PBNF-transform as a formulation of Propositional Calculus, I

TL;DR

This paper introduces the PBNF-transform, a novel algebraic polynomial-based approach to propositional calculus that simplifies logical reasoning by eliminating the need for axioms or truth tables.

Contribution

It presents a new polynomial formulation of propositional calculus that is geometrically motivated and bijectively maps logical statements to polynomial families, simplifying logical inference.

Findings

Polynomials form a geometrization of logical connectives

Statements can be mapped bijectively into polynomial families

The approach simplifies propositional calculus methods

Abstract

Here, in a series of articles, we show methods for calculating propositional statements using algebraic polynomials as symbols for the connectives, which are named operators. These polynomials originate from the transformation between the principles of duality and the Disjunctive Boolean Normal Form, DBNF, and they appear if we use a geometrization in the unit square and simple algebraic methods, modulo 2. This we call the PBNF-transform. PBNF stands for Polynomial Boolean Normal Form as these families are based on DBNF involved here. In the first paper in this series, we show that statements can be mapped bijectively into different polynomial families g(p,q) belonging to H(g)$, which we call the The House of PBNF. We can also replace the connectives of logic with PBNF, as the polynomials are, in fact, a geometrization of these connectives; the systems are isomorphic. The benefit of this formulation of the Propositional Calculus(PC) is a near trivialization of the methods. No axioms are needed, no truth tables, just a list of polynomials (which in themselves are self-explanatory), the only law of inference is the rule of Substitution.