PBNF-transform as a formulation of Propositional Calculus, II

TL;DR

This paper introduces a novel algebraic polynomial-based formulation of propositional calculus called PBNF, transforming propositional statements into polynomial families in a dual space to simplify logical reasoning.

Contribution

It presents the PBNF framework, utilizing algebraic polynomials and a substitution rule to represent propositional logic, enabling new theorem discovery and generalization.

Findings

PBNF simplifies propositional calculus using polynomial families.

The method allows for the derivation of new theorems.

It generalizes classical propositional logic results.

Abstract

Here we show, in the second paper in a series of articles, methods to calculate propositional statements with algebraic polyno mials as symbols for the connectives, which here are named operators. In the first article, we explained this formulation of the Propositional Calculus. In short, we transform to a dual space, which we here refer to as a polynomial family, which is another shape of DBNF. We name the polynomial families as PBNF, which stands for Polynomial Boolean Normal Form. We just use the one law of inference, the rule of Substi tution. We can use different polynomial families in the House of PBNF, depending on the statement form, making it even simpler. It is also pos sible to find new theorems and generalize older ones, for example, those given by Church and Barkley Rosser (see follow-up article) concerning duality.