Copyright © Cengage Learning. All rights reserved.
1
Logic
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Copyright © Cengage Learning. All rights reserved.
1.5
Analyzing Arguments
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Objectives
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Valid Arguments
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Valid Arguments
When someone makes a sequence of statements and draws some conclusion from them, he or she is presenting an argument.
An argument consists of two components: the initial statements, or premises, and the final statement, or conclusion.
When presented with an argument, a listener or reader may ask, “Does this person have a logical argument? Does his or her conclusion necessarily follow from the given statements?”
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Valid Arguments
An argument is valid if the conclusion of the argument is guaranteed under its given set of premises. (That is, the conclusion is inescapable in all instances.)
For example,
“All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.”
is a valid argument.
the premises
the conclusion
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Valid Arguments
Given the premises, the conclusion is guaranteed. The term valid does not mean that all the statements are true but merely that the conclusion was reached via a proper deductive process. The argument
“All doctors are men.
My mother is a doctor.
Therefore, my mother is a man.”
is also a valid argument.
the premises
the conclusion
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Valid Arguments
Even though the conclusion is obviously false, the conclusion is guaranteed, given the premises.
The premises in a given logical argument may consist of several interrelated statements, each containing negations, conjunctions, disjunctions, and conditionals.
By joining all the premises in the form of a conjunction, we can form a single conditional that represents the entire argument.
That is, if an argument has n premises P1, P2, . . . , Pn and conclusion C, the argument will have the form �“if (P1 and P2 . . . and Pn), then C.”
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Valid Arguments
If the conditional representation of an argument is always true (regardless of the actual truthfulness of the individual statements), the argument is valid. If there is at least one instance in which the conditional is false, the argument is
invalid.
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Example 1 – Using a Truth Table to Analyze an Argument
Determine whether the following argument is valid:
If he is illiterate, he cannot fill out the application.
He can fill out the application.
Therefore, he is not illiterate.”
Solution:
First, number the premises and separate them from the conclusion with a line:
1. If he is illiterate, he cannot fill out the application.
2. He can fill out the application.
Therefore, he is not illiterate.
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Example 1 – Solution
Now use symbols to represent each different component in the statements:
p: He is illiterate.
q: He can fill out the application.
We could have defined q as “He cannot fill out the application” (as stated in premise 1), but it is customary to define the symbols with a positive sense. �
cont’d
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Example 1 – Solution
Symbolically, the argument has the form
1. p → ~q�
2. q�
� ∴~p
�and is represented by the conditional [(p → ~q) q] → ~p. The symbol ∴ is read “therefore.”
cont’d
the premises
conclusion
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Example 1 – Solution
To construct a truth table for this conditional, we need 22= 4 rows. A column is required for the following: each negation, each premise, the conjunction of the premises, the conclusion, and the conditional representation of the argument.
The initial setup is shown in Figure 1.63.
Figure 1.63
Required columns in the truth table
cont’d
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Example 1 – Solution
Fill in the truth table as follows:
~q: A negation has the opposite truth values; enter a T in rows 2 and 4 and an F in rows 1 and 3.
Premise 1: A conditional is false only when its antecedent is true and its consequent is false; enter an F in row 1 and Ts elsewhere.
Premise 2: Recopy the q column.
1 2: A conjunction is true only when both components are true; enter a T in row 3 and Fs elsewhere.
cont’d
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Example 1 – Solution
Conclusion C: A negation has the opposite truth values; enter an F in rows 1 and 2 and a T in rows 3 and 4.
At this point, all that remains is the final column
(see Figure 1.64).
cont’d
Figure1.64
Truth values of the expressions.
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Example 1 – Solution
The last column in the truth table is the conditional that represents the entire argument.
A conditional is false only when its antecedent is true and its consequent is false. The only instance in which the premise (1 2) is true is row 3. Corresponding to this entry, the conclusion ~p is also true.
Consequently, the conditional (1 2) → C is true in row 3. Because the premise (1 2) is false in rows 1, 2, and 4, the conditional (1 2) → C is automatically true in those rows as well.
cont’d
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Example 1 – Solution
The completed truth table is shown in Figure 1.65.
cont’d
Figure 1.65
Truth table for the argument [(p → ~ q) q] → ~p.
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Example 1 – Solution
The completed truth table shows that the conditional
[(p → ~q) q] → ~p is always true.
The conditional represents the argument “If he is illiterate, he cannot fill out the application. He can fill out the application. Therefore, he is not illiterate.”
Thus, the argument is valid.
cont’d
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Tautologies
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Tautologies
A tautology is a statement that is always true.
For example, the statement
“(a + b)2 = a2 + 2ab + b2”
is a tautology.
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Example 2 – Determining Whether a Statement is a Tautology
Determine whether the statement (p q) → (p q) is a tautology
Solution:
We need to construct a truth table for the statement. Because there are two letters, the table must have 22 = 4 rows.
We need a column for (p q), one for (p q), and one for (p q) → (p q).
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Example 2 – Solution
The completed truth table is shown in Figure 1.66.
Because (p q) → (p q) is always true, it is a tautology.
Figure 1.66
cont’d
Truth table for the statement (p q) → (p q).
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Tautologies
As we have seen, an argument can be represented by a single conditional. If this conditional is always true, the argument is valid (and vice versa).
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Example 3 – Using a Truth Table to Analyze an Argument
Determine whether the following argument is valid:
“If the defendant is innocent, the defendant does not go to jail. The defendant does not go to jail. Therefore, the defendant is innocent.”
Solution:
Separating the premises from the conclusion, we have
1. If the defendant is innocent, the defendant does not go
to jail.
2. The defendant does not go to jail.
Therefore, the defendant is innocent.
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Example 3 – Solution
Now we define symbols to represent the various components of the statements:
p: The defendant is innocent.
q: The defendant goes to jail.
Symbolically, the argument has the form
1. p → ~q
2. ~q�
∴ p
and is represented by the conditional [(p → ~q) ~q] → p.
cont’d
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Example 3 – Solution
Now we construct a truth table with four rows, along with the necessary columns.
The completed table is shown in Figure 1.67.
Truth table for the argument [(p → ~ q) ~ q] → p.
cont’d
Figure 1.67
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Example 3 – Solution
The column representing the argument has an F in row 4; therefore, the conditional representation of the argument is not a tautology.
In particular, the conclusion does not logically follow the premises when both p and q are false (row 4).The argument is not valid.
Let us interpret the circumstances expressed in row 4, the row in which the argument breaks down.
Both p and q are false—that is, the defendant is guilty and the defendant does not go to jail.
cont’d
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Example 3 – Solution
Unfortunately, this situation can occur in the real world; guilty people do not always go to jail! As long as it is�possible for a guilty person to avoid jail, the argument is invalid.
cont’d
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