## Introduction

Propositional calculus, or propositional logic, is a fundamental branch
of logic that focuses on the manipulation and combination of
propositions statements that can be definitively declared as true or
false. It lays the groundwork for understanding more complex logical
systems and finds applications across various disciplines.

## Propositions

Propositions are declarative sentences that assert a fact about the
world, which can either be true (T) or false (F), such as "It is
raining".

## Truth Tables

Truth tables are systematic methods for determining the truth value of
logical expressions based on the truth values of their constituent
propositions, offering a clear visual representation of logical
operations. They can look like the following:

p |
q |
p → q |

F |
F |
T |

F |
T |
T |

T |
F |
F |

T |
T |
T |

## Logical Operators

Logical operators are symbols used to connect propositions or alter
their truth values, forming the basis for constructing complex logical
expressions. The primary operators include:

### NOT (¬)

Negates the truth value of a proposition.

### AND (∧)

True if both propositions it combines are true.

p |
q |
p ∧ q |

F |
F |
F |

F |
T |
F |

T |
F |
F |

T |
T |
T |

### OR (∨)

True if at least one of the combined propositions is true.

p |
q |
p ∨ q |

F |
F |
F |

F |
T |
T |

T |
F |
T |

T |
T |
T |

### IMPLIES (→)

True except when the first proposition is true and the second is false.

p |
q |
p → q |

F |
F |
T |

F |
T |
T |

T |
F |
F |

T |
T |
T |

### BICONDITIONAL (↔)

True if both propositions are equally true or false.

p |
q |
p ↔ q |

F |
F |
T |

F |
T |
F |

T |
F |
F |

T |
T |
T |

## Expressions

Expressions are more complex statements formed by joining propositions
with logical operators, allowing for the representation of nuanced
logical relationships.

## Logical Equivalences

Logical equivalences are expressions that hold the same truth value
under all possible conditions. They include fundamental laws like the
Law of Identity, the Law of Noncontradiction, and De Morgan’s laws.

## Proofs

Proofs in propositional calculus involve demonstrating the truth of a
proposition based on axioms (assumed truths), previously established
truths, and rules of inference. They are crucial for validating logical
arguments and theorems.

## Applications

Propositional calculus is not just a theoretical framework but also has
practical applications in computer science for software verification, in
mathematics for formalizing proofs, and in philosophy for analyzing
arguments. Its principles underpin the study of more advanced logical
systems, such as predicate logic, and play a vital role in the
development of logical reasoning and critical thinking skills.