## Propositional Logic

## Syntactically Well-Formed Formulas

**Propositions**: The letters 'A', 'B', 'C', ... 'Z' are syntactically well-formed formulas.
They stand for propositions.

**Negation**: If 'A' is a syntactically well-formed formula, then 'not (A)' is a syntactically well-formed formula too.
The bracket is required.

**Conjunction**: If 'A' and 'B' are syntactically well-formed formulas, then '(A and B)' is a syntactically well-formed formula too.

**Disjunction (inclusive or)**: If 'A' and 'B' are syntactically well-formed formulas, then '(A or B)' is a syntactically well-formed formula too.

**Exclusive or**: If 'A' and 'B' are syntactically well-formed formulas, then '(A xor B)' is a syntactically well-formed formula too.

**Implication**: If 'A' and 'B' are syntactically well-formed formulas, then '(A -> B)' is a syntactically well-formed formula too.

**Equivalence**: If 'A' and 'B' are syntactically well-formed formulas, then '(A <-> B)' is a syntactically well-formed formula too.

## Examples of Syntactically Well-Formed Formulas

'((A and B) or not (C))' is a syntactically well-formed formula.

'((A or B) and not ((C and D)))' is a syntactically well-formed formula.
Two brackets are required around 'C and D'.
One bracket is required for the conjunction of C and D.
Another bracket is required for the negation of the conjunction.

## Truth Valuations of Syntactically Well-Formed Formulas

### The Truth Table for the Negation

A | not (A) |
---|---|

true | false |

false | true |

### The Truth Table for the Conjunction

A | B | (A and B) |
---|---|---|

true | true | true |

true | false | false |

false | true | false |

false | false | false |

### The Truth Table for the Disjunction

A | B | (A or B) |
---|---|---|

true | true | true |

true | false | true |

false | true | true |

false | false | false |

### The Truth Table for the Exclusive 'Or'

A | B | (A xor B) |
---|---|---|

true | true | false |

true | false | true |

false | true | true |

false | false | false |

### The Truth Table for the Implication

A | B | (A -> B) |
---|---|---|

true | true | true |

true | false | false |

false | true | true |

false | false | true |

### The Truth Table for the Equivalence

A | B | (A <-> B) |
---|---|---|

true | true | true |

true | false | false |

false | true | false |

false | false | true |

## Rules to Draw Logically Inferences Pertaining to Negation, Conjunction, and Disjunction

**Unpacking a Double Negation**: If 'not ( not (A) )' is a true formula, then 'A' is a true formula too.

**Unpacking a Conjunction**: If '( A and B )' is a true formula, then both 'A' and 'B' are true formulas too.

**Unpacking the Negation of a Disjunction**: If 'not ( A or B )' is a true formula, then 'not ( A )' and 'not ( B )' are true formulas too.

**Unpacking a Disjunction**: If '( A or B )' is a true formula, then at least one of the formulas 'A' and 'B' is a true formula too.

**Unpacking the Negation of a Conjunction**: If 'not ( A and B )' is a true formula, then at least one of the formulas 'not ( A )' and 'not ( B )' is a true formula too.

## Rules to Draw Logically Inferences Pertaining to Negation, Implication, and Equivalence

**Unpacking an Implication**: If '( A -> B )' is a true formula, then at least one of the formulas 'not ( A )' and 'B' is a true formula too.

**Unpacking the Negation of an Implication**: If 'not ( A -> B )' is a true formula, then 'A' and 'not ( B )' are true formulas too.

**Unpacking an Equivalence**: If '( A <-> B )' is a true formula, then either 'A' and 'B' are true formulas or 'not ( A )' and 'not ( B )' are true formulas.

**Unpacking the Negation of an Equivalence**: If 'not ( A <-> B )' is a true formula, then either 'A' and 'not ( B )' are true formulas or 'not ( A )' and 'B' are true formulas.