In logic and reasoning, **truth tables** play an important role in analysing and evaluating the truth values of propositions.

Truth tables take compound statements (e.g. “A and B”) and arguments (e.g. “If A then B, A, therefore B”) and break them down into their constituent parts (e.g. “A”, “B”, “A and B”, “if A then B”). This provides a systematic way to explore the different combinations of truth and falsehood for a given set of propositions, which can be used to define logical connectives and determine whether arguments are *valid* or *invalid*.

This blog post will provide a simple overview of truth tables. First, we’ll use truth tables to define the basic logical connectives. Then, we will look at how we can use truth tables to determine the validity of arguments.

## Truth Tables for Basic Logical Connectives

**Logical connectives** combine or relate propositions, enabling us to form more complex statements by connecting simpler statements together. These connectives serve as building blocks for constructing logical expressions and analysing their truth values.

Below are truth tables that define the basic logical connectives:

- Conjunction (AND)
- Disjunction (OR)
- Negation (NOT)
- Implication (IF… THEN)
- Biconditional (IF AND ONLY IF)

##### Conjunction (AND)

Let’s begin with an example of a **conjunction** (AND) between two propositions, P and Q.

Here’s the truth table representing all possible truth value combinations for P **and** Q:

P | Q | P AND Q |
---|---|---|

true |
true |
true |

true |
false |
false |

false |
true |
false |

false |
false |
false |

The truth table for conjunction shows us that the result is true **only when both propositions being combined are true**. If either proposition is false, the conjunction is false. This defines the behaviour of the “and” connective.

##### Disjunction (OR)

Next, let’s explore the truth table for a disjunction (OR) between two propositions, P and Q.

Here’s the truth table representing all possible truth value combinations for P **or** Q:

P | Q | P OR Q |
---|---|---|

true |
true |
true |

true |
false |
true |

false |
true |
true |

false |
false |
false |

The truth table for disjunction tells us that the result is true **as long as at least one of the propositions being combined is true.** It is false only when both propositions are false. This defines the behaviour of the “or” connective.

*Note: More specifically, this is an “inclusive OR”, which means it is true when either or both statements is true. It can be contrasted with an “exclusive OR”, which is true when one and only one (not both) statements is true.*

##### Negation (NOT)

Now, let’s examine the truth table for negation (NOT) of a proposition, P.

Here’s the truth table representing all possible truth value combinations for **not P**:

P | NOT P |
---|---|

true |
false |

false |
true |

The truth table for negation shows us that it **reverses the truth value of a proposition**. If the original proposition is true, the negation is false, and vice versa. This defines the behaviour of the “not” connective.

##### Implication (IF… THEN)

Moving on, let’s explore the truth table for an implication (IF…THEN) between two propositions, P and Q.

Here’s the truth table representing all possible truth value combinations for** if **P** then **Q:

P | Q | P → Q |
---|---|---|

true |
true |
true |

true |
false |
false |

false |
true |
true |

false |
false |
true |

The truth table for implication reveals that it is true unless the antecedent (the first proposition) is true and the consequent (the second proposition) is false. It captures the idea that if the antecedent is true, the consequent must also be true.

##### Biconditional (IF AND ONLY IF)

Lastly, let’s examine the truth table for a biconditional (IF AND ONLY IF) between two propositions, P and Q.

Here’s the truth table representing all possible truth value combinations for Q** if and only if **P:

P | Q | P ↔ Q |
---|---|---|

true |
true |
true |

true |
false |
false |

false |
true |
false |

false |
false |
true |

The truth table for biconditional demonstrates that it is true only when both propositions have the same truth value. If they differ in truth value, the biconditional is false. This helps us understand how “if and only if” works.

## Evaluating Validity of Arguments Using Truth Tables

Having defined our connectives, we can now use them to form **arguments**. And truth tables can also be used to determine the **validity** of these arguments.

When evaluating arguments using truth tables, we examine the truth values of the premises and the conclusion to see if there are any scenarios in which the premises can be true and the conclusion false. **If we find such a scenario, it means the argument is invalid** because there exists a counterexample where the premises do not logically guarantee the truth of the conclusion. **If there are no such scenarios, then the argument is valid.**

##### Example 1: Modus Ponens

Modus Ponens is a *valid* argument format that takes the form:

If A, then B. A. Therefore, B.

In the truth table below, we evaluate the truth values of the premises (“A” and “If A, then B”) and the conclusion (“B”) for each possible combination of truth values for A and B:

A | B | If A, then B | A | Conclusion (B) |
---|---|---|---|---|

true |
true |
true | true | true |

true |
false |
false | true | false |

false |
true |
true | false | true |

false |
false |
true | false | false |

What we observe is that whenever:

- Proposition “A” is true, and
- “If A, then B” is true
- The conclusion (proposition “B”) is
*also true*.

This proves that **Modus Ponens is a valid argument format.**

*Note: The other rows of the truth table, where A is false, are not relevant for evaluating the validity of Modus Ponens. This is because Modus Ponens only requires the premises to be true, and it does not make any claims about what happens when the premises are false.*

##### Example 2: Affirming the Consequent

Affirming the Consequent is an *invalid* argument format that takes the form:

If A, then B. B. Therefore, A.

Like before, we can put all the possible combinations of truth values for A and B into a truth table like the one below:

A | B | If A, then B | B | Conclusion (A) |
---|---|---|---|---|

true |
true |
true | true | true |

true |
false |
false | false | true |

false |
true |
true |
true |
false |

false |
false |
true | false | false |

In this case, what we observe is that there is an instance** (row 3)** where:

- The premises are true (such as in the row of the truth table where both “If A, then B” and “B” are true)
- But the conclusion A is
*false*.

This proves that **affirming the consequent is an invalid argument format.**

*Note: Material conditionals (i.e. “if A, then B”) are only considered false when the antecedent (i.e. “A”) is true and the consequent (i.e. “B”) is false. In other words, when the antecedent of a conditional statement is false, then the conditional statement as a whole is considered true. *

##### Example 3: Modus Tollens

Modus Tollens is a *valid* argument format that takes the form:

If A, then B. Not B. Therefore, not A.

Below is the truth table for Modus Tollens:

A | B | If A, then B | Not B | Not A |
---|---|---|---|---|

true |
true |
true | false | false |

true |
false |
false | true | false |

false |
true |
true | false | true |

false |
false |
true | true | true |

Like with Modus Ponens above, there is no combination where the premises of the argument are true and the conclusion false. So, this proves that **Modus Tollens is also valid argument format.**

##### Example 4: Denying the Antecedent

Denying the Antecedent is an *invalid* argument format that takes the form:

If A, then B. Not A. Therefore, not B.

And here’s the truth table:

A | B | If A, then B | Not A | Not B |
---|---|---|---|---|

true |
true |
true | false | false |

true |
false |
false | false | true |

false |
true |
true | true | false |

false |
false |
true | true | true |

Like Affirming the Consequent above, the truth table here shows possible combinations where the premises are true and the conclusion false (row 3), which demonstrates that **Denying the Antecedent is an invalid argument format.**

##### Example 5: Disjunctive Syllogism

Disjunctive Syllogism is a *valid* argument format that takes the form:

A or B. Not A. Therefore, B.

And here’s the truth table that proves it:

A | B | A or B | Not A | B |
---|---|---|---|---|

true |
true |
true | false | true |

true |
false |
true | false | false |

false |
true |
true | true | true |

false |
false |
false | true | false |

What we see is that in all cases where the premises are true, the conclusion is always true. This demonstrates that **Disjunctive Syllogism is a valid argument format**.

##### Example 6: Contraposition

Contraposition is a *valid* argument format that takes the form:

If A, then B. Therefore, if not B, then not A.

Here’s the truth table:

A | B | If A, then B | Not B | Not A | If not B, then not A |
---|---|---|---|---|---|

true |
true |
true | false | false | true |

true |
false |
false | true | false | false |

false |
true |
true | false | true | true |

false |
false |
true | true | true | true |

Again, whenever the premises are true, the conclusion is also true. This shows that **Contraposition is a valid argument format**.

##### Example 7: Hypothetical Syllogism

Hypothetical Syllogism is a *valid* argument format that takes the form:

If A, then B. If B, then C. Therefore, if A, then C.

The truth tables we’ve looked at so far have involved at most two statements: A and B. But as Hypothetical Syllogism involves a third statement – C – our truth table needs 8 rows instead of 4:

A | B | C | If A, then B | If B, then C | If A, then C |
---|---|---|---|---|---|

true |
true |
true |
true | true | true |

true |
true |
false |
true | false | false |

true |
false |
true |
false | true | true |

true |
false |
false |
false | true | false |

false |
true |
true |
true | true | true |

false |
true |
false |
true | false | true |

false |
false |
true |
true | true | true |

false |
false |
false |
true | true | true |

But even with all these rows, there are no examples where the premises are true and the conclusion is false. So, this truth table demonstrates that **Hypothetical Syllogism is a valid argument format**.

## Summary

Truth tables are a powerful tool for evaluating logical arguments. By systematically examining all possible combinations of truth values, we can determine the validity of an argument and uncover the inherent logical relationships between propositions.