2.7 Logical Arguments

Learning Objectives

After completing this section, you should be able to:

1. Apply the law of detachment to determine the conclusion of a pair of statements.
2. Apply the law of denying the consequent to determine the conclusion for pairs of statements.
3. Apply the chain rule to determine valid conclusions for pairs of true statements.

Law of Detachment

The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, $p\to q$, and the hypothesis, $p$, are true, then the conclusion $q$ must also be true. The law of detachment is also called affirming the hypothesis (or antecedent) and modus ponens. Symbolically, it has the form $\left(\left(p\to q\right)\wedge p\right)\to q$.

Looking at the truth table for the conditional statement, the only time the conditional is true is when the hypothesis $p$ is also true. The only place this happens is in the first row, where $q$ is also true, confirming that the law of detachment is a valid argument.

Another way to verify that the law of detachment is a valid argument is to construct a truth table for the argument $\left(\left(p\to q\right)\wedge p\right)\to q$ and verify that it is a tautology.

Venn diagrams may also be used to verify deductive arguments, which include conditional premises. Consider the statement $p\to q:$ “If you play guitar, then you are a musician.” The set of guitarists is a subset of the set of musicians, $p\subset q.$ To verify that an argument is valid using a Venn diagram, draw the Venn diagram representing all the premises in the argument only, as shown in Figure 2.16. Then verify if the conclusion is also represented by the Venn diagram of the premises. If it is, the argument is valid. If it is not, the argument is not valid. The set of guitarists is drawn as a subset of the set of musicians to represent the premise, $p\to q.$ The $\times$ represents the premise: $p$ is true. This completes the drawing of the premises.

Now, examine the Venn diagram to verify if the conclusion is included in the picture. The conclusion is $q$. Because the $\times$ is in the set $p$, and $p$ is a subset of $q$, $\times$ is also in $q$; therefore, the law of detachment is a valid argument.

Law of Denying the Consequent

Another form of a valid conditional argument is called the law of denying the consequent, or modus tollens. Recall, that the conditional statement, $p\to q$, is logically equivalent to the contrapositive, $~q\to ~p.$ So, if the conditional statement is true, then the contrapositive statement is also true. By the law of detachment, if $~q$ is also true, then it follows that $~p$ must also be true. Symbolically, it has the form $\left(\left(p\to q\right)\wedge ~q\right)\to ~p$.

To verify if the law of denying the consequent is a valid argument, construct a truth table for the argument, $\left(\left(p\to q\right)\wedge ~q\right)\to ~p$, and verify that it is a tautology.

To verify an argument of this form using a Venn diagram, again consider the premise: $p\to q:$ “If you play guitar, then you are a musician.” We will change the second premise to $~q.$ In this case, the $\times$ represents the premise, $~q.$ So, it will be placed inside the universal set of all people, but outside the set of musicians, as depicted in the Venn diagram in Figure 2.17.

Because the $\times$ is also outside the set of guitarists, the statement $~p$ follows from the premises and the argument is valid.

Chain Rule for Conditional Arguments

The chain rule for conditional arguments is another form of a valid conditional argument. It is also called hypothetical syllogism or the transitivity of implication. Recall that the conditional statement $p\to q$ can also be read as $p$ implies $q$. This is where the name transitivity of implication comes from. The transitive property for numbers states that, if $3<4$ and $4<5,$ then it follows that $3<5.$ The chain rule extends this property to conditional statements. If the premises of the argument consist of two conditional statements, with the form “$p\to q$” and “$q\to r,$” then it follows that $p\to r.$ Symbolically, it has the form $$\left(\left(p\to q\right)\wedge \left(q\to r\right)\right)\to \left(p\to r\right)$$.

To verify the chain rule for conditional arguments, construct a truth table for the argument, $$\left(\left(p\to q\right)\wedge \left(q\to r\right)\right)\to \left(p\to r\right)$$, and verify that it is a tautology.

To verify an argument of this form using a Venn diagram, again consider the premise $p\to q:$ “If you play guitar, then you are a musician,” but change the second premise to $q\to r:$ “If you are a musician, then you are an artist.” In this case, the set $p$ of guitarists is a subset of the set $r$ of artists, and it follows that if you are a guitarist, then you are an artist. Therefore, the conclusion $p\to r$ follows from the premises and the chain rule for logical arguments is valid. See Figure 2.18.

Projects

Key Terms

• sound
• fallacy
• deductive arguments
• law of detachment
• law of denying the consequent
• chain rule for conditional arguments

Key Concepts

• A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true.
• The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, $p\to q$ is true and the hypothesis, $p$ is true, then the conclusion $q$ must also be true.

  Law of Detachment
  Premise:	$p\to q$
  Premise:	$p$
  Conclusion:	$\therefore \mathrm{}q$

• Know how to apply the law of detachment to determine the conclusion of a pair of statements.
• The law of denying the consequent is a valid form of a conditional argument that asserts that if both the conditional, $p\to q$ is true and the negation of the conclusion, $~q$ is true, then the negation of the hypothesis $~p$ must also be true.

  Law of Denying the Consequent
  Premise:	$p\to q$
  Premise:	$~q$
  Conclusion:	$\therefore \mathrm{}~p$

• Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements.
• The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, $p\to q$ and $q\to r$, then it follows that $p\to r.$

  Chain Rule for Conditional Arguments
  Premise:	$p\to q$
  Premise:	$q\to r$
  Conclusion:	$\therefore \mathrm{}p\to r$

• Know how to apply the chain rule to determine valid conclusions for pairs of true statements.

Video

• Logic Part 14: Common Argument Forms like Modus Ponens and Tollens