Algebraic Properties of Propositional Calculus

TL;DR

This paper explores the algebraic structure of propositional calculus by representing formulas as elements of Boolean algebra, introducing 'logical primes' to simplify derivations of logical properties.

Contribution

It introduces the concept of 'logical primes' within Boolean algebra to algebraically analyze propositional calculus properties, offering a new algebraic perspective.

Findings

Logical formulas can be represented as elements of Boolean algebra.

Algebraic manipulations can derive propositional calculus properties.

'Logical primes' provide a unique algebraic representation of formulas.

Abstract

In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to substitute lengthy logical considerations by schematic algebraic manipulations.