# What is propositional calculus?

by Stephen M. Walker II, Co-Founder / CEO

## What is propositional calculus?

Propositional calculus, also known as propositional logic, statement logic, sentential calculus, or sentential logic, is a branch of logic that deals with propositions and the relationships between them.

A proposition is a declarative statement that is either true or false, but not both. Propositions can be simple or compound. Simple propositions, also known as atomic propositions, contain no logical connectives, while compound propositions are formed by connecting propositions using logical connectives.

The logical connectives used in propositional calculus include "AND", "OR", "NOT", and "implies". These connectives allow for the construction of complex logical statements from simpler ones. For example, if P and Q are propositions, then "P AND Q", "P OR Q", "NOT P", and "P implies Q" are all compound propositions.

Propositional calculus also involves the use of truth tables to determine the truth value of compound propositions based on the truth values of their constituent propositions. For instance, the truth value of the compound proposition "P AND Q" is true if and only if both P and Q are true.

In addition to truth tables, propositional calculus involves the use of formal grammar and inference rules to define well-formed formulas and to derive new propositions from existing ones. The set of axioms, or distinguished formulas, and the set of inference rules can vary among different formulations of propositional calculus, but they are all designed to capture the same underlying logic.

Propositional calculus is foundational to many areas including computer science, where it is used in the design of logic gates and in the formal specification and verification of software and hardware systems. It also forms the basis for first-order logic and higher-order logics, which extend propositional logic by introducing quantifiers and predicates.

## What is a proposition?

A **proposition** is a declarative sentence that is either true or false, but not both. It's a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. For instance, the sentence "The sky is blue" denotes the proposition that the sky is blue.

## What is the truth value of a proposition?

The **truth value** of a proposition is the attribute of a proposition as to whether the proposition is true or false. For example, the truth value for "7 is odd" is true, which can be denoted as T. The truth value of "1 + 1 = 3" is false, which can be denoted as F.

## What is the negation of a proposition?

The **negation** of a proposition p, denoted by ¬p, is the proposition that is false when p is true and true when p is false. For example, if p is the statement “I understand this”, then its negation would be “I do not understand this” or “It is not the case that I understand this”.

## What is the conjunction of two propositions?

The **conjunction** of two propositions p and q, denoted by p∧q, is a compound proposition which consists of two propositions joined by the connective “and”. The conjunction p∧q is true only when both p and q are true. For example, if p is "This book is interesting" and q is "I am staying at home", then p∧q would be "This book is interesting, and I am staying at home".

## What is the disjunction of two propositions?

The **disjunction** of two propositions p and q, denoted by pvq, is a compound proposition which consists of two propositions joined by the connective “or”. The disjunction pvq is false only if both propositions are false. In other words, it is true if at least one of the propositions is true.