Lesson 3.4: Testing Arguments with Truth Tables

Introduction

Welcome to today's lesson! In this session, we are going to dive into the concept of truth tables and how they help us test the validity of arguments. By the end of this lesson, you (students) will understand how to represent arguments symbolically, use truth tables to determine validity, and identify where an argument fails if it is invalid.

Learning Objectives:

• Represent an argument as premises and a conclusion in symbols.
• Understand the truth-table test for validity (there should be no row with true premises and a false conclusion).
• Use the short-cut (indirect) truth-table method effectively.
• Learn to diagnose exactly where an invalid argument fails.
• Understand the link between validity and the conditional that represents the argument.

What is a Truth Table?

A truth table is a tool that helps us visualize the truth values of propositions and their combinations. It lists all possible truth values for the given premises and conclusions, allowing us to analyze the validity of an argument. To illustrate, let's consider a simple argument:

Example Argument

1. If it rains, then the ground is wet. (Premise 1)
2. It is raining. (Premise 2)
3. Therefore, the ground is wet. (Conclusion)

We can represent this argument using symbols:

• Let $p$ be "It rains."
• Let $q$ be "The ground is wet."

The argument can be expressed as:

1. $p \to q$ (if $p$, then $q$)
2. $p$ (premise)
3. Therefore, $q$ (conclusion)

Constructing a Truth Table

Now let's create a truth table for our argument. The truth table will consist of the columns for $p$, $q$, $p \to q$, and the conclusion $q$.

| $p$ | $q$ | $p \to q$ | Conclusion ($q$) |

|-----|-----|----------|------------------|

| T | T | T | T |

| T | F | F | F |

| F | T | T | T |

| F | F | T | F |

Analyzing the Truth Table

From the truth table:

• We see that there is one row where both premises are true (the first row), and in that row, the conclusion is also true.
• This tells us that the argument is valid.

The Truth-Table Test for Validity

The truth-table test for validity states that an argument is valid if there is no row in the truth table where all premises are true and the conclusion is false. If we find such a row, the argument is invalid.

Using the Short-Cut (Indirect) Truth-Table Method

Sometimes, constructing a full truth table can be time-consuming, especially for more complex arguments. This is where the short-cut method comes in!

Example

Let’s consider the argument:

1. If $p$, then $q$.
2. $q$ is false.
3. Therefore, $p$ is false.

We can write this as:

1. $p \to q$

eg q

1. Therefore,

eg p

Indirect Truth-Table

To perform an indirect truth table, start by assuming that the argument's conclusion is true. Then check for any contradictions under that assumption.

• Assume $p$ is true.
• If $p$ is true, then $q$ must also be true by premise 1 ($p \to q$).
• But

eg q contradicts this assumption, meaning our initial assumption was incorrect! Thus, if the conclusion is supposed to be true and it leads to a contradiction, then the argument is invalid.

Diagnosing Invalid Arguments

When an argument is invalid, it helps to pinpoint where the failure occurred.

Example:

Let’s examine:

1. Either $p$ or $q$. (Premise 1)
2. $q$ is false. (Premise 2)
3. Therefore, $p$ is true. (Conclusion)

When we analyze by truth table:

| $p$ | $q$ | $p \lor q$ | Conclusion ($p$) |

|-----|-----|----------|------------------|

| T | T | T | T |

| T | F | T | T |

| F | T | T | F |

| F | F | F | F |

From the truth table:

• There is a row where both premises are true ($p$ can be false), leading to a false conclusion.
• Therefore, we could diagnose that the argument fails because it does not account for the possibility of both being false.

Conclusion

In this lesson, students, we explored the concept of truth tables and how to apply them to test the validity of arguments. We learned how to represent arguments symbolically, constructed truth tables, used the indirect truth-table method, and diagnosed where arguments fail. Understanding these concepts is crucial, especially in fields like mathematics, philosophy, and law, where logical reasoning is essential.

Study Notes

• A truth table shows all possible truth values for premises and conclusions.
• An argument is valid if no row has true premises and a false conclusion.
• The short-cut method can help evaluate arguments without full tables.
• Understanding the point of failure in invalid arguments is essential for proper reasoning.
• Familiarize yourself with common logical connectives: and ($\land$), or ($\lor$), not (

eg$), implies ($$\to$).