# The Semantics of Propositional Logic

So far we have discussed propositional logical connectives and formulas. We noted that formulas derive from statements. We would now like to determine whether a formula is true or false based on the truth/falsehood of the statements that comprise the formula. We will use the letter $T$ to denote “true” and the letter $F$ to denote “false”.

The semantics of propositional logic are the rules we give for our propositional connectives. They are defined in the “truth” tables below.

## Negation

$P$ | $\neg P$ |
---|---|

T | F |

F | T |

*If $P$ is true then $\neg P$ is false.*

*If $P$ is false then $\neg P$ is true.*

## Conjunction

$P$ | $Q$ | $P \wedge Q$ |
---|---|---|

T | T | T |

T | F | F |

F | T | F |

F | F | F |

*If $P$ is true and $Q$ is true then $P \wedge Q$ is true.*

*If $P$ is true and $Q$ is false then $P \wedge Q$ is false.*

*If $P$ is false and $Q$ is true then $P \wedge Q$ is false.*

*If $P$ is false and $Q$ is false then $P \wedge Q$ is false.*

## Disjunction

$P$ | $Q$ | $P \vee Q$ |
---|---|---|

T | T | T |

T | F | T |

F | T | T |

F | F | F |

*If $P$ is true and $Q$ is true then $P \vee Q$ is true.*

*If $P$ is true and $Q$ is false then $P \vee Q$ is true.*

*If $P$ is false and $Q$ is true then $P \vee Q$ is true.*

*If $P$ is false and $Q$ is false then $P \vee Q$ is false.*

## Exclusive Disjunction / Exclusive Or

$P$ | $Q$ | $P \veebar Q$ |
---|---|---|

T | T | F |

T | F | T |

F | T | T |

F | F | F |

*If $P$ is true and $Q$ is true then $P \veebar Q$ is false.*

*If $P$ is true and $Q$ is false then $P \veebar Q$ is true.*

*If $P$ is false and $Q$ is true then $P \veebar Q$ is true.*

*If $P$ is false and $Q$ is false then $P \veebar Q$ is false.*

## Implication

$P$ | $Q$ | $P \rightarrow Q$ |
---|---|---|

T | T | T |

T | F | F |

F | T | T |

F | F | T |

*If $P$ is true and $Q$ is true then $P \rightarrow Q$ is true.*

*If $P$ is true and $Q$ is false then $P \rightarrow Q$ is false.*

*If $P$ is false and $Q$ is true then $P \rightarrow Q$ is true.*

*If $P$ is false and $Q$ is false then $P \rightarrow Q$ is true.*

## Biconditional

$P$ | $Q$ | $P \leftrightarrow Q$ |
---|---|---|

T | T | T |

T | F | F |

F | T | F |

F | F | T |

*If $P$ is true and $Q$ is true then $P \leftrightarrow Q$ is true.*

*If $P$ is true and $Q$ is false then $P \leftrightarrow Q$ is false.*

*If $P$ is false and $Q$ is true then $P \leftrightarrow Q$ is false.*

*If $P$ is false and $Q$ is false then $P \leftrightarrow Q$ is true.*

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