# Vacuous Truths

Something that has been bothering me greatly about Zeroth Order Logic was that the truth table of the material implication results in `true`

regardless of the **consequent** whenever the **antecedent** is `false`

. That is, **p => q** is always true when **p** is false, regardless of the value of ** q**:

p | q | p => q |
---|---|---|

F | F | T |

F | T | T |

T | F | F |

T | T | T |

This is bothersome because it means that you can construct implications that are true whenever you know something is absolutely false. For example:

- If
**the earth is flat**then**dinosaurs are alive** - If
**California is in Brazil**then**I’m the smartest person in the world** - If
**red is blue**then**mathematics is wrong**

These are all **true** statements, because the antecedents are known to be **false** making the entire implication **true**.

It also bothered me that if you know something **p** to be **false** and you asked whether the implication **p => q** was true, you’d always get **true** regardless of the **q** you used.

So, I was really happy when a friend taught me the concept of vacuous truths because it describes exactly what I was struggling with, and more.

Specifically, vacuous truths also appear in **universal quantifications** of empty sets.

For example: **all my grandkids are tall** is true if I don’t have any **grandkids**, as would the otherwise apparently conflicting conjunction **all my grankids are tall and short**.

Now that I know there is a name to what I was feeling, I can dig into the ways people have addressed this! What a relief!

More to follow!