Lesson 3.3: Truth Tables

Introduction

Welcome to Lesson 3.3 on Truth Tables! 🎉 In this lesson, we will explore the fascinating world of propositional logic through truth tables. By the end of this lesson, you, students, will be able to construct truth tables, analyze compound formulas, and determine whether statements are logically equivalent. Let's dive in!

Learning Outcomes

By the end of this lesson, you should be able to:

• Construct a truth table for a logical connective and for a compound formula.
• Determine the number of rows based on the number of variables.
• Identify tautologies, contradictions, and contingent statements.
• Test if two statements are logically equivalent.
• Read a complex truth table accurately and systematically.

What is a Truth Table?

A truth table is a mathematical table used in logic—especially in propositional logic—to determine the truth value of a statement based on its logical components. It deconstructs complex logical expressions to reveal their validity under different conditions or combinations of true (T) and false (F).

Here’s a simple truth table for a single propositional variable, $ P $:

| P | Truth Value |

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

| T | True |

| F | False |

Constructing a Truth Table

Let's start with defining logical connectives. The most common are:

• AND (Conjunction, $ \land $)
• OR (Disjunction, $ \lor $)
• NOT (Negation, $ \lnot $)
• IF...THEN (Implication,

ightarrow )

• IF AND ONLY IF (Biconditional, $ \leftrightarrow $)

Example: Constructing a Truth Table for $ P \land Q $

To create a truth table for the compound statement $ P \land Q $, we need to find all combinations of truth values for $ P $ and $ Q $. Since there are two variables, each can be either True (T) or False (F). Hence, we set up 4 rows:

| P | Q | $ P \land Q $ |

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

| T | T | T |

| T | F | F |

| F | T | F |

| F | F | F |

In this example, the only time $ P \land Q $ is true is when both $ P $ and $ Q $ are true.

Determining the Number of Rows

The number of rows in a truth table is determined by the number of different propositional variables involved. The formula to calculate this is $ 2^n $, where $ n $ is the number of variables.

Example: For three variables $ P, Q, R $:

• $ n = 3 $
• Number of rows = $ 2^3 = 8 $

Hence, the truth table for $ P, Q, R $ would have 8 rows!

Tautologies, Contradictions, and Contingent Statements

1. Tautology: A statement that is always true, regardless of the truth values of its components.

• Example: The statement $ P \lor \lnot P $ (either $ P $ is true, or $ P $ is not true) is a tautology. Its truth table:

| P | $ \lnot P $ | $ P \lor \lnot P $ |

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

| T | F | T |

| F | T | T |

1. Contradiction: A statement that is always false.

• Example: The statement $ P \land \lnot P $ is a contradiction. Its truth table:

| P | $ \lnot P $ | $ P \land \lnot P $ |

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

| T | F | F |

| F | T | F |

1. Contingent Statement: A statement that can be either true or false depending on the truth values of the variables.

• Example: The statement $ P \land Q $ can be contingent, having true or false values based on $ P $ and $ Q $.

Testing Logical Equivalence

Two statements are logically equivalent if they yield the same truth value in all possible scenarios. To test equivalence, construct truth tables for both statements and compare the output columns.

Example: Testing $ P \lor Q $ and $ Q \lor P $:

| P | Q | $ P \lor Q $ | $ Q \lor P $ |

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

| T | T | T | T |

| T | F | T | T |

| F | T | T | T |

| F | F | F | F |

Since both columns are identical, $ P \lor Q $ is logically equivalent to $ Q \lor P $.

Conclusion

In this lesson, we covered the construction of truth tables, determining the number of rows based on variables, identifying tautologies, contradictions, and contingent statements, and testing logical equivalence through truth tables. Understanding these concepts is crucial for mastering propositional logic and critical thinking!

Study Notes

• A truth table displays the truth values of a statement based on its components.
• The number of rows is given by $ 2^n $, where $ n $ is the number of variables.
• Tautologies are always true, contradictions are always false, and contingent statements depend on variable values.
• Test logical equivalence by comparing truth values from truth tables of different statements.