – Solution
The completed truth table is shown in Figure 1.59. Examining the entries in Figure 1.59.
cont’d
Figure1.59
Truth table for a conditional and its variations.
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Example 3 – Solution
We can see that the columns under p → q and ~q → ~p are identical; each has an F in row 2 and Ts elsewhere. Consequently, a conditional and its contrapositive �are equivalent:
Likewise, we notice that q → p and ~p → ~q have identical truth values; each has an F in row 3 and Ts elsewhere. Thus, the converse and the inverse of a conditional are equivalent:
cont’d
p → q ≡ ~q → ~p.
q → p ≡ ~p → ~q.
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Equivalent Conditionals
We have seen that different “if . . . then . . .” statements can convey the same meaning—that is, that certain variations of a conditional are equivalent (see Figure1.60).
Equivalent “if . . . then . . .” statements.
Figure 1.60
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Equivalent Conditionals
For example, the compound statements “If you are�compassionate, then you contribute to charities” and “If you�do not contribute to charities, then you are not�compassionate” convey the same meaning.
(The second conditional is the contrapositive of the first.) �Regardless of its specific contents (p, q, ~p, or ~q), every�“if . . . then . . .” statement has an equivalent variation formed by negating and interchanging the antecedent and the consequent of the given conditional statement.
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Example 4 – Creating a Contrapositive
Given the statement “Being a doctor is necessary for being�a surgeon,” express the contrapositive in terms of the
following:
a. a sufficient condition
b. a necessary condition
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Example 4 – Solution
a. We know that a necessary condition is the consequent of � a conditional, we can rephrase the statement “Being a � doctor is necessary for being a surgeon” as follows:
Therefore, by negating and interchanging the antecedent � and consequent, the contrapositive is
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Example 4 – Solution
We know that a sufficient condition is the antecedent of a � conditional, we can phrase the contrapositive of the � original statement as “Not being a doctor is sufficient for � not being a surgeon.”
b. From part (a), the contrapositive of the original statement
is the conditional statement
cont’d
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Example 4 – Solution
Because a necessary condition is the consequent of a� conditional, the contrapositive of the (original) statement� “Being a doctor is necessary for being a surgeon” can be� expressed as “Not being a surgeon is necessary for not� being a doctor.”
cont’d
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The “Only If” Connective
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The “Only If” Connective
Consider the statement “A prisoner is paroled only if the prisoner obeys the rules.” What is the antecedent, and what is the consequent? Rather than using p and q (which might bias our investigation), we define
r: A prisoner is paroled.
s: A prisoner obeys the rules.
The given statement is represented by “r only if s.” Now, “r only if s” means that r can happen only if s happens. In other words, if s does not happen, then r does not happen, or ~s → ~r.
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The “Only If” Connective
We have seen that ~s → ~r is equivalent to r → s.
Consequently, “r only if s” is equivalent to the conditional r → s.
The antecedent of the statement “A prisoner is paroled only if the prisoner obeys the rules” is “A prisoner is paroled,” and the consequent is “The prisoner obeys the rules.”
The conditional p → q can be phrased “p only if q.” Even though the word if precedes q, q is not the antecedent. Whatever follows the connective “only if” is the consequent of the conditional.
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Example 5 – Analyzing an “Only If” Statement
For the compound statement “You receive a federal grant
only if your artwork is not obscene,” do the following:
a. Determine the antecedent and the consequent.
b. Rewrite the compound statement in the standard “if . . . then . . .” form.
c. Interpret the conditions that make the statement false.
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Example 5 – Solution
a. Because the compound statement contains an “only if ” connective, the statement that follows “only if ” is the consequent of the conditional.
The antecedent is “You receive a federal grant.” The �consequent is “Your artwork is not obscene.”
b. The given compound statement can be rewritten as “If you receive a grant, then your artwork is not obscene.”
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Example 5 – Solution
c. First we define the symbols.
p: You receive a federal grant.
q: Your artwork is obscene.
Then the statement has the symbolic representation
p → ~q. The truth table for p → ~q is given In Figure 1.61.
cont’d
Truth table for the conditional p → ~q.
Figure 1.61
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Example 5 – Solution
The expression p → ~q is false under the conditions listed in row 1 (when p and q are both true).
Therefore, the statement “You receive a federal grant only if your artwork is not obscene” is false when an artist does receive a federal grant and the artist’s artwork is obscene.
cont’d
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The Biconditional p ↔ q
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The Biconditional p ↔ q
What do the words bicycle, binomial, and bilingual have in�common? Each word begins with the prefix bi, meaning�“two.” Just as the word bilingual means “two languages,” the word biconditional means “two conditionals.”
In everyday speech, conditionals often get “hooked�together” in a circular fashion. For instance, someone�might say, “If I am rich, then I am happy, and if I am happy, then I am rich.”
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The Biconditional p ↔ q
Notice that this compound statement is actually the�Conjunction (and) of a conditional (if rich, then happy)�and its converse (if happy, then rich).�
Such a statement is referred to as a biconditional. A�biconditional is a statement of the form (p → q) (q → p) and is symbolized as p ↔ q.��The symbol p ↔ q is read “p if and only if q” and is �frequently abbreviated “p iff q.” A biconditional�is equivalent to the conjunction of two conversely�related conditionals:
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The Biconditional p ↔ q
In addition to the phrase “if and only if,” a biconditional can�also be expressed by using “necessary” and “sufficient”�terminology.
The statement “p is sufficient for q” can be rephrased as “if p then q” (and symbolized as p → q), whereas the statement “p is necessary for q” can be rephrased as “if q then p” (and symbolized as q → p).
Therefore, the biconditional “p if and only if q” can also be�phrased as “p is necessary and sufficient for q.”
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Example 6 – Analyzing a Biconditional Statement
Express the biconditional “A citizen is eligible to vote if and�only if the citizen is at least eighteen years old” as the �conjunction of two conditionals.
Solution:
The given biconditional is equivalent to “If a citizen is �eligible to vote, then the citizen is at least eighteen years old, and if a citizen is at least eighteen years old, then the citizen is eligible to vote.”
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The Biconditional p ↔ q
Under what circumstances is the biconditional p ↔ q true,�and when is it false? To find the answer, we must�construct a truth table.
Utilizing the equivalence we get�the completed table shown in Figure 1.62.
Figure 1.62
Truth table for a biconditional p ↔ q.
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The Biconditional p ↔ q
(We know that a conditional is false only when its antecedent is true and its consequent is false and that a conjunction is true only when both components are true.)
We can see that a biconditional is true only when the two components p and q have the same truth value—that is, when p and q are both true or when p and q are both false.
On the other hand, a biconditional is false when the two components p and q have opposite truth value—that is, when p is true and q is false or vice versa.
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The Biconditional p ↔ q
Many theorems in mathematics can be expressed as biconditionals.
For example, when solving a quadratic equation, we have the following: “The equation ax2 + bx + c = 0 has exactly one solution if and only if the discriminant b2 – 4ac = 0.” Recall that the solutions of a quadratic equation are
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The Biconditional p ↔ q
This biconditional is equivalent to “If the equation ax2 + bx + c = 0 has exactly one solution, then the discriminant �b2 – 4ac = 0, and if the discriminant b2 – 4ac = 0, then the equation ax2 + bx + c = 0 has exactly one solution”—that is, one condition implies the other.
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