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Truth-tables are powerful tools with a variety of applications in logical analysis. In the first part of this course you will learn to construct truth-tables for logical formulae. Next, you will learn to use these truth-tables to identify the logical properties of statements. Here’s an example:

**“It is currently raining in Paris or it is not currently raining in Paris.”**

Intuitively, we know that this sentence must be true, and we can prove this using a truth-table. The first step is to analyse the sentence. Here’s what the entire analyzed sentence looks like with the truth-values filled out under the simple sentences:

**P = “It is raining in Paris”**

(P |
∨ |
~ |
P) |

T | T | ||

F | F |

And, here’s how it looks when we fill out the entire truth-table:

(P |
∨ |
~ |
P) |

T | T |
F | T |

F | T |
T | F |

Now, focus on the truth-values under the operator with the broadest scope, “∨”. The truth-value under the broadest operator on a given row is the truth-value of the entire formula on that row. And as we can see, the truth-value under the main operator “∨” is T on both rows. This means that (P∨~P) is always true, no matter what the truth-value P happens to be. Thus, (P∨~P) is a necessary truth.

Let’s look at another example:

**“It is currently raining in Paris AND it is not currently raining in Paris.”**

Intuitively, we know that this sentence must be false, and we can prove that this using a truth-table. Again, the first step is to analyze the sentence. Here’s what the entire formula looks like with the truth-values filled out under the simple sentences:

**P = “It is raining in Paris”**

(P |
& |
~ |
P) |

T | T | ||

F | F |

And, here’s how it looks when we fill out the entire truth-table:

(P |
& |
~ |
P) |

T | F | F | T |

F | F | T | F |

Now, focus again on the truth-values under the operator with the broadest scope, which this time around is “&”. The truth-value under the broadest operator on a given row is the truth-value of the entire formula on that row, and as we can see, the truth-value under the main operator “∨” is F on both rows. This means that (P&~P) is a necessary falsehood. It is false no matter what the truth-value of P.

Once we have mastered the use of truth-tables for identifying logical properties, we will learn to use truth-tables to test for relations of consistency, contradiction, equivalence and entailment within pairs of formulae. In **Level 4: Using Truth-Tables To Test For Validity**, we will once again return to the topic of truth-tables, this time using them to test argument forms or *sequents* for validity.

Here is a complete list of lessons for this course:

**Lesson 1: Mastering the Logical Operators AVAILABLE AS A FREE SAMPLE LESSON!**

**Lesson 2: Truth-Tables for Logical Formulae**

**Lesson 3: Logical Properties**

**Lesson 4: Equivalence**

**Lesson 5: Contradiction & Consistency**

**Lesson 6: Entailment**

**Lesson 7: All Four Relations**