When working with logic in discrete math appliations there are a plethora of rules you can use for working with the well formed formulas. Remembering them all can be a daunting task, which is why I like to have a cheat sheet available. As such, here’s a simple one that I like to use when working with these problems.

## Equivalence Rules

Expression | Equivalent To | Name of the Rule |
---|---|---|

$$ P \lor Q $$ | $$ Q \lor P $$ | Commutative - comm. |

$$ P \land Q $$ | $$ Q \land P $$ | Commutative - comm. |

$$\left(P\lor Q\right)\lor R$$ | $$P \lor\left(Q\lor R\right)$$ | Associative - ass. |

$$\left(P\land Q\right)\land R$$ | $$P\land\left(Q\land R\right)$$ | Associative - ass. |

$$\left(P\lor Q\right)'$$ | $$P'\land Q'$$ | DeMorgan’s Law |

$$\left(P\land Q\right)'$$ | $$P'\lor Q'$$ | DeMorgan’s Law |

$$P\rightarrow Q$$ | $$P'\lor Q$$ | Implication - imp. |

$$P$$ | $$\left(P'\right)'$$ | Double Negation - dn. |

$$P\iff Q$$ | $$\left(P\rightarrow Q\right)\land\left(Q\rightarrow P\right)$$ | Definition of Equivalence - equ. |

## Inference Rules

Expression | Can Derive | Name of the Rule |
---|---|---|

$$P,\ P\rightarrow Q$$ | $$Q$$ | Modus Ponens - mp. |

$$P\rightarrow Q,\ Q'$$ | $$P'$$ | Modus Tollens - mt. |

$$P,\ Q$$ | $$P\land Q$$ | Conjunction - con. |

$$P\land Q$$ | $$P,\ Q$$ | Simplification - simp. |

$$P$$ | $$P\lor Q$$ | Addition - add. |

$$P\rightarrow Q,\ Q\rightarrow R$$ | $$P\rightarrow R$$ | Hypothetical Syllogism - hs |

$$P\lor Q,\ P'$$ | $$Q$$ | Disjunctive Syllogism - ds. |

$$P\rightarrow Q$$ | $$Q'\rightarrow P'$$ | Contraposition - cont. |

$$Q'\rightarrow P'$$ | $$P\rightarrow Q$$ | Contraposition - cont. |

$$P$$ | $$P\land P$$ | Self Reference - self. |

$$P\lor P$$ | $$P$$ | Self Reference - self. |

$$\left(P\land Q\right)\rightarrow R$$ | $$P\rightarrow\left(Q\rightarrow R\right)$$ | Exportation - exp. |

$$P,\ P'$$ | $$Q$$ | Inconsistency - inc. |

## Derivation Rules

The general workflow for using derivation rules is:

- Strip off the quantifiers
- Work with the independent well formed formulas
- Insert the quantifiers back in

**Universal Instantiation – ui.:**

From $$\left(\forall x\right)P\left(x\right)$$ we can deduce $$P\left(t\right)$$

*Note: t must not already appear as a variable in the expression for*

$$P(x)$$

**Existential Instantiation – ei.:**

From

$$\left(\exists x\right)P\left(x\right)$$ we can deduce $$P\left(t\right)$$

*Note: t must be introduced for the first time. As such, you will want to do these early in proofs.*

**Universal Generalization – ug.:**

From

$$P\left(x\right)$$ we can deduce $$\left(\forall x\right)P\left(x\right)$$

*Notes:*

- P(x) hasn’t been deduced by existential instantiation from any hypothesis in which x was a free variable.
- P(x) hasn’t been deduced by existential instantiation from another wff in which x was a free variable.

**Existential Generalization – eg.:**

From

$$P\left(a\right)$$ we can deduce $$\left(\exists x\right)P\left(x\right)$$

**Note: x must not appear in**

$$ P(a) $$