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Section 2.2 Three Forms of Valid Arguments

Three especially important forms of valid arguments, used repeatedly in logic, are discussed next.

Subsection 2.2.1 Law of Detachment (Direct Reasoning): \([((p \rightarrow q) \wedge p)] \rightarrow q\)

The Law of Detachment is the most commonly used principle of deductive reasoning. In words, this law says that whenever a conditional statement and its hypothesis are true, the conclusion is also true. That is, the conclusion can be “detached” from the conditional (see Example 2.2.1).

  • If the units digit of a number is zero, then the number is a multiple of \(10\text{.}\)
  • The units digit in the number \(40\) is zero.
  • Therefore, \(40\) is a multiple of \(10\text{.}\)

Special types of diagrams, called Euler (pronounced “oiler”) diagrams, can also be used to help determine the validity of arguments. The argument in Example 2.2.1 can be visualized using an Euler diagram as indicated in Figure 2.2.2.

Euler diagram for direct reasoning
Figure 2.2.2. Euler diagram for direct reasoning

Subsection 2.2.2 Law of Syllogism (Transitive Reasoning): \([(p \rightarrow q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)\)

The Law of Syllogism is also called transitive reasoning or the chain rule. Examples of the Law of Syllogism occur repeatedly in mathematics. The following argument is an application of this law.

  • If a number is a multiple of eight, then it is a multiple of four.
  • If a number is a multiple of four, then it is a multiple of two.
  • Therefore, if a number is a multiple of eight, then it is a multiple of two.

An Euler diagram for this argument is given in Figure 2.2.4. Notice that if \(x\) is any number that is a multiple of eight, then \(x\) is also a multiple of four. Then, since \(x\) is a multiple of four, \(x\) must also be a multiple of two.

Euler diagram for transitive reasoning
Figure 2.2.4. Euler diagram for transitive reasoning

Subsection 2.2.3 Law of Contraposition (Indirect Reasoning): \([(p \rightarrow q) \wedge \negate q] \rightarrow \negate p\)

Since the contrapositive of a conditional is logically equivalent to the original conditional, \(\negate q \rightarrow \negate p\) is logically equivalent to \(p \rightarrow q\text{.}\) Then, by applying the Law of Detachment to the contrapositive of \(p \rightarrow q\text{,}\) we may deduce \(\negate p\text{.}\)

  • If a number is a power of \(3\text{,}\) then its units digit is \(1\text{,}\) \(3\text{,}\) \(7\text{,}\) or \(9\text{.}\)
  • The units digit ins \(3{,}124\) is not \(1\text{,}\) \(3\text{,}\) \(7\text{,}\) or \(9\text{.}\)
  • Therefore, \(3{,}124\) is not a power of \(3\text{.}\)

In words, the Law of Contraposition says that whenever a conditional is true and its conclusion is false, then the hypothesis is also false. (In other words, if a conditional is true and the negation of its conclusion is also true, then the negation of its hypothesis is true.) Again, an Euler diagram may be used to help determine the validity of the argument (Figure 2.2.6).

Euler diagram for indirect reasoning
Figure 2.2.6. Euler diagram indirect reasoning

Exercises 2.2.4 Exercises

1.

Determine the validity of the following arguments. Justify your thinking.

    • All equilateral triangles are equiangular.
    • All equiangular triangles are isosceles.
    • Therefore, all equilateral triangles are isosceles.
    • All equilateral triangles are equiangular.
    • All equiangular triangles are isosceles.
    • Therefore all isosceles triangles are equilateral.
    • If you study every day, then you will be successful.
    • You do not study every day.
    • Therefore, you will not be successful.
    • If you study every day, then you will be successful.
    • You are not successful.
    • Therefore, you did not study every day.
    • Some females are doctors.
    • All doctors are college graduates.
    • Therefore, all females are college graduates.
    • If the alarm goes off, then I will call the police.
    • I called the police.
    • So the alarm went off.
    • If the alarm goes off, then I will call the police.
    • If I call the police, then I will file a report.
    • The alarm went off, so I will file a report.
    • If today is Friday, then tomorrow is Saturday.
    • Tomorrow is Monday, so today is not Friday.
    • All teachers are smart.
    • Some teachers are funny.
    • Therefore, some smart people are funny.
    • If a student is a freshman, then the student takes English.
    • Mark is a junior.
    • Therefore, Mark does not take English.
2.

Determine a valid conclusion that follows from each of the following statements and explain your reasoning.

  1. If you come to class every day, then you will be successful. You come to class every day.
  2. If Jana does not fall, then she will win the race. Jana does not win the race.
  3. Every square is a rectangle. Some parallelograms are rhombuses. Every rectangle is a parallelogram.
3.

Suppose that \(r \rightarrow s\) is false. Determine the truth values (true, false, or cannot be determined) of each of the following statements.

  1. \(\displaystyle s \rightarrow r\)
  2. \(\displaystyle s \vee r\)
  3. \(\displaystyle s \wedge r\)
  4. \(\displaystyle \negate r\)
  5. \(\displaystyle \negate s \rightarrow r\)
  6. \(\displaystyle \negate s \wedge r\)
4.

What conclusions can be deduced from these sets of hypotheses? (Let \(f\) stand for a statement that is false.)

  1. Hypotheses: \(p\) or \(q\)
    \(\negate p\)
    Conclusion ?
  2. Hypotheses: \(\negate p \rightarrow f\)
    Conclusion ?
  3. Hypotheses: \((p \wedge q) \rightarrow r\)
    \(p\)
    Conclusion ? (Give a conditional statement.)
5.

There are three forms of invalid reasoning which commonly occur.

  • Fallacy of the converse.
    If \(p\text{,}\) then \(q\text{.}\)
    \(q\)
    \(p\) (invalid)
  • Fallacy of the inverse.
    If \(p\text{,}\) then \(q\text{.}\)
    \(\negate p\)
    \(\negate q\) (invalid)
  • False transitivity.
    If \(p\text{,}\) then \(q\text{.}\)
    If \(p\text{,}\) then \(r\text{.}\)
    If \(q\text{,}\) then \(r\text{.}\) (invalid)

Which fallacies occur in the following arguments?

  1. If I am a good person, nothing bad will happen to me. Nothing happened to me. Therefore, I am a good person.
  2. If you work hard, you will be wealthy and wise. Therefore, if you are wealthy, then you will be wise.