## Maths in a minute: Truth tables

Submitted by Marianne on July 4, 2018In standard mathematical logic every statement — "the cat is white", "the dog is black", "I am hungry" — is considered to be either true or false. Given two statements *P* and *Q*, you can make more complicated statements using logical connectives such as AND and OR.

For example, the statement *P* AND *Q* (eg "the cat is white and the dog is black") is only considered true if both *P* and *Q* are true, otherwise it is false. This can be summarised in a truth table:

P | Q | P AND Q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

The table lists every combination of truth values for *P* and *Q* and then tells you what the corresponding truth value for *P* AND *Q* is.

Similarly, the OR connective is defined by the following table:

P | Q | P OR Q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

There is also a truth table that defines NOT *P*, the negation of a statement *P* (if *P* is "the cat is white" then NOT *P* is "the cat is not white"). Unsurprisingly, NOT *P* is true when *P* is false and vice versa:

P | NOT P |

T | F |

F | T |

Using the OR and the NOT operators, we can derive the *law of the excluded middle*, which says that *P* OR NOT *P* is always true:

P | NOT P | P OR NOT P |

T | F | T |

F | T | T |

Using truth tables you can figure out how the truth values of more complex statements, such as

*P* AND (*Q* OR NOT R)

depend on the truth values of its components. We have filled in part of the truth table for our example below, and leave it up to you to fill in the rest.

P | Q | R | NOT R | Q OR NOT R | P AND (Q OR NOT R) |

T | T | T | F | T | |

T | T | F | T | T | |

T | F | T | F | F | |

F | T | T | F | T | |

T | F | F | T | T | |

F | T | F | T | T | |

F | F | T | F | F | |

F | F | F | T | T |

If you have enjoyed doing this, you could also define your own logical connectives using truth tables. Or you could read a text book on Boolean logic or propositional logic.

### About this article

Marianne Freiberger is Editor of *Plus*.

This article is part of our *Who's watching? The physics of observers* project, run in collaboration with FQXi. Click here to see more articles about constructivism.